research article integrated multiscale latent variable...

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2013 Article ID 730456 17 pageshttpdxdoiorg1011552013730456

Research ArticleIntegrated Multiscale Latent Variable Regression andApplication to Distillation Columns

Muddu Madakyaru1 Mohamed N Nounou1 and Hazem N Nounou2

1 Chemical Engineering Program Texas AampM University at Qatar Doha Qatar2 Electrical and Computer Engineering Program Texas AampM University at Qatar Doha Qatar

Correspondence should be addressed to Mohamed N Nounou mohamednounouqatartamuedu

Received 14 November 2012 Revised 20 February 2013 Accepted 20 February 2013

Academic Editor Guowei Wei

Copyright copy 2013 Muddu Madakyaru et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Proper control of distillation columns requires estimating some key variables that are challenging to measure online (such ascompositions) which are usually estimated using inferential models Commonly used inferential models include latent variableregression (LVR) techniques such as principal component regression (PCR) partial least squares (PLS) and regularized canonicalcorrelation analysis (RCCA) Unfortunately measured practical data are usually contaminated with errors which degrade theprediction abilities of inferential models Therefore noisy measurements need to be filtered to enhance the prediction accuracy ofthesemodels Multiscale filtering has been shown to be a powerful feature extraction tool In this work the advantages of multiscalefiltering are utilized to enhance the prediction accuracy of LVR models by developing an integrated multiscale LVR (IMSLVR)modeling algorithm that integratesmodeling and feature extractionThe idea behind the IMSLVRmodeling algorithm is to filter theprocess data at different decomposition levels model the filtered data from each level and then select the LVRmodel that optimizesa model selection criterion The performance of the developed IMSLVR algorithm is illustrated using three examples one usingsynthetic data one using simulated distillation column data and one using experimental packed bed distillation column data Allexamples clearly demonstrate the effectiveness of the IMSLVR algorithm over the conventional methods

1 Introduction

In the chemical process industry models play a key role invarious process operations such as process control monitor-ing and scheduling For example the control of a distillationcolumn requires the availability of the distillate and bottomstream compositions Measuring compositions online is verychallenging and costly therefore these compositions are usu-ally estimated (using inferential models) from other processvariables which are easier to measure such as temperaturepressure flow rates heat duties and others However thereare several challenges that can affect the accuracy of theseinferential models which include the presence of collinearity(or redundancy among the variables) and the presence ofmeasurement noise in the data

The presence of collinearity which is due to the largenumber of variables associated with inferential modelsincreases the uncertainty about the estimated model param-eters and degrades its prediction accuracy Latent variable

regression (LVR) which is a commonly used framework ininferential modeling deals with collinearity among the vari-ables by transforming the variables so that most of the datainformation is captured in a smaller number of variables thatcan be used to construct the model In fact LVR models per-form regression on a small number of latent variables that arelinear combinations of the original variables This generallyresults in well-conditioned models and good predictions [1]LVR model estimation techniques include principal compo-nent regression (PCR) [2 3] partial least squares (PLS) [24 5] and regularized canonical correlation analysis (RCCA)[6ndash9] PCR is performed in two main steps transform theinput variables using principal component analysis (PCA)and then construct a simple model relating the output tothe transformed inputs (principal components) using ordi-nary least squares (OLS) Thus PCR completely ignoresthe output(s) when determining the principal componentsPartial least squares (PLS) on the other hand transform thevariables taking the input-output relationship into account

2 Modelling and Simulation in Engineering

by maximizing the covariance between the output and thetransformed input variablesThat is why PLS has been widelyutilized in practice such as in the chemical industry to esti-mate distillation column compositions [10ndash13] Other LVRmodel estimationmethods include regularized canonical cor-relation analysis (RCCA) RCCA is an extension of anotherestimation technique called canonical correlation analysis(CCA) which determines the transformed input variables bymaximizing the correlation between the transformed inputsand the output(s) [6 14] Thus CCA also takes the input-output relationship into account when transforming thevariables CCA however requires computing the inverses ofthe input covariance matrix Thus in the case of collinearityamong the variables or rank deficiency regularization ofthese matrices is performed to enhance the conditioning ofthe estimated model and thus is referred to as regularizedCCA (RCCA) Since the covariance and correlation of thetransformed variables are related RCCA reduces to PLSunder certain assumptions

The other challenge in constructing inferential models isthe presence of measurement noise in the data Measuredprocess data are usually contaminated by random and grosserrors due to normal fluctuations disturbances instrumentdegradation and human errors Such errors mask the impor-tant features in the data and degrade the prediction abilityof the estimated inferential model Therefore measurementnoise needs to be filtered for improved modelrsquos predic-tion Unfortunately measured data are usually multiscale innature whichmeans that they contain features and noisewithvarying contributions over both time and frequency [15] Forexample an abrupt change in the data spans a wide rangein the frequency domain and a small range in the timedomain while a slow change spans a wide range in the timedomain and a small range in the frequency domain Filteringsuch data using conventional low pass filters such as themean filter (MF) or exponentially weighted moving average(EWMA) filter does not usually provide a good noise-featureseparation because these filtering techniques classify noise ashigh frequency features and filter the data by removing allfeatures having frequencies higher than a defined thresholdThus modeling multiscale data requires developing multi-scalemodeling techniques that can take thismultiscale natureof the data into account

