Research ArticleTrace Ratio Criterion-Based Kernel DiscriminantAnalysis for Fault Diagnosis of Rolling Element BearingsUsing Binary Immune Genetic Algorithm

Wen-An Yang1 Maohua Xiao2 Wei Zhou3 Yu Guo1 Wenhe Liao1 and Gang Shen4

1College of Mechanical and Electrical Engineering Nanjing University of Aeronautics and Astronautics Nanjing 210016 China2College of Engineering Nanjing Agricultural University Nanjing 210031 China3Nanjing Surveying and Mapping Instrument Factory Nanjing 210003 China4School of Mechatronic Engineering China University of Mining and Technology Xuzhou 221116 China

Correspondence should be addressed to Wen-An Yang dreamflownuaaeducn

Received 3 July 2015 Revised 19 October 2015 Accepted 27 October 2015

Academic Editor Dong Wang

Copyright copy 2016 Wen-An Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The rolling element bearing is a core component of many systems such as aircraft train steamboat and machine tool andtheir failure can lead to reduced capability downtime and even catastrophic breakdowns Due to misoperation manufacturingdeficiencies or the lack ofmonitoring andmaintenance it is often found to be themost unreliable component within these systemsTherefore effective and efficient fault diagnosis of rolling element bearings has an important role in ensuring the continued safe andreliable operation of their host systems This study presents a trace ratio criterion-based kernel discriminant analysis (TR-KDA)for fault diagnosis of rolling element bearings The binary immune genetic algorithm (BIGA) is employed to solve the trace ratioproblem in TR-KDA The numerical results obtained using extensive simulation indicate that the proposed TR-KDA using BIGA(called TR-KDA-BIGA) can effectively and efficiently classify different classes of rolling element bearing data while also providingthe capability of real-time visualization that is very useful for the practitioners to monitor the health status of rolling elementbearings Empirical comparisons show that the proposed TR-KDA-BIGA performs better than existing methods in classifyingdifferent classes of rolling element bearing dataThe proposed TR-KDA-BIGAmay be a promising tool for fault diagnosis of rollingelement bearings

1 Introduction

The rolling element bearing is a core component of manysystems such as aircraft train steamboat and machine tooland their failure can lead to reduced capability downtimeand even catastrophic breakdowns [1ndash6] Due to misopera-tion manufacturing deficiencies or the lack of monitoringand maintenance it is often found to be the most unreliablecomponent within these systems Therefore effective andefficient fault diagnosis of rolling element bearings has animportant role in ensuring the continued safe and reliableoperation of their host systems

Over the past few years much research effort has beendevoted to developing approaches to fault diagnosis of rollingelement bearings When faults occur in rolling element

bearings vibration signals in the relevant timefrequency-domain have been demonstrated to deviate from their nor-mal ones because of the increased friction and impulsiveforces [7ndash10] Usually several dozens or even hundredsof timefrequency-domain features are calculated from thebearing vibration signals to represent the different health sta-tus In the current study 9 time-domain features and 6 time-frequency-domain features are extracted from the bearingvibration signals to jointly construct a 15-dimension featurevector In that way fault diagnosis of rolling element bearingsis usually solved as a high-dimensional pattern recognitionproblem However for high-dimensional data the intrinsicdimensionmay be small For example the number of featuresresponsible for a certain type of fault pattern may be smallMoreover projection of high-dimensional data onto 2- or

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 8631639 15 pageshttpdxdoiorg10115520168631639

2 Shock and Vibration

3-dimension subspace can provide real-time visualizationwhich is convenient for the user to monitor the healthstatus of rolling element bearings In addition projectionof high-dimensional data onto low dimension subspacealso plays a part of data compression which is helpful forefficient storage and retrievalThus dimensionality reductiontechniques are often used to project the high-dimensionalfeature space to a lower-dimensional space while preservingmost of ldquointrinsic informationrdquo contained in the data prop-erties [11ndash15] Upon performing dimensionality reductionon the data its compact representation can be utilizedfor succeeding tasks (eg visualization and classification)Among various dimensionality reduction methods [16ndash24]principal component analysis (PCA) and linear discriminantanalysis (LDA) are the two most common methods [21]The former is an unsupervised method which pursues thedirection of maximum variance for optimal reconstructionThe latter is a supervised method which aims to maximizethe between-class scatter while minimizing the within-classscatter Owning to the utilization of labeled information thelatter generally outperforms the former if sufficient labeledsamples are provided [21] In the past few years a seriesof studies have been conducted to formulate the LDAs forpattern recognition by Fukunaga [21] Wang et al [22] Sunand Chen [25] Guo et al [26] Zhao et al [27] Jin et al[28] Jia et al [29] and so on Generally the formulation ofLDAs is based on the ratio trace criterion but not trace ratiocriterion because the ratio trace problem is more tractablethan the trace ratio problem Nevertheless as pointed outby Wang et al [22] solutions obtained based on ratiotrace criterion may deviate from the original intent of thetrace ratio problems To improve the behaviour of LDAimplementation Wang et al [22] Guo et al [26] Zhao etal [27] Jin et al [28] and Jia et al [29] presented varioustrace ratio criterion-based LDAs (TR-LDAs) in which thenumerator and denominator of the criterion directly reflectEuclidean distances between of inter- and intraclass samplesAnother advantage of trace ratio criterion is that the calcu-lated projection matrix is orthogonal which can eliminatethe redundancy between different projection directions Inaddition the orthogonal projection can thus preserve suchsimilarities without any change when using Euclidean dis-tance to evaluate the similarity between data points [22]Although the above TR-LDA formulation methods have theaforementioned advantages they are criticized due to theirincapability of dealing with the redundancy among eigen-vectors For example if the most discriminative eigenvectoris duplicated several times the above TR-LDA formulationmethods are prone to selecting all of themThis is problematicfor selection of an optimal subset of eigenvectors becauseother discriminative and complementary eigenvectors willbe missed A classifier with the eigenvectors selected inthis way can give rise to poor classification performanceTherefore the issue of TR-LDA formulation has remainedunresolved

A review of the related literature also indicates thatmost of the previous work in the area of applying LDAor TR-LDA to fault diagnosis assumed that samples ineach class follow a linear distribution However in many

fault diagnosis practices samples in each class that mayfollow a nonlinear distribution cannot satisfy the assump-tion Without this assumption the separation of differentclasses may not be well characterized by the scatter matricescausing the classification results to be degraded [21] Tosolve this problem kernel trick [30ndash32] which is to extendmany linear methods to its nonlinear kernel version canbe used to extend TR-LDA to handle nonlinear problemThus this study develops a nonlinear kernel version of TR-LDA that is trace ratio criterion-based kernel discriminantanalysis (TR-KDA) for fault diagnosis of rolling elementbearings However like many other TR-LDAmodels the TR-KDA model presented in this study shares the trace ratioproblem in the formulation of projection matrix Althoughthe above TR-LDA formulation methods have the aforemen-tioned advantages they are criticized due to the inability tohandle redundancy in eigenvector selection For exampleif the most discriminative eigenvector is duplicated severaltimes the above TR-LDA formulation methods are proneto selecting all of them This is problematic for selectingthe best set of eigenvectors because other discriminativeand complementary eigenvectors will be missed A classifierwith the eigenvectors selected in such a way can leadto a poor classification performance Fortunately immunegenetic algorithm (IGA) a novel evolutionary computa-tion technique developed by Jiao and Wang [33] has thepotential to determine a set of discriminative and mutuallyirredundant eigenvectors In this study we propose a methodcalled TR-KDA-BIGA that uses binary IGA (BIGA) to for-mulate TR-KDA for dimensionality reduction of statisticaland wavelet features extracted from the vibration signalsand gives rise to effective and efficient fault diagnosis ofrolling element bearings In particular the contributions areto

(i) use immune evolutionary computation techniquesuch as BIGA to obtain a reduced set of discriminativeand mutually irredundant eigenvectors for TR-KDA-BIGA formulation

(ii) provide the capability of two-dimensional represen-tation of bearing data that is very useful for thepractitioners to monitor the health status of bearings

(iii) build a TR-KDA-BIGA model architecture for thevibration measurements for effective and efficientfault diagnosis of rolling element bearings

The rest of this study is structured as follows Section 2briefly reviews the basic concepts of TR-LDA and kernelextension Section 3 presents a TR-KDA-BIGA methodSection 4 discusses its convergence and initializationSection 5 conducts performance evaluations of the proposedTR-KDA-BIGA on benchmark problems Section 6 describesan overall flowchart of the proposed TR-KDA-BIGA for faultdiagnosis of rolling element bearings Section 7 summarizesthe conclusions drawn from this study

Shock and Vibration 3

2 Review of TR-LDA and Kernel Extension

21 Review of TR-LDA Suppose we are given a set of 119899 119889-dimensional samples x

1 x2 x

119899 belonging to 119897 different

classes The goal of LDA tries to obtain a linear projectionmatrix W isin R119889times119896 that can map the original 119889-dimensionaldata x

119894onto the 119896-dimensional data y

119894(usually 119896 ≪ 119889)

by maximizing the between-class scatter and meanwhileminimizing the within-class scatter The between-class scat-ter matrix S

119861and the within-class scatter matrix S

119882are

expressed as follows

S119861=

119897

sum

119894=1

119899119894(m(119894) minusm) (m(119894) minusm)

T

S119882=

119897

sum

119894=1

(

119899119894

sum

119895=1

(x(119894)119895minusm(119894)) (x(119894)

119895minusm(119894))

T)

(1)

where m represents the total sample mean vector 119899119894repre-

sents the number of samples in the 119894th class m(119894) representsthe average vector of the 119894th class and x(119894)

119895represents the119895th

sample in the 119894th class The new mapped feature vectors 119910119894isin

R119896 can then be expressed as y119894= WTx

119894The original LDA for-

mulation known as the Fisher LDA [21] only handles binaryclassification problemsHowevermany practical applicationsinvolve multiclass classification In order to overcome thisissue a number of researchers have proposed optimizationcriteria for extending the Fisher LDA to handle multiclassclassification problems The first optimization criterion is ina ratio trace form (referred to as RT-LDA)

Wlowast = arg maxWTW=I

Tr(WTS119861W

WTS119882W) (2)

where Tr[sdot] denotes the matrix trace I is an identity matrixIn order to achieve a set of orthogonal normalized vectorsit usually adds the constraint WTW = I to (2) The secondoptimization criterion is in a trace ratio form (referred to asTR-LDA)

Wlowast = arg maxWTW=I

Tr (WTS119861W)

Tr (WTS119882W)

(3)

The optimization problem in (3) can be solved directlythrough the generalized eigenvalue decomposition (GED)method [22]

S119861w119894= 120591119894S119882w119894 (4)

where 120591119894is the 119894th largest eigenvaluew

119894is the eigenvector cor-

responding to 120591119894 and w

119894constitutes the 119894th column vector of

the matrixW Although a closed-form solution for (3) can beapproximately obtained with the GED it does not necessarilyguarantee best trace ratio optimization Thus this approxi-mation of ratio trace optimization to trace ratio optimizationmay lead to classification capability loss of the derived opti-mal low-dimensional feature space Moreover the physicalmeaning of the trace ratio form is clearer than that of

the ratio trace form However the optimization problem in(3) is generally nonconvex and a closed-form solution for itdoes not exist Fortunately a recent study conducted by Guoet al [26] showed that using the trace difference function119911(120582) = maxWTW=ITr[WT

(S119861minus 120582S119882)W] the trace ratio prob-

lem can be solved equivalently by finding zero points of theequation 119911(120582) = 0 Following up Guo et alrsquos work Wang et alpresented an iterative method named ITR algorithm to solvethe trace ratio problem [22]The ITR algorithm optimizes theobjective function 120582 = maxWTW=ITr[WTS

119861W]Tr[WTS

119882W]

in an iterative and incremental mannerTheW in the 119905th iter-ation step (referred to asW

119905) is obtained through solving the

trace difference problem argmaxWTW=ITr[WT(S119861minus120582119905S119882)W]

where 120582119905represents the trace ratio value derived from the

W in the previous iteration step (referred to as W119905minus1

)However the initialization for theW influences substantiallythe convergence performance of the ITR algorithm A goodinitialization can generally make the ITR algorithm yield aquick convergence A bad initialization usually increases thenumber of iterationsMoreover in ITR algorithm although itseems that theW formedwith these eigenvectors correspond-ing to the 119896 largest eigenvalues of S

119861minus120582119905S119882canmaximize the

trace difference Tr[WT(S119861minus 120582119905S119882)W] it cannot necessarily

maximize the trace ratio Tr[WTS119861W]Tr[WTS

119882W] On the

other hand from the perspective of fault diagnosis the aimis mainly to find a set of projection vectors that can pose thehighest levels of discrimination in the different fault patternsThus these eigenvectors with the largest eigenvalues are notnecessarily representative for discriminating one class fromothers as previously mentioned in Section 1 To overcomethe above shortcomings this study presents a BIGA-basedsolution method for trace ratio criterion (to be in detaildiscussed in Section 3)

22 Kernel Extension In some applications it is insufficientto model the data using the TR-LDA which is a lineardiscriminatingmethod To address the issue of nonlinearitiesin the data this section presents a nonlinear discriminatingmethod using kernel trick [30ndash32] that is TR-KDA Theso-called kernel trick is to map the original data to a high-dimensional Hilbert space through a nonlinear mappingfunction 120601 Let 120601(X) denote the data matrix in the Hilbertspace 120601(X) = [120601(x

1) 120601(x2) 120601(x

119899)] The function form

of the mapping does not need to be known since it isimplicitly defined by the choice of kernel function119870(x

119894 x119895) =

120601(x119894)T120601(x119895) that is the inner product in the kernel-induced

feature space The kernel function 119870 may be any positivekernel satisfying Mercerrsquos condition Radial basis function(RBF) kernel function one of the most popular kernelfunctions employed in various kernelled learning algorithmsis adopted in this study Then (3) in Hilbert space can bewritten as follows

W120601lowast

= arg max(W120601)TW120601=I

Tr ((W120601)TS120601119861W120601)

Tr ((W120601)T S120601119882W120601)

(5)

where W120601 S120601119861 and S120601

119882are the matrices in Hilbert space

corresponding toW S119861 and S

119882in (3) respectively Notably

4 Shock and Vibration

we can show that matrices S119861and S119882in (3) can be essentially

expressed as S119861

= XL119861XT and S

119882= XL

119882XT through

simple manipulation respectively X is the vector where X =

[x1 x2 x

119899] Matrices L

119861and L

119882are the graph Laplacian

matrices [34] of theweighted undirected graphs reflecting thebetween-class and within-class relationship of the samplesConsider

S119861=

119897

sum

119894=1

119899119894(m(119894) minusm) (m(119894) minusm)

T

= (

119897

sum

119894=1

119899119894m(119894) (m(119894))

T) minusm(

119897

sum

119894=1

119899119894(m(119894))

T)

minus (

119897

sum

119894=1

119899119894m(119894))mT

+ (

119897

sum

119894=1

119899119894)mmT

= (

119897

sum

119894=1

1

119899119894

(x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

sdot (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) minus 2119899mmT

+ 119899mmT

= (

119897

sum

119894=1

119899119894

sum

119895119902=1

1

119899119894

x(119894)119895(x(119894)119902)

T) minus 119899mmT

= XGXT

minus 119899mmT= XGXT

minus X (

1

119899

eeT)XT= X (G minus

1

119899

sdot eeT)XT

(6)

where e = [1 1 1]T is an 119899-dimensional vector We can

simplify the above equation even further by defining that

(G)119894119895

=

1

119899119902

if samples x119894and x

119895belong to the 119902th class

0 otherwise

(7)

L119861= G minus

1

119899

eeT (8)

Thus we get

S119861= XL119861XT

(9)

Then the matrix S119882can similarly be computed as follows

S119882=

119897

sum

119894=1

(

119899119894

sum

119895=1

(x(119894)119895minusm(119894)) (x(119894)

119895minusm(119894))

T)

=

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

Tminusm(119894) (x(119894)

119895)

Tminus x(119894)119895(m(119894))

T

+m(119894) (m(119894))T) =

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

T

minus 119899119894m(119894) (m(119894))

T) =

119897

sum

119894=1

(X119894XT119894minus

1

119899119894

(x(119894)1+ x(119894)2

+ sdot sdot sdot + x(119894)119899119894) (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) =

119897

sum

119894=1

(X119894XT119894

minus

1

119899119894

X119894(e119894eT119894)XT119894) =

119897

sum

119894=1

(X119894(I minus 1

119899119894

e119894eT119894)XT119894)

=

119897

sum

119894=1

X119894L119894XT119894

(10)

where X119894= [x(119894)

1 x(119894)2 x(119894)

119899119894] is a 119889 times 119899

119894matrix e

119894=

[1 1 1]T is an 119899

119894-dimensional vector I is the identity

matrix L119894= I minus 1119899

119894e119894eT119894is an 119899

119894times 119899119894matrix and X

119894L119894XT119894

is the data covariance matrix of the 119894th class Based on (7)the above equation can be simplified similarly by defining

L119882= I minus G (11)

Thus we get

S119882= XL119882XT

(12)

Using the definitions in (9) and (12) (5) can be rewritten asfollows

W120601lowast

= arg max(W120601)TW120601=I

Tr ((W120601)T(120601 (X) L

119861(120601 (X))T)W120601)

Tr ((W120601)T (120601 (X) L119882(120601 (X))T)W120601)

(13)

In order to pursue thematrixW120601lowast

solving the above equationinvolves decomposition of 120601(X) into an orthogonal matrixQ(satisfyingQQT

= I) and a right triangularmatrixR such thatQ lowast R = 120601(X) We have

(120601 (X))T 120601 (X) = RTR (14)

Let us map W120601 into the span of Q Q is currently anorthogonal basis of 120601(X) so we have

W120601 = QV120601 (15)

where V120601 = R119896times119899 is an orthogonal matrix satisfying(V120601)TV120601 = I Using the definitions in (14) and (15) (13) canbe further rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TRL119861RTV120601)

Tr ((V120601)T RL119882RTV120601)

(16)

