research article time-frequency fault feature extraction...

Research ArticleTime-Frequency Fault Feature Extraction forRolling Bearing Based on the Tensor Manifold Method

Fengtao Wang1 Shouhai Chen1 Jian Sun1 Dawen Yan2 Lei Wang1 and Lihua Zhang3

1 Institute of Vibration Engineering School of Mechanical Engineering Dalian University of Technology Dalian 116024 China2 School of Mathematical Sciences Dalian University of Technology Dalian 116024 China3 Institute of Microelectromechanical Systems and Precision Engineering School of Mechanical EngineeringDalian University of Technology Dalian 116024 China

Correspondence should be addressed to Fengtao Wang wangftdluteducn

Received 2 May 2014 Revised 20 June 2014 Accepted 10 July 2014 Published 4 August 2014

Academic Editor Weihua Li

Copyright copy 2014 Fengtao Wang 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

Rolling-bearing faults can be effectively reflected using time-frequency characteristics However there are inevitable interferenceand redundancy components in the conventional time-frequency characteristics Therefore it is critical to extract the sensitiveparameters that reflect the rolling-bearing state from the time-frequency characteristics to accurately classify rolling-bearing faultsThus a new tensor manifold method is proposed First we apply the Hilbert-Huang transform (HHT) to rolling-bearing vibrationsignals to obtain the HHT time-frequency spectrum which can be transformed into the HHT time-frequency energy histogramThen the tensor manifold time-frequency energy histogram is extracted from the traditional HHT time-frequency spectrumusing the tensor manifold method Five time-frequency characteristic parameters are defined to quantitatively depict the failurecharacteristics Finally the tensor manifold time-frequency characteristic parameters and probabilistic neural network (PNN) arecombined to effectively classify the rolling-bearing failure samples Engineering data are used to validate the proposed methodCompared with traditional HHT time-frequency characteristic parameters the information redundancy of the time-frequencycharacteristics is greatly reduced using the tensor manifold time-frequency characteristic parameters and different rolling-bearingfault states are more effectively distinguished when combined with the PNN

1 Introduction

Rolling bearings are widely used inmodern rotatingmachin-ery and their failure is one of the most common causes ofmachine breakdowns and accidents [1ndash3] Therefore faultdiagnosis of rolling bearings is necessary to ensure thesafe and efficient operation of machines in engineeringapplications The main aspects of bearing fault diagnosis areclassification and pattern recognition where feature extrac-tion directly affects the accuracy and reliability of the faultdiagnosis [4] Rolling-bearing fault features can be generallydivided into three categories time-domain characteristicsfrequency-domain characteristics and time- and frequency-domain characteristics [5 6]

Time-domain characteristics are fairly intuitive howeverthey fluctuate significantly and lack quantitative judgingcriteria Thus they cannot be directly used to diagnosebearing faults In contrast frequency-domain characteristics

can be used to diagnose bearing fault conditions more accu-rately because different bearing faults correspond to differentcharacteristic frequencies However there are typically noiseand modulation components in the bearing fault signalsThus direct application of the frequency-domain methodwill submerge the fault characteristic frequency in noise orfalse frequency components because of improper selection ofthe demodulation parameters Furthermore signal denoisingand demodulation must be conducted before extractingthe bearing fault characteristic frequencies In the processof signal denoising and demodulation parameters such asthe denoising parameters demodulation center and filterbandwidth should be properly selected based on experienceand a satisfactory selection is only obtained after numerousadjustments

The time- and frequency-domain characteristics whichhave the intuitive feature of the time-domain characteristics

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 198362 15 pageshttpdxdoiorg1011552014198362

2 Mathematical Problems in Engineering

and good time-frequency aggregation can simultaneouslyreflect the time-domain and frequency-domain character-istics of a signal [7ndash10] Therefore extracting the time-frequency fault characteristics is important for fault diag-nosis Wang and Hu used the principle of time-frequencyimage analysis to diagnose gearbox faults in 1993 [11] thiseffort was the first application of time-frequency image forthe fault diagnosis of machinery and equipment Zhang et alsubsequently used time-frequency images to classify dieselengine faults under complex vibration conditions [12] Zhuet al used short-time Fourier transform to extract time-frequency features for fault diagnosis [13] and satisfactoryresults were achieved However the aforementioned time-frequency characteristics are not adaptive and can only beused for reciprocating machinery To overcome the limi-tations of the above methods Huang et al proposed theHHT time-frequency spectrum which is self-adaptive [14]The HHT time-frequency spectrum is suitable for analyzingnonstationary signals because of its frequency instantaneity[15] However mode mixing is inevitable for signals withinstantaneous frequency trajectory crossings [16 17] Li etal used the geometric center of the HHT time-frequencyspectrum as a feature vector [18 19] in combination withSVM and classified rolling-bearing fault signals Howeverbecause the geometric center requires a considerable amountof calculations and lacks corresponding physical meaning itcan only provide qualitative classification criteria Manifoldlearning has recently emerged in nonlinear-feature extractionbecause of its capability of effectively identifying hiddenlow-dimensional nonlinear structures in high-dimensionaldata He [20] proposed a time-frequency manifold feature bycombining the time-frequency distribution and the nonlin-ear manifold for an effective quantitative representation ofmachinery health pattern

This paper proposes a new tensor manifold time-frequency feature extraction method to overcome the weak-ness of traditional HHT time-frequency characteristics TheHHT time-frequency spectrum which contains a consider-able amount of failure information is used as the researchobject The tensor manifold learning method is applied toextract the tensor manifold time-frequency characteristicsof the HHT time-frequency spectrum The two-dimensionaltime-frequency information does not need to be convertedinto a one-dimensional vector when calculating the tensormanifold and the information loss is significantly reducedOn this basis five time-frequency characteristic parametersare defined The tensor manifold time-frequency charac-teristic parameters can distinguish different rolling-bearingfault states more effectively than traditional HHT time-frequency characteristic parameters Combined with PNNthe tensor manifold time-frequency characteristic parame-ters can effectively distinguish different rolling-bearing faultstates Engineering vibration signals were used to evaluate theefficiency of the proposed method

The remainder of this paper is organized as followsThe theory basis is introduced in Section 2 and thetensor manifold time-frequency fault feature extractionmethod is described in Section 3 Section 4 presents theadaption of the proposed method to rolling-bearing fault

classification Rolling-bearing fault classification is imple-mented in Section 5 Finally conclusions are drawn inSection 6

2 Theory Basis

21 HHT Time-Frequency Spectrum Based on the defini-tion of instantaneous frequency and EMD the HHT time-frequency spectrum is analytically derived as follows

Apply the EMD to signal119883(119905) to obtain the IMFs of119883(119905)Then the analytical form of 119883(119905) can be expressed as

119883 (119905) = Re119899

sum

119894=1

119860119894(119905) e119895 int120596119894(119905)119889119905 (1)

where Re is the real part of the selected signal 119860119894(119905) is the

instantaneous amplitude of the ith IMF and 120596119894(119905) is the

corresponding instantaneous frequencyThe time frequency and amplitude of the signals can

be combined to form the three-dimensional time-frequencyspace Then the amplitude distribution on time-frequencyplane is referred to as the HHT time-frequency spectrumwhich is expressed as

119867(119905 120596) = Re119899

sum

119894=1

119887119894119860119894(119905) e119895 int120596119894(119905)119889119905 (2)

where Re is the real part of the selected signal and 119887119894is the

indicator variable When 120596119894= 120596 119887

119894= 1 and when 120596

119894= 120596

119887119894= 0The HHT time-frequency analysis is a decomposition

method based on signal local characteristics which pro-vides a physical basis for the concept of instantaneousfrequency and sets this method apart from conventionalmethods through its use of numerous harmonic componentsto describe complex nonlinear and nonstationary signalsTherefore from the concept definition and the nature ofsignal analysis theHHT time-frequency spectrumeliminatesthe limitations of Fourier transform and can accuratelydescribe nonstationary signal characteristics

22 Tensor Manifold Algorithm

221 Locality Preserving Projection (LPP) Manifold LearningAlgorithm The LPP manifold learning algorithm aims atfinding the linear transformation matrix W to reduce thedimensionality of high-dimensional dataThere are 119897 trainingsamples x

119894119897

119894=1isin R119898 andW can be obtained by minimizing

the following objective function

minW

(sum

119894119895

(W119879x119894minus W119879x

119895)2

119878119894119895) (3)

where 119878119894119895is the similarity measure among objects and can be

defined using the k-nearest-neighbor method

119878119894119895

=

exp(minus

10038171003817100381710038171003817x119894minus x119895

10038171003817100381710038171003817

2

119891

119905

) if x119894isin 119874(119896 x

119894) or x

119895isin 119874(119896 x

119895)