Many investigators have used multiscale techniques toimprove the accuracy of estimated empirical models [16ndash27] For example in [17] the authors used multiscale rep-resentation of data to design wavelet prefilters for modelingpurposes In [16] on the other hand the author discussedthe advantages of usingmultiscale representation in empiricalmodeling and in [18] he developed amultiscale PCAmodel-ing technique and used it in process monitoring Also in [1920 23] the authors used multiscale representation to reducecollinearity and shrink the large variations in FIR modelparameters Furthermore in [21 24] multiscale representa-tion was utilized to enhance the prediction and parsimonyof fuzzy and ARX models respectively In [22] the authorextends the classic single-scale system identification tools tothe description of multiscale systems In [25] the authorsdeveloped a multiscale latent variable regression (MSLVR)

modeling algorithm by decomposing the input-output dataat multiple scales using wavelet and scaling functions andthen constructing multiple latent variable regression modelsat multiple scales using the scaled signal approximations ofthe data Note that in this MSLVR approach [25] the LVRmodels are estimated using only the scaled signals and thusneglect the effect of any significant wavelet coefficients onthe model input-output relationship Later the same authorsextended the same principle to construct nonlinear modelsusingmultiscale representation [26] Finally in [27] waveletswere used asmodulating functions for control-related systemidentification Unfortunately the advantages of multiscalefiltering have not been fully utilized to enhance the predictionaccuracy of the general class of latent variable regression(LVR)models (eg PCR PLS and RCCA) which is the focusof this paper

The objective of this paper is to utilize wavelet-basedmul-tiscale filtering to enhance the prediction accuracy of LVRmodels by developing a modeling technique that integratesmultiscale filtering and LVR model estimation The soughttechnique should provide improvement over conventionalLVR methods as well as those obtained by prefiltering theprocess data (using low pass or multiscale filters)

The remainder of this paper is organized as followsIn Section 2 a statement of the problem addressed in thiswork is presented followed by descriptions of several com-monly used LVR model estimation techniques in Section 3In Section 4 brief descriptions of low pass and multiscalefiltering techniques are presented Then in Section 5 theadvantages of utilizing multiscale filtering in empirical mod-eling are discussed followed by a description of an algorithmcalled integrated multiscale LVR modeling (IMSLVR) thatintegrates multiscale filtering and LVR modeling Thenin Section 6 the performance of the developed IMSLVRmodeling technique is assessed through three examplestwo simulated examples using synthetic data and distillationcolumn data and one experimental example using practicalpacked bed distillation column data Finally concludingremarks are presented in Section 7

2 Problem Statement

This work addresses the problem of enhancing the predictionaccuracy of linear inferential models (that can be used toestimate or infer key process variables that are difficult orexpensive to measure from more easily measured ones)using multiscale filtering All variables inputs and outputsare assumed to be contaminated with additive zero-meanGaussian noise Also it is assumed that there exists a strongcollinearity among the variables Thus given noisy measure-ments of the input and output data it is desired to constructa linear model with enhanced prediction ability (comparedto existing LVR modeling methods) using multiscale datafiltering A general form of a linear inferential model can beexpressed as

y = Xb + 120598 (1)

Modelling and Simulation in Engineering 3

where X isin R119899times119898 is the input matrix y isin R119899times1 is the outputvector b isin R119898times1 is the unknown model parameter vectorand 120598 isin R119899times1 is the model error respectively

Multiscale filtering has great feature extraction propertiesas will be discussed in Sections 4 and 5 However modelingprefiltered data may result in the elimination of model-relevant information from the filtered input-output dataTherefore the developed multiscale modeling technique isexpected to integrate multiscale filtering and LVRmodel esti-mation to enhance the prediction ability of the estimated LVRmodel Some of the conventional LVRmodeling methods aredescribed next

3 Latent Variable Regression (LVR) Modeling

One main challenge in developing inferential models is thepresence of collinearity among the large number of processvariables associated with these models which affects theirprediction ability Multivariate statistical projection methodssuch as PCR PLS and RCCA can be utilized to deal withthis issue by performing regression on a smaller number oftransformed variables called latent variables (or principalcomponents) which are linear combinations of the originalvariables This approach which is called latent variable reg-ression (LVR) generally results in well-conditioned parame-ter estimates and good model predictions [1]

In this section descriptions of some of the well-knownLVR modeling techniques which include PCR PLS andRCCA are presented However before we describe thesetechniques let us introduce some definitions Let the matrixD be defined as the augmented scaled input and output datathat is D = [Xy] Note that scaling the data is performedby making each variable (input and output) zero-mean witha unit variance Then the covariance of D can be defined asfollows [9]

C = 119864 (DD119879) = 119864 ([Xy]119879 [Xy])

= [

119864 (X119879X) 119864 (X119879y)119864 (y119879X) 119864 (y119879y)

] = [

CXX CXyCyX Cyy

]