Let S119861= RL119861RT and S

119882= RL119882RT then (16) can be further

rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TS119861V120601)

Tr ((V120601)T S119882V120601)

(17)

Shock and Vibration 5

After the matrix V120601lowast is obtained with the BIGA-basedsolution method (to be in detail discussed in Section 3) theoutput points in the reduced data space can thus be expressedas

Y120601lowast

= (W120601lowast

)

T120601 (X) = (V120601

lowast

)

TQTQR = (V120601

lowast

)

TR (18)

3 The Proposed TR-KDA-BIGA

As previously mentioned construction of TR-KDA needsto select 119896 out of 119889 eigenvectors to form the matrix V120601for dimensionality reduction However finding a subset ofeigenvectors based on the trace ratio criterion is not an easytask since the space of possible subsets is very large especiallywhen 119889 is a large number Thus it is not impractical to useexhaustive search to find an optimal subset of 119896 eigenvectorsInstead in this study the BIGA is utilized to select 119896 outof 119889 eigenvectors of V120601

119905as the bases for projection matrix

formulation based on the trace ratio criterion such thatthe trace ratio value 120582 = Tr[(V120601)TS

119861V120601]Tr[(V120601)TS

119882V120601]

can be maximized Immune genetic algorithm originallydeveloped by Jiao andWang [33] is a novel genetic algorithmbased on the biological immune theory which combined theimmune mechanism with the evolutionary mechanism Inwhat follows further discussion of the proposed TR-KDA-BIGA is carried out

31 Chromosome Encoding Encoding a solution of a probleminto a chromosome is an important issue when using BIGAsIn this study every chromosome in a BIGA corresponds to adiscrete binary selector u = [119906

1 1199062 119906

119889] where each gene

in the chromosome is ldquo1rdquo indicating an eigenvector k120601119894(119894 =

1 2 119889) of S119861minus 120582119905S119882appearing in forming the projection

matrixV120601 of the 119905th step while ldquo0rdquo denotes its absenceThusthe length of the chromosome is 119889

32 Genetic Operators Genetic operators give every chro-mosome the chance to become the fittest chromosome ofits generation If it is difficult to reach the target of traceratio optimization crossover and mutation may introducedegeneracy into generations of chromosomes

321 Crossover Operator Crossover operator in a BIGAis employed to generate two new children chromosomesbased on two existing parent chromosomes selected from thecurrent population in terms of a prespecified crossover rateIn this study ldquoone-pointrdquo crossover operator was adopted torandomly select a cut point to exchange the parts betweenthe cut point and the end of the string of the parentchromosomes Specifically suppose that two parent chromo-somes 119875

1and 119875

2selected randomly from the population are

undergoing the crossover operation at a randomly selectedcrossover point 119892 (1 le 119892 le 119889) where

1198751= (1199061198941 1199061198942 119906

119894119889)

1198752= (1199061198951 1199061198952 119906

119895119889)

(19)

Consequently the offspring is generated by one-pointcrossover on the genes of two parents selected randomlyfrom the population We can thus get the two offspringchromosomes 119862

1and 119862

2

1198621= (1199061198941 1199061198942 119906

119894119892minus1 119906119894119892 119906119895119892+1

119906119895119889)

1198622= (1199061198951 1199061198952 119906

119895119892minus1 119906119895119892 119906119894119892+1

119906119894119889)

(20)

However the exchange procedure is not simply exchangingtheir genetic information between gene segments after thecrossover pointsWemust keep the number of eigenvectors tobe included in the subset equal to 119896 In this study therefore asimple but effective crossover operator strategy in this study isperformed in order to ensure that the crossover operator doesnot change the total number of ldquo1rdquo genes in chromosomes

Let

1198991198751=

119889

sum

119902=119892+1

119906119894119902

1198991198752=

119889

sum

119902=119892+1

119906119895119902

(21)

When 1198991198751is not equal to 119899

1198752 the following retention criterion

will be conducted(1) If 119899

1198751is larger than 119899

1198752 randomly select (119899

1198751minus

1198991198752) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo1-bitrdquo and then randomly select (1198991198751

minus

1198991198752) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo0-bitrdquo(2) If 119899

1198751is smaller than 119899

1198752 randomly select (119899

1198752minus

1198991198751) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo0-bitrdquo and then randomly select (1198991198752

minus

1198991198751) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo1-bitrdquo

322 Mutation Operator Mutation operator in a BIGA isused primarily as a mechanism for maintaining diversityin the population For each gene in a chromosome that isundergoing the mutation a real-valued number is randomlyselected within the range of [0 1] If the real-valued numberis less than the prespecified mutation rate then the genewill change from ldquo0-bitrdquo to ldquo1-bitrdquo and vice versa Uponadding (or removing) one eigenvector in that way we shallrandomly remove (or add) a different one such that thenumber of eigenvectors to be included in the subset is equalto 119896 The mutation operator helps the chromosomes to guidethe search in new areas

33 Immune Operators The immune ability of BIGAs is real-ized through two kinds of immune operators a vaccinationand an immune selection The vaccination is responsiblefor improving individualsrsquo overall fitness levels The immuneselection is responsible for prevention of deterioration

6 Shock and Vibration

331 Vaccination Operator Given a chromosome u vac-cination operation in a BIGA is employed to modify thegenes on some bits according to a priori knowledge such thatindividuals with higher fitness have a greater probability ofbeing selected Let U = (u

1 u2 u

1198990) be a population

the vaccination operation on U means that the operationis performed on 119899

120572= 120572119899

0chromosomes selected from

U according to the proportion of 120572 where 1198990represents

the population size of a BIGA A vaccine is abstractedfrom the prior knowledge of the pending problem whoseinformation amount and validity play an important role inthe performance of the algorithm

332 Immune Selection Operator The immune selectionoperation consists of the following two steps The first step isthe immunity test if the fitness of a chromosome u is smallerthan that of the parent chromosome which indicates thatdegeneration occurred during crossover and mutation thenthe parent chromosomewill be used for the next competitionThe second step is the annealing selection [35] a chromo-some u

119894is selected from the current offspring population U

119896

to join with the new parents with the probability as follows

119875 (u119894) =

exp (119891 (u119894) 119879119896)

sum1198990

119894=1exp (119891 (u

119894) 119879119896)

(22)

where 119891(u119894) is the fitness of the individual u

119894and 119879

119896 is the

temperature-controlled series tending towards 0

34 Fitness Evaluation Fitness evaluation plays a criticalrole in selecting offspring chromosomes from the currentpopulation for the next generation In this study the fitnessfunction for eigenvector selection is defined as

119891 (u) = 1199061ℎ1+ 1199062ℎ2+ sdot sdot sdot + 119906

119889ℎ119889

11990611198921+ 11990621198922+ sdot sdot sdot + 119906

119889119892119889

=

uhT

ugT (23)

where ℎ119894denotes the vT

119894S119861v119894value for the 119894th eigenvector

119892119894denotes the vT

119894S119882v119894value for the 119894th eigenvector h =

[ℎ1 ℎ2 ℎ

119889] g = [119892

1 1198922 119892

119889] u = [119906

1 1199062 119906

119889]

119906119894isin 0 1 u1T = 119896 and 119894 = 1 2 119889 Notably u is

called the binary selector and 119896 is the desired lower featuredimension Finally according to the evolved binary selectoru we can thus form the projection matrix V120601 of the 119905thstep by choosing the 119896 eigenvectors with 119906

119894= 1 (119894 =

1 2 119889) The procedures of the proposed TR-KDA-BIGAare summarized in the procedures of the proposed TR-KDA-BIGA part The computational flow of the BIGA obtainedusing the aforementioned genetic and immune operators isalso provided in the computational flow of the BIGA part

The Procedures of the Proposed TR-KDA-BIGA The proce-dures are as follows

(1) Construct the kernel matrix K = (120601(X))T120601(X)(2) Perform Cholesky decomposition to the kernel

matrix K = RTR(3) Form the kernel scatter matrixes as S

119861= RL119861RT and

S119882= RL119882RT

(4) Set iterations number 119905 to 1(5) Set the initial trace ratio value 120582

119905to Tr(S

119861)Tr(S

119882)

(6) Compute the eigendecomposition of S119861minus 120582119905S119882

as(S119861minus 120582119905S119882)k120601119894= 120591119894k120601119894 where k120601

119894(119894 = 1 2 119889) is

the eigenvector of S119861minus 120582119905S119882

(7) Calculate ℎ119894= (k120601119894)TS119861k120601119894and 119892

119894= (k120601119894)TS119882k120601119894for

each eigenvector k120601119894(119894 = 1 2 119889)

(8) Generate a population of BIGA selectors(9) Evolve the population where the fitness of a BIGA

selector u is measured as 119891(u) = uhTugT(10) ulowast is the evolved best BIGA selector

(11) Form the projection matrix V120601119905by choosing the 119896

eigenvectors k120601119894with 119906lowast

119894= 1 (119894 = 1 2 119889)

(12) Update the trace ratio value 120582119905+1

=

Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] 119905 = 119905 + 1 and go to

step (6) Repeat this procedure until a convergencecondition was established when the trace ratio valuedoes not increase in consecutive 5 iterations

(13) Output Y120601lowast = (V120601lowast)TR

TheComputational Flow of the BIGAThe computational flowis as follows

(1) Set 119897 (time of generation) to 1(2) Initialize randomly the original population 119860

119897

(3) Evaluate each chromosome in the original population119860119897

(4) Abstract vaccines according to the prior knowledge(5) Check for termination criteria If the fixed number

of generations is not reached or the optimal chromo-some found thus far is not satisfied then go to the nextstep Otherwise output the optimal chromosome asthe final solutions for further decision-making

(6) Perform crossover operation on the 119860119897and then

generate the population 119861119897

(7) Perform mutation operation on the 119861119897and then

generate the population results 119862119897

(8) Perform vaccination operation on the 119862119897and then

generate the population119863119897

(9) Perform immune selection operation on the 119863119897and

then generate the next generational population 119860119897+1

Go to step (3)

4 The Convergence of theProposed TR-KDA-BIGA

In this section we analyze the convergence of the proposedTR-LDA-BIGA Before doing this task it should be worthnoting that the BIGA is convergent It has been demonstratedby Jiao and Wang [33] that as long as enough iteration has

Shock and Vibration 7

been completed the immune genetic population convergestowards the true optimum with probability one

Recall the trace difference function

119911 (120582) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582S119882)V120601] (24)

it follows that

119911 (120582119905+1) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582119905+1S119882)V120601] 997904rArr

119911 (120582119905+1) = Tr [(V120601

119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

(25)

Since 120582119905+1

= Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] as previously

mentioned we get

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (26)

Consider the inequality

Tr [(V120601119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

ge Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905]

(27)

and the equation

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (28)

and we have

119911 (120582119905+1) ge 0

Tr [(V120601119905+1)

TS119861V120601119905+1] minus 120582119905+1

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 0

(29)

Consequently

Tr [(V120601119905+1)

TS119861V120601119905+1]

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 120582119905+1

997904rArr

120582119905+2

ge 120582119905+1

(30)

Substituting the subscript 119905 + 1 by 119905 yields

120582119905+1

ge 120582119905 (31)

So we obtain the following inequality which gives the firstexpression of convergence of the proposed TR-KDA-BIGA

Further suppose that 120582lowast is the optimal trace ratio valueit follows that

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

] = 0 (32)

Table 1 Specification of benchmark problems

Data set samples dim classHeart-statlog 270 13 2Ionosphere 351 34 2Iris 150 4 3Wine 178 13 3Waveform 5000 40 3Balance 625 4 3SCCTS 600 60 6

where V120601lowast is the optimal projection matrix We thereforehave

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

]

ge Tr [(V120601119905)

T(S119861minus 120582lowastS119882)V120601119905]

= Tr [(V120601119905)

T(S119861minus 120582119905+1

S119882)V120601119905]

+ (120582119905+1

minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

= 119911 (120582119905+1) + (120582

119905+1minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

(33)

Consider 119911(120582lowast) = 0 119911(120582119905+1) ge 0 and S

119882is semipositive

definite we have

120582119905+1

minus 120582lowastle 0 (34)

So we obtain the following inequality which gives the secondexpression of convergence of the proposed TR-KDA-BIGA

120582119905+1

le 120582lowast (35)

We conclude therefore that for a particular initial trace ratiovalue 120582

119905 the updated value 120582

119905+1can always satisfy (1) 120582

119905+1ge

120582119905and (2) 120582

119905+1le 120582lowast

5 Performance Evaluation onBenchmark Problems

In order to extensively verify the performance of the proposedTR-KDA-BIGA it is first tested on wide types of commonlyused benchmark problems taken from the UCI machinelearning repository and evaluated with the classificationrate (ie the number of correctly identified training exam-plestotal number of training examples) by comparison withother existing methods such as PCA LDA KPCA [30 31]KDA [32] and TR-LDA These data sets include Heart-statlog Ionosphere Iris Wine Waveform Balance and Syn-thetic Control Chart Time Series (SCCTS) data sets (Table 1)which are of small sizes low dimensions large sizes andorhigh dimensions For comparative study we randomly select50 data points from each data set as training set and therest of the data points as test set All methods use training set

8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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2 Shock and Vibration

3-dimension subspace can provide real-time visualizationwhich is convenient for the user to monitor the healthstatus of rolling element bearings In addition projectionof high-dimensional data onto low dimension subspacealso plays a part of data compression which is helpful forefficient storage and retrievalThus dimensionality reductiontechniques are often used to project the high-dimensionalfeature space to a lower-dimensional space while preservingmost of ldquointrinsic informationrdquo contained in the data prop-erties [11ndash15] Upon performing dimensionality reductionon the data its compact representation can be utilizedfor succeeding tasks (eg visualization and classification)Among various dimensionality reduction methods [16ndash24]principal component analysis (PCA) and linear discriminantanalysis (LDA) are the two most common methods [21]The former is an unsupervised method which pursues thedirection of maximum variance for optimal reconstructionThe latter is a supervised method which aims to maximizethe between-class scatter while minimizing the within-classscatter Owning to the utilization of labeled information thelatter generally outperforms the former if sufficient labeledsamples are provided [21] In the past few years a seriesof studies have been conducted to formulate the LDAs forpattern recognition by Fukunaga [21] Wang et al [22] Sunand Chen [25] Guo et al [26] Zhao et al [27] Jin et al[28] Jia et al [29] and so on Generally the formulation ofLDAs is based on the ratio trace criterion but not trace ratiocriterion because the ratio trace problem is more tractablethan the trace ratio problem Nevertheless as pointed outby Wang et al [22] solutions obtained based on ratiotrace criterion may deviate from the original intent of thetrace ratio problems To improve the behaviour of LDAimplementation Wang et al [22] Guo et al [26] Zhao etal [27] Jin et al [28] and Jia et al [29] presented varioustrace ratio criterion-based LDAs (TR-LDAs) in which thenumerator and denominator of the criterion directly reflectEuclidean distances between of inter- and intraclass samplesAnother advantage of trace ratio criterion is that the calcu-lated projection matrix is orthogonal which can eliminatethe redundancy between different projection directions Inaddition the orthogonal projection can thus preserve suchsimilarities without any change when using Euclidean dis-tance to evaluate the similarity between data points [22]Although the above TR-LDA formulation methods have theaforementioned advantages they are criticized due to theirincapability of dealing with the redundancy among eigen-vectors For example if the most discriminative eigenvectoris duplicated several times the above TR-LDA formulationmethods are prone to selecting all of themThis is problematicfor selection of an optimal subset of eigenvectors becauseother discriminative and complementary eigenvectors willbe missed A classifier with the eigenvectors selected inthis way can give rise to poor classification performanceTherefore the issue of TR-LDA formulation has remainedunresolved

A review of the related literature also indicates thatmost of the previous work in the area of applying LDAor TR-LDA to fault diagnosis assumed that samples ineach class follow a linear distribution However in many

fault diagnosis practices samples in each class that mayfollow a nonlinear distribution cannot satisfy the assump-tion Without this assumption the separation of differentclasses may not be well characterized by the scatter matricescausing the classification results to be degraded [21] Tosolve this problem kernel trick [30ndash32] which is to extendmany linear methods to its nonlinear kernel version canbe used to extend TR-LDA to handle nonlinear problemThus this study develops a nonlinear kernel version of TR-LDA that is trace ratio criterion-based kernel discriminantanalysis (TR-KDA) for fault diagnosis of rolling elementbearings However like many other TR-LDAmodels the TR-KDA model presented in this study shares the trace ratioproblem in the formulation of projection matrix Althoughthe above TR-LDA formulation methods have the aforemen-tioned advantages they are criticized due to the inability tohandle redundancy in eigenvector selection For exampleif the most discriminative eigenvector is duplicated severaltimes the above TR-LDA formulation methods are proneto selecting all of them This is problematic for selectingthe best set of eigenvectors because other discriminativeand complementary eigenvectors will be missed A classifierwith the eigenvectors selected in such a way can leadto a poor classification performance Fortunately immunegenetic algorithm (IGA) a novel evolutionary computa-tion technique developed by Jiao and Wang [33] has thepotential to determine a set of discriminative and mutuallyirredundant eigenvectors In this study we propose a methodcalled TR-KDA-BIGA that uses binary IGA (BIGA) to for-mulate TR-KDA for dimensionality reduction of statisticaland wavelet features extracted from the vibration signalsand gives rise to effective and efficient fault diagnosis ofrolling element bearings In particular the contributions areto

(i) use immune evolutionary computation techniquesuch as BIGA to obtain a reduced set of discriminativeand mutually irredundant eigenvectors for TR-KDA-BIGA formulation

(ii) provide the capability of two-dimensional represen-tation of bearing data that is very useful for thepractitioners to monitor the health status of bearings

(iii) build a TR-KDA-BIGA model architecture for thevibration measurements for effective and efficientfault diagnosis of rolling element bearings

The rest of this study is structured as follows Section 2briefly reviews the basic concepts of TR-LDA and kernelextension Section 3 presents a TR-KDA-BIGA methodSection 4 discusses its convergence and initializationSection 5 conducts performance evaluations of the proposedTR-KDA-BIGA on benchmark problems Section 6 describesan overall flowchart of the proposed TR-KDA-BIGA for faultdiagnosis of rolling element bearings Section 7 summarizesthe conclusions drawn from this study