0 Otherwise

(4)

Mathematical Problems in Engineering 3

where 119874(119896 x119894) denotes the 119896 nearest neighbor of x

119894and 119905 is a

positive constant Both 119896 and 119905 can be determined empiricallyEquation (3) demonstrates the feature space after dimen-

sion reduction canmaintain the local structure of the originalhigh-dimensional space We apply an algebraic transforma-tion to (3) as follows

1

2sum

119894119895

(W119879x119894minus W119879x

119895)2

119878119894119895

= sum

119894119895

W119879x119894119863119894119894x119879119894W minus sum

119894119895

W119879x119894119878119894119895x119879119894W

= W119879X (D minus S)X119879W = W119879XLX119879W

(5)

where X = [x1 x2 x

119897]D denotes an 119897 times 119897 diagonal matrix

where the diagonal element 119863119894119894

= sum119894119878119894119895 S = (119878

119894119895)119897times119897

andL=Dminus S

Then the problemof solving for the optimal vectorW canbe transformed into the following eigenvalue problem

XLX119879W = 120582XDX119879W (6)

222 Tensor LPP Manifold Learning Algorithm The LPPmanifold learning algorithm [21] can only be regarded asa one-dimensional manifold feature extraction algorithmHowever the number of training images in the two-dimensional (eg time-frequency spectrum) image featureextraction process is notably small compared to the dimen-sions of the image vectors which results in a singularity ofXDX119879 and failure of the LPP algorithm To alleviate thedrawback of the LPP this paper uses a new tensor LPPmanifold learning algorithm (Ten-LoPP) [22] to extract thetime-frequency spectrum fault characteristics

There are 119897 two-dimensional training images A119894119897

119894=1isin

R119898times119899 where 120596 denotes an 119899-dimensional unitization columnvector The main objective of the tensor manifold algorithmis to make each 119898 times 119899 image matrix A

119894project onto 120596 using

a linear transformation y119894

= A119894120596 In this manner an 119898-

dimensional column vector can be obtained and considered aprojection feature vector of image A

119894 The objective function

of the tensor manifold algorithm is expressed as follows

1

2sum

119894119895

(A119894120596 minus A

119895120596)2

119878119894119895

= sum

119894119895

W119879A119879119894119863119894119894I119898A119894120596 minus sum

119894119895

W119879A119879119894119878119894119895I119898A119894120596

= W119879A119879 [(D minus S) otimes I119898]AW = W119879A119879 (L otimes I

119898)AW

(7)

where A = [A1A2 Al] the definitions of D and L are

identical to those in the LPP manifold learning method andotimes denotes the Kronecker product

Then the problem of solving for the optimal vector 120596 istransformed into the following eigenvalue problem

A119879 (L otimes I119898)A120596 = 120582A119879 (D otimes I

119898)A120596 (8)

where 120596 is comprised of 119889 feature vectors that correspond tothe smallest nonzero eigenvalues that is there are 119889 optimal

projection vectors 120596 which can form the projection matrixW = [120596

11205962 120596

119889] For any image A

119909 there is

y119909119894

= A119909120596119894 119894 = 1 2 119889 (9)

where y1199091

y1199092

y119909119889

are the projection feature vectors ofthe sample image A

119909and y

119909= [y1199091

y1199092

y119909119889

] which iscomprised of projection feature vectors is the characteristicmatrix of the sample image A

119909

23 Probabilistic Neural Network (PNN) The neural com-position structure and elements of the PNN are shown inFigure 1

In the PNN characteristic parameters were transportedinto each node on the pattern layer through the inputlayer Then we apply layer nonlinear mapping to the inputparameters in each node of the PNNpattern and complete thecomparison between an unknown type with a known typeFinally the characteristic parameters that represent the typesare input to the next layer for processing The node structureof the layers is used to be called the RBF center and the nodeoutput is expressed as follows

O119894= R119894(1003817100381710038171003817X minus 120596

119894

1003817100381710038171003817) (10)

where the 119894th center vector is 120596119894 which is the same size as the

input vector 119877119894(sdot) denotes the radial basis function which is

typically a Gaussian function that is

exp(minus

1003817100381710038171003817X minus 120596119894

1003817100381710038171003817

2

21205902

119894

) (11)

where 120590119894denotes the shape parameter that corresponds to the

119894th component of the radial basis functionTo facilitate the calculation X and 120596

119894are processed with

mathematical regularization and unit Assume that z119894= Xsdot120596

119894

Then the above expression is expressed as follows

119892 (z119894) = exp[

(z119894minus 1)

1205902] (12)

Finally through the output layer (decision-making layer)the characteristic parameters which are derived from thepattern layer are accumulated to provide the category featurevector which is

119891119860(X) =

119873

sum

119895=1

119892 (z119894) (13)

The PNN has the following characteristics (1) the train-ing convergence speed is high making the PNN suitable forthe real-time processing of various data types (2) the patternunit can form any nonlinear mapping judgment surfacewhich is closest to the optimal judgment surface bayes (3)the selection of the RBF center kernel function has diversityand the form of the kernel function has a small effect on therecognition results and (4) the number of neuron nodes ineach PNN layer is relatively stable the hardware processing isconvenient and the fault tolerance is highThe PNNhas beenwidely used in pattern recognition prediction estimationand filtering denoising


Input unit Pattern unit Sum unit Output unit

(a)

11

1minus1

g(zi)zi = X middot wi

(b)

+1

minus1

sum

(c)

Figure 1 Internal composition structure and elements of the PNN (a) system structure of the PNN (b) pattern unit of the PNN and (c)output unit of the PNN

3 Tensor Manifold Time-Frequency FaultFeature Extraction Method

31 Description of the Proposed Method Themanifold learn-ing method is a nonlinear dimension reduction method toextract low-dimensional nonlinear characteristics fromhigh-dimensional data Unlike the conventional linear dimensionreductionmethods such asmultidimensional scaling (MDS)principal component analysis (PCA) and linear discriminantanalysis (LDA) this method is a nonlinear method toaddress the part before the whole By satisfying the entireoptimization the manifold learning method can preservethe partial manifold characteristics and effectively extractthe nonlinear manifold characteristics that are inherent inthe high-dimensional characteristic set However the man-ifold learning algorithm suffers from information loss anderror that are caused by the transformation from a set oftwo-dimensional time-frequency characteristics to a one-dimensional vector

To alleviate the drawback of information loss and errorthis section presents a tensor manifold time-frequency faultfeature extraction method based on the tensor manifoldalgorithm to extract the set of low-dimensional time-frequency characteristics from the set of high-dimensionaltime-frequency characteristics Then five tensor manifoldtime-frequency characteristic parameters were defined and

combined with the PNN to classify the rolling-bearing failuresamples

The tensor manifold time-frequency fault feature extrac-tion method is described as follows and Figure 2 presents itsflow chart

(1) Group the rolling-bearing vibration signal samplesto be classified and for training and then calculatethe HHT time-frequency spectrum To hasten thecalculation of the tensor manifold algorithm grid thetime-frequency regions integrate the energy value oftheHHT time-frequency spectrum of eachmesh andconvert theHHT time-frequency spectrum intoHHTtime-frequency energy histograms

(2) The HHT time-frequency energy histograms areessentially two-dimensional matrices Use the HHTtime-frequency energy histograms that correspondto signal samples to form a set of high-dimensionaltime-frequency characteristics

(3) Apply the tensor manifold algorithm to extract theset of low-dimensional time-frequency characteris-tics from the set of high-dimensional time-frequencycharacteristics In this manner the tensor manifoldtime-frequency energy histograms are obtained

(4) Based on the result of step (3) define different tensormanifold time-frequency characteristic parameters


Failure A HHT spectrum

Failure A

energy histograms

Failure B HHT spectrum

Failure C HHT spectrum

Failure D HHT spectrum

Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms

Tensor LPP algorithm

Failure A

energy histograms

Failure A

Use probabilistic neural network to classify

Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms

Failure B tensor manifoldtime-frequency

characteristic parameters

tensor manifoldtime-frequency


Failure C Failure D tensor manifoldtime-frequency





HHT time-frequency HHT time-frequency HHT time-frequency HHT time-frequency




Figure 2 Flow chart of the time-frequency characteristic extraction method based on a tensor manifold

Input the defined parameters of the training signalsamples into the PNN for the rolling-bearing faultclassification

(5) Input the tensor manifold time-frequency character-istic parameters of the to-be-classified signal samplesinto the trained PNN to classify the rolling-bearingfaults