(2)

where the matricesCXXCXyCyX andCyy are of dimensions(119898 times 119898) (119898 times 1) (1 times 119898) and (1 times 1) respectively

Since the latent variable model will be developed usingtransformed (latent) variables let us define the transformedinputs as follows

z119894= Xa119894 (3)

where z119894is the 119894th latent input variable (119894 = 1 119898) and a

119894

is the 119894th input loading vector which is of dimension (119898times 1)

31 Principal Component Regression (PCR) PCRaccounts forcollinearity in the input variables by reducing their dimensionusing principal component analysis (PCA) which utilizessingular value decomposition (SVD) to compute the latentvariables or principal components Then it constructs a sim-ple linear model between the latent variables and the output

using ordinary least square (OLS) regression [2 3] There-fore PCR can be formulated as two consecutive estimationproblems First the loading vectors are estimated by maxi-mizing the variance of the estimated principal componentsas follows

a119894= arg max

a119894var (z

119894) (119894 = 1 119898)

st a119879119894a119894= 1 z

119894= Xa119894

(4)

which (because the data are mean centered) can also beexpressed in terms of the input covariance matrix CXX asfollows

a119894= arg max

a119894a119879119894CXX a

119894(119894 = 1 119898)

st a119879119894a119894= 1

(5)

The solution of the optimization problem (5) can be obtainedusing the method of Lagrangian multiplier which results inthe following eigenvalue problem [3 28]

CXX a119894= 120582119894a119894 (6)

which means that the estimated loading vectors are theeigenvectors of the matrix CXX

Secondly after the principal components (PCs) are com-puted a subset (or all) of these PCs (which correspond tothe largest eigenvalues) are used to construct a simple linearmodel (that relates these PCs to the output) using OLS Letthe subset of PCs used to construct the model be defined asZ = [z

1sdot sdot sdot z119901] where 119901 le 119898 then the model parameters

relating these PCs to the output can be estimated using thefollowing optimization problem

120573 = arg min

120573

(

1003817100381710038171003817

Z120573 minus y1003817100381710038171003817

2

2) (7)

which has the following closed-from solution

120573 = (Z119879Z)

minus1

Z119879y (8)

Note that if all the estimated principal components are usedin constructing the inferential model (ie 119901 = 119898) then PCRreduces to OLS Note also that all principal components inPCR are estimated at the same time (using (6)) and withouttaking the model output into account Other methods thattake the input-output relationship into consideration whenestimating the principal components include partial leastsquares (PLS) and regularized canonical correlation analysis(RCCA) which are presented next

32 Partial Least Squares (PLS) PLS computes the inputloading vectors a

119894 by maximizing the covariance between

the estimated latent variable z119894and model output y that is

[14 29]

a119894= arg max

a119894cov (z

119894 y)

st a119879119894a119894= 1 z

119894= Xa119894

(9)


where 119894 = 1 119901 119901 le 119898 Since z119894= Xa

119894and the data

are mean centered (9) can also be expressed in terms of thecovariance matrix CXy as follows

a119894= arg max

a119894a119879119894CXy

st a119879119894a119894= 1

(10)

The solution of the optimization problem (10) can beobtained using the method of Lagrangian multiplier whichleads to the following eigenvalue problem [3 28]

CXy CyX a119894 = 120582

2

119894a119894

(11)

which means that the estimated loading vectors are theeigenvectors of the matrix (CXyCyX)

Note that PLS utilizes an iterative algorithm [14 30] toestimate the latent variables used in the model where onelatent variable or principal component is added iteratively tothe model After the inclusion of a latent variable the inputand output residuals are computed and the process is repeatedusing the residual data until a cross-validation error criterionis minimized [2 3 30 31]

33 Regularized Canonical Correlation Analysis (RCCA)RCCA is an extension of a method called canonical correla-tion analysis (CCA) which was first proposed by Hotelling[6] CCA reduces the dimension of the model input spaceby exploiting the correlation among the input and outputvariables The assumption behind CCA is that the inputand output data contain some joint information that can berepresented by the correlation between these variables ThusCCA computes the model loading vectors by maximizing thecorrelation between the estimated principal components andthe model output [6ndash9] that is

a119894= arg max

a119894corr (z

119894 y)

st z119894= Xa119894

(12)

where 119894 = 1 119901 119901 le 119898 Since the correlation betweentwo variables is the covariance divided by the product ofthe variances of the individual variables (12) can be writtenin terms of the covariance between z

119894and y subject to the

following two additional constraints a119879119894CXX a

119894= 1 and

Cyy = 1 Thus the CCA formulation can be expressed asfollows

a119894= arg max

a119894cov (z

119894 y)

st z119894= Xa119894 a119879119894CXX a

119894= 1

(13)

Note that the constraint (Cyy = 1) is omitted from (13)because it is satisfied by scaling the data to have zero-meanand unit variance as described in Section 3 Since the data aremean centered (13) can be written in terms of the covariancematrix CXy as follows

a119894= arg max

a119894a119879119894CXy

st a119879119894CXX a

119894= 1

(14)