Shock and Vibration 3

2 Review of TR-LDA and Kernel Extension

21 Review of TR-LDA Suppose we are given a set of 119899 119889-dimensional samples x

1 x2 x

119899 belonging to 119897 different

classes The goal of LDA tries to obtain a linear projectionmatrix W isin R119889times119896 that can map the original 119889-dimensionaldata x

119894onto the 119896-dimensional data y

119894(usually 119896 ≪ 119889)

by maximizing the between-class scatter and meanwhileminimizing the within-class scatter The between-class scat-ter matrix S

119861and the within-class scatter matrix S

119882are

expressed as follows

S119861=

119897

sum

119894=1

119899119894(m(119894) minusm) (m(119894) minusm)

T

S119882=

119897

sum

119894=1

(

119899119894

sum

119895=1

(x(119894)119895minusm(119894)) (x(119894)

119895minusm(119894))

T)

(1)

where m represents the total sample mean vector 119899119894repre-

sents the number of samples in the 119894th class m(119894) representsthe average vector of the 119894th class and x(119894)

119895represents the119895th

sample in the 119894th class The new mapped feature vectors 119910119894isin

R119896 can then be expressed as y119894= WTx

119894The original LDA for-

mulation known as the Fisher LDA [21] only handles binaryclassification problemsHowevermany practical applicationsinvolve multiclass classification In order to overcome thisissue a number of researchers have proposed optimizationcriteria for extending the Fisher LDA to handle multiclassclassification problems The first optimization criterion is ina ratio trace form (referred to as RT-LDA)

Wlowast = arg maxWTW=I

Tr(WTS119861W

WTS119882W) (2)

where Tr[sdot] denotes the matrix trace I is an identity matrixIn order to achieve a set of orthogonal normalized vectorsit usually adds the constraint WTW = I to (2) The secondoptimization criterion is in a trace ratio form (referred to asTR-LDA)

Wlowast = arg maxWTW=I

Tr (WTS119861W)

Tr (WTS119882W)

(3)

The optimization problem in (3) can be solved directlythrough the generalized eigenvalue decomposition (GED)method [22]

S119861w119894= 120591119894S119882w119894 (4)

where 120591119894is the 119894th largest eigenvaluew

119894is the eigenvector cor-

responding to 120591119894 and w

119894constitutes the 119894th column vector of

the matrixW Although a closed-form solution for (3) can beapproximately obtained with the GED it does not necessarilyguarantee best trace ratio optimization Thus this approxi-mation of ratio trace optimization to trace ratio optimizationmay lead to classification capability loss of the derived opti-mal low-dimensional feature space Moreover the physicalmeaning of the trace ratio form is clearer than that of

the ratio trace form However the optimization problem in(3) is generally nonconvex and a closed-form solution for itdoes not exist Fortunately a recent study conducted by Guoet al [26] showed that using the trace difference function119911(120582) = maxWTW=ITr[WT

(S119861minus 120582S119882)W] the trace ratio prob-

lem can be solved equivalently by finding zero points of theequation 119911(120582) = 0 Following up Guo et alrsquos work Wang et alpresented an iterative method named ITR algorithm to solvethe trace ratio problem [22]The ITR algorithm optimizes theobjective function 120582 = maxWTW=ITr[WTS

119861W]Tr[WTS

119882W]

in an iterative and incremental mannerTheW in the 119905th iter-ation step (referred to asW

119905) is obtained through solving the

trace difference problem argmaxWTW=ITr[WT(S119861minus120582119905S119882)W]

where 120582119905represents the trace ratio value derived from the

W in the previous iteration step (referred to as W119905minus1

)However the initialization for theW influences substantiallythe convergence performance of the ITR algorithm A goodinitialization can generally make the ITR algorithm yield aquick convergence A bad initialization usually increases thenumber of iterationsMoreover in ITR algorithm although itseems that theW formedwith these eigenvectors correspond-ing to the 119896 largest eigenvalues of S

119861minus120582119905S119882canmaximize the

trace difference Tr[WT(S119861minus 120582119905S119882)W] it cannot necessarily

maximize the trace ratio Tr[WTS119861W]Tr[WTS

119882W] On the

other hand from the perspective of fault diagnosis the aimis mainly to find a set of projection vectors that can pose thehighest levels of discrimination in the different fault patternsThus these eigenvectors with the largest eigenvalues are notnecessarily representative for discriminating one class fromothers as previously mentioned in Section 1 To overcomethe above shortcomings this study presents a BIGA-basedsolution method for trace ratio criterion (to be in detaildiscussed in Section 3)

22 Kernel Extension In some applications it is insufficientto model the data using the TR-LDA which is a lineardiscriminatingmethod To address the issue of nonlinearitiesin the data this section presents a nonlinear discriminatingmethod using kernel trick [30ndash32] that is TR-KDA Theso-called kernel trick is to map the original data to a high-dimensional Hilbert space through a nonlinear mappingfunction 120601 Let 120601(X) denote the data matrix in the Hilbertspace 120601(X) = [120601(x

1) 120601(x2) 120601(x

119899)] The function form

of the mapping does not need to be known since it isimplicitly defined by the choice of kernel function119870(x

119894 x119895) =

120601(x119894)T120601(x119895) that is the inner product in the kernel-induced

feature space The kernel function 119870 may be any positivekernel satisfying Mercerrsquos condition Radial basis function(RBF) kernel function one of the most popular kernelfunctions employed in various kernelled learning algorithmsis adopted in this study Then (3) in Hilbert space can bewritten as follows

W120601lowast

= arg max(W120601)TW120601=I

Tr ((W120601)TS120601119861W120601)

Tr ((W120601)T S120601119882W120601)

(5)

where W120601 S120601119861 and S120601

119882are the matrices in Hilbert space

corresponding toW S119861 and S

119882in (3) respectively Notably

4 Shock and Vibration

we can show that matrices S119861and S119882in (3) can be essentially

expressed as S119861

= XL119861XT and S

119882= XL

119882XT through

simple manipulation respectively X is the vector where X =

[x1 x2 x

119899] Matrices L

119861and L

119882are the graph Laplacian

matrices [34] of theweighted undirected graphs reflecting thebetween-class and within-class relationship of the samplesConsider

S119861=

119897

sum

119894=1

119899119894(m(119894) minusm) (m(119894) minusm)

T

= (

119897

sum

119894=1

119899119894m(119894) (m(119894))

T) minusm(

119897

sum

119894=1

119899119894(m(119894))

T)

minus (

119897

sum

119894=1

119899119894m(119894))mT

+ (

119897

sum

119894=1

119899119894)mmT

= (

119897

sum

119894=1

1

119899119894

(x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

sdot (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) minus 2119899mmT

+ 119899mmT

= (

119897

sum

119894=1

119899119894

sum

119895119902=1

1

119899119894

x(119894)119895(x(119894)119902)

T) minus 119899mmT

= XGXT

minus 119899mmT= XGXT

minus X (

1

119899

eeT)XT= X (G minus

1

119899

sdot eeT)XT

(6)

where e = [1 1 1]T is an 119899-dimensional vector We can

simplify the above equation even further by defining that

(G)119894119895

=

1

119899119902

if samples x119894and x

119895belong to the 119902th class

0 otherwise

(7)

L119861= G minus

1

119899

eeT (8)

Thus we get

S119861= XL119861XT

(9)

Then the matrix S119882can similarly be computed as follows

S119882=

119897

sum

119894=1

(

119899119894

sum

119895=1

(x(119894)119895minusm(119894)) (x(119894)

119895minusm(119894))

T)

=

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

Tminusm(119894) (x(119894)

119895)

Tminus x(119894)119895(m(119894))

T

+m(119894) (m(119894))T) =

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

T

minus 119899119894m(119894) (m(119894))

T) =

119897

sum

119894=1

(X119894XT119894minus

1

119899119894

(x(119894)1+ x(119894)2

+ sdot sdot sdot + x(119894)119899119894) (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) =

119897

sum

119894=1

(X119894XT119894

minus

1

119899119894

X119894(e119894eT119894)XT119894) =

119897

sum

119894=1

(X119894(I minus 1

119899119894

e119894eT119894)XT119894)

=

119897

sum

119894=1

X119894L119894XT119894

(10)

where X119894= [x(119894)

1 x(119894)2 x(119894)

119899119894] is a 119889 times 119899

119894matrix e

119894=

[1 1 1]T is an 119899

119894-dimensional vector I is the identity

matrix L119894= I minus 1119899

119894e119894eT119894is an 119899

119894times 119899119894matrix and X

119894L119894XT119894

is the data covariance matrix of the 119894th class Based on (7)the above equation can be simplified similarly by defining

L119882= I minus G (11)

Thus we get

S119882= XL119882XT

(12)

Using the definitions in (9) and (12) (5) can be rewritten asfollows

W120601lowast

= arg max(W120601)TW120601=I

Tr ((W120601)T(120601 (X) L

119861(120601 (X))T)W120601)

Tr ((W120601)T (120601 (X) L119882(120601 (X))T)W120601)

(13)

In order to pursue thematrixW120601lowast

solving the above equationinvolves decomposition of 120601(X) into an orthogonal matrixQ(satisfyingQQT

= I) and a right triangularmatrixR such thatQ lowast R = 120601(X) We have

(120601 (X))T 120601 (X) = RTR (14)

Let us map W120601 into the span of Q Q is currently anorthogonal basis of 120601(X) so we have

W120601 = QV120601 (15)

where V120601 = R119896times119899 is an orthogonal matrix satisfying(V120601)TV120601 = I Using the definitions in (14) and (15) (13) canbe further rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TRL119861RTV120601)

Tr ((V120601)T RL119882RTV120601)

(16)

Let S119861= RL119861RT and S

119882= RL119882RT then (16) can be further

rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TS119861V120601)

Tr ((V120601)T S119882V120601)

(17)

Shock and Vibration 5

After the matrix V120601lowast is obtained with the BIGA-basedsolution method (to be in detail discussed in Section 3) theoutput points in the reduced data space can thus be expressedas

Y120601lowast

= (W120601lowast

)

T120601 (X) = (V120601

lowast

)

TQTQR = (V120601

lowast

)

TR (18)

3 The Proposed TR-KDA-BIGA

As previously mentioned construction of TR-KDA needsto select 119896 out of 119889 eigenvectors to form the matrix V120601for dimensionality reduction However finding a subset ofeigenvectors based on the trace ratio criterion is not an easytask since the space of possible subsets is very large especiallywhen 119889 is a large number Thus it is not impractical to useexhaustive search to find an optimal subset of 119896 eigenvectorsInstead in this study the BIGA is utilized to select 119896 outof 119889 eigenvectors of V120601

119905as the bases for projection matrix

formulation based on the trace ratio criterion such thatthe trace ratio value 120582 = Tr[(V120601)TS

119861V120601]Tr[(V120601)TS

119882V120601]

can be maximized Immune genetic algorithm originallydeveloped by Jiao andWang [33] is a novel genetic algorithmbased on the biological immune theory which combined theimmune mechanism with the evolutionary mechanism Inwhat follows further discussion of the proposed TR-KDA-BIGA is carried out

31 Chromosome Encoding Encoding a solution of a probleminto a chromosome is an important issue when using BIGAsIn this study every chromosome in a BIGA corresponds to adiscrete binary selector u = [119906

1 1199062 119906

119889] where each gene

in the chromosome is ldquo1rdquo indicating an eigenvector k120601119894(119894 =

1 2 119889) of S119861minus 120582119905S119882appearing in forming the projection

matrixV120601 of the 119905th step while ldquo0rdquo denotes its absenceThusthe length of the chromosome is 119889

32 Genetic Operators Genetic operators give every chro-mosome the chance to become the fittest chromosome ofits generation If it is difficult to reach the target of traceratio optimization crossover and mutation may introducedegeneracy into generations of chromosomes

321 Crossover Operator Crossover operator in a BIGAis employed to generate two new children chromosomesbased on two existing parent chromosomes selected from thecurrent population in terms of a prespecified crossover rateIn this study ldquoone-pointrdquo crossover operator was adopted torandomly select a cut point to exchange the parts betweenthe cut point and the end of the string of the parentchromosomes Specifically suppose that two parent chromo-somes 119875

1and 119875

2selected randomly from the population are

undergoing the crossover operation at a randomly selectedcrossover point 119892 (1 le 119892 le 119889) where

1198751= (1199061198941 1199061198942 119906

119894119889)

1198752= (1199061198951 1199061198952 119906

119895119889)

(19)

Consequently the offspring is generated by one-pointcrossover on the genes of two parents selected randomlyfrom the population We can thus get the two offspringchromosomes 119862

1and 119862

2

1198621= (1199061198941 1199061198942 119906

119894119892minus1 119906119894119892 119906119895119892+1

119906119895119889)

1198622= (1199061198951 1199061198952 119906

119895119892minus1 119906119895119892 119906119894119892+1

119906119894119889)

(20)

However the exchange procedure is not simply exchangingtheir genetic information between gene segments after thecrossover pointsWemust keep the number of eigenvectors tobe included in the subset equal to 119896 In this study therefore asimple but effective crossover operator strategy in this study isperformed in order to ensure that the crossover operator doesnot change the total number of ldquo1rdquo genes in chromosomes

Let

1198991198751=

119889

sum

119902=119892+1

119906119894119902

1198991198752=

119889

sum

119902=119892+1

119906119895119902

(21)

When 1198991198751is not equal to 119899

1198752 the following retention criterion

will be conducted(1) If 119899

1198751is larger than 119899

1198752 randomly select (119899

1198751minus

1198991198752) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo1-bitrdquo and then randomly select (1198991198751

minus

1198991198752) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo0-bitrdquo(2) If 119899

1198751is smaller than 119899

1198752 randomly select (119899

1198752minus

1198991198751) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo0-bitrdquo and then randomly select (1198991198752

minus

1198991198751) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo1-bitrdquo

322 Mutation Operator Mutation operator in a BIGA isused primarily as a mechanism for maintaining diversityin the population For each gene in a chromosome that isundergoing the mutation a real-valued number is randomlyselected within the range of [0 1] If the real-valued numberis less than the prespecified mutation rate then the genewill change from ldquo0-bitrdquo to ldquo1-bitrdquo and vice versa Uponadding (or removing) one eigenvector in that way we shallrandomly remove (or add) a different one such that thenumber of eigenvectors to be included in the subset is equalto 119896 The mutation operator helps the chromosomes to guidethe search in new areas

33 Immune Operators The immune ability of BIGAs is real-ized through two kinds of immune operators a vaccinationand an immune selection The vaccination is responsiblefor improving individualsrsquo overall fitness levels The immuneselection is responsible for prevention of deterioration

6 Shock and Vibration

331 Vaccination Operator Given a chromosome u vac-cination operation in a BIGA is employed to modify thegenes on some bits according to a priori knowledge such thatindividuals with higher fitness have a greater probability ofbeing selected Let U = (u

1 u2 u

1198990) be a population

the vaccination operation on U means that the operationis performed on 119899

120572= 120572119899

0chromosomes selected from

U according to the proportion of 120572 where 1198990represents

the population size of a BIGA A vaccine is abstractedfrom the prior knowledge of the pending problem whoseinformation amount and validity play an important role inthe performance of the algorithm

332 Immune Selection Operator The immune selectionoperation consists of the following two steps The first step isthe immunity test if the fitness of a chromosome u is smallerthan that of the parent chromosome which indicates thatdegeneration occurred during crossover and mutation thenthe parent chromosomewill be used for the next competitionThe second step is the annealing selection [35] a chromo-some u

119894is selected from the current offspring population U

119896

to join with the new parents with the probability as follows

119875 (u119894) =

exp (119891 (u119894) 119879119896)

sum1198990

119894=1exp (119891 (u

119894) 119879119896)

(22)

where 119891(u119894) is the fitness of the individual u

119894and 119879

119896 is the

temperature-controlled series tending towards 0

34 Fitness Evaluation Fitness evaluation plays a criticalrole in selecting offspring chromosomes from the currentpopulation for the next generation In this study the fitnessfunction for eigenvector selection is defined as

119891 (u) = 1199061ℎ1+ 1199062ℎ2+ sdot sdot sdot + 119906

119889ℎ119889

11990611198921+ 11990621198922+ sdot sdot sdot + 119906

119889119892119889

=

uhT

ugT (23)

where ℎ119894denotes the vT

119894S119861v119894value for the 119894th eigenvector

119892119894denotes the vT

119894S119882v119894value for the 119894th eigenvector h =

[ℎ1 ℎ2 ℎ

119889] g = [119892

1 1198922 119892

119889] u = [119906

1 1199062 119906

119889]

119906119894isin 0 1 u1T = 119896 and 119894 = 1 2 119889 Notably u is

called the binary selector and 119896 is the desired lower featuredimension Finally according to the evolved binary selectoru we can thus form the projection matrix V120601 of the 119905thstep by choosing the 119896 eigenvectors with 119906

119894= 1 (119894 =

1 2 119889) The procedures of the proposed TR-KDA-BIGAare summarized in the procedures of the proposed TR-KDA-BIGA part The computational flow of the BIGA obtainedusing the aforementioned genetic and immune operators isalso provided in the computational flow of the BIGA part

The Procedures of the Proposed TR-KDA-BIGA The proce-dures are as follows

(1) Construct the kernel matrix K = (120601(X))T120601(X)(2) Perform Cholesky decomposition to the kernel

matrix K = RTR(3) Form the kernel scatter matrixes as S

119861= RL119861RT and

S119882= RL119882RT

(4) Set iterations number 119905 to 1(5) Set the initial trace ratio value 120582

119905to Tr(S

119861)Tr(S

119882)

(6) Compute the eigendecomposition of S119861minus 120582119905S119882

as(S119861minus 120582119905S119882)k120601119894= 120591119894k120601119894 where k120601

119894(119894 = 1 2 119889) is

the eigenvector of S119861minus 120582119905S119882

(7) Calculate ℎ119894= (k120601119894)TS119861k120601119894and 119892

119894= (k120601119894)TS119882k120601119894for

each eigenvector k120601119894(119894 = 1 2 119889)