32 Definition of the Time-Frequency Characteristic Parame-ters The tensor manifold time-frequency energy histogramis a nonlinear time-frequency fault feature and can effectivelydifferentiate different rolling-bearing fault signals Howeverit is equal to a two-dimensional matrix which makes itunsuitable for direct application in fault classification In thissection several parameters are presented to quantitativelymeasure the difference among the tensor manifold time-frequency energy histograms Their definitions are providedas follows

321 Energy Entropy Entropy is proposed to measure thedata complexity and the probability to generate the new signalmodel Here the energy entropy is defined as follows

119867 = minus

119899

sum

119894=1

119901119894log (119901

119894) 119901

119894=

119890119894

sum119899

119894=1119890119894

(14)

where 119867 is the energy entropy 119890119894is the value of the time

frequency energy histogram and 119901119894is the proportion of each

119890119894in the total sum

119899

119894=1119890119894 In addition the energy entropy can

reflect the uncertainty in the energy distribution

322 Energy Correlation Coefficient Divide the time-frequency energy histogram into 119898 sections by frequencyand mark E

1198911E1198912

E119891119898

and E119905= E1198911

+ E1198912

+ sdot sdot sdot + E119891119898

Because each E

119891119894varies with different time-frequency energy

histograms we can analyze the relevance of E119891119894

and E119905

to measure the difference in the time-frequency energyhistogram The energy correlation coefficient vector isdefined as follows

Ecoef = [corcoef(1) corcoef(2) sdot sdot sdot corcoef(119898)]119879

(15)

where corcoef(119894) = corcoef(E119891119894E119905) and corcoef(sdot) is the

cross-correlation function

323 Energy Sparsity The signal energy distribution of thetime-frequency energy histogram varies more significantlyas it approaches zero The sparsity expresses the sparsedistribution of energy and the purpose of estimating thesparsity is to obtain a function 119902(x) x isin R119899 If x is sparse then119902(x) is relatively large and vice versa Generally the norm 119871

119901

of vector x is used to quantitatively estimate the sparsityHere


we define the 119871119901norm of the standardized form of vector x

as follows

119902minus1

(x) =

x119901

1198991119901minus12 sdot x2

=1

1198991119901minus12sdot

(sum119899

119896=1x119901119896)1119901

(sum119899

119896=1x2119896)12

(16)

where 1 le 119901 lt infin and we select 119901 = 1 such that 1198711can

accurately reflect the energy distribution of the histogram

324 Energy Mutual Information Mutual information isproposed to measure the degree of independence amongrandom variables The mutual information of multiple vari-ables is defined as the 119870119871 divergence of the multivariatejoint probability density and its marginal probability densityproduct

119868 (119909) = 119870119871(119901 (119909)

119873

prod

119894=1

119901119894(119909119894))

= int119901 (119909) log(119901 (119909)

prod119873

119894=1119901119894(119909119894)

)119889119909

(17)

where 119870119871 is the divergence 119909 = 1199091 1199092 119909119873 119901(119909)

is the multivariate joint probability density function and119901119894(119909119894) (119894 = 1 sim 119873) is the marginal probability density

function Then 119870119871 is defined as follows

119870119871 [119901 (119909) 119902 (119909)] = int119901 (119909) log119901 (119909)

119902 (119909)119889119909 (18)

where 119901(119909) 119902(119909) are two different probability density func-tions of a random vector x The energy mutual informationcan be calculated for each histogram according to (18)

325 Energy Kurtosis Kurtosis is a physical parameter thatis proposed to measure the degree of Gaussian distribu-tion of a random variable A larger energy kurtosis in thetime-frequency energy histogram corresponds to weakerGaussianity of the energy distribution whereas a smallerkurtosis indicates stronger Gaussianity If the Gaussianityof the energy distribution is strong the energy distributionpresents the ldquomiddle big two sides smallrdquo phenomenon Theenergy values are mainly within the middle range and largeror smaller values are less likely to occur For the sequenceX = 119909

1 1199092 119909119873

of energy values the overall kurtosis isdefined as

Kurt (11990912119873

) =sum119873

119894=1(119909119894minus 120583) 119873

1205904minus 3 (19)

4 Application of the Proposed Method toRolling-Bearing Fault Classification

41 Rolling-Bearing Fault Data To verify the effectiveness ofthe proposed method the new method was used to analyzebearing fault data from the Bearing Data Center of CaseWestern Reserve University [23]

The test rig which is shown in Figure 3 was constructedfor the run-to-failure testing of the rolling bearing A 15 kW

Figure 3 Diagram of the experimental test rig

3-phase inductionmotorwas connected to a powermeter andtorque sensor by self-calibration coupling which drove thefan The load was adjusted using the fan Data were collectedusing a vibration acceleration sensor which was verticallyfixed above the chassis of the drive end bearings of theinduction motor The bearings are deep-groove ball bearingsof the type SKF6205-2RS JEMThere is a single point of failurein the inner ball and outer surface of themachining spark thefailure sizes are 018mm in diameter and 028mm deep Theexperimental data were collected with a sample frequency of12000Hz and a shaft running speed of 2953Hz (1772 rpm)The corresponding ball pass frequency inner-race (BPFI) ballrotation frequency (BS) and ball pass frequency outer-race(BPFO) were estimated to be 15993 13919 and 10587Hzrespectively

Figures 4(a)ndash4(d) depict a group of normal inner-racefault ball fault and outer-race fault time-domain signalsAlthough there are several differences among the four typesof signals in the time-domain wave nature it is difficult todistinguish the rolling-bearing fault conditions using theseintuitive qualitative differences Therefore the fault featuresthat quantitatively represent the differences of differentrolling-bearing fault statuses must be studied

42 HHT Time-Frequency Characteristics of Rolling BearingsFirst the HHT time-frequency spectra of four states werecalculated using the HHT method Then the HHT time-frequency spectrum was divided into 64 regions of identicalsize The histogram of the HHT time-frequency spectrumwas obtained via the integral of the energy amplitude for eachregion Different types of signal HHT time-frequency spectraand their histograms are shown in Figure 5

Figure 5 describes the HHT time-frequency spectrumand corresponding time-frequency energy histograms ofthe normal rolling-bearing vibration signal As shown inFigure 5 the time-frequency energy is mainly distributedin the low-frequency region and decreases with increasingfrequency The amplitude ranges from 0 to 20 g2

Figure 6 presents the HHT time-frequency spectrum andcorresponding time-frequency energy histogram with aninner-race fault As shown in Figure 6 the time-frequencyenergy is widely distributed in both the low- and high-frequency regions with a lull in the mid-frequency regionThe amplitude ranges from 0 to 40 g2 the maximum value ofwhich is greater than that in the normal case

Figure 7 presents the HHT time-frequency spectrum andcorresponding time-frequency energy histograms with a ball


0

02

Am

plitu

de (g

)

minus02

Time (s)0 01 02 03 04 05 06

(a)

Am

plitu

de (g

)

0

2

minus2

Time (s)0 01 02 03 04 05 06

(b)

Am

plitu

de (g

)

0

05

minus05

Time (s)0 01 02 03 04 05 06

(c)

Am

plitu

de (g

)

minus2

0

2

Time (s)0 01 02 03 04 05 06

(d)

Figure 4 Four heterogeneous rolling-bearing fault data (a) normal (b) inner-race faults (c) ball faults and (d) outer-race faults

0002

004006

008

0

2000

4000

0

005

01

015

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

( )quency

(a)

1 2 3 4 5 6 7 8

87

65

43

21

0

10

20

Sample number

Sample number

Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

l number

Sample number

(b)

Figure 5 (a) Normal HHT time-frequency spectrum and (b) normal time-frequency energy histogram

fault As shown in Figure 7 the time-frequency energy ismainly distributed in the high-frequency region and exhibitsa less significant yet stable distribution in the low-frequencyregionThe amplitude of the energy histogram ranges from 0to 30 g2

Figure 8 presents the HHT time-frequency spectrum andcorresponding time-frequency energy histograms with anouter-race fault As shown in Figure 8 the time-frequencyenergy is centered in the high-frequency region and exhibitsa lull in the low-frequency region The distribution trendbegins at a rather low frequency and increases abruptly ata certain high frequency The magnitude ranges from 0 to100 g2

43 Extraction of the Tensor Manifold Time-Frequency Char-acteristic Parameters For convenience and conciseness weonly discuss the tensor manifold time-frequency characteris-tic parameters of four types of rolling-bearing faults in thissection in other situations such as different damage degreesin the same fault type or different fault types with differentdamage degrees the classification result will also be discussedin subsequent Section 5