The solution of the optimization problem (14) can be obtainedusing themethod of Lagrangianmultiplier which leads to thefollowing eigenvalue problem [14 28]

Cminus1XXCXyCyX a119894 = 120582

2

119894a119894 (15)

which means that the estimated loading vector is the eigen-vector of the matrix Cminus1XXCXyCyX

Equation (15) shows that CCA requires inverting thematrixCXX to obtain the loading vector a

119894 In the case of col-

linearity in the model input space the matrix CXX becomesnearly singular which results in poor estimation of the load-ing vectors and thus a poor model Therefore a regularizedversion ofCCA (calledRCCA)has been developed to accountfor this drawback of CCA [14]The formulation of RCCA canbe expressed as follows

a119894= arg max

a119894a119879119894CXy

st a119879119894((1 minus 120591

119886)CXX + 120591

119886I) a119894= 1

(16)

The solution of the optimization problem (16) can be obtainedusing themethod of Lagrangianmultiplier which leads to thefollowing eigenvalue problem [14]

[(1 minus 120591119886)CXX + 120591

119886I]minus1CXyCyX a

119894= 120582

2

119894a119894 (17)

whichmeans that the estimated loading vectors are the eigen-vectors of the matrix ([(1 minus 120591

119886)CXX + 120591

119886I]minus1CXyCyX) Note

from (17) that RCCA deals with possible collinearity in themodel input space by inverting a weighted sum of the matrixCXX and the identitymatrix that is [(1minus120591

119886)CXX+120591119886I] instead

of inverting the matrix CXX itself However this requiresknowledge of the weighting or regularization parameter 120591

119886

We know however that when 120591119886

= 0 the RCCA solution(17) reduces to the CCA solution (15) and when 120591

119886= 1 the

RCCA solution (17) reduces to the PLS solution (11) sinceCyyis a scalar

331 Optimizing the RCCA Regularization Parameter Theabove discussion shows that depending on the value of 120591

119886

where 0 le 120591119886le 1 RCCA provides a solution that converges

to CCA or PLS at the two end points 0 or 1 respectivelyIn [14] it has been shown that RCCA can provide betterresults than PLS for some intermediate values of 120591

119886between

0 and 1 Therefore in this section we propose to optimizethe performance of RCCA by optimizing its regularizationparameter by solving the following nested optimization pro-blem to find the optimum value of 120591

119886

119886= arg min

120591119886

(y minus y)119879 (y minus y)

st y = RCCA model prediction(18)

The inner loop of the optimization problem shown in (18)solves for the RCCA model prediction given the value ofthe regularization parameter 120591

119886 and the outer loop selects

the value of 120591119886that provides the least cross-validation mean

square error using unseen testing data


Note that RCCA solves for the latent variable regressionmodel in an iterative fashion similar to PLS where onelatent variable is estimated in each iteration [14] Then thecontributions of the latent variable and its correspondingmodel prediction are subtracted from the input and outputdata and the process is repeated using the residual datauntil an optimum number of principal components or latentvariables are used according to some cross-validation errorcriterion

4 Data Filtering

In this section brief descriptions of some of the filteringtechniques which will be used later to enhance the predictionof LVRmodels are presentedThese techniques include linear(or low pass) as well as multiscale filtering techniques

41 LinearData Filtering Linear filtering techniques filter thedata by computing aweighted sumof previousmeasurementsin a window of finite or infinite length and are called finiteimpulse response (FIR) and infinite impulse response (IIR)filters A linear filter can be written as follows

119896=

119873minus1

sum

119894=0

119908119894119910119896minus119894

(19)

where sum119894119908119894= 1 and 119873 is the filter length Well-known FIR

and IIR filters include the mean filer (MF) and the exponen-tially weighted moving average (EWMA) filter respectivelyThe mean filter uses equal weights that is 119908

119894= 1119873 while

the exponentially weighted moving average (EWMA) filteraverages all the previous measurements The EWMA filtercan also be implemented recursively as follows

119896= 120572119910119896+ (1 minus 120572)

119896minus1 (20)

where 119910119896and

119896are the measured and filtered data samples

at time step (119896) The parameter 120572 is an adjustable smoothingparameter lying between 0 and 1 where a value of 1 corres-ponds to no filtering and a value of zero corresponds tokeeping only the first measured point A more detailed dis-cussion of different types of filters is presented in [32]

In linear filtering the basis functions representing rawmeasured data have a temporal localization equal to thesampling interval This means that linear filters are singlescale in nature since all the basis functions have the samefixedtime-frequency localization Consequently these methodsface a tradeoff between accurate representation of temporallylocalized changes and efficient removal of temporally globalnoise [33] Therefore simultaneous noise removal and accu-rate feature representation of measured signals containingmultiscale features cannot be effectively achieved by single-scale filtering methods [33] Enhanced denoising can beachieved using multiscale filtering as will be described next

42 Multiscale Data Filtering In this section a brief descrip-tion of multiscale filtering is presented However sincemultiscale filtering relies on multiscale representation of datausing wavelets and scaling functions a brief introduction tomultiscale representation is presented first