(8) Generate a population of BIGA selectors(9) Evolve the population where the fitness of a BIGA

selector u is measured as 119891(u) = uhTugT(10) ulowast is the evolved best BIGA selector

(11) Form the projection matrix V120601119905by choosing the 119896

eigenvectors k120601119894with 119906lowast

119894= 1 (119894 = 1 2 119889)

(12) Update the trace ratio value 120582119905+1

=

Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] 119905 = 119905 + 1 and go to

step (6) Repeat this procedure until a convergencecondition was established when the trace ratio valuedoes not increase in consecutive 5 iterations

(13) Output Y120601lowast = (V120601lowast)TR

TheComputational Flow of the BIGAThe computational flowis as follows

(1) Set 119897 (time of generation) to 1(2) Initialize randomly the original population 119860

119897

(3) Evaluate each chromosome in the original population119860119897

(4) Abstract vaccines according to the prior knowledge(5) Check for termination criteria If the fixed number

of generations is not reached or the optimal chromo-some found thus far is not satisfied then go to the nextstep Otherwise output the optimal chromosome asthe final solutions for further decision-making

(6) Perform crossover operation on the 119860119897and then

generate the population 119861119897

(7) Perform mutation operation on the 119861119897and then

generate the population results 119862119897

(8) Perform vaccination operation on the 119862119897and then

generate the population119863119897

(9) Perform immune selection operation on the 119863119897and

then generate the next generational population 119860119897+1

Go to step (3)

4 The Convergence of theProposed TR-KDA-BIGA

In this section we analyze the convergence of the proposedTR-LDA-BIGA Before doing this task it should be worthnoting that the BIGA is convergent It has been demonstratedby Jiao and Wang [33] that as long as enough iteration has

Shock and Vibration 7

been completed the immune genetic population convergestowards the true optimum with probability one

Recall the trace difference function

119911 (120582) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582S119882)V120601] (24)

it follows that

119911 (120582119905+1) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582119905+1S119882)V120601] 997904rArr

119911 (120582119905+1) = Tr [(V120601

119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

(25)

Since 120582119905+1

= Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] as previously

mentioned we get

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (26)

Consider the inequality

Tr [(V120601119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

ge Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905]

(27)

and the equation

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (28)

and we have

119911 (120582119905+1) ge 0

Tr [(V120601119905+1)

TS119861V120601119905+1] minus 120582119905+1

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 0

(29)

Consequently

Tr [(V120601119905+1)

TS119861V120601119905+1]

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 120582119905+1

997904rArr

120582119905+2

ge 120582119905+1

(30)

Substituting the subscript 119905 + 1 by 119905 yields

120582119905+1

ge 120582119905 (31)

So we obtain the following inequality which gives the firstexpression of convergence of the proposed TR-KDA-BIGA

Further suppose that 120582lowast is the optimal trace ratio valueit follows that

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

] = 0 (32)

Table 1 Specification of benchmark problems

Data set samples dim classHeart-statlog 270 13 2Ionosphere 351 34 2Iris 150 4 3Wine 178 13 3Waveform 5000 40 3Balance 625 4 3SCCTS 600 60 6

where V120601lowast is the optimal projection matrix We thereforehave

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

]

ge Tr [(V120601119905)

T(S119861minus 120582lowastS119882)V120601119905]

= Tr [(V120601119905)

T(S119861minus 120582119905+1

S119882)V120601119905]

+ (120582119905+1

minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

= 119911 (120582119905+1) + (120582

119905+1minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

(33)

Consider 119911(120582lowast) = 0 119911(120582119905+1) ge 0 and S

119882is semipositive

definite we have

120582119905+1

minus 120582lowastle 0 (34)

So we obtain the following inequality which gives the secondexpression of convergence of the proposed TR-KDA-BIGA

120582119905+1

le 120582lowast (35)

We conclude therefore that for a particular initial trace ratiovalue 120582

119905 the updated value 120582

119905+1can always satisfy (1) 120582

119905+1ge

120582119905and (2) 120582

119905+1le 120582lowast

5 Performance Evaluation onBenchmark Problems

In order to extensively verify the performance of the proposedTR-KDA-BIGA it is first tested on wide types of commonlyused benchmark problems taken from the UCI machinelearning repository and evaluated with the classificationrate (ie the number of correctly identified training exam-plestotal number of training examples) by comparison withother existing methods such as PCA LDA KPCA [30 31]KDA [32] and TR-LDA These data sets include Heart-statlog Ionosphere Iris Wine Waveform Balance and Syn-thetic Control Chart Time Series (SCCTS) data sets (Table 1)which are of small sizes low dimensions large sizes andorhigh dimensions For comparative study we randomly select50 data points from each data set as training set and therest of the data points as test set All methods use training set

8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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Shock and Vibration 3

2 Review of TR-LDA and Kernel Extension

21 Review of TR-LDA Suppose we are given a set of 119899 119889-dimensional samples x

1 x2 x

119899 belonging to 119897 different

classes The goal of LDA tries to obtain a linear projectionmatrix W isin R119889times119896 that can map the original 119889-dimensionaldata x

119894onto the 119896-dimensional data y

119894(usually 119896 ≪ 119889)

by maximizing the between-class scatter and meanwhileminimizing the within-class scatter The between-class scat-ter matrix S

119861and the within-class scatter matrix S

119882are

expressed as follows

S119861=

119897

sum

119894=1

119899119894(m(119894) minusm) (m(119894) minusm)

T

S119882=

119897

sum

119894=1

(

119899119894

sum

119895=1

(x(119894)119895minusm(119894)) (x(119894)

119895minusm(119894))

T)

(1)

where m represents the total sample mean vector 119899119894repre-

sents the number of samples in the 119894th class m(119894) representsthe average vector of the 119894th class and x(119894)

119895represents the119895th

sample in the 119894th class The new mapped feature vectors 119910119894isin

R119896 can then be expressed as y119894= WTx

119894The original LDA for-

mulation known as the Fisher LDA [21] only handles binaryclassification problemsHowevermany practical applicationsinvolve multiclass classification In order to overcome thisissue a number of researchers have proposed optimizationcriteria for extending the Fisher LDA to handle multiclassclassification problems The first optimization criterion is ina ratio trace form (referred to as RT-LDA)

Wlowast = arg maxWTW=I

Tr(WTS119861W

WTS119882W) (2)

where Tr[sdot] denotes the matrix trace I is an identity matrixIn order to achieve a set of orthogonal normalized vectorsit usually adds the constraint WTW = I to (2) The secondoptimization criterion is in a trace ratio form (referred to asTR-LDA)

Wlowast = arg maxWTW=I

Tr (WTS119861W)

Tr (WTS119882W)

(3)

The optimization problem in (3) can be solved directlythrough the generalized eigenvalue decomposition (GED)method [22]

S119861w119894= 120591119894S119882w119894 (4)

where 120591119894is the 119894th largest eigenvaluew

119894is the eigenvector cor-

responding to 120591119894 and w

119894constitutes the 119894th column vector of

the matrixW Although a closed-form solution for (3) can beapproximately obtained with the GED it does not necessarilyguarantee best trace ratio optimization Thus this approxi-mation of ratio trace optimization to trace ratio optimizationmay lead to classification capability loss of the derived opti-mal low-dimensional feature space Moreover the physicalmeaning of the trace ratio form is clearer than that of

the ratio trace form However the optimization problem in(3) is generally nonconvex and a closed-form solution for itdoes not exist Fortunately a recent study conducted by Guoet al [26] showed that using the trace difference function119911(120582) = maxWTW=ITr[WT

(S119861minus 120582S119882)W] the trace ratio prob-

lem can be solved equivalently by finding zero points of theequation 119911(120582) = 0 Following up Guo et alrsquos work Wang et alpresented an iterative method named ITR algorithm to solvethe trace ratio problem [22]The ITR algorithm optimizes theobjective function 120582 = maxWTW=ITr[WTS

119861W]Tr[WTS

119882W]

in an iterative and incremental mannerTheW in the 119905th iter-ation step (referred to asW

119905) is obtained through solving the

trace difference problem argmaxWTW=ITr[WT(S119861minus120582119905S119882)W]

where 120582119905represents the trace ratio value derived from the

W in the previous iteration step (referred to as W119905minus1

)However the initialization for theW influences substantiallythe convergence performance of the ITR algorithm A goodinitialization can generally make the ITR algorithm yield aquick convergence A bad initialization usually increases thenumber of iterationsMoreover in ITR algorithm although itseems that theW formedwith these eigenvectors correspond-ing to the 119896 largest eigenvalues of S

119861minus120582119905S119882canmaximize the

trace difference Tr[WT(S119861minus 120582119905S119882)W] it cannot necessarily

maximize the trace ratio Tr[WTS119861W]Tr[WTS

119882W] On the

other hand from the perspective of fault diagnosis the aimis mainly to find a set of projection vectors that can pose thehighest levels of discrimination in the different fault patternsThus these eigenvectors with the largest eigenvalues are notnecessarily representative for discriminating one class fromothers as previously mentioned in Section 1 To overcomethe above shortcomings this study presents a BIGA-basedsolution method for trace ratio criterion (to be in detaildiscussed in Section 3)

22 Kernel Extension In some applications it is insufficientto model the data using the TR-LDA which is a lineardiscriminatingmethod To address the issue of nonlinearitiesin the data this section presents a nonlinear discriminatingmethod using kernel trick [30ndash32] that is TR-KDA Theso-called kernel trick is to map the original data to a high-dimensional Hilbert space through a nonlinear mappingfunction 120601 Let 120601(X) denote the data matrix in the Hilbertspace 120601(X) = [120601(x

1) 120601(x2) 120601(x

119899)] The function form

of the mapping does not need to be known since it isimplicitly defined by the choice of kernel function119870(x

119894 x119895) =

120601(x119894)T120601(x119895) that is the inner product in the kernel-induced

feature space The kernel function 119870 may be any positivekernel satisfying Mercerrsquos condition Radial basis function(RBF) kernel function one of the most popular kernelfunctions employed in various kernelled learning algorithmsis adopted in this study Then (3) in Hilbert space can bewritten as follows

W120601lowast

= arg max(W120601)TW120601=I

Tr ((W120601)TS120601119861W120601)

Tr ((W120601)T S120601119882W120601)

(5)

where W120601 S120601119861 and S120601

119882are the matrices in Hilbert space

corresponding toW S119861 and S

119882in (3) respectively Notably

4 Shock and Vibration

we can show that matrices S119861and S119882in (3) can be essentially

expressed as S119861

= XL119861XT and S

119882= XL

119882XT through

simple manipulation respectively X is the vector where X =

[x1 x2 x

119899] Matrices L

119861and L

119882are the graph Laplacian

matrices [34] of theweighted undirected graphs reflecting thebetween-class and within-class relationship of the samplesConsider

S119861=

119897

sum

119894=1

119899119894(m(119894) minusm) (m(119894) minusm)

T

= (

119897

sum

119894=1

119899119894m(119894) (m(119894))

T) minusm(

119897

sum

119894=1

119899119894(m(119894))

T)

minus (

119897

sum

119894=1

119899119894m(119894))mT

+ (

119897

sum

119894=1

119899119894)mmT

= (

119897

sum

119894=1

1

119899119894

(x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

sdot (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) minus 2119899mmT

+ 119899mmT

= (

119897

sum

119894=1

119899119894

sum

119895119902=1

1

119899119894

x(119894)119895(x(119894)119902)

T) minus 119899mmT

= XGXT

minus 119899mmT= XGXT

minus X (

1

119899

eeT)XT= X (G minus

1

119899

sdot eeT)XT

(6)

where e = [1 1 1]T is an 119899-dimensional vector We can

simplify the above equation even further by defining that

(G)119894119895

=

1

119899119902

if samples x119894and x

119895belong to the 119902th class

0 otherwise

(7)

L119861= G minus

1

119899

eeT (8)

Thus we get

S119861= XL119861XT

(9)

Then the matrix S119882can similarly be computed as follows

S119882=

119897

sum

119894=1

(

119899119894

sum

119895=1

(x(119894)119895minusm(119894)) (x(119894)

119895minusm(119894))

T)

=

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

Tminusm(119894) (x(119894)

119895)

Tminus x(119894)119895(m(119894))

T

+m(119894) (m(119894))T) =

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

T

minus 119899119894m(119894) (m(119894))

T) =

119897

sum

119894=1

(X119894XT119894minus

1

119899119894

(x(119894)1+ x(119894)2

+ sdot sdot sdot + x(119894)119899119894) (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) =

119897

sum

119894=1

(X119894XT119894

minus

1

119899119894

X119894(e119894eT119894)XT119894) =

119897

sum

119894=1

(X119894(I minus 1

119899119894

e119894eT119894)XT119894)

=

119897

sum

119894=1

X119894L119894XT119894

(10)

where X119894= [x(119894)

1 x(119894)2 x(119894)

119899119894] is a 119889 times 119899

119894matrix e

119894=

[1 1 1]T is an 119899

119894-dimensional vector I is the identity

matrix L119894= I minus 1119899

119894e119894eT119894is an 119899

119894times 119899119894matrix and X

119894L119894XT119894

is the data covariance matrix of the 119894th class Based on (7)the above equation can be simplified similarly by defining

L119882= I minus G (11)

Thus we get

S119882= XL119882XT

(12)

Using the definitions in (9) and (12) (5) can be rewritten asfollows

W120601lowast

= arg max(W120601)TW120601=I

Tr ((W120601)T(120601 (X) L

119861(120601 (X))T)W120601)

Tr ((W120601)T (120601 (X) L119882(120601 (X))T)W120601)

(13)

In order to pursue thematrixW120601lowast

solving the above equationinvolves decomposition of 120601(X) into an orthogonal matrixQ(satisfyingQQT

= I) and a right triangularmatrixR such thatQ lowast R = 120601(X) We have

(120601 (X))T 120601 (X) = RTR (14)

Let us map W120601 into the span of Q Q is currently anorthogonal basis of 120601(X) so we have

W120601 = QV120601 (15)

where V120601 = R119896times119899 is an orthogonal matrix satisfying(V120601)TV120601 = I Using the definitions in (14) and (15) (13) canbe further rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TRL119861RTV120601)

Tr ((V120601)T RL119882RTV120601)

(16)

Let S119861= RL119861RT and S

119882= RL119882RT then (16) can be further

rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TS119861V120601)

Tr ((V120601)T S119882V120601)

(17)

Shock and Vibration 5

After the matrix V120601lowast is obtained with the BIGA-basedsolution method (to be in detail discussed in Section 3) theoutput points in the reduced data space can thus be expressedas

Y120601lowast

= (W120601lowast

)

T120601 (X) = (V120601

lowast

)

TQTQR = (V120601

lowast

)

TR (18)

3 The Proposed TR-KDA-BIGA

As previously mentioned construction of TR-KDA needsto select 119896 out of 119889 eigenvectors to form the matrix V120601for dimensionality reduction However finding a subset ofeigenvectors based on the trace ratio criterion is not an easytask since the space of possible subsets is very large especiallywhen 119889 is a large number Thus it is not impractical to useexhaustive search to find an optimal subset of 119896 eigenvectorsInstead in this study the BIGA is utilized to select 119896 outof 119889 eigenvectors of V120601

119905as the bases for projection matrix

formulation based on the trace ratio criterion such thatthe trace ratio value 120582 = Tr[(V120601)TS

119861V120601]Tr[(V120601)TS

119882V120601]

can be maximized Immune genetic algorithm originallydeveloped by Jiao andWang [33] is a novel genetic algorithmbased on the biological immune theory which combined theimmune mechanism with the evolutionary mechanism Inwhat follows further discussion of the proposed TR-KDA-BIGA is carried out

31 Chromosome Encoding Encoding a solution of a probleminto a chromosome is an important issue when using BIGAsIn this study every chromosome in a BIGA corresponds to adiscrete binary selector u = [119906

1 1199062 119906

119889] where each gene

in the chromosome is ldquo1rdquo indicating an eigenvector k120601119894(119894 =

1 2 119889) of S119861minus 120582119905S119882appearing in forming the projection

matrixV120601 of the 119905th step while ldquo0rdquo denotes its absenceThusthe length of the chromosome is 119889

32 Genetic Operators Genetic operators give every chro-mosome the chance to become the fittest chromosome ofits generation If it is difficult to reach the target of traceratio optimization crossover and mutation may introducedegeneracy into generations of chromosomes

321 Crossover Operator Crossover operator in a BIGAis employed to generate two new children chromosomesbased on two existing parent chromosomes selected from thecurrent population in terms of a prespecified crossover rateIn this study ldquoone-pointrdquo crossover operator was adopted torandomly select a cut point to exchange the parts betweenthe cut point and the end of the string of the parentchromosomes Specifically suppose that two parent chromo-somes 119875

1and 119875

2selected randomly from the population are

undergoing the crossover operation at a randomly selectedcrossover point 119892 (1 le 119892 le 119889) where

1198751= (1199061198941 1199061198942 119906

119894119889)

1198752= (1199061198951 1199061198952 119906

119895119889)

(19)

Consequently the offspring is generated by one-pointcrossover on the genes of two parents selected randomlyfrom the population We can thus get the two offspringchromosomes 119862

1and 119862

2

1198621= (1199061198941 1199061198942 119906

119894119892minus1 119906119894119892 119906119895119892+1

119906119895119889)

1198622= (1199061198951 1199061198952 119906

119895119892minus1 119906119895119892 119906119894119892+1

119906119894119889)

(20)

However the exchange procedure is not simply exchangingtheir genetic information between gene segments after thecrossover pointsWemust keep the number of eigenvectors tobe included in the subset equal to 119896 In this study therefore asimple but effective crossover operator strategy in this study isperformed in order to ensure that the crossover operator doesnot change the total number of ldquo1rdquo genes in chromosomes

Let

1198991198751=

119889

sum

119902=119892+1

119906119894119902

1198991198752=

119889

sum

119902=119892+1

119906119895119902

(21)

When 1198991198751is not equal to 119899

1198752 the following retention criterion

will be conducted(1) If 119899

1198751is larger than 119899

1198752 randomly select (119899

1198751minus

1198991198752) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo1-bitrdquo and then randomly select (1198991198751

minus

1198991198752) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo0-bitrdquo(2) If 119899