We consider 20 normal inner-race fault ball fault andouter-race fault time-domain signals (each sample datasethas 1024 points and 80 samples are used for training) TheHHT time-frequency spectrum and HHT time-frequencyenergy histogram of each signal are obtained using the


0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40

Sample numberSample number

Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21

Sample numb le number

(b)

Figure 6 (a) HHT time-frequency spectrum with an inner-race fault and (b) time-frequency energy histogram with an inner-race fault

0002

004006

008

0

2000

4000

0

01

02

03

04

Time (s)Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)

Figure 7 (a) HHT time-frequency spectrum with a ball fault and (b) time-frequency energy histogram with a ball fault

aforementioned method Then a high-dimensional time-frequency feature combination with 80 samples is obtainedThe tensor manifold algorithm is applied to extract thelow-dimensional tensor manifold features from the high-dimensional characteristic set The optimal projected vectorsW = [120596

11205962 120596

119889] are obtained and the parameter is

defined as 6 because of the distribution of eigenvalues Finallywe project the 8 lowast 8 matrix energy histogram onto W andobtain the 8 lowast 6 tensor manifold energy histograms

According to the aforementioned definition of the fivetime-frequency characteristic parameters we take the abso-lute value of the elements of the obtained 8lowast6 tensormanifoldenergy histograms and calculate the tensor manifold time-frequency characteristic parameters of the tensor manifoldenergy histograms

Five tensor manifold time-frequency characteristic pa-rameters of the above 80 samples are obtained Below only10 samples of the above four different types of rolling-bearingsignals are considered for clarity in the graphics

The results are as follows

431 Manifold Energy Entropy Samples 1ndash10 correspond tothe normal signals samples 11ndash20 correspond to the inner-race fault signals samples 21ndash30 correspond to the ball faultsignals and samples 31ndash40 correspond to the outer-racefault signals The manifold energy entropy of each tensormanifold time-frequency energy histogram is calculated anddepicted in Figure 9(a) The energy entropy of the HHTtime-frequency energy histogram without a tensor manifoldanalysis is presented in Figure 9(b) Compared to Figure 9(a)


0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)

Figure 8 (a) HHT time-frequency spectrum with outer-race fault and (b) time-frequency energy histogram with outer-race fault

012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)

Figure 9 (a) Manifold energy entropy of the four fault signals and (b) energy entropy of the four fault signals

the energy entropy shown in Figure 9(b) cannot provide aclear distinction between inner-race faults and ball faultsThus the manifold energy entropy is more appropriate forclassifying rolling-bearing faults

432 Manifold Energy Correlation Coefficient The manifoldenergy correlation coefficient (manifold energy corcoef(119894)for short) is obtained by calculating the manifold energycorcoef(119894) between E

1198911E1198912

E1198916

and E119905 The results are

shown in Figure 10As shown in Figure 10 the manifold energy corcoef(119894)

can generally distinguish different fault signals but differentmanifold energies corcoef(119894) have different abilities Firstcorcoef(1) can generally distinguish four rolling-bearingfailures corcoef(2) is also suitable for distinguishing failuresexcept for the normal and ball fault samples corcoef(3) failedto distinguish the ball fault and outer-race fault corcoef(4)failed to distinguish the inner-race fault and ball fault andcorcoef(5) and corcoef(6) failed to distinguish all faultsThuscorcoef(1) is accepted as the parameter that is best able todistinguish different rolling-bearing failures

Figure 11 presents the energy correlation coefficient (here-after denoted as ldquoenergy corcoef(119894)rdquo) which is calculatedusing six large energy bands of the HHT time-frequency

energy histogram without manifold analysis As shown inFigure 11 the energy corcoef(119894) where 119894 = 1 6 cannotprovide clear distinctions and thus is not suitable for classify-ing different rolling-bearing faults

433 Manifold Energy Sparsity The energy distributions ofdifferent fault signals are different as are the energy dis-tributions of different regions in the time-frequency energyhistogram Figure 12(a) presents themanifold energy sparsityof four rolling-bearing signals As shown in Figure 12(a) themanifold energy sparsity can effectively distinguish differentfault samples and can be used to classify different rolling-bearing faults Figure 12(b) presents the energy sparsity offour types of signal samples Although the energy sparsitycan distinguish different rolling-bearing samples the energysparsity within the sample fluctuations and the differencein energy sparsity of the inter-class sample is not obviousTherefore the energy sparsity is not a rolling-bearing faultparameter

434 Manifold Energy Mutual Information We divide thetensor manifold time-frequency energy histogram and HHTtime-frequency energy histogram of the four types of signal


0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Figure 10 Manifold energy correlation coefficient of the four faultsignals

samples into 6 regions based on the frequency Then wecalculate the corresponding mutual manifold energy infor-mation and mutual energy information

As described in Figure 13(a) different rolling-bearingfaults can be accurately distinguished using the mutual man-ifold energy information which is clearly different amongthe fault samples thus the mutual manifold energy infor-mation can be used as the rolling-bearing fault characteristicparameter Figure 13(b) illustrates that the mutual energyinformation of normal and ball fault samples is similar andthus these two fault types cannot be distinguishedThereforethe energy mutual information is not suitable for use as therolling-bearing fault characteristic parameter

435 Manifold Energy Kurtosis We calculate the manifoldenergy kurtosis and energy kurtosis based on the correspond-ing energy histograms Figure 14(a) presents the manifold

0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Figure 11 Energy correlation coefficient of the four fault signals

energy kurtosis of the four different samples the manifoldenergy kurtosis varies significantly and thus the mutualenergy manifold information can be used to classify differ-ent rolling-bearing faults Figure 14(b) presents the energykurtosis of the four different samples as shown all samplesexhibit highly similar energy kurtosis values In particularthe values of the normal fault ball fault and outer-racefault are extremely similar Therefore we cannot distinguishdifferent rolling-bearing faults using energy kurtosis

44 Discussion The merits of the proposed method forextracting the tensor manifold time-frequency characteristicparameters are mainly based on the fact that the tensormanifold time-frequency feature explores the time-varyingcharacteristic of the nonstationary fault signals The tensormanifold utilizes the HHT time-frequency fault feature ofthe rolling-bearing fault vibration signalsThus the advancedfeature is suitable for nonstationary vibration signals More-over there are simple features that are widely used forclassification in rolling-bearing fault diagnosis such as time-domain features (eg kurtosis variance) frequency-domain


0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)

Figure 12 (a) Manifold energy sparsity of the four fault signals and (b) energy sparsity of the four fault signals

010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)

Figure 13 (a) Manifold energy mutual information of the four fault signals and (b) energy mutual information of the four fault signals

features (eg subband energy) and time-frequency domainfeatures (eg HHT time-frequency spectrum) These simplefeatures are not as advantageous as the tensor manifold time-frequency features for capturing synthetic signal informationThus we use the tensor manifold time-frequency parametersfor rolling-bearing fault diagnosis in this paper

To demonstrate the benefit of the proposed parameterssimple features based on the HHT time-frequency spectrumare also conducted to analyze the four types of rolling-bearingfault signals The test results are presented in Figures 9ndash13These simple features perform worse in classification thando the tensor-manifold-based features To avoid possiblemistakes in pattern identification it is necessary to improvethe classification capability for reliable pattern diagnosis byexploring advanced features which is the purpose of thispaper

5 Rolling-Bearing Fault Classification

The preceding analysis demonstrates that the five parame-ters (manifold energy entropy manifold energy correlationcoefficientmanifold energy sparsitymanifold energymutualinformation and manifold energy kurtosis) can efficientlydistinguish the rolling-bearing fault states Thus they areused as the PNN input parameters for the bearing faultclassification

To verify the effectiveness of the manifold feature foridentifying the four bearing faults 20 samples of four types(normal inner-race faults ball faults and outer-race faults)were used as training samples The other 20 samples ofeach type were used for classification purposes Each samplewas extracted for the five aforementioned manifold featureparametersThe characteristic parameters of 80 training sam-ples were used to train the PNN and the numbers of nodes in

the four PNNs were 5 30 4 and 4 Finally the characteristicparameters of the 80 to-be-classified samples were input intothe PNN for classification The PNN classification results ofthe four bearing faults are shown in Table 1

Table 1 illustrates that when the tensor manifold time-frequency characteristic parameters are used as inputs forthe PNN four types of rolling-bearing fault samples can beeffectively distinguished and each of the 20 to-be-classifiedsamples for each type of fault can be correctly classifiedThe normal sample classification exhibits the best resultswhereas the minimum components of the category vectorsof the inner-race fault samples ball fault samples and outer-race fault samples are 092 093 and 092 respectively Theclassification results of the PNN indicate that the rolling-bearing fault condition can be effectively described using thetensor manifold time-frequency characteristic parametersand that the rolling-bearing fault type can be accuratelyidentified with the PNN