421 Multiscale Representation of Data Any square-integ-rable signal (or data vector) can be represented at multiplescales by expressing the signal as a superposition of waveletsand scaling functions as shown in Figure 1 The signals inFigures 1(b) 1(d) and 1(f) are at increasingly coarser scalescompared to the original signal shown in Figure 1(a) Thesescaled signals are determined by filtering the data using a lowpass filter of length 119903 hf = [ℎ

1 ℎ2 ℎ

119903] which is equivalent

to projecting the original signal on a set of orthonormalscaling functions of the form

120601119895119896

(119905) =radic2

minus119895120601 (2

minus119895

119905 minus 119896) (21)

On the other hand the signals in Figures 1(c) 1(e) and 1(g)which are called the detail signals capture the details betweenany scaled signal and the scaled signal at the finer scaleThesedetailed signals are determined by projecting the signal on aset of wavelet basis functions of the form

120595119895119896

(119905) =radic2

minus119895120595 (2

minus119895

119905 minus 119896) (22)

or equivalently by filtering the scaled signal at the finer scaleusing a high pass filter of length 119903 gf = [119892

1 1198922 119892

119903] that

is derived from the wavelet basis functions Therefore theoriginal signal can be represented as the sum of all detailedsignals at all scales and the scaled signal at the coarsest scaleas follows

119909 (119905) =

1198992minus119869

sum

119896=1

a119869119896120601119869119896

(119905) +

119869

sum

119895=1

1198992minus119895

sum

119896=1

d119895119896120595119895119896

(119905) (23)

where 119895 119896 119869 and 119899 are the dilation parameter translationparameter maximum number of scales (or decompositiondepth) and the length of the original signal respectively[27 34ndash36]

Fast wavelet transform algorithms with 119874(119899) complexityfor a discrete signal of dyadic length have been developed[37] For example the wavelet and scaling function coeffi-cients at a particular scale (119895) a

119895and d119895 can be computed in a

compact fashion by multiplying the scaling coefficient vectorat the finer scale a

119895minus1 by thematricesH

119895andG

119895 respectively

that isa119895= H119895a119895minus1

d119895= G119895a119895minus1

(24)where

H119895=

[

[

[

[

ℎ1

sdot ℎ119903

sdot sdot

0 ℎ1

sdot ℎ119903

0

0 0 sdot sdot sdot

0 0 ℎ1

sdot ℎ119903

]

]

]

]1198992119895times1198992119895

G119895=

[

[

[

[

1198921

119892119903

0 1198921

119892119903

0

0 0

0 0 1198921

119892119903

]

]

]

]1198992119895times1198992119895

(25)Note that the length of the scaled and detailed signals

decreases dyadically at coarser resolutions (higher 119895) In otherwords the length of scaled signal at scale (119895) is half the lengthof scaled signal at the finer scale (119895 minus 1) This is due todownsampling which is used in discrete wavelet transform


Second scaledsignal

Third scaledsignal

(c)

(e)

(g)

Originaldata

(a)

First scaledsignal

(b)

(d)

(f)

First detailed signal

Second detailed signal

Third detailed signal

H

H

H

G

G

G

Figure 1 Multiscale decomposition of a heavy-sine signal using Haar

422 Multiscale Data Filtering Algorithm Multiscale filter-ing using wavelets is based on the observation that randomerrors in a signal are present over all wavelet coefficientswhile deterministic changes get captured in a small numberof relatively large coefficients [16 38ndash41] Thus stationaryGaussian noise may be removed by a three-step method [40]

(i) Transform the noisy signal into the time-frequencydomain by decomposing the signal on a selected setof orthonormal wavelet basis functions

(ii) Threshold the wavelet coefficients by suppressing anycoefficients smaller than a selected threshold value

(iii) Transform the thresholded coefficients back into theoriginal time domain

Donoho and coworkers have studied the statistical prop-erties of wavelet thresholding and have shown that for a noisysignal of length 119899 the filtered signal will have an error within119874(log 119899) of the error between the noise-free signal and thesignal filtered with a priori knowledge of the smoothness ofthe underlying signal [39]

Selecting the proper value of the threshold is a criticalstep in this filtering process and several methods have beendevised For good visual quality of the filtered signal theVisushrink method determines the threshold as [42]

119905119895= 120590119895radic2 log 119899 (26)

where 119899 is the signal length and 120590119895is the standard deviation of

the errors at scale 119895 which can be estimated from the waveletcoefficients at that scale using the following relation

120590119895=

1

06745

median

10038161003816100381610038161003816

119889119895119896

10038161003816100381610038161003816

(27)

Othermethods for determining the value of the threshold aredescribed in [43]

5 Multiscale LVR Modeling

In this section multiscale filtering will be utilized to enhancethe prediction accuracy of various LVR modeling techniquesin the presence of measurement noise in the data It isimportant to note that in practical process data features andnoise span wide ranges over time and frequency In otherwords features in the input-output data may change at ahigh frequency over a certain time span but at a much lowerfrequency over a different time span Also noise (especiallycolored or correlated) may have varying frequency contentsover time In modeling such multiscale data the modelestimation technique should be capable of extracting theimportant features in the data and removing the undesirablenoise and disturbance to minimize the effect of these distur-bances on the estimated model