1198751is smaller than 119899

1198752 randomly select (119899

1198752minus

1198991198751) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo0-bitrdquo and then randomly select (1198991198752

minus

1198991198751) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo1-bitrdquo

322 Mutation Operator Mutation operator in a BIGA isused primarily as a mechanism for maintaining diversityin the population For each gene in a chromosome that isundergoing the mutation a real-valued number is randomlyselected within the range of [0 1] If the real-valued numberis less than the prespecified mutation rate then the genewill change from ldquo0-bitrdquo to ldquo1-bitrdquo and vice versa Uponadding (or removing) one eigenvector in that way we shallrandomly remove (or add) a different one such that thenumber of eigenvectors to be included in the subset is equalto 119896 The mutation operator helps the chromosomes to guidethe search in new areas

33 Immune Operators The immune ability of BIGAs is real-ized through two kinds of immune operators a vaccinationand an immune selection The vaccination is responsiblefor improving individualsrsquo overall fitness levels The immuneselection is responsible for prevention of deterioration

6 Shock and Vibration

331 Vaccination Operator Given a chromosome u vac-cination operation in a BIGA is employed to modify thegenes on some bits according to a priori knowledge such thatindividuals with higher fitness have a greater probability ofbeing selected Let U = (u

1 u2 u

1198990) be a population

the vaccination operation on U means that the operationis performed on 119899

120572= 120572119899

0chromosomes selected from

U according to the proportion of 120572 where 1198990represents

the population size of a BIGA A vaccine is abstractedfrom the prior knowledge of the pending problem whoseinformation amount and validity play an important role inthe performance of the algorithm

332 Immune Selection Operator The immune selectionoperation consists of the following two steps The first step isthe immunity test if the fitness of a chromosome u is smallerthan that of the parent chromosome which indicates thatdegeneration occurred during crossover and mutation thenthe parent chromosomewill be used for the next competitionThe second step is the annealing selection [35] a chromo-some u

119894is selected from the current offspring population U

119896

to join with the new parents with the probability as follows

119875 (u119894) =

exp (119891 (u119894) 119879119896)

sum1198990

119894=1exp (119891 (u

119894) 119879119896)

(22)

where 119891(u119894) is the fitness of the individual u

119894and 119879

119896 is the

temperature-controlled series tending towards 0

34 Fitness Evaluation Fitness evaluation plays a criticalrole in selecting offspring chromosomes from the currentpopulation for the next generation In this study the fitnessfunction for eigenvector selection is defined as

119891 (u) = 1199061ℎ1+ 1199062ℎ2+ sdot sdot sdot + 119906

119889ℎ119889

11990611198921+ 11990621198922+ sdot sdot sdot + 119906

119889119892119889

=

uhT

ugT (23)

where ℎ119894denotes the vT

119894S119861v119894value for the 119894th eigenvector

119892119894denotes the vT

119894S119882v119894value for the 119894th eigenvector h =

[ℎ1 ℎ2 ℎ

119889] g = [119892

1 1198922 119892

119889] u = [119906

1 1199062 119906

119889]

119906119894isin 0 1 u1T = 119896 and 119894 = 1 2 119889 Notably u is

called the binary selector and 119896 is the desired lower featuredimension Finally according to the evolved binary selectoru we can thus form the projection matrix V120601 of the 119905thstep by choosing the 119896 eigenvectors with 119906

119894= 1 (119894 =

1 2 119889) The procedures of the proposed TR-KDA-BIGAare summarized in the procedures of the proposed TR-KDA-BIGA part The computational flow of the BIGA obtainedusing the aforementioned genetic and immune operators isalso provided in the computational flow of the BIGA part

The Procedures of the Proposed TR-KDA-BIGA The proce-dures are as follows

(1) Construct the kernel matrix K = (120601(X))T120601(X)(2) Perform Cholesky decomposition to the kernel

matrix K = RTR(3) Form the kernel scatter matrixes as S

119861= RL119861RT and

S119882= RL119882RT

(4) Set iterations number 119905 to 1(5) Set the initial trace ratio value 120582

119905to Tr(S

119861)Tr(S

119882)

(6) Compute the eigendecomposition of S119861minus 120582119905S119882

as(S119861minus 120582119905S119882)k120601119894= 120591119894k120601119894 where k120601

119894(119894 = 1 2 119889) is

the eigenvector of S119861minus 120582119905S119882

(7) Calculate ℎ119894= (k120601119894)TS119861k120601119894and 119892

119894= (k120601119894)TS119882k120601119894for

each eigenvector k120601119894(119894 = 1 2 119889)

(8) Generate a population of BIGA selectors(9) Evolve the population where the fitness of a BIGA

selector u is measured as 119891(u) = uhTugT(10) ulowast is the evolved best BIGA selector

(11) Form the projection matrix V120601119905by choosing the 119896

eigenvectors k120601119894with 119906lowast

119894= 1 (119894 = 1 2 119889)

(12) Update the trace ratio value 120582119905+1

=

Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] 119905 = 119905 + 1 and go to

step (6) Repeat this procedure until a convergencecondition was established when the trace ratio valuedoes not increase in consecutive 5 iterations

(13) Output Y120601lowast = (V120601lowast)TR

TheComputational Flow of the BIGAThe computational flowis as follows

(1) Set 119897 (time of generation) to 1(2) Initialize randomly the original population 119860

119897

(3) Evaluate each chromosome in the original population119860119897

(4) Abstract vaccines according to the prior knowledge(5) Check for termination criteria If the fixed number

of generations is not reached or the optimal chromo-some found thus far is not satisfied then go to the nextstep Otherwise output the optimal chromosome asthe final solutions for further decision-making

(6) Perform crossover operation on the 119860119897and then

generate the population 119861119897

(7) Perform mutation operation on the 119861119897and then

generate the population results 119862119897

(8) Perform vaccination operation on the 119862119897and then

generate the population119863119897

(9) Perform immune selection operation on the 119863119897and

then generate the next generational population 119860119897+1

Go to step (3)

4 The Convergence of theProposed TR-KDA-BIGA

In this section we analyze the convergence of the proposedTR-LDA-BIGA Before doing this task it should be worthnoting that the BIGA is convergent It has been demonstratedby Jiao and Wang [33] that as long as enough iteration has

Shock and Vibration 7

been completed the immune genetic population convergestowards the true optimum with probability one

Recall the trace difference function

119911 (120582) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582S119882)V120601] (24)

it follows that

119911 (120582119905+1) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582119905+1S119882)V120601] 997904rArr

119911 (120582119905+1) = Tr [(V120601

119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

(25)

Since 120582119905+1

= Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] as previously

mentioned we get

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (26)

Consider the inequality

Tr [(V120601119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

ge Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905]

(27)

and the equation

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (28)

and we have

119911 (120582119905+1) ge 0

Tr [(V120601119905+1)

TS119861V120601119905+1] minus 120582119905+1

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 0

(29)

Consequently

Tr [(V120601119905+1)

TS119861V120601119905+1]

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 120582119905+1

997904rArr

120582119905+2

ge 120582119905+1

(30)

Substituting the subscript 119905 + 1 by 119905 yields

120582119905+1

ge 120582119905 (31)

So we obtain the following inequality which gives the firstexpression of convergence of the proposed TR-KDA-BIGA

Further suppose that 120582lowast is the optimal trace ratio valueit follows that

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

] = 0 (32)

Table 1 Specification of benchmark problems

Data set samples dim classHeart-statlog 270 13 2Ionosphere 351 34 2Iris 150 4 3Wine 178 13 3Waveform 5000 40 3Balance 625 4 3SCCTS 600 60 6

where V120601lowast is the optimal projection matrix We thereforehave

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

]

ge Tr [(V120601119905)

T(S119861minus 120582lowastS119882)V120601119905]

= Tr [(V120601119905)

T(S119861minus 120582119905+1

S119882)V120601119905]

+ (120582119905+1

minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

= 119911 (120582119905+1) + (120582

119905+1minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

(33)

Consider 119911(120582lowast) = 0 119911(120582119905+1) ge 0 and S

119882is semipositive

definite we have

120582119905+1

minus 120582lowastle 0 (34)

So we obtain the following inequality which gives the secondexpression of convergence of the proposed TR-KDA-BIGA

120582119905+1

le 120582lowast (35)

We conclude therefore that for a particular initial trace ratiovalue 120582

119905 the updated value 120582

119905+1can always satisfy (1) 120582

119905+1ge

120582119905and (2) 120582

119905+1le 120582lowast

5 Performance Evaluation onBenchmark Problems

In order to extensively verify the performance of the proposedTR-KDA-BIGA it is first tested on wide types of commonlyused benchmark problems taken from the UCI machinelearning repository and evaluated with the classificationrate (ie the number of correctly identified training exam-plestotal number of training examples) by comparison withother existing methods such as PCA LDA KPCA [30 31]KDA [32] and TR-LDA These data sets include Heart-statlog Ionosphere Iris Wine Waveform Balance and Syn-thetic Control Chart Time Series (SCCTS) data sets (Table 1)which are of small sizes low dimensions large sizes andorhigh dimensions For comparative study we randomly select50 data points from each data set as training set and therest of the data points as test set All methods use training set

8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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4 Shock and Vibration

we can show that matrices S119861and S119882in (3) can be essentially

expressed as S119861

= XL119861XT and S

119882= XL

119882XT through

simple manipulation respectively X is the vector where X =

[x1 x2 x

119899] Matrices L

119861and L

119882are the graph Laplacian

matrices [34] of theweighted undirected graphs reflecting thebetween-class and within-class relationship of the samplesConsider

S119861=

119897

sum

119894=1

119899119894(m(119894) minusm) (m(119894) minusm)

T

= (

119897

sum

119894=1

119899119894m(119894) (m(119894))

T) minusm(

119897

sum

119894=1

119899119894(m(119894))

T)

minus (

119897

sum

119894=1

119899119894m(119894))mT

+ (

119897

sum

119894=1

119899119894)mmT

= (

119897

sum

119894=1

1

119899119894

(x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

sdot (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) minus 2119899mmT

+ 119899mmT

= (

119897

sum

119894=1

119899119894

sum

119895119902=1

1

119899119894

x(119894)119895(x(119894)119902)

T) minus 119899mmT

= XGXT

minus 119899mmT= XGXT

minus X (

1

119899

eeT)XT= X (G minus

1

119899

sdot eeT)XT

(6)

where e = [1 1 1]T is an 119899-dimensional vector We can

simplify the above equation even further by defining that

(G)119894119895

=

1

119899119902

if samples x119894and x

119895belong to the 119902th class

0 otherwise

(7)

L119861= G minus

1

119899

eeT (8)

Thus we get

S119861= XL119861XT

(9)

Then the matrix S119882can similarly be computed as follows

S119882=

119897

sum

119894=1

(

119899119894

sum

119895=1

(x(119894)119895minusm(119894)) (x(119894)

119895minusm(119894))

T)

=

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

Tminusm(119894) (x(119894)

119895)

Tminus x(119894)119895(m(119894))

T

+m(119894) (m(119894))T) =

119897

sum

119894=1

(

119899119894

sum

119895=1

x(119894)119895(x(119894)119895)

T

minus 119899119894m(119894) (m(119894))

T) =

119897

sum

119894=1

(X119894XT119894minus

1

119899119894

(x(119894)1+ x(119894)2

+ sdot sdot sdot + x(119894)119899119894) (x(119894)1+ x(119894)2+ sdot sdot sdot + x(119894)

119899119894)

T) =

119897

sum

119894=1

(X119894XT119894

minus

1

119899119894

X119894(e119894eT119894)XT119894) =

119897

sum

119894=1

(X119894(I minus 1

119899119894

e119894eT119894)XT119894)

=

119897

sum

119894=1

X119894L119894XT119894

(10)

where X119894= [x(119894)

1 x(119894)2 x(119894)

119899119894] is a 119889 times 119899

119894matrix e

119894=

[1 1 1]T is an 119899

119894-dimensional vector I is the identity

matrix L119894= I minus 1119899

119894e119894eT119894is an 119899

119894times 119899119894matrix and X

119894L119894XT119894

is the data covariance matrix of the 119894th class Based on (7)the above equation can be simplified similarly by defining

L119882= I minus G (11)

Thus we get

S119882= XL119882XT

(12)

Using the definitions in (9) and (12) (5) can be rewritten asfollows

W120601lowast

= arg max(W120601)TW120601=I

Tr ((W120601)T(120601 (X) L

119861(120601 (X))T)W120601)

Tr ((W120601)T (120601 (X) L119882(120601 (X))T)W120601)

(13)

In order to pursue thematrixW120601lowast

solving the above equationinvolves decomposition of 120601(X) into an orthogonal matrixQ(satisfyingQQT

= I) and a right triangularmatrixR such thatQ lowast R = 120601(X) We have

(120601 (X))T 120601 (X) = RTR (14)

Let us map W120601 into the span of Q Q is currently anorthogonal basis of 120601(X) so we have

W120601 = QV120601 (15)

where V120601 = R119896times119899 is an orthogonal matrix satisfying(V120601)TV120601 = I Using the definitions in (14) and (15) (13) canbe further rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TRL119861RTV120601)

Tr ((V120601)T RL119882RTV120601)

(16)

Let S119861= RL119861RT and S

119882= RL119882RT then (16) can be further

rewritten as follows

V120601lowast

= arg max(V120601)TV120601=I

Tr ((V120601)TS119861V120601)

Tr ((V120601)T S119882V120601)

(17)

Shock and Vibration 5

After the matrix V120601lowast is obtained with the BIGA-basedsolution method (to be in detail discussed in Section 3) theoutput points in the reduced data space can thus be expressedas

Y120601lowast

= (W120601lowast

)

T120601 (X) = (V120601

lowast

)

TQTQR = (V120601

lowast

)

TR (18)

3 The Proposed TR-KDA-BIGA

As previously mentioned construction of TR-KDA needsto select 119896 out of 119889 eigenvectors to form the matrix V120601for dimensionality reduction However finding a subset ofeigenvectors based on the trace ratio criterion is not an easytask since the space of possible subsets is very large especiallywhen 119889 is a large number Thus it is not impractical to useexhaustive search to find an optimal subset of 119896 eigenvectorsInstead in this study the BIGA is utilized to select 119896 outof 119889 eigenvectors of V120601

119905as the bases for projection matrix

formulation based on the trace ratio criterion such thatthe trace ratio value 120582 = Tr[(V120601)TS

119861V120601]Tr[(V120601)TS

119882V120601]

can be maximized Immune genetic algorithm originallydeveloped by Jiao andWang [33] is a novel genetic algorithmbased on the biological immune theory which combined theimmune mechanism with the evolutionary mechanism Inwhat follows further discussion of the proposed TR-KDA-BIGA is carried out

31 Chromosome Encoding Encoding a solution of a probleminto a chromosome is an important issue when using BIGAsIn this study every chromosome in a BIGA corresponds to adiscrete binary selector u = [119906

1 1199062 119906

119889] where each gene

in the chromosome is ldquo1rdquo indicating an eigenvector k120601119894(119894 =

1 2 119889) of S119861minus 120582119905S119882appearing in forming the projection

matrixV120601 of the 119905th step while ldquo0rdquo denotes its absenceThusthe length of the chromosome is 119889

32 Genetic Operators Genetic operators give every chro-mosome the chance to become the fittest chromosome ofits generation If it is difficult to reach the target of traceratio optimization crossover and mutation may introducedegeneracy into generations of chromosomes

321 Crossover Operator Crossover operator in a BIGAis employed to generate two new children chromosomesbased on two existing parent chromosomes selected from thecurrent population in terms of a prespecified crossover rateIn this study ldquoone-pointrdquo crossover operator was adopted torandomly select a cut point to exchange the parts betweenthe cut point and the end of the string of the parentchromosomes Specifically suppose that two parent chromo-somes 119875

1and 119875

2selected randomly from the population are

undergoing the crossover operation at a randomly selectedcrossover point 119892 (1 le 119892 le 119889) where

1198751= (1199061198941 1199061198942 119906

119894119889)

1198752= (1199061198951 1199061198952 119906

119895119889)

(19)

Consequently the offspring is generated by one-pointcrossover on the genes of two parents selected randomlyfrom the population We can thus get the two offspringchromosomes 119862

1and 119862

2

1198621= (1199061198941 1199061198942 119906

119894119892minus1 119906119894119892 119906119895119892+1

119906119895119889)

1198622= (1199061198951 1199061198952 119906

119895119892minus1 119906119895119892 119906119894119892+1

119906119894119889)

(20)

However the exchange procedure is not simply exchangingtheir genetic information between gene segments after thecrossover pointsWemust keep the number of eigenvectors tobe included in the subset equal to 119896 In this study therefore asimple but effective crossover operator strategy in this study isperformed in order to ensure that the crossover operator doesnot change the total number of ldquo1rdquo genes in chromosomes

Let

1198991198751=

119889

sum

119902=119892+1

119906119894119902

1198991198752=

119889

sum

119902=119892+1

119906119895119902

(21)

When 1198991198751is not equal to 119899

1198752 the following retention criterion

will be conducted(1) If 119899

1198751is larger than 119899

1198752 randomly select (119899

1198751minus

1198991198752) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo1-bitrdquo and then randomly select (1198991198751

minus

1198991198752) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo0-bitrdquo(2) If 119899

1198751is smaller than 119899

1198752 randomly select (119899

1198752minus

1198991198751) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo0-bitrdquo and then randomly select (1198991198752

minus

1198991198751) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo1-bitrdquo

322 Mutation Operator Mutation operator in a BIGA isused primarily as a mechanism for maintaining diversityin the population For each gene in a chromosome that isundergoing the mutation a real-valued number is randomlyselected within the range of [0 1] If the real-valued numberis less than the prespecified mutation rate then the genewill change from ldquo0-bitrdquo to ldquo1-bitrdquo and vice versa Uponadding (or removing) one eigenvector in that way we shallrandomly remove (or add) a different one such that thenumber of eigenvectors to be included in the subset is equalto 119896 The mutation operator helps the chromosomes to guidethe search in new areas