To compare the proposedmethodwith traditional extrac-tion methods we extract the five defined parameters ofthe same training and to-be-classified samples using thetraditional HHT time-frequency method as the PNN inputparameters for the bearing fault classification The results areshown in Table 2

Table 2 illustrates that when the HHT time-frequencycharacteristic parameters are used as inputs to the PNNfour types of rolling-bearing fault samples can generally bedistinguished but the distinction is not adequateThe normalsample classification exhibits the best results whereas theminimum components of the category vector of the inner-race fault samples ball fault samples and outer-race faults are069 076 and 073 respectively

In Table 1 the minimum components of the categoryvectors of the four bearing faults are 092 In contrast inTable 2 except the normal-state the components of the


0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)

Figure 14 (a) Manifold energy kurtosis of the four fault signals and (b) energy kurtosis of the four fault signals

Table 1 Classification results of the four bearing faults using the PNN with the proposed tensor manifold features

Sample Category vector of thenormal-state samples

Category vector of theinner-race fault samples

Category vector of theball fault samples

Category vector of theouter-race fault samples

1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092

category vectors of the inner-race fault the ball fault andthe outer-race fault above 085 account for 65 80 and60 respectively Compared with the results of the PNNwhich uses tensor manifold time-frequency characteristicparameters as inputs the classification performance withtraditional HHT time-frequency features is relatively poor

To verify the effectiveness of the manifold feature foridentifying different damage degrees in the same fault typeof rolling-bearing status the proposed method was used toclassify the inner-race fault samples with different damagedegrees 20 samples of four degrees (normal mild damagemoderate damage and severe damage) were used as trainingsamples The other 10 samples of each degree were used forclassification purposes The PNN classification results of theto-be-classified inner-race fault samples are shown in Table 3

To compare the proposedmethodwith traditional extrac-tion methods we extract the five defined parameters of thesame training and to-be-classified inner-race fault sampleswith different damage degrees using the traditional HHT

time-frequency method as the PNN input parameters for thebearing fault classification The results are shown in Table 4

Table 4 depicts that when the HHT time-frequencycharacteristic parameters are used as inputs to the PNNinner-race fault samples with different damage degrees cangenerally be distinguished but the distinction is not adequateThe minimum components of the category vector of theinner-race mild-damage fault samples moderate-damagefault samples and severe-damage fault samples are 083 086and 078 respectively

In Table 3 the minimum components of the categoryvectors of the four bearing faults are 093 In contrast inTable 4 except the normal-state the components of thecategory vectors of the inner-race fault the ball fault and theouter-race fault above 085 account for 90 100 and 60respectively Compared with the results in Table 3 the resultsof the PNN which uses the traditional HHT time-frequencycharacteristic parameters as inputs indicate a relatively poorclassification performance


Table 2 Classification results of the four bearing faults using the PNN with traditional HHT time-frequency features





1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082

Table 3 Classification results of the inner-race fault using the PNN with the proposed tensor manifold features


Category vector of theinner-race

mild-damagefault samples


moderate-damagefault samples


severe-damagefault samples

1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion

This paper studies the problem of rolling-bearing fault fea-ture extraction A time-frequency feature extraction methodbased on tensor manifolds for rolling bearings was proposedto overcome the deficiencies of the traditional HHT time-frequency feature extraction methods and to remove redun-dant time-frequency feature information The HHT time-frequency energy histograms of the rolling-bearing fault sig-nal were used to compose high-dimensional time-frequencyfault feature sets On this basis the signal time-frequency

characteristicswere extracted using tensormanifold learningFive tensor manifold time-frequency characteristic parame-ters were defined manifold energy entropy manifold energycorrelation coefficient manifold energy sparsity manifoldenergy mutual information and manifold energy kurtosisThese characteristic parameters and a PNN were combinedto accurately classify rolling-bearing fault samplesThe tensormanifold method can realize the nonlinear fusion of thetime-frequency information which can effectively extract theintrinsic nonlinear characteristics of high-dimensional time-frequency combination and avoid the loss of information


Table 4 Classification results of the inner-race fault using the PNN with traditional HHT time-frequency features








1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083

caused by traditional manifold-learningmethods Comparedwith the HHT time-frequency characteristic parameters thetensor manifold time-frequency characteristic parameterscan more effectively distinguish the four bearing faultsdifferent damage degrees in the same fault type and differentfault types with different damage degrees because of itsstrong nonlinearity and reduced information redundancyThe effectiveness of the proposed method was verified usingreal rolling-bearing fault signals Thus this paper providesan important method to solve the rolling-bearing featureextraction problems

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (no 51375067) the Aviation ScienceFoundation of China (no 20132163010) and the FundamentalResearch Funds for the Central Universities of China (noDUT13JS08)

References

[1] Q Liu F Chen Z Zhou and QWei ldquoFault diagnosis of rollingbearing based on wavelet package transform and ensembleempirical mode decompositionrdquo Advances in Mechanical Engi-neering vol 2013 Article ID 792584 6 pages 2013

[2] X S Lou and K A Loparo ldquoBearing fault diagnosis based onwavelet transform and fuzzy inferencerdquoMechanical Systems andSignal Processing vol 18 no 5 pp 1077ndash1095 2004

[3] F Cong J Chen G Dong and M Pecht ldquoVibration modelof rolling element bearings in a rotor-bearing system for faultdiagnosisrdquo Journal of Sound and Vibration vol 332 no 8 pp2081ndash2097 2013

[4] L Jiang J Xuan and T Shi ldquoFeature extraction based on semi-supervised kernel Marginal Fisher analysis and its application

in bearing fault diagnosisrdquo Mechanical Systems and SignalProcessing vol 41 no 1-2 pp 113ndash126 2013

[5] Y Lei Z He and Y Zi ldquoA new approach to intelligentfault diagnosis of rotating machineryrdquo Expert Systems withApplications vol 35 no 4 pp 1593ndash1600 2008

[6] Z K Peng P W Tse and F L Chu ldquoA comparison studyof improved Hilbert-Huang transform and wavelet transformapplication to fault diagnosis for rolling bearingrdquo MechanicalSystems and Signal Processing vol 19 no 5 pp 974ndash988 2005

[7] A Papandreou-SuppappolaApplications in TimeFrequency Sig-nal Processing vol 10 CRC Press Boca Raton Fla USA 2003

[8] B Boashash Time Frequency Signal Analysis and ProcessingPrentice Hall New York NY USA 2002

[9] K Grochenig Foundations of TimendashFrequency AnalysisBirkhauser Boston Mass USA 2000

[10] S Stankovic ldquoTime-frequency analysis and its application indigital watermarkingrdquo Eurasip Journal on Advances in SignalProcessing vol 2010 Article ID 579295 2010

[11] ZWWang and C Y Hu ldquoShock spectra and damage boundarycurves for nonlinear package cushioning systemrdquo PackagingTechnology and Science vol 12 no 5 pp 207ndash217 1999

[12] J Y Zhang Y Y Zhang and Y B Xie ldquoApplications of time-frequency analysis based on wavelet packet to incipient impulsefaults diagnosisrdquo Journal of Vibration Engineering vol 13 no 2pp 66ndash72 2000

[13] L M Zhu X W Niu B L Zhong and H Ding ldquoApproachfor extracting the time-frequency feature of a signal with appli-cation to machine condition monitoringrdquo Journal of VibrationEngineering vol 17 no 4 pp 71ndash76 2004

[14] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[15] M Gandetto M Guainazzo and C S Regazzoni ldquoUse of time-frequency analysis and neural networks for mode identificationin a wireless software-defined radio approachrdquo EURASIP Jour-nal on Advances in Signal Processing vol 2004 no 12 pp 1778ndash1790 2004

[16] Y G Lei Z J He and Y Y Zi ldquoEEMD method and WNN forfault diagnosis of locomotive roller bearingsrdquo Expert Systemswith Applications vol 38 no 6 pp 7334ndash7341 2011


[17] P Shan and M Li ldquoNonlinear time-varying spectral analysisHHT and MODWPTrdquo Mathematical Problems in Engineeringvol 2010 Article ID 618231 14 pages 2010

[18] H Li P Zhou and Z Zhang ldquoAn investigation into machinepattern recognition based on time-frequency image featureextraction using a support vector machinerdquo Journal of Mechan-ical Engineering Science vol 224 no 4 pp 981ndash994 2010

[19] H K Li S Zhou and W Z Huang ldquoTime-frequency imagefeature extraction for machine condition classification and itsapplicationrdquo Journal of Vibration and Shock vol 29 no 7 pp184ndash188 2010