51 Advantages of Multiscale Filtering in LVRModeling Sincepractical process data are usuallymultiscale in naturemodel-ing such data requires a multiscale modeling technique thataccounts for this type of data Below is a description ofsome of the advantages of multiscale filtering in LVR modelestimation [44]


(i) The presence of noise in measured data can consider-ably affect the accuracy of estimated LVRmodelsThiseffect can be greatly reduced by filtering the data usingwavelet-based multiscale filtering which provideseffective separation of noise from important featuresto improve the quality of the estimated models Thisnoise-feature separation can be visually seen fromFigure 1 which shows that the scaled signals are lessnoise corrupted at coarser scales

(ii) Another advantage of multiscale representation isthat correlated noise (within each variable) getsapproximately decorrelated at multiple scales Cor-related (or colored) noise arises in situations wherethe source of error is not completely independent andrandom such asmalfunctioning sensors or erroneoussensor calibrationHaving correlated noise in the datamakesmodelingmore challenging because such noiseis interpreted as important features in the data whileit is in fact noiseThis property ofmultiscale represen-tation is really useful in practice where measurementerrors are not always random [33]

These advantages will be utilized to enhance the accuracyof LVR models by developing an algorithm that integratesmultiscale filtering and LVR model estimation as describednext

52 Integrated Multiscale LVR (IMSLVR) Modeling The ideabehind the developed integrated multiscale LVR (IMSLVR)modeling algorithm is to combine the advantages of multi-scale filtering and LVR model estimation to provide inferen-tial models with improved predictions Let the time domaininput and output data be X and y and let the filtereddata (using the multiscale filtering algorithm described inSection 422) at a particular scale (119895) be X

119895and y

119895 then

the inferential model (which is estimated using these filtereddata) can be expressed as follows

y119895= X119895b119895+ 120598119895 (28)

where X119895isin R119899times119898 is the filtered input data matrix at scale (119895)

y119895isin R119899times1 is the filtered output vector at scale (119895) b isin R119898times1 is

the estimated model parameter vector using the filtered dataat scale (119895) and 120598

119895isin R119899times1 is the model error when the filtered

data at scale (119895) are used respectivelyBefore we present the formulations of the LVR modeling

techniques using the multiscale filtered data let us define thefollowing Let the matrix D

119895be defined as the augmented

scaled and filtered input and output data that isD119895= [X119895y119895]

Then the covariance ofD119895can be defined as follows [9]

C119895= 119864 (D

119895D119879119895) = 119864 ([X

119895y119895]

119879

[X119895y119895]) = [

CX119895X119895 CX119895y119895Cy119895X119895 Cy119895y119895

]

(29)

Also since the LVR models are developed using trans-formed variables the transformed input variables using thefiltered inputs at scale (119895) can be expressed as follows

z119894119895

= X119895a119894119895 (30)

where z119894119895

is the 119894th latent input variable (119894 = 1 119898) anda119894119895

is the 119894th input loading vector which is estimated usingthe filtered data at scale (119895) using any of the LVR modelingtechniques that is PCR PLS or RCCAThus the LVRmodelestimation problem (using themultiscale filtered data at scale(119895)) can be formulated as follows

521 LVR Modeling Using Multiscale Filtered Data The PCRmodel can be estimated using the multiscale filtered data atscale (119895) as follows

a119894119895

= arg maxa119894119895

a119879119894119895CX119895X119895a119894119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(31)

Similarly the PLS model can be estimated using the multi-scale filtered data at scale (119895) as follows

a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(32)

And finally the RCCAmodel can be estimated using themul-tiscale filtered data at scale (119895) as follows

a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895

((1 minus 120591119886119895

)CX119895X119895 + 120591119886119895I) a119894119895

= 1

(33)

522 Integrated Multiscale LVR Modeling Algorithm It isimportant to note that multiscale filtering enhances the qual-ity of the data and the accuracy of the LVR models estimatedusing these data However filtering the input and output dataa priori without taking the relationship between these twodata sets into account may result in the removal of featuresthat are important to the model Thus multiscale filteringneeds to be integrated with LVR model for proper noiseremoval This is what is referred to as integrated multiscaleLVR (IMSLVR) modeling One way to accomplish this integ-ration between multiscale filtering and LVR modeling isusing the following IMSLVR modeling algorithm which isschematically illustrated in Figure 2

(i) split the data into two sets training and testing(ii) scale the training and testing data sets(iii) filter the input and output training data at different

scales (decomposition depths) using the algorithmdescribed in Section 422

(iv) using the filtered training data from each scale con-struct an LVR model The number of principal com-ponents is optimized using cross-validation

(v) use the estimated model from each scale to predictthe output for the testing data and compute the cross-validation mean square error


Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data

Modelselectioncriterion

Integratedmultiscale

LVR modelScale 119895

Figure 2 A schematic diagram of the integrated multiscale LVR (IMSLVR) modeling algorithm

(vi) select the LVR with the least cross-validation meansquare error as the IMSLVR model