33 Immune Operators The immune ability of BIGAs is real-ized through two kinds of immune operators a vaccinationand an immune selection The vaccination is responsiblefor improving individualsrsquo overall fitness levels The immuneselection is responsible for prevention of deterioration

6 Shock and Vibration

331 Vaccination Operator Given a chromosome u vac-cination operation in a BIGA is employed to modify thegenes on some bits according to a priori knowledge such thatindividuals with higher fitness have a greater probability ofbeing selected Let U = (u

1 u2 u

1198990) be a population

the vaccination operation on U means that the operationis performed on 119899

120572= 120572119899

0chromosomes selected from

U according to the proportion of 120572 where 1198990represents

the population size of a BIGA A vaccine is abstractedfrom the prior knowledge of the pending problem whoseinformation amount and validity play an important role inthe performance of the algorithm

332 Immune Selection Operator The immune selectionoperation consists of the following two steps The first step isthe immunity test if the fitness of a chromosome u is smallerthan that of the parent chromosome which indicates thatdegeneration occurred during crossover and mutation thenthe parent chromosomewill be used for the next competitionThe second step is the annealing selection [35] a chromo-some u

119894is selected from the current offspring population U

119896

to join with the new parents with the probability as follows

119875 (u119894) =

exp (119891 (u119894) 119879119896)

sum1198990

119894=1exp (119891 (u

119894) 119879119896)

(22)

where 119891(u119894) is the fitness of the individual u

119894and 119879

119896 is the

temperature-controlled series tending towards 0

34 Fitness Evaluation Fitness evaluation plays a criticalrole in selecting offspring chromosomes from the currentpopulation for the next generation In this study the fitnessfunction for eigenvector selection is defined as

119891 (u) = 1199061ℎ1+ 1199062ℎ2+ sdot sdot sdot + 119906

119889ℎ119889

11990611198921+ 11990621198922+ sdot sdot sdot + 119906

119889119892119889

=

uhT

ugT (23)

where ℎ119894denotes the vT

119894S119861v119894value for the 119894th eigenvector

119892119894denotes the vT

119894S119882v119894value for the 119894th eigenvector h =

[ℎ1 ℎ2 ℎ

119889] g = [119892

1 1198922 119892

119889] u = [119906

1 1199062 119906

119889]

119906119894isin 0 1 u1T = 119896 and 119894 = 1 2 119889 Notably u is

called the binary selector and 119896 is the desired lower featuredimension Finally according to the evolved binary selectoru we can thus form the projection matrix V120601 of the 119905thstep by choosing the 119896 eigenvectors with 119906

119894= 1 (119894 =

1 2 119889) The procedures of the proposed TR-KDA-BIGAare summarized in the procedures of the proposed TR-KDA-BIGA part The computational flow of the BIGA obtainedusing the aforementioned genetic and immune operators isalso provided in the computational flow of the BIGA part

The Procedures of the Proposed TR-KDA-BIGA The proce-dures are as follows

(1) Construct the kernel matrix K = (120601(X))T120601(X)(2) Perform Cholesky decomposition to the kernel

matrix K = RTR(3) Form the kernel scatter matrixes as S

119861= RL119861RT and

S119882= RL119882RT

(4) Set iterations number 119905 to 1(5) Set the initial trace ratio value 120582

119905to Tr(S

119861)Tr(S

119882)

(6) Compute the eigendecomposition of S119861minus 120582119905S119882

as(S119861minus 120582119905S119882)k120601119894= 120591119894k120601119894 where k120601

119894(119894 = 1 2 119889) is

the eigenvector of S119861minus 120582119905S119882

(7) Calculate ℎ119894= (k120601119894)TS119861k120601119894and 119892

119894= (k120601119894)TS119882k120601119894for

each eigenvector k120601119894(119894 = 1 2 119889)

(8) Generate a population of BIGA selectors(9) Evolve the population where the fitness of a BIGA

selector u is measured as 119891(u) = uhTugT(10) ulowast is the evolved best BIGA selector

(11) Form the projection matrix V120601119905by choosing the 119896

eigenvectors k120601119894with 119906lowast

119894= 1 (119894 = 1 2 119889)

(12) Update the trace ratio value 120582119905+1

=

Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] 119905 = 119905 + 1 and go to

step (6) Repeat this procedure until a convergencecondition was established when the trace ratio valuedoes not increase in consecutive 5 iterations

(13) Output Y120601lowast = (V120601lowast)TR

TheComputational Flow of the BIGAThe computational flowis as follows

(1) Set 119897 (time of generation) to 1(2) Initialize randomly the original population 119860

119897

(3) Evaluate each chromosome in the original population119860119897

(4) Abstract vaccines according to the prior knowledge(5) Check for termination criteria If the fixed number

of generations is not reached or the optimal chromo-some found thus far is not satisfied then go to the nextstep Otherwise output the optimal chromosome asthe final solutions for further decision-making

(6) Perform crossover operation on the 119860119897and then

generate the population 119861119897

(7) Perform mutation operation on the 119861119897and then

generate the population results 119862119897

(8) Perform vaccination operation on the 119862119897and then

generate the population119863119897

(9) Perform immune selection operation on the 119863119897and

then generate the next generational population 119860119897+1

Go to step (3)

4 The Convergence of theProposed TR-KDA-BIGA

In this section we analyze the convergence of the proposedTR-LDA-BIGA Before doing this task it should be worthnoting that the BIGA is convergent It has been demonstratedby Jiao and Wang [33] that as long as enough iteration has

Shock and Vibration 7

been completed the immune genetic population convergestowards the true optimum with probability one

Recall the trace difference function

119911 (120582) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582S119882)V120601] (24)

it follows that

119911 (120582119905+1) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582119905+1S119882)V120601] 997904rArr

119911 (120582119905+1) = Tr [(V120601

119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

(25)

Since 120582119905+1

= Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] as previously

mentioned we get

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (26)

Consider the inequality

Tr [(V120601119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

ge Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905]

(27)

and the equation

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (28)

and we have

119911 (120582119905+1) ge 0

Tr [(V120601119905+1)

TS119861V120601119905+1] minus 120582119905+1

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 0

(29)

Consequently

Tr [(V120601119905+1)

TS119861V120601119905+1]

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 120582119905+1

997904rArr

120582119905+2

ge 120582119905+1

(30)

Substituting the subscript 119905 + 1 by 119905 yields

120582119905+1

ge 120582119905 (31)

So we obtain the following inequality which gives the firstexpression of convergence of the proposed TR-KDA-BIGA

Further suppose that 120582lowast is the optimal trace ratio valueit follows that

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

] = 0 (32)

Table 1 Specification of benchmark problems

Data set samples dim classHeart-statlog 270 13 2Ionosphere 351 34 2Iris 150 4 3Wine 178 13 3Waveform 5000 40 3Balance 625 4 3SCCTS 600 60 6

where V120601lowast is the optimal projection matrix We thereforehave

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

]

ge Tr [(V120601119905)

T(S119861minus 120582lowastS119882)V120601119905]

= Tr [(V120601119905)

T(S119861minus 120582119905+1

S119882)V120601119905]

+ (120582119905+1

minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

= 119911 (120582119905+1) + (120582

119905+1minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

(33)

Consider 119911(120582lowast) = 0 119911(120582119905+1) ge 0 and S

119882is semipositive

definite we have

120582119905+1

minus 120582lowastle 0 (34)

So we obtain the following inequality which gives the secondexpression of convergence of the proposed TR-KDA-BIGA

120582119905+1

le 120582lowast (35)

We conclude therefore that for a particular initial trace ratiovalue 120582

119905 the updated value 120582

119905+1can always satisfy (1) 120582

119905+1ge

120582119905and (2) 120582

119905+1le 120582lowast

5 Performance Evaluation onBenchmark Problems

In order to extensively verify the performance of the proposedTR-KDA-BIGA it is first tested on wide types of commonlyused benchmark problems taken from the UCI machinelearning repository and evaluated with the classificationrate (ie the number of correctly identified training exam-plestotal number of training examples) by comparison withother existing methods such as PCA LDA KPCA [30 31]KDA [32] and TR-LDA These data sets include Heart-statlog Ionosphere Iris Wine Waveform Balance and Syn-thetic Control Chart Time Series (SCCTS) data sets (Table 1)which are of small sizes low dimensions large sizes andorhigh dimensions For comparative study we randomly select50 data points from each data set as training set and therest of the data points as test set All methods use training set

8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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Shock and Vibration

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Shock and Vibration 5

After the matrix V120601lowast is obtained with the BIGA-basedsolution method (to be in detail discussed in Section 3) theoutput points in the reduced data space can thus be expressedas

Y120601lowast

= (W120601lowast

)

T120601 (X) = (V120601

lowast

)

TQTQR = (V120601

lowast

)

TR (18)

3 The Proposed TR-KDA-BIGA

As previously mentioned construction of TR-KDA needsto select 119896 out of 119889 eigenvectors to form the matrix V120601for dimensionality reduction However finding a subset ofeigenvectors based on the trace ratio criterion is not an easytask since the space of possible subsets is very large especiallywhen 119889 is a large number Thus it is not impractical to useexhaustive search to find an optimal subset of 119896 eigenvectorsInstead in this study the BIGA is utilized to select 119896 outof 119889 eigenvectors of V120601

119905as the bases for projection matrix

formulation based on the trace ratio criterion such thatthe trace ratio value 120582 = Tr[(V120601)TS

119861V120601]Tr[(V120601)TS

119882V120601]

can be maximized Immune genetic algorithm originallydeveloped by Jiao andWang [33] is a novel genetic algorithmbased on the biological immune theory which combined theimmune mechanism with the evolutionary mechanism Inwhat follows further discussion of the proposed TR-KDA-BIGA is carried out

31 Chromosome Encoding Encoding a solution of a probleminto a chromosome is an important issue when using BIGAsIn this study every chromosome in a BIGA corresponds to adiscrete binary selector u = [119906

1 1199062 119906

119889] where each gene

in the chromosome is ldquo1rdquo indicating an eigenvector k120601119894(119894 =

1 2 119889) of S119861minus 120582119905S119882appearing in forming the projection

matrixV120601 of the 119905th step while ldquo0rdquo denotes its absenceThusthe length of the chromosome is 119889

32 Genetic Operators Genetic operators give every chro-mosome the chance to become the fittest chromosome ofits generation If it is difficult to reach the target of traceratio optimization crossover and mutation may introducedegeneracy into generations of chromosomes

321 Crossover Operator Crossover operator in a BIGAis employed to generate two new children chromosomesbased on two existing parent chromosomes selected from thecurrent population in terms of a prespecified crossover rateIn this study ldquoone-pointrdquo crossover operator was adopted torandomly select a cut point to exchange the parts betweenthe cut point and the end of the string of the parentchromosomes Specifically suppose that two parent chromo-somes 119875

1and 119875

2selected randomly from the population are

undergoing the crossover operation at a randomly selectedcrossover point 119892 (1 le 119892 le 119889) where

1198751= (1199061198941 1199061198942 119906

119894119889)

1198752= (1199061198951 1199061198952 119906

119895119889)

(19)

Consequently the offspring is generated by one-pointcrossover on the genes of two parents selected randomlyfrom the population We can thus get the two offspringchromosomes 119862

1and 119862

2

1198621= (1199061198941 1199061198942 119906

119894119892minus1 119906119894119892 119906119895119892+1

119906119895119889)

1198622= (1199061198951 1199061198952 119906

119895119892minus1 119906119895119892 119906119894119892+1

119906119894119889)

(20)

However the exchange procedure is not simply exchangingtheir genetic information between gene segments after thecrossover pointsWemust keep the number of eigenvectors tobe included in the subset equal to 119896 In this study therefore asimple but effective crossover operator strategy in this study isperformed in order to ensure that the crossover operator doesnot change the total number of ldquo1rdquo genes in chromosomes

Let

1198991198751=

119889

sum

119902=119892+1

119906119894119902

1198991198752=

119889

sum

119902=119892+1

119906119895119902

(21)

When 1198991198751is not equal to 119899

1198752 the following retention criterion

will be conducted(1) If 119899

1198751is larger than 119899

1198752 randomly select (119899

1198751minus

1198991198752) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo1-bitrdquo and then randomly select (1198991198751

minus

1198991198752) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198751minus 1198991198752) selected

genes to ldquo0-bitrdquo(2) If 119899

1198751is smaller than 119899

1198752 randomly select (119899

1198752minus

1198991198751) genes with ldquo1-bitrdquo from the current offspring

chromosome 1198621and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo0-bitrdquo and then randomly select (1198991198752

minus

1198991198751) genes with ldquo0-bitrdquo from the current offspring

chromosome 1198622and reset these (119899

1198752minus 1198991198751) selected

genes to ldquo1-bitrdquo

322 Mutation Operator Mutation operator in a BIGA isused primarily as a mechanism for maintaining diversityin the population For each gene in a chromosome that isundergoing the mutation a real-valued number is randomlyselected within the range of [0 1] If the real-valued numberis less than the prespecified mutation rate then the genewill change from ldquo0-bitrdquo to ldquo1-bitrdquo and vice versa Uponadding (or removing) one eigenvector in that way we shallrandomly remove (or add) a different one such that thenumber of eigenvectors to be included in the subset is equalto 119896 The mutation operator helps the chromosomes to guidethe search in new areas

33 Immune Operators The immune ability of BIGAs is real-ized through two kinds of immune operators a vaccinationand an immune selection The vaccination is responsiblefor improving individualsrsquo overall fitness levels The immuneselection is responsible for prevention of deterioration

6 Shock and Vibration

331 Vaccination Operator Given a chromosome u vac-cination operation in a BIGA is employed to modify thegenes on some bits according to a priori knowledge such thatindividuals with higher fitness have a greater probability ofbeing selected Let U = (u

1 u2 u

1198990) be a population

the vaccination operation on U means that the operationis performed on 119899

120572= 120572119899

0chromosomes selected from

U according to the proportion of 120572 where 1198990represents

the population size of a BIGA A vaccine is abstractedfrom the prior knowledge of the pending problem whoseinformation amount and validity play an important role inthe performance of the algorithm

332 Immune Selection Operator The immune selectionoperation consists of the following two steps The first step isthe immunity test if the fitness of a chromosome u is smallerthan that of the parent chromosome which indicates thatdegeneration occurred during crossover and mutation thenthe parent chromosomewill be used for the next competitionThe second step is the annealing selection [35] a chromo-some u

119894is selected from the current offspring population U

119896

to join with the new parents with the probability as follows

119875 (u119894) =

exp (119891 (u119894) 119879119896)

sum1198990

119894=1exp (119891 (u

119894) 119879119896)

(22)

where 119891(u119894) is the fitness of the individual u

119894and 119879

119896 is the

temperature-controlled series tending towards 0

34 Fitness Evaluation Fitness evaluation plays a criticalrole in selecting offspring chromosomes from the currentpopulation for the next generation In this study the fitnessfunction for eigenvector selection is defined as

119891 (u) = 1199061ℎ1+ 1199062ℎ2+ sdot sdot sdot + 119906

119889ℎ119889

11990611198921+ 11990621198922+ sdot sdot sdot + 119906

119889119892119889

=

uhT

ugT (23)

where ℎ119894denotes the vT

119894S119861v119894value for the 119894th eigenvector

119892119894denotes the vT

119894S119882v119894value for the 119894th eigenvector h =

[ℎ1 ℎ2 ℎ

119889] g = [119892

1 1198922 119892

119889] u = [119906

1 1199062 119906

119889]

119906119894isin 0 1 u1T = 119896 and 119894 = 1 2 119889 Notably u is

called the binary selector and 119896 is the desired lower featuredimension Finally according to the evolved binary selectoru we can thus form the projection matrix V120601 of the 119905thstep by choosing the 119896 eigenvectors with 119906

119894= 1 (119894 =

1 2 119889) The procedures of the proposed TR-KDA-BIGAare summarized in the procedures of the proposed TR-KDA-BIGA part The computational flow of the BIGA obtainedusing the aforementioned genetic and immune operators isalso provided in the computational flow of the BIGA part

The Procedures of the Proposed TR-KDA-BIGA The proce-dures are as follows

(1) Construct the kernel matrix K = (120601(X))T120601(X)(2) Perform Cholesky decomposition to the kernel

matrix K = RTR(3) Form the kernel scatter matrixes as S

119861= RL119861RT and

S119882= RL119882RT

(4) Set iterations number 119905 to 1(5) Set the initial trace ratio value 120582

119905to Tr(S

119861)Tr(S

119882)

(6) Compute the eigendecomposition of S119861minus 120582119905S119882

as(S119861minus 120582119905S119882)k120601119894= 120591119894k120601119894 where k120601

119894(119894 = 1 2 119889) is

the eigenvector of S119861minus 120582119905S119882

(7) Calculate ℎ119894= (k120601119894)TS119861k120601119894and 119892

119894= (k120601119894)TS119882k120601119894for

each eigenvector k120601119894(119894 = 1 2 119889)

(8) Generate a population of BIGA selectors(9) Evolve the population where the fitness of a BIGA

selector u is measured as 119891(u) = uhTugT(10) ulowast is the evolved best BIGA selector

(11) Form the projection matrix V120601119905by choosing the 119896

eigenvectors k120601119894with 119906lowast

119894= 1 (119894 = 1 2 119889)

(12) Update the trace ratio value 120582119905+1

=

Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] 119905 = 119905 + 1 and go to

step (6) Repeat this procedure until a convergencecondition was established when the trace ratio valuedoes not increase in consecutive 5 iterations

(13) Output Y120601lowast = (V120601lowast)TR

TheComputational Flow of the BIGAThe computational flowis as follows

(1) Set 119897 (time of generation) to 1(2) Initialize randomly the original population 119860

119897

(3) Evaluate each chromosome in the original population119860119897

(4) Abstract vaccines according to the prior knowledge(5) Check for termination criteria If the fixed number

of generations is not reached or the optimal chromo-some found thus far is not satisfied then go to the nextstep Otherwise output the optimal chromosome asthe final solutions for further decision-making