[20] Q He ldquoTime-frequency manifold for nonlinear feature extrac-tion in machinery fault diagnosisrdquo Mechanical Systems andSignal Processing vol 35 no 1-2 pp 200ndash218 2013

[21] F Dornaika and A Assoum ldquoEnhanced and parameterlesslocality preserving projections for face recognitionrdquoNeurocom-puting vol 99 pp 448ndash457 2013

[22] M Ravishankar and D R Rameshbabu ldquoTen-LoPP tensorlocality preserving projections approach for moving objectdetection and trackingrdquo in Proceedings of the 9th InternationalConference on Computing and InformationTechnology pp 291ndash300 Springer Berlin Germany 2013

[23] ldquoBearings Vibration Data Set Case Western Reserve Univer-sityrdquo httpcsegroupscaseedubearingdatacenterhome

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and good time-frequency aggregation can simultaneouslyreflect the time-domain and frequency-domain character-istics of a signal [7ndash10] Therefore extracting the time-frequency fault characteristics is important for fault diag-nosis Wang and Hu used the principle of time-frequencyimage analysis to diagnose gearbox faults in 1993 [11] thiseffort was the first application of time-frequency image forthe fault diagnosis of machinery and equipment Zhang et alsubsequently used time-frequency images to classify dieselengine faults under complex vibration conditions [12] Zhuet al used short-time Fourier transform to extract time-frequency features for fault diagnosis [13] and satisfactoryresults were achieved However the aforementioned time-frequency characteristics are not adaptive and can only beused for reciprocating machinery To overcome the limi-tations of the above methods Huang et al proposed theHHT time-frequency spectrum which is self-adaptive [14]The HHT time-frequency spectrum is suitable for analyzingnonstationary signals because of its frequency instantaneity[15] However mode mixing is inevitable for signals withinstantaneous frequency trajectory crossings [16 17] Li etal used the geometric center of the HHT time-frequencyspectrum as a feature vector [18 19] in combination withSVM and classified rolling-bearing fault signals Howeverbecause the geometric center requires a considerable amountof calculations and lacks corresponding physical meaning itcan only provide qualitative classification criteria Manifoldlearning has recently emerged in nonlinear-feature extractionbecause of its capability of effectively identifying hiddenlow-dimensional nonlinear structures in high-dimensionaldata He [20] proposed a time-frequency manifold feature bycombining the time-frequency distribution and the nonlin-ear manifold for an effective quantitative representation ofmachinery health pattern

This paper proposes a new tensor manifold time-frequency feature extraction method to overcome the weak-ness of traditional HHT time-frequency characteristics TheHHT time-frequency spectrum which contains a consider-able amount of failure information is used as the researchobject The tensor manifold learning method is applied toextract the tensor manifold time-frequency characteristicsof the HHT time-frequency spectrum The two-dimensionaltime-frequency information does not need to be convertedinto a one-dimensional vector when calculating the tensormanifold and the information loss is significantly reducedOn this basis five time-frequency characteristic parametersare defined The tensor manifold time-frequency charac-teristic parameters can distinguish different rolling-bearingfault states more effectively than traditional HHT time-frequency characteristic parameters Combined with PNNthe tensor manifold time-frequency characteristic parame-ters can effectively distinguish different rolling-bearing faultstates Engineering vibration signals were used to evaluate theefficiency of the proposed method

The remainder of this paper is organized as followsThe theory basis is introduced in Section 2 and thetensor manifold time-frequency fault feature extractionmethod is described in Section 3 Section 4 presents theadaption of the proposed method to rolling-bearing fault

classification Rolling-bearing fault classification is imple-mented in Section 5 Finally conclusions are drawn inSection 6

2 Theory Basis

21 HHT Time-Frequency Spectrum Based on the defini-tion of instantaneous frequency and EMD the HHT time-frequency spectrum is analytically derived as follows

Apply the EMD to signal119883(119905) to obtain the IMFs of119883(119905)Then the analytical form of 119883(119905) can be expressed as

119883 (119905) = Re119899

sum

119894=1

119860119894(119905) e119895 int120596119894(119905)119889119905 (1)

where Re is the real part of the selected signal 119860119894(119905) is the

instantaneous amplitude of the ith IMF and 120596119894(119905) is the

corresponding instantaneous frequencyThe time frequency and amplitude of the signals can

be combined to form the three-dimensional time-frequencyspace Then the amplitude distribution on time-frequencyplane is referred to as the HHT time-frequency spectrumwhich is expressed as

119867(119905 120596) = Re119899

sum

119894=1

119887119894119860119894(119905) e119895 int120596119894(119905)119889119905 (2)

where Re is the real part of the selected signal and 119887119894is the

indicator variable When 120596119894= 120596 119887

119894= 1 and when 120596

119894= 120596

119887119894= 0The HHT time-frequency analysis is a decomposition

method based on signal local characteristics which pro-vides a physical basis for the concept of instantaneousfrequency and sets this method apart from conventionalmethods through its use of numerous harmonic componentsto describe complex nonlinear and nonstationary signalsTherefore from the concept definition and the nature ofsignal analysis theHHT time-frequency spectrumeliminatesthe limitations of Fourier transform and can accuratelydescribe nonstationary signal characteristics

22 Tensor Manifold Algorithm

221 Locality Preserving Projection (LPP) Manifold LearningAlgorithm The LPP manifold learning algorithm aims atfinding the linear transformation matrix W to reduce thedimensionality of high-dimensional dataThere are 119897 trainingsamples x

119894119897

119894=1isin R119898 andW can be obtained by minimizing

the following objective function

minW

(sum

119894119895

(W119879x119894minus W119879x

119895)2

119878119894119895) (3)

where 119878119894119895is the similarity measure among objects and can be

defined using the k-nearest-neighbor method

119878119894119895

=

exp(minus

10038171003817100381710038171003817x119894minus x119895

10038171003817100381710038171003817

2

119891

119905

) if x119894isin 119874(119896 x

119894) or x

119895isin 119874(119896 x

119895)

0 Otherwise

(4)



119894and 119905 is a



1

2sum

119894119895

(W119879x119894minus W119879x

119895)2

119878119894119895

= sum

119894119895

W119879x119894119863119894119894x119879119894W minus sum

119894119895

W119879x119894119878119894119895x119879119894W


(5)

where X = [x1 x2 x



= sum119894119878119894119895 S = (119878

119894119895)119897times119897

andL=Dminus S


XLX119879W = 120582XDX119879W (6)



119894=1isin








1

2sum

119894119895

(A119894120596 minus A

119895120596)2

119878119894119895

= sum

119894119895

W119879A119879119894119863119894119894I119898A119894120596 minus sum

119894119895

W119879A119879119894119878119894119895I119898A119894120596


119898)AW

(7)





119898)A120596 (8)



11205962 120596


119909 there is

y119909119894

= A119909120596119894 119894 = 1 2 119889 (9)

where y1199091

y1199092

y119909119889


119909and y

119909= [y1199091

y1199092

y119909119889


119909



O119894= R119894(1003817100381710038171003817X minus 120596

119894

1003817100381710038171003817) (10)




exp(minus

1003817100381710038171003817X minus 120596119894

1003817100381710038171003817

2

21205902

119894

) (11)





119894


119892 (z119894) = exp[

(z119894minus 1)

1205902] (12)


119891119860(X) =

119873

sum

119895=1

119892 (z119894) (13)




(a)

11

1minus1


(b)

+1

minus1

sum

(c)













Failure A

energy histograms




Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms


Failure A

energy histograms

Failure A


Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms



















119867 = minus

119899

sum

119894=1

119901119894log (119901

119894) 119901

119894=

119890119894

sum119899

119894=1119890119894

(14)




119899




1198911E1198912

E119891119898

and E119905= E1198911

+ E1198912


Because each E



and E119905



(15)




119901




as follows

119902minus1

(x) =

x119901

1198991119901minus12 sdot x2

=1

1198991119901minus12sdot

(sum119899

119896=1x119901119896)1119901

(sum119899

119896=1x2119896)12

(16)




119868 (119909) = 119870119871(119901 (119909)

119873

prod

119894=1

119901119894(119909119894))

= int119901 (119909) log(119901 (119909)

prod119873

119894=1119901119894(119909119894)

)119889119909

(17)




119870119871 [119901 (119909) 119902 (119909)] = int119901 (119909) log119901 (119909)

119902 (119909)119889119909 (18)



1 1199092 119909119873


Kurt (11990912119873

) =sum119873

119894=1(119909119894minus 120583) 119873

1205904minus 3 (19)












0

02

Am

plitu

de (g

)

minus02

Time (s)0 01 02 03 04 05 06

(a)