6 Illustrative Examples

In this section the performances of the IMSLVR modelingalgorithm described in Section 522 is illustrated and com-pared with those of the conventional LVRmodeling methodsas well as the models obtained by prefiltering the data (usingeither multiscale filtering or low pass filtering) This com-parison is performed through three examples The first twoexamples are simulated examples one using synthetic dataand the other using simulated distillation column data Thethird example is a practical example that uses experimentalpacked bed distillation column data In all examples theestimated models are optimized and compared using cross-validation byminimizing the output predictionmean squareerror (MSE) using unseen testing data as follow

MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)

where 119910(119896) and (119896) are the measured and predicted outputsat time step (119896) and 119899 is the total number of testing mea-surements Also the number of retained latent variables (orprincipal components) by the various LVR modeling tech-niques (RCCA PLS and PCR) is optimized using cross-validation Note that the data (inputs and output) are scaled(by subtracting the mean and dividing by the standarddeviation) before constructing the LVR models to enhancetheir prediction abilities

61 Example 1 Inferential Modeling of Synthetic Data In thisexample the performances of the various LVR modeling

techniques are compared by modeling synthetic data consist-ing of ten input variables and one output variable

611 Data Generation The data are generated as followsThe first two input variables are ldquoblockrdquo and ldquoheavy-sinerdquosignals and the other input variables are computed as linearcombinations of the first two inputs as follows

x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)

which means that the input matrix X is of rank 2 Then theoutput is computed as a weighed sum of all inputs as follows

y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894

= 007 003 minus005 004 002 minus11 minus004 minus002

001 minus003 for 119894 = 1 10 The total number of generateddata samples is 512 All variables inputs and output whichare assumed to be noise-free are then contaminated withadditive zero-mean Gaussian noise Different levels of noisewhich correspond to signal-to-noise ratios (SNR) of 5 10and 20 are used to illustrate the performances of the various


0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25

Figure 3 Sample output data set used in example 1 for the casewhere SNR = 10 (solid line noise-free data dots noisy data)

methods at different noise contributions The SNR is definedas the variance of the noise-free data divided by the varianceof the contaminating noise A sample of the output datawhere SNR = 10 is shown in Figure 3

612 Selection of Decomposition Depth and Optimal FilterParameters The decomposition depth used in multiscalefiltering and the parameters of the low pass filters (ie thelength of the mean filter and the value of the smoothingparameter120572) are optimized using a cross-validation criterionwhich was proposed in [43] The idea here is to split thedata into two sets odd (y

119900) and even (y

119890) filter the odd set

compute estimates of the even numbered data from thefiltered odd data by averaging the two adjacent filtered sam-ples that is y

119890119894= (12)(y

119900119894+ y119900119894+1

) and then compute thecross-validationMSE (CVMSE) with respect to the even datasamples as follows

CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)

The same process is repeated using the even numberedsamples as the training data and then the optimum filterparameters are selected by minimizing the sum of cross-validation mean squared errors using both the odd and evendata samples

613 Simulation Results In this section the performance ofthe IMSLVR modeling algorithm is compared to those ofthe conventional LVR algorithms (RCCA PLS and PCR)and those obtained by prefiltering the data using multiscalefiltering mean filtering (MF) and EWMA filtering In multi-scale filtering the Daubechies wavelet filter of order three isused and the filtering parameters for all filtering techniquesare optimized using cross-validation To obtain statisticallyvalid conclusions a Monte Carlo simulation using 1000realizations is performed and the results are shown inTable 1

0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910

Figure 4 Comparison of the model predictions using the variousLVR (RCCA) modeling techniques in example 1 for the case whereSNR = 10 (solid blue line model prediction solid red line noise-free data black dots noisy data)


The results in Table 1 clearly show that modeling prefiltereddata (using multiscale filtering (MSF+LVR) EWMA filtering(EWMA+LVR) or mean filtering (MF+LVR)) provides a sig-nificant improvement over the conventional LVR modelingtechniques This advantage is much clearer for multiscalefiltering over the single-scale (low pass) filtering techniquesHowever the IMSLVR algorithm provides a further improve-ment over multiscale prefiltering (MSF+LVR) for all noiselevels This is because the IMSLVR algorithm integratesmodeling and feature extraction to retain features in the datathat are important to the model which improves the modelprediction ability Finally the results in Table 1 also showthat the advantages of the IMSLVR algorithm are clearer forlarger noise contents that is smaller SNR As an examplethe performances of all estimated models using RCCA aredemonstrated in Figure 4 for the case where SNR = 10which clearly shows the advantages of IMSLVR over otherLVR modeling techniques

614 Effect of Wavelet Filter on Model Prediction The choiceof the wavelet filter has a great impact on the performance ofthe estimated model using the IMSLVR modeling algorithmTo study the effect of the wavelet filter on the performanceof the estimated models in this example we repeated thesimulations using different wavelet filters (Haar Daubechiessecond and third order filters) and results of a Monte Carlosimulation using 1000 realizations are shown in Figure 5Thesimulation results clearly show that the Daubechies thirdorder filter is the best filter for this example which makessense because it is smoother than the other two filters andthus it fits the nature of the data better