(6) Perform crossover operation on the 119860119897and then

generate the population 119861119897

(7) Perform mutation operation on the 119861119897and then

generate the population results 119862119897

(8) Perform vaccination operation on the 119862119897and then

generate the population119863119897

(9) Perform immune selection operation on the 119863119897and

then generate the next generational population 119860119897+1

Go to step (3)

4 The Convergence of theProposed TR-KDA-BIGA

In this section we analyze the convergence of the proposedTR-LDA-BIGA Before doing this task it should be worthnoting that the BIGA is convergent It has been demonstratedby Jiao and Wang [33] that as long as enough iteration has

Shock and Vibration 7

been completed the immune genetic population convergestowards the true optimum with probability one

Recall the trace difference function

119911 (120582) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582S119882)V120601] (24)

it follows that

119911 (120582119905+1) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582119905+1S119882)V120601] 997904rArr

119911 (120582119905+1) = Tr [(V120601

119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

(25)

Since 120582119905+1

= Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] as previously

mentioned we get

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (26)

Consider the inequality

Tr [(V120601119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

ge Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905]

(27)

and the equation

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (28)

and we have

119911 (120582119905+1) ge 0

Tr [(V120601119905+1)

TS119861V120601119905+1] minus 120582119905+1

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 0

(29)

Consequently

Tr [(V120601119905+1)

TS119861V120601119905+1]

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 120582119905+1

997904rArr

120582119905+2

ge 120582119905+1

(30)

Substituting the subscript 119905 + 1 by 119905 yields

120582119905+1

ge 120582119905 (31)

So we obtain the following inequality which gives the firstexpression of convergence of the proposed TR-KDA-BIGA

Further suppose that 120582lowast is the optimal trace ratio valueit follows that

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

] = 0 (32)

Table 1 Specification of benchmark problems

Data set samples dim classHeart-statlog 270 13 2Ionosphere 351 34 2Iris 150 4 3Wine 178 13 3Waveform 5000 40 3Balance 625 4 3SCCTS 600 60 6

where V120601lowast is the optimal projection matrix We thereforehave

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

]

ge Tr [(V120601119905)

T(S119861minus 120582lowastS119882)V120601119905]

= Tr [(V120601119905)

T(S119861minus 120582119905+1

S119882)V120601119905]

+ (120582119905+1

minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

= 119911 (120582119905+1) + (120582

119905+1minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

(33)

Consider 119911(120582lowast) = 0 119911(120582119905+1) ge 0 and S

119882is semipositive

definite we have

120582119905+1

minus 120582lowastle 0 (34)

So we obtain the following inequality which gives the secondexpression of convergence of the proposed TR-KDA-BIGA

120582119905+1

le 120582lowast (35)

We conclude therefore that for a particular initial trace ratiovalue 120582

119905 the updated value 120582

119905+1can always satisfy (1) 120582

119905+1ge

120582119905and (2) 120582

119905+1le 120582lowast

5 Performance Evaluation onBenchmark Problems

In order to extensively verify the performance of the proposedTR-KDA-BIGA it is first tested on wide types of commonlyused benchmark problems taken from the UCI machinelearning repository and evaluated with the classificationrate (ie the number of correctly identified training exam-plestotal number of training examples) by comparison withother existing methods such as PCA LDA KPCA [30 31]KDA [32] and TR-LDA These data sets include Heart-statlog Ionosphere Iris Wine Waveform Balance and Syn-thetic Control Chart Time Series (SCCTS) data sets (Table 1)which are of small sizes low dimensions large sizes andorhigh dimensions For comparative study we randomly select50 data points from each data set as training set and therest of the data points as test set All methods use training set

8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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6 Shock and Vibration

331 Vaccination Operator Given a chromosome u vac-cination operation in a BIGA is employed to modify thegenes on some bits according to a priori knowledge such thatindividuals with higher fitness have a greater probability ofbeing selected Let U = (u

1 u2 u

1198990) be a population

the vaccination operation on U means that the operationis performed on 119899

120572= 120572119899

0chromosomes selected from

U according to the proportion of 120572 where 1198990represents

the population size of a BIGA A vaccine is abstractedfrom the prior knowledge of the pending problem whoseinformation amount and validity play an important role inthe performance of the algorithm

332 Immune Selection Operator The immune selectionoperation consists of the following two steps The first step isthe immunity test if the fitness of a chromosome u is smallerthan that of the parent chromosome which indicates thatdegeneration occurred during crossover and mutation thenthe parent chromosomewill be used for the next competitionThe second step is the annealing selection [35] a chromo-some u

119894is selected from the current offspring population U

119896

to join with the new parents with the probability as follows

119875 (u119894) =

exp (119891 (u119894) 119879119896)

sum1198990

119894=1exp (119891 (u

119894) 119879119896)

(22)

where 119891(u119894) is the fitness of the individual u

119894and 119879

119896 is the

temperature-controlled series tending towards 0

34 Fitness Evaluation Fitness evaluation plays a criticalrole in selecting offspring chromosomes from the currentpopulation for the next generation In this study the fitnessfunction for eigenvector selection is defined as

119891 (u) = 1199061ℎ1+ 1199062ℎ2+ sdot sdot sdot + 119906

119889ℎ119889

11990611198921+ 11990621198922+ sdot sdot sdot + 119906

119889119892119889

=

uhT

ugT (23)

where ℎ119894denotes the vT

119894S119861v119894value for the 119894th eigenvector

119892119894denotes the vT

119894S119882v119894value for the 119894th eigenvector h =

[ℎ1 ℎ2 ℎ

119889] g = [119892

1 1198922 119892

119889] u = [119906

1 1199062 119906

119889]

119906119894isin 0 1 u1T = 119896 and 119894 = 1 2 119889 Notably u is

called the binary selector and 119896 is the desired lower featuredimension Finally according to the evolved binary selectoru we can thus form the projection matrix V120601 of the 119905thstep by choosing the 119896 eigenvectors with 119906

119894= 1 (119894 =

1 2 119889) The procedures of the proposed TR-KDA-BIGAare summarized in the procedures of the proposed TR-KDA-BIGA part The computational flow of the BIGA obtainedusing the aforementioned genetic and immune operators isalso provided in the computational flow of the BIGA part

The Procedures of the Proposed TR-KDA-BIGA The proce-dures are as follows

(1) Construct the kernel matrix K = (120601(X))T120601(X)(2) Perform Cholesky decomposition to the kernel

matrix K = RTR(3) Form the kernel scatter matrixes as S

119861= RL119861RT and

S119882= RL119882RT

(4) Set iterations number 119905 to 1(5) Set the initial trace ratio value 120582

119905to Tr(S

119861)Tr(S

119882)

(6) Compute the eigendecomposition of S119861minus 120582119905S119882

as(S119861minus 120582119905S119882)k120601119894= 120591119894k120601119894 where k120601

119894(119894 = 1 2 119889) is

the eigenvector of S119861minus 120582119905S119882

(7) Calculate ℎ119894= (k120601119894)TS119861k120601119894and 119892

119894= (k120601119894)TS119882k120601119894for

each eigenvector k120601119894(119894 = 1 2 119889)

(8) Generate a population of BIGA selectors(9) Evolve the population where the fitness of a BIGA

selector u is measured as 119891(u) = uhTugT(10) ulowast is the evolved best BIGA selector

(11) Form the projection matrix V120601119905by choosing the 119896

eigenvectors k120601119894with 119906lowast

119894= 1 (119894 = 1 2 119889)

(12) Update the trace ratio value 120582119905+1

=

Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] 119905 = 119905 + 1 and go to

step (6) Repeat this procedure until a convergencecondition was established when the trace ratio valuedoes not increase in consecutive 5 iterations

(13) Output Y120601lowast = (V120601lowast)TR

TheComputational Flow of the BIGAThe computational flowis as follows

(1) Set 119897 (time of generation) to 1(2) Initialize randomly the original population 119860

119897

(3) Evaluate each chromosome in the original population119860119897

(4) Abstract vaccines according to the prior knowledge(5) Check for termination criteria If the fixed number

of generations is not reached or the optimal chromo-some found thus far is not satisfied then go to the nextstep Otherwise output the optimal chromosome asthe final solutions for further decision-making

(6) Perform crossover operation on the 119860119897and then

generate the population 119861119897

(7) Perform mutation operation on the 119861119897and then

generate the population results 119862119897

(8) Perform vaccination operation on the 119862119897and then

generate the population119863119897

(9) Perform immune selection operation on the 119863119897and

then generate the next generational population 119860119897+1

Go to step (3)

4 The Convergence of theProposed TR-KDA-BIGA

In this section we analyze the convergence of the proposedTR-LDA-BIGA Before doing this task it should be worthnoting that the BIGA is convergent It has been demonstratedby Jiao and Wang [33] that as long as enough iteration has

Shock and Vibration 7

been completed the immune genetic population convergestowards the true optimum with probability one

Recall the trace difference function

119911 (120582) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582S119882)V120601] (24)

it follows that

119911 (120582119905+1) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582119905+1S119882)V120601] 997904rArr

119911 (120582119905+1) = Tr [(V120601

119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

(25)

Since 120582119905+1

= Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] as previously

mentioned we get

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (26)

Consider the inequality

Tr [(V120601119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

ge Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905]

(27)

and the equation

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (28)

and we have

119911 (120582119905+1) ge 0

Tr [(V120601119905+1)

TS119861V120601119905+1] minus 120582119905+1

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 0

(29)

Consequently

Tr [(V120601119905+1)

TS119861V120601119905+1]

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 120582119905+1

997904rArr

120582119905+2

ge 120582119905+1

(30)

Substituting the subscript 119905 + 1 by 119905 yields

120582119905+1

ge 120582119905 (31)

So we obtain the following inequality which gives the firstexpression of convergence of the proposed TR-KDA-BIGA

Further suppose that 120582lowast is the optimal trace ratio valueit follows that

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

] = 0 (32)

Table 1 Specification of benchmark problems

Data set samples dim classHeart-statlog 270 13 2Ionosphere 351 34 2Iris 150 4 3Wine 178 13 3Waveform 5000 40 3Balance 625 4 3SCCTS 600 60 6

where V120601lowast is the optimal projection matrix We thereforehave

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

]

ge Tr [(V120601119905)

T(S119861minus 120582lowastS119882)V120601119905]

= Tr [(V120601119905)

T(S119861minus 120582119905+1

S119882)V120601119905]

+ (120582119905+1

minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

= 119911 (120582119905+1) + (120582

119905+1minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

(33)

Consider 119911(120582lowast) = 0 119911(120582119905+1) ge 0 and S

119882is semipositive

definite we have

120582119905+1

minus 120582lowastle 0 (34)

So we obtain the following inequality which gives the secondexpression of convergence of the proposed TR-KDA-BIGA

120582119905+1

le 120582lowast (35)

We conclude therefore that for a particular initial trace ratiovalue 120582

119905 the updated value 120582

119905+1can always satisfy (1) 120582

119905+1ge

120582119905and (2) 120582

119905+1le 120582lowast

5 Performance Evaluation onBenchmark Problems

In order to extensively verify the performance of the proposedTR-KDA-BIGA it is first tested on wide types of commonlyused benchmark problems taken from the UCI machinelearning repository and evaluated with the classificationrate (ie the number of correctly identified training exam-plestotal number of training examples) by comparison withother existing methods such as PCA LDA KPCA [30 31]KDA [32] and TR-LDA These data sets include Heart-statlog Ionosphere Iris Wine Waveform Balance and Syn-thetic Control Chart Time Series (SCCTS) data sets (Table 1)which are of small sizes low dimensions large sizes andorhigh dimensions For comparative study we randomly select50 data points from each data set as training set and therest of the data points as test set All methods use training set

8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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Shock and Vibration

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Shock and Vibration 7

been completed the immune genetic population convergestowards the true optimum with probability one

Recall the trace difference function

119911 (120582) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582S119882)V120601] (24)

it follows that

119911 (120582119905+1) = max(V120601)TV120601=I

Tr [(V120601)T(S119861minus 120582119905+1S119882)V120601] 997904rArr

119911 (120582119905+1) = Tr [(V120601

119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

(25)

Since 120582119905+1

= Tr[(V120601119905)TS119861V120601119905]Tr[(V120601

119905)TS119882V120601119905] as previously

mentioned we get

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (26)

Consider the inequality

Tr [(V120601119905+1)

T(S119861minus 120582119905+1

S119882)V120601119905+1]

ge Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905]

(27)

and the equation

Tr [(V120601119905)

T(S119861minus 120582119905+1S119882)V120601119905] = 0 (28)

and we have

119911 (120582119905+1) ge 0

Tr [(V120601119905+1)

TS119861V120601119905+1] minus 120582119905+1

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 0

(29)

Consequently

Tr [(V120601119905+1)

TS119861V120601119905+1]

Tr [(V120601119905+1)

TS119882V120601119905+1]

ge 120582119905+1

997904rArr

120582119905+2

ge 120582119905+1

(30)

Substituting the subscript 119905 + 1 by 119905 yields

120582119905+1

ge 120582119905 (31)

So we obtain the following inequality which gives the firstexpression of convergence of the proposed TR-KDA-BIGA

Further suppose that 120582lowast is the optimal trace ratio valueit follows that

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

] = 0 (32)

Table 1 Specification of benchmark problems

Data set samples dim classHeart-statlog 270 13 2Ionosphere 351 34 2Iris 150 4 3Wine 178 13 3Waveform 5000 40 3Balance 625 4 3SCCTS 600 60 6

where V120601lowast is the optimal projection matrix We thereforehave

119911 (120582lowast) = Tr [(V120601

lowast

)

T(S119861minus 120582lowastS119882)V120601lowast

]

ge Tr [(V120601119905)

T(S119861minus 120582lowastS119882)V120601119905]

= Tr [(V120601119905)

T(S119861minus 120582119905+1

S119882)V120601119905]

+ (120582119905+1

minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

= 119911 (120582119905+1) + (120582

119905+1minus 120582lowast)Tr [(V120601

119905)

TS119882V120601119905]

(33)

Consider 119911(120582lowast) = 0 119911(120582119905+1) ge 0 and S

119882is semipositive

definite we have

120582119905+1

minus 120582lowastle 0 (34)

So we obtain the following inequality which gives the secondexpression of convergence of the proposed TR-KDA-BIGA

120582119905+1

le 120582lowast (35)

We conclude therefore that for a particular initial trace ratiovalue 120582

119905 the updated value 120582

119905+1can always satisfy (1) 120582

119905+1ge

120582119905and (2) 120582

119905+1le 120582lowast

5 Performance Evaluation onBenchmark Problems

In order to extensively verify the performance of the proposedTR-KDA-BIGA it is first tested on wide types of commonlyused benchmark problems taken from the UCI machinelearning repository and evaluated with the classificationrate (ie the number of correctly identified training exam-plestotal number of training examples) by comparison withother existing methods such as PCA LDA KPCA [30 31]KDA [32] and TR-LDA These data sets include Heart-statlog Ionosphere Iris Wine Waveform Balance and Syn-thetic Control Chart Time Series (SCCTS) data sets (Table 1)which are of small sizes low dimensions large sizes andorhigh dimensions For comparative study we randomly select50 data points from each data set as training set and therest of the data points as test set All methods use training set

8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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8 Shock and Vibration

Table 2 Results of the classification rate for the benchmark problems (mean plusmn derivation)

Data set PCA KPCA LDA KDA TR-LDA TR-KDA-BIGAHeart-statlog 597 plusmn 46 600 plusmn 42 755 plusmn 33 742 plusmn 27 761 plusmn 31 758 plusmn 23

Ionosphere 744 plusmn 40 749 plusmn 38 747 plusmn 57 751 plusmn 47 752 plusmn 59 767 plusmn 45

Iris 910 plusmn 40 910 plusmn 42 910 plusmn 35 918 plusmn 30 923 plusmn 32 929 plusmn 36

Wine 683 plusmn 52 690 plusmn 49 777 plusmn 84 770 plusmn 69 784 plusmn 85 786 plusmn 63

Waveform 729 plusmn 17 731 plusmn 15 733 plusmn 17 730 plusmn 23 745 plusmn 13 749 plusmn 28

Balance 634 plusmn 42 664 plusmn 47 778 plusmn 49 784 plusmn 37 783 plusmn 42 792 plusmn 32

SCCTS 800 plusmn 48 801 plusmn 50 805 plusmn 54 811 plusmn 62 812 plusmn 56 829 plusmn 67

Table 3 Time-domain statistical features

Feature Eq Feature Eq

Standard deviation 119909std =radicsum119873

119894=1(119909(119894) minus 119909)

2

119873

Clearance factor CLF =119909119901

119909smr

Peak 119909119901= max |119909(119894)| Shape factor SF =

119909rms

(1119873)sum119873

119894=1|119909(119894)|

Skewness 119909ske =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

3

1199093

stdImpact factor IF =

119909119901

(1119873)sum119873

119894=1|119909(119894)|

Kurtosis 119909kur =(1119873)sum

119873

119894=1(119909(119894) minus 119909)

4

1199094

stdSquare mean root 119909smr = (

1

119873

119873

sum

119894=1

radic|119909(119894)|)

2

Crest factor CF =119909119901

119909rms

where 119909(119894) is a digital signal series 119894 = 1 2 119873119873 is the number of elements of the digital signal and 119909 = sum119873119894=1119909(119894)119873 and 119909rms = radicsum

119873

119894=1119909(119894)2119873 are the

mean value and root-mean-square value of the digital signal series respectively

in the output reduced space to train one nearest neigh-borhood (1NN) classifier for evaluating the classificationrate of test set To restrict the influence of random effectsthe experiments of PCA LDA KPCA KDA TR-LDA andTR-KDA-BIGA compared on each benchmark problem areindependently performed for 20 runs Table 2 compares theclassification rate for benchmark problems of the proposedTR-KDA-BIGAwith that of the PCA LDA KPCA KDA andTR-LDA As seen in Table 2 the proposed TR-KDA-BIGAcan perform better than all the compared methods except inthe case of Heart-statlog

The results obtained demonstrate the ability of theproposed TR-KDA-BIGA in classifying different classeswell Thus the proposed TR-KDA-BIGA may be effectivelyemployed for fault diagnosis of rolling element bearings