Am

plitu

de (g

)

0

2

minus2

Time (s)0 01 02 03 04 05 06

(b)

Am

plitu

de (g

)

0

05

minus05

Time (s)0 01 02 03 04 05 06

(c)

Am

plitu

de (g

)

minus2

0

2

Time (s)0 01 02 03 04 05 06

(d)


0002

004006

008

0

2000

4000

0

005

01

015

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

( )quency

(a)

1 2 3 4 5 6 7 8

87

65

43

21

0

10

20

Sample number

Sample number

Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

l number

Sample number

(b)







0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40


Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21


(b)


0002

004006

008

0

2000

4000

0

01

02

03

04


Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)



11205962 120596








0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894and 119905 is a



1

2sum

119894119895

(W119879x119894minus W119879x

119895)2

119878119894119895

= sum

119894119895

W119879x119894119863119894119894x119879119894W minus sum

119894119895

W119879x119894119878119894119895x119879119894W


(5)

where X = [x1 x2 x



= sum119894119878119894119895 S = (119878

119894119895)119897times119897

andL=Dminus S


XLX119879W = 120582XDX119879W (6)



119894=1isin








1

2sum

119894119895

(A119894120596 minus A

119895120596)2

119878119894119895

= sum

119894119895

W119879A119879119894119863119894119894I119898A119894120596 minus sum

119894119895

W119879A119879119894119878119894119895I119898A119894120596


119898)AW

(7)





119898)A120596 (8)



11205962 120596


119909 there is

y119909119894

= A119909120596119894 119894 = 1 2 119889 (9)

where y1199091

y1199092

y119909119889


119909and y

119909= [y1199091

y1199092

y119909119889


119909



O119894= R119894(1003817100381710038171003817X minus 120596

119894

1003817100381710038171003817) (10)




exp(minus

1003817100381710038171003817X minus 120596119894

1003817100381710038171003817

2

21205902

119894

) (11)





119894


119892 (z119894) = exp[

(z119894minus 1)

1205902] (12)


119891119860(X) =

119873

sum

119895=1

119892 (z119894) (13)




(a)

11

1minus1


(b)

+1

minus1

sum

(c)













Failure A

energy histograms




Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms


Failure A

energy histograms

Failure A


Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms



















119867 = minus

119899

sum

119894=1

119901119894log (119901

119894) 119901

119894=

119890119894

sum119899

119894=1119890119894

(14)




119899




1198911E1198912

E119891119898

and E119905= E1198911

+ E1198912


Because each E



and E119905



(15)




119901




as follows

119902minus1

(x) =

x119901

1198991119901minus12 sdot x2

=1

1198991119901minus12sdot

(sum119899

119896=1x119901119896)1119901

(sum119899

119896=1x2119896)12

(16)




119868 (119909) = 119870119871(119901 (119909)

119873

prod

119894=1

119901119894(119909119894))

= int119901 (119909) log(119901 (119909)

prod119873

119894=1119901119894(119909119894)

)119889119909

(17)




119870119871 [119901 (119909) 119902 (119909)] = int119901 (119909) log119901 (119909)

119902 (119909)119889119909 (18)



1 1199092 119909119873


Kurt (11990912119873

) =sum119873

119894=1(119909119894minus 120583) 119873

1205904minus 3 (19)












0

02

Am

plitu

de (g

)

minus02

Time (s)0 01 02 03 04 05 06

(a)

Am

plitu

de (g

)

0

2

minus2

Time (s)0 01 02 03 04 05 06

(b)

Am

plitu

de (g

)

0

05

minus05

Time (s)0 01 02 03 04 05 06

(c)

Am

plitu

de (g

)

minus2

0

2

Time (s)0 01 02 03 04 05 06

(d)


0002

004006

008

0

2000

4000

0

005

01

015

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

( )quency

(a)

1 2 3 4 5 6 7 8

87

65

43

21

0

10

20

Sample number

Sample number

Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

l number

Sample number

(b)







0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40


Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21


(b)


0002

004006

008

0

2000

4000

0

01

02

03

04


Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)



11205962 120596








0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a)

11

1minus1


(b)

+1

minus1

sum

(c)













Failure A

energy histograms




Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms


Failure A

energy histograms

Failure A


Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms



















119867 = minus

119899

sum

119894=1

119901119894log (119901

119894) 119901

119894=

119890119894

sum119899

119894=1119890119894

(14)




119899




1198911E1198912

E119891119898

and E119905= E1198911

+ E1198912


Because each E



and E119905



(15)




119901




as follows

119902minus1

(x) =

x119901

1198991119901minus12 sdot x2

=1

1198991119901minus12sdot

(sum119899

119896=1x119901119896)1119901

(sum119899

119896=1x2119896)12

(16)




119868 (119909) = 119870119871(119901 (119909)

119873

prod

119894=1

119901119894(119909119894))

= int119901 (119909) log(119901 (119909)

prod119873

119894=1119901119894(119909119894)

)119889119909

(17)




119870119871 [119901 (119909) 119902 (119909)] = int119901 (119909) log119901 (119909)

119902 (119909)119889119909 (18)



1 1199092 119909119873


Kurt (11990912119873

) =sum119873

119894=1(119909119894minus 120583) 119873

1205904minus 3 (19)












0

02

Am

plitu

de (g

)

minus02

Time (s)0 01 02 03 04 05 06

(a)

Am

plitu

de (g

)

0

2

minus2

Time (s)0 01 02 03 04 05 06

(b)

Am

plitu

de (g

)

0

05

minus05

Time (s)0 01 02 03 04 05 06

(c)

Am

plitu

de (g

)

minus2

0

2

Time (s)0 01 02 03 04 05 06

(d)


0002

004006

008

0

2000

4000

0

005

01

015

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

( )quency

(a)

1 2 3 4 5 6 7 8

87

65

43

21

0

10

20

Sample number

Sample number

Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

l number

Sample number

(b)







0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40


Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21


(b)


0002

004006

008

0

2000

4000

0

01

02

03

04


Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)



11205962 120596








0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Failure A

energy histograms




Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms


Failure A

energy histograms

Failure A


Failure B

energy histograms

Failure C

energy histograms

Failure D

energy histograms



















119867 = minus

119899

sum

119894=1

119901119894log (119901

119894) 119901

119894=

119890119894

sum119899

119894=1119890119894

(14)




119899




1198911E1198912

E119891119898

and E119905= E1198911

+ E1198912


Because each E



and E119905



(15)




119901




as follows

119902minus1

(x) =

x119901

1198991119901minus12 sdot x2

=1

1198991119901minus12sdot

(sum119899

119896=1x119901119896)1119901

(sum119899

119896=1x2119896)12

(16)




119868 (119909) = 119870119871(119901 (119909)

119873

prod

119894=1

119901119894(119909119894))

= int119901 (119909) log(119901 (119909)

prod119873

119894=1119901119894(119909119894)

)119889119909

(17)




119870119871 [119901 (119909) 119902 (119909)] = int119901 (119909) log119901 (119909)

119902 (119909)119889119909 (18)



1 1199092 119909119873


Kurt (11990912119873

) =sum119873

119894=1(119909119894minus 120583) 119873

1205904minus 3 (19)












0

02

Am

plitu

de (g

)

minus02

Time (s)0 01 02 03 04 05 06

(a)

Am

plitu

de (g

)

0

2

minus2

Time (s)0 01 02 03 04 05 06

(b)

Am

plitu

de (g

)

0

05

minus05

Time (s)0 01 02 03 04 05 06

(c)

Am

plitu

de (g

)

minus2

0

2

Time (s)0 01 02 03 04 05 06

(d)


0002

004006

008

0

2000

4000

0

005

01

015

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

( )quency

(a)

1 2 3 4 5 6 7 8

87

65

43

21

0

10

20

Sample number

Sample number

Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

l number

Sample number

(b)







0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40


Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21


(b)


0002

004006

008

0

2000

4000

0

01

02

03

04


Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)



11205962 120596








0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











as follows

119902minus1

(x) =

x119901

1198991119901minus12 sdot x2

=1

1198991119901minus12sdot

(sum119899

119896=1x119901119896)1119901

(sum119899

119896=1x2119896)12

(16)




119868 (119909) = 119870119871(119901 (119909)

119873

prod

119894=1

119901119894(119909119894))

= int119901 (119909) log(119901 (119909)

prod119873

119894=1119901119894(119909119894)

)119889119909

(17)




119870119871 [119901 (119909) 119902 (119909)] = int119901 (119909) log119901 (119909)

119902 (119909)119889119909 (18)



1 1199092 119909119873


Kurt (11990912119873

) =sum119873

119894=1(119909119894minus 120583) 119873

1205904minus 3 (19)