62 Example 2 Inferential Modeling of Distillation Col-umn Data In this example the prediction abilities of thevarious modeling techniques (ie IMSLVR MSF+LVREWMA+LVR MF+LVR and LVR) are compared throughtheir application to model the distillate and bottom streamcompositions of a distillation columnThedynamic operationof the distillation column which consists of 32 theoreticalstages (including the reboiler and a total condenser) is sim-ulated using Aspen Tech 72 The feed stream which is abinary mixture of propane and isobutene enters the columnat stage 16 as a saturated liquid having a flow rate of 1 kmols atemperature of 322K and compositions of 40molepropaneand 60 mole isobutene The nominal steady state operatingconditions of the column are presented in Table 2

621 Data Generation The data used in this modeling pro-blem are generated by perturbing the flow rates of the feedand the reflux streams from their nominal operating valuesFirst step changes of magnitudes plusmn2 in the feed flow ratearound its nominal condition are introduced and in eachcase the process is allowed to settle to a new steady state Afterattaining the nominal conditions again similar step changesof magnitudes plusmn2 in the reflux flow rate around its nominalcondition are introduced These perturbations are used togenerate training and testing data (each consisting of 512 datapoints) to be used in developing the various models These

IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR

Figure 5 Comparison of the MSEs for various wavelet filters inexample 1 for the case where SNR = 10

perturbations (in the training and testing data sets) are shownin Figures 6(e) 6(f) 6(g) and 6(h)


0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)

Figure 6 The dynamic input-output data used for training and testing the models in the simulated distillation column example for the casewhere the noise SNR = 10 (solid red line noise-free data blue dots noisy data)

In this simulated modeling problem the input variablesconsist of ten temperatures at different trays of the columnin addition to the flow rates of the feed and reflux streamsThe output variables on the other hand are the compositionsof the light component (propane) in the distillate and the bot-tom streams (ie 119909

119863and119909119861 resp)The dynamic temperature

and composition data generated using the Aspen simulator(due to the perturbations in the feed and reflux flow rates) areassumed to be noise-free which are then contaminated withzero-mean Gaussian noise To assess the robustness of thevariousmodeling techniques to different noise contributionsdifferent levels of noise (which correspond to signal-to-noiseratios of 5 10 and 20) are used Sample training and testing

data sets showing the effect of the perturbations on thecolumn compositions are shown in Figures 6(a) 6(b) 6(c)and 6(d) for the case where the signal-to-noise ratio is 10

622 Simulation Results In this section the performance ofthe IMSLVR algorithm is compared to the conventional LVRmodels as well as the models estimated using prefiltered dataTo obtain statistically valid conclusions a Monte Carlo simu-lation of 1000 realizations is performed and the results arepresented in Tables 3 and 4 for the estimation of top andbottom distillation column compositions that is 119909

119863and

119909119861 respectively As in the first example the results in both


Table 1 Comparison of the Monte Carlo MSEs for the various modeling techniques in example 1

Model type IMSLVR MSF+LVR EWMA+LVR MF+LVR LVRSNR = 5

RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508

Table 2 Steady state operating conditions of the distillation column

Process variable Value Process variable ValueFeed

F 1 kg molesec P 17022 times 10

6 PaT 322 K 119909

1198630979

P 17225 times 10

6 Pa Reboiler drum119911119865

04 B 05979 kg molesecReflux drum Q 27385 times 10

7WattsD 040206 kg molesec T 366 KT 325K P 172362 times 10

6 PaReflux 626602 kgsec 119909

119861001

Table 3 Comparison of the Monte Carlo MSErsquos for 119909119863in the simulated distillation column example

Model type IMSLVR MSF+LVR EWMA+LVR MF+LVR LVRtimes10

minus4 SNR = 5RCCA 00197 00205 00274 00286 00987PLS 00202 00210 00288 00303 00984PCR 00204 00212 00288 00357 00983times10


minus5 SNR = 20RCCA 00785 00791 01071 01157 03012PLS 00844 00849 01130 01218 03017PCR 00801 00803 01112 01200 03040

Tables 3 and 4 show that modeling prefiltered data signifi-cantly improves the prediction accuracy of the estimated LVRmodels over the conventional model estimation methodsThe IMSLVR algorithm however improves the prediction ofthe estimated LVR model even further especially at highernoise contents that is at smaller SNR To illustrate the relativeperformances of the various LVRmodeling techniques as anexample the performances of the estimated RCCA models

for the top composition (119909119863) in the case of SNR = 10 are

shown in Figure 7

63 Example 3 Dynamic LVR Modeling of an Experimen-tal Packed Bed Distillation Column In this example thedeveloped IMSLVR modeling algorithm is used to modela practical packed bed distillation column with a recycle







50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863

Figure 7 Comparison of the RCCA model predictions of 119909119863using the various LVR (RCCA) modeling techniques for the simulated dis-

tillation column example and the case where the noise SNR = 10 (solid blue line model prediction black dots noisy data solid red linenoise-free data)


Reflux drum

Condenser

119879 temperature measurement sensor