6 The Proposed TR-KDA Using BIGA forFault Diagnosis of Rolling Element Bearings

In this section the proposed TR-KDA-BIGA is applied tofault diagnosis of rolling element bearings Vibration signalsresulting from rolling element bearings are first filtered byusing a low-pass filter Then the filtered vibration signalsare divided into sections of equal window length One set ofrelevant features obtained from eachwindow is used for char-acterizing to some extent the health status of the rolling ele-ment bearingsMost of the faults occurring in rolling element

bearings will introduce the increased friction and impulsiveforces when bearings are rotating which generally lead thevibration signals in time-domain frequency-domain andortime-frequency-domain to vary (become different) from thenormal ones In this study 9 time-domain statistical features(Table 3) are extracted from the vibration signal All of these9 time-domain statistical features reflect the characteristicsof time series data in the time-domain Moreover 6 time-frequency-domain wavelet features about the percentages ofenergy corresponding to wavelet coefficients are extractedfrom the vibration signal by using Daubechies-4 (db4)wavelet to decompose the vibration signal into five levels [32]Wavelet features extracted in such a way can to the greatestextent reflect the vibration energy distribution in the time-frequency-domain Thus 9 time-domain statistical featurestogether with 6 time-frequency-domain wavelet features areused to represent each windowrsquos vibration signals

61 Experimental Setup In order to demonstrate the per-formance of the proposed TR-KDA-BIGA rolling elementbearing data obtained from the Bearing Data Centre CaseWestern Reserve University [36] are used The test rollingelement bearings were SKF 6205 JEM a type of deep grooveball bearing Single-point faults were seeded into the driveend ball bearing using electrodischarge machining Faultsoccurring in rolling element bearings introduced impact-likevibration signals when bearings were rotating An accelerom-eter was mounted on the drive end of the motor housing

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

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Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 9

(a)

Induction motor Load motor

Drive endFan end

Torque transducerAccelerometer Drive end bearing

(b)

Figure 1 Experimental setup (a) test rig (b) schematic description of the test rig

to detect such impacts that behaved like damped oscillationsVibration signals were captured from four different healthstatuses of bearing that is normal bearings (Normal) innerrace fault (IR) ball fault (BA) and outer race fault (OR)For each of the three abnormal statuses (IR BA and OR)there are three different levels of severity with fault diam-eters (0007 inches 0014 inches and 0021 inches) All theexperiments were done for three different load conditions(1HP 2HP and 3HP) Figure 1 illustrates the experimentalsetup Experimental data were collected from the drive endball bearing of an induction motor (Reliance Electric 2HPIQPreAlert) driven test rig Table 4 gives a short descriptionof rolling element bearing data

62 Experiment Results

621 Visualization of Bearing Data Visualization perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA using simu-lations where KPCA and KDA are the kernel extensions toPCA and LDA respectively The two-dimensional visualiza-tion results of bearing data for three different load conditions(1 2 and 3HP) obtained with PCA LDA KPCA KDA TR-LDA and the proposed TR-KDA-BIGA are summarized inFigures 2 3 and 4 respectively As seen in Figures 2 3 and 4the proposed TR-KDA-BIGA outperforms all the comparedmethods in not only closely conglomerating bearing databelonging to the same class but also clearly separating bearingdata belonging to different classes of three different loadconditions (1 2 and 3HP) Compared with the unsupervisedmethods (ie PCA and KPCA) the supervised methods(ie LDA KDA TR-LDA and TR-KDA-BIGA) can preservemore discriminative information embedded in bearing dataand obtain clearer and less overlapped boundaries It canalso be concluded from Figures 2 3 and 4 that the methodsusing kernel trick (ie KPCA KDA and TR-KDA-BIGA)performed better than themethodswithout using kernel trick(ie PCA LDA and TR-LDA) in separating the discrimina-tive propertymdashsamples from different classes in the learnedsubspace

622 Classification of Bearing Data Classification perfor-mances of the proposed TR-KDA-BIGA are compared withthose of PCA LDA KPCA KDA and TR-LDA In orderto show the robustness of the proposed TR-KDA-BIGA we

Table 4 Description of rolling element bearing data

samples dimension class samples per class800 15 10 80

perform 4 independent experiments for each load conditionin terms of 4 different data partitions In this study 10 2030 and 40 samples per class in bearing data set are randomlyselected from each class in bearing data as the training setand the remaining samples as the test set Then each methoduses the training set to train a 1NN classifier in order toclassify different health status in test set Tables 5 6 and 7summarize the average classification results of PCA LDAKPCA KDA TR-LDA and the proposed TR-KDA-BIGAwith various numbers of training samples for 1HP 2HPand 3HP load conditions respectively It can be observedthat the overall average performance of the classification ofhealth status is fairly good Tables 5 6 and 7 demonstrate thatthe proposed TR-KDA-BIGA performs remarkably betterthan the compared methods (PCA LDA KPCA KDA andTR-LDA) It should be noted that the proposed model canalso provide the capability of real-time visualization thatis very useful for the practitioners to monitor the healthstatus of rolling element bearings Tables 5 6 and 7 alsodemonstrate that the number of training samples does sig-nificantly affect the classification accuracy for bearing healthstatus

7 Conclusions

The rolling element bearing is a core component of manysystems and their failure can lead to reduced capabilitydowntime and even catastrophic breakdowns Effective andefficient fault diagnosis of rolling element bearings plays anextremely important role in the safe and reliable operationof their host systems In the current study fault diagnosisof rolling element bearings is done in a pattern recogni-tion way by calculating a high-dimensional feature dataset from vibration signals which represents the differentstatus of bearings Specifically the TR-KDA is presented forfault diagnosis of rolling element bearings and the BIGAis employed to solve the trace ratio problem in TR-KDAThe numerical results obtained using extensive simulationindicate that the proposed TR-KDA-BIGA can effectively

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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Advances inOptoElectronics

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 2 Two-dimensional representation of bearing data under 1HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 3 Two-dimensional representation of bearing data under 2HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(a)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(b)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(c)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(d)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(e)

NormalInch 07 (IR)Inch 07 (BA)Inch 07 (OR)Inch 14 (IR)

Inch 14 (BA)Inch 14 (OR)Inch 21 (IR)Inch 21 (BA)Inch 21 (OR)

(f)Figure 4 Two-dimensional representation of bearing data under 3HP motor load by each method (a) PCA (b) KPCA (c) LDA (d) KDA(e) TR-LDA (f) TR-KDA-BIGA

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

Table 5 Classification accuracy of bearing data under 1HP motor load

Method Number of training samples Average10 20 30 40

PCA 947793 959016 967564 977785 9630395KPCA 949203 959007 965873 984867 9647375LDA 962741 971929 980017 988438 9757813KDA 967026 971957 981504 991511 9779995TR-LDA 971914 979510 990587 997006 9847543TR-KDA-BIGA 975793 988971 995416 998068 9895620

Table 6 Classification accuracy of bearing data under 2HP motor load

Method Number of training samples Average10 20 30 40

PCA 944654 954452 963670 976571 9598368KPCA 942155 965207 973084 987273 9669298LDA 960477 969640 976314 990035 9741165KDA 963333 975962 972495 994105 9764738TR-LDA 973188 980867 987036 997777 9847170TR-KDA-BIGA 989123 986209 997737 999311 9930950

Table 7 Classification accuracy of bearing data under 3HP motor load

Method Number of training samples Average10 20 30 40

PCA 961337 973051 976719 979738 9727113KPCA 967442 970369 976968 982854 9744083LDA 970421 985900 992875 994996 9860480KDA 974820 988408 991516 996082 9877065TR-LDA 988961 993457 999190 996082 9944225TR-KDA-BIGA 992918 999153 999103 999291 9976163

classify different classes of rolling element bearing data whilealso providing the capability of real-time visualization that isvery useful for the practitioners to monitor the health statusof rolling element bearings Empirical comparisons show thatthe proposed TR-KDA-BIGA performs better than existingmethods in classifying different rolling element bearing dataThe proposed TR-KDA-BIGA may be a promising tool forfault diagnosis of rolling element bearings

Three research directions are worth pursuing Firstalthough this study considers the specific fault diagnosisof rolling element bearings the proposed method can bemodified and extended to address the fault diagnosis of gear-boxes [37 38] and cutting tools [39 40] Second frequency-domain information can be utilized for fault diagnosis ofrolling element bearings [41 42] it would thus be interest-ing to integrate frequency-domain features to time-domainand time-frequency-domain features Third empirical modedecomposition is a very powerful tool for nonlinear andnonstationary signal processing [43ndash45] it would be alsointeresting to employ the empirical mode decomposition toextract periodic components and random transient com-ponents from the bearing vibration signal mixture whichmay be very helpful for extraction of fault signatures from acollected bearing vibration signal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded partially by the National Sci-ence Foundation of China (51405239) National DefenseBasic Scientific Research Program of China (A2620132010A2520110003) Jiangsu Provincial Natural Science Founda-tion of China (BK20150745 BK20140727) Jiangsu ProvinceScience and Technology Support Program (BE2014134) Fun-damental Research Funds for the Central Universities (1005-YAH15055) and Jiangsu Postdoctoral Science Foundation ofChina (1501024C) The authors would like to express sincereappreciation to Professor KA Loparo and Case WesternReserve University for their efforts to make bearing data setavailable and permission to use data set

References

[1] X Jin E W M Ma L L Cheng and M Pecht ldquoHealthmonitoring of cooling fans based onmahalanobis distance withmRMR feature selectionrdquo IEEETransactions on Instrumentationand Measurement vol 61 no 8 pp 2222ndash2229 2012

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

[2] X Jin and T W S Chow ldquoAnomaly detection of cooling fanand fault classification of induction motor using MahalanobisndashTaguchi systemrdquo Expert Systems with Applications vol 40 no15 pp 5787ndash5795 2013

[3] J Zarei ldquoInductionmotors bearing fault detection using patternrecognition techniquesrdquo Expert Systems with Applications vol39 no 1 pp 68ndash73 2012

[4] J-B Yu ldquoBearing performance degradation assessment usinglocality preserving projectionsrdquo Expert Systems with Applica-tions vol 38 no 6 pp 7440ndash7450 2011

[5] D Wang P W Tse and Y L Tse ldquoA morphogram withthe optimal selection of parameters used in morphologicalanalysis for enhancing the ability in bearing fault diagnosisrdquoMeasurement Science and Technology vol 23 no 6 Article ID065001 2012

[6] W Wang and M Pecht ldquoEconomic analysis of canary-basedprognostics and health managementrdquo IEEE Transactions onIndustrial Electronics vol 58 no 7 pp 3077ndash3089 2011

[7] R B Randall and J Antoni ldquoRolling element bearingdiagnosticsmdasha tutorialrdquoMechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[8] Y Yang Y Liao G Meng and J Lee ldquoA hybrid feature selectionscheme for unsupervised learning and its application in bearingfault diagnosisrdquo Expert Systems with Applications vol 38 no 9pp 11311ndash11320 2011

[9] J Rafiee M A Rafiee and P W Tse ldquoApplication of motherwavelet functions for automatic gear and bearing fault diagno-sisrdquo Expert Systems with Applications vol 37 no 6 pp 4568ndash4579 2010

[10] W He Z-N Jiang and K Feng ldquoBearing fault detectionbased on optimal wavelet filter and sparse code shrinkagerdquoMeasurement vol 42 no 7 pp 1092ndash1102 2009

[11] J B Tenenbaum V de Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[12] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[13] M D Prieto G Cirrincione A G Espinosa J A Ortegaand H Henao ldquoBearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neu-ral networksrdquo IEEE Transactions on Industrial Electronics vol60 no 8 pp 3398ndash3407 2013

[14] M B Zhao Z Zhang and T W S Chow ldquoTrace ratio criterionbased generalized discriminative learning for semi-superviseddimensionality reductionrdquo Pattern Recognition vol 45 no 4pp 1482ndash1499 2012

[15] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005

[16] K Feng Z Jiang W He and B Ma ldquoA recognition and noveltydetection approach based on Curvelet transform nonlinearPCAand SVMwith application to indicator diagramdiagnosisrdquoExpert SystemswithApplications vol 38 no 10 pp 12721ndash127292011

[17] Q Jiang M Jia J Hu and F Xu ldquoMachinery fault diagnosisusing supervised manifold learningrdquo Mechanical Systems andSignal Processing vol 23 no 7 pp 2301ndash2311 2009

[18] E G Strangas S Aviyente and S S H Zaidi ldquoTime-frequencyanalysis for efficient fault diagnosis and failure prognosis for

interior permanent-magnet AC motorsrdquo IEEE Transactions onIndustrial Electronics vol 55 no 12 pp 4191ndash4199 2008

[19] YWang EWMMa TW S Chow andK-L Tsui ldquoA two-stepparametric method for failure prediction in hard disk drivesrdquoIEEE Transactions on Industrial Informatics vol 10 no 1 pp419ndash430 2014

[20] J Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[21] K Fukunaga Introduction to Statistical Pattern RecognitionAcademic Press 1990

[22] H Wang S Yan D Xu X Tang and T Huang ldquoTrace ratiovs ratio trace for dimensionality reductionrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[23] M B Zhao R H M Chan P Tang T W S Chow and S WH Wong ldquoTrace ratio linear discriminant analysis for medicaldiagnosis a case study of dementiardquo IEEE Signal ProcessingLetters vol 20 no 5 pp 431ndash434 2013

[24] L Zhou L Wang and C H Shen ldquoFeature selection withredundancy-constrained class separabilityrdquo IEEE Transactionson Neural Networks vol 21 no 5 pp 853ndash858 2010

[25] T Sun and S Chen ldquoClass label versus sample label-basedCCArdquoAppliedMathematics and Computation vol 185 no 1 pp272ndash283 2007

[26] Y-F Guo S-J Li J-Y Yang T-T Shu and L-D Wu ldquoA gen-eralized Foley-Sammon transform based on generalized fisherdiscriminant criterion and its application to face recognitionrdquoPattern Recognition Letters vol 24 no 1ndash3 pp 147ndash158 2003

[27] M B Zhao X H Jin Z Zhang and B Li ldquoFault diagnosis ofrolling element bearings via discriminative subspace learningvisualization and classificationrdquo Expert Systems with Applica-tions vol 41 no 7 pp 3391ndash3401 2014

[28] X H Jin M B Zhao T W S Chow and M S Pecht ldquoMotorbearing fault diagnosis using trace ratio linear discriminantanalysisrdquo IEEE Transactions on Industrial Electronics vol 61 no5 pp 2441ndash2451 2014

[29] Y Q Jia F P Nie and C S Zhang ldquoTrace ratio problemrevisitedrdquo IEEE Transactions on Neural Networks vol 20 no4 pp 729ndash735 2009

[30] J Yang A F Frangi J-Y Yang D Zhang and Z Jin ldquoKPCAplus LDA a complete kernel Fisher discriminant frameworkfor feature extraction and recognitionrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 27 no 2 pp 230ndash244 2005

[31] C Zhang F Nie and S Xiang ldquoA general kernelizationframework for learning algorithms based on kernel PCArdquoNeurocomputing vol 73 no 4ndash6 pp 959ndash967 2010

[32] S Ji and J Ye ldquoKernel uncorrelated and regularized discrim-inant analysis a theoretical and computational studyrdquo IEEETransactions on Knowledge and Data Engineering vol 20 no10 pp 1311ndash1321 2008

[33] L C Jiao and L Wang ldquoA novel genetic algorithm basedon immunityrdquo IEEE Transactions on Systems Man andCyberneticsmdashPart A Systems and Humans vol 30 no 5 pp552ndash561 2000

[34] F R K Chung Spectral Graph Theory CBMS Regional Con-ference Series in Mathematics No 92 American MathematicalSociety 1997

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

[35] J S Zhang Z B Xu and Y Liang ldquoThe whole annealing ge-netic algorithms and their sufficient and necessary conditions ofconvergencerdquo Science of China vol 27 no 2 pp 154ndash164 1997

[36] K A Loparo ldquoBearings vibration data setrdquo Case Western Re-serve University httpcsegroupscaseedubearingdatacenterpageswelcome-case-western-reserve-university-bearing-data-center-website

[37] D Wang Q Miao and R Kang ldquoRobust health evaluation ofgearbox subject to tooth failure with wavelet decompositionrdquoJournal of Sound and Vibration vol 324 no 3ndash5 pp 1141ndash11572009

[38] D Wang P W Tse W Guo and Q Miao ldquoSupport vectordata description for fusion of multiple health indicators forenhancing gearbox fault diagnosis and prognosisrdquo Measure-ment Science and Technology vol 22 no 2 Article ID 0251022011

[39] F J Alonso and D R Salgado ldquoAnalysis of the structure ofvibration signals for tool wear detectionrdquo Mechanical Systemsand Signal Processing vol 22 no 3 pp 735ndash748 2008

[40] K P Zhu G S Hong and Y S Wong ldquoA comparativestudy of feature selection for hidden Markov model-basedmicro-milling tool wear monitoringrdquo Machining Science andTechnology vol 12 no 3 pp 348ndash369 2008

[41] DWang QMiao X F Fan andH-Z Huang ldquoRolling elementbearing fault detection using an improved combination ofHilbert and wavelet transformsrdquo Journal of Mechanical Scienceand Technology vol 23 no 12 pp 3292ndash3301 2010

[42] Y G Lei M J Zuo Z J He and Y Y Zi ldquoA multidimensionalhybrid intelligent method for gear fault diagnosisrdquo ExpertSystems with Applications vol 37 no 2 pp 1419ndash1430 2010

[43] D Wang W Guo and P W Tse ldquoAn enhanced empirical modedecomposition method for blind component separation of asingle-channel vibration signal mixturerdquo Journal of Vibrationand Control 2014

[44] Y G Lei M J Zuo and M R Hoseini ldquoThe use of ensembleempirical mode decomposition to improve bispectral analysisfor fault detection in rotating machineryrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 224 no 8 pp 1759ndash1769 2010

[45] YG Lei Z JHe andYY Zi ldquoApplication of the EEMDmethodto rotor fault diagnosis of rotating machineryrdquo MechanicalSystems and Signal Processing vol 23 no 4 pp 1327ndash1338 2009

International Journal of

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RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of