0

02

Am

plitu

de (g

)

minus02

Time (s)0 01 02 03 04 05 06

(a)

Am

plitu

de (g

)

0

2

minus2

Time (s)0 01 02 03 04 05 06

(b)

Am

plitu

de (g

)

0

05

minus05

Time (s)0 01 02 03 04 05 06

(c)

Am

plitu

de (g

)

minus2

0

2

Time (s)0 01 02 03 04 05 06

(d)


0002

004006

008

0

2000

4000

0

005

01

015

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

( )quency

(a)

1 2 3 4 5 6 7 8

87

65

43

21

0

10

20

Sample number

Sample number

Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

l number

Sample number

(b)







0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40


Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21


(b)


0002

004006

008

0

2000

4000

0

01

02

03

04


Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)



11205962 120596








0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

Am

plitu

de (g

)

minus02

Time (s)0 01 02 03 04 05 06

(a)

Am

plitu

de (g

)

0

2

minus2

Time (s)0 01 02 03 04 05 06

(b)

Am

plitu

de (g

)

0

05

minus05

Time (s)0 01 02 03 04 05 06

(c)

Am

plitu

de (g

)

minus2

0

2

Time (s)0 01 02 03 04 05 06

(d)


0002

004006

008

0

2000

4000

0

005

01

015

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

( )quency

(a)

1 2 3 4 5 6 7 8

87

65

43

21

0

10

20

Sample number

Sample number

Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

l number

Sample number

(b)







0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40


Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21


(b)


0002

004006

008

0

2000

4000

0

01

02

03

04


Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)



11205962 120596








0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0002

004006

008

0

2000

4000

0

05

1

15

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

000

e (s)requenc

(a)

1 2 3 4 56 7 8

87

65

43

21

0

20

40


Am

plitu

de (g

2)

2 3 4 56 7 8

87

65

43

21


(b)


0002

004006

008

0

2000

4000

0

01

02

03

04


Am

plitu

de (g

2)

002004

0060

2000

000

me (s)requenc

(a)

1 23 4 5 6 7 8

87

65

43

21

0

10

20

30


Am

plitu

de (g

2)

23 4 5 6 7 8

87

65

43

21

Sample numb number

(b)



11205962 120596








0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0002

004006

008

0

2000

4000

0

05

1

15

2

25

3

Time (s)

Frequency (Hz)

Am

plitu

de (g

2)

002004

0060

2000

00

(s)

requency

(a)

12 3 4 5 6 7 8

87

65

43

21

0

50

100

150


Am

plitu

de (g

2)

2 3 4 5 6 7 8

87

65

43

21

Sample numbe number

(b)


012345

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Man

ifold

ener

gy

entro

py

Sample number

(a)

Sample number

0

1

2

3

4

1 5 9 13 17 21 25 29 33 37

Ener

gy en

tropy

(b)




1198911E1198912

E1198916









0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(6)

0

09Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

09Corcoef(4)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(3)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(2)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Corcoef(1)

0

09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0

09 Corcoef(1)

0

09 Corcoef(2)

0

09 Corcoef(3)

0

09 Corcoef(4)

0

09

0

09 Corcoef(6)

Corcoef(5)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number





0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0005

01015

02025

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

Man

ifold

ener

gysp

arsit

y

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0005

01015

02025

Ener

gy sp

arsit

y

(b)


010203040

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

mut

ual i

nfor

mat

ion

Sample number

(a)

1 5 9 13 17 21 25 29 33 37Sample number

05

101520

Ener

gy m

utua

l in

form

atio

n

(b)













0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0204060

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Man

ifold

ener

gy

kurt

osis

Sample number

(a)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Sample number

0

5

10

Ener

gy k

urto

sis

(b)







1 1 0 0 0 0 097 001 002 0 001 099 0 0 001 002 0972 1 0 0 0 0 099 0 001 0 002 097 001 0 005 003 0923 1 0 0 0 0 095 003 002 0 002 097 001 0 003 002 0954 1 0 0 0 0 098 001 001 0 003 095 002 0 002 005 0935 1 0 0 0 0 093 005 002 0 002 098 0 0 002 002 0966 1 0 0 0 0 097 002 001 0 001 096 003 0 005 003 0927 1 0 0 0 0 095 002 003 0 002 097 001 0 002 001 0978 1 0 0 0 0 092 003 005 0 003 093 004 0 004 003 0939 1 0 0 0 0 094 002 004 0 002 095 003 0 003 002 09510 1 0 0 0 0 096 002 002 0 001 097 002 0 001 001 09811 1 0 0 0 0 093 004 003 0 001 098 001 0 003 001 09612 1 0 0 0 0 096 003 001 0 001 099 0 0 002 003 09513 1 0 0 0 0 097 001 002 0 002 096 003 0 001 002 09714 1 0 0 0 0 097 001 002 0 002 097 001 0 001 003 09615 1 0 0 0 0 095 003 002 0 002 095 003 0 006 002 09216 1 0 0 0 0 098 001 001 0 0 098 002 0 003 004 09317 1 0 0 0 0 096 001 003 0 003 096 001 0 003 002 09518 1 0 0 0 0 097 002 001 0 001 097 002 0 003 001 09619 1 0 0 0 0 093 003 004 0 002 096 002 0 003 004 09320 1 0 0 0 0 096 001 003 0 003 093 004 0 003 005 092













1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1 1 0 0 0 0 090 005 005 0 010 089 001 0 006 007 0832 1 0 0 0 0 092 003 005 0 012 087 001 0 005 006 0893 1 0 0 0 0 089 005 006 0 012 079 009 0 007 008 0854 1 0 0 0 0 090 006 004 0 013 085 012 0 005 004 0915 1 0 0 0 0 087 006 007 0 010 078 012 0 006 008 0866 1 0 0 0 0 092 004 008 0 013 076 011 0 015 006 0797 1 0 0 0 0 091 005 004 0 007 087 006 0 006 007 0878 1 0 0 0 0 082 008 01 0 002 093 005 0 015 012 0739 1 0 0 0 0 087 008 005 0 008 085 007 0 007 008 07510 1 0 0 0 0 093 005 002 0 007 087 006 0 005 007 08811 1 0 0 0 0 069 014 017 0 001 088 011 0 006 008 08612 1 0 0 0 0 074 012 014 0 020 079 001 0 007 008 08513 1 0 0 0 0 080 012 008 0 008 088 004 0 003 006 09114 1 0 0 0 0 086 009 005 0 007 087 006 0 007 004 08915 1 0 0 0 0 075 023 002 0 006 089 005 0 008 009 08316 1 0 0 0 0 078 010 012 0 003 093 002 0 009 010 08117 1 0 0 0 0 086 012 012 0 006 087 007 0 007 004 08918 1 0 0 0 0 087 006 007 0 009 089 002 0 003 008 08919 1 0 0 0 0 083 013 014 0 002 091 007 0 007 001 08320 1 0 0 0 0 092 005 003 0 005 092 003 0 007 011 082









1 1 0 0 0 0 097 001 002 0 001 098 001 0 001 002 0972 1 0 0 0 0 096 002 002 0 002 097 001 0 002 002 0963 1 0 0 0 0 098 001 001 0 001 097 002 0 003 002 0954 1 0 0 0 0 095 002 003 0 003 095 002 0 002 002 0965 1 0 0 0 0 096 002 002 0 001 098 001 0 001 003 0966 1 0 0 0 0 097 001 002 0 001 096 003 0 005 002 0937 1 0 0 0 0 094 005 002 0 002 095 003 0 003 002 0958 1 0 0 0 0 095 003 002 0 002 097 001 0 001 001 0989 1 0 0 0 0 094 002 004 0 004 094 002 0 003 003 09410 1 0 0 0 0 093 002 005 0 004 093 003 0 003 001 096

6 Conclusion












1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















1 1 0 0 0 0 092 005 003 0 011 087 002 0 012 005 0832 1 0 0 0 0 087 006 007 0 006 089 005 0 005 006 0893 1 0 0 0 0 088 006 006 0 007 091 002 0 006 006 0884 1 0 0 0 0 091 007 002 0 003 092 005 0 006 002 0925 1 0 0 0 0 083 009 008 0 006 090 004 0 003 006 0916 1 0 0 0 0 089 005 006 0 009 086 005 0 005 006 0897 1 0 0 0 0 092 004 004 0 005 091 004 0 006 003 0918 1 0 0 0 0 090 006 004 0 003 092 005 0 005 012 0839 1 0 0 0 0 091 004 005 0 009 086 005 0 015 007 07810 1 0 0 0 0 092 003 005 0 005 089 006 0 011 006 083




Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article time-frequency fault feature extraction...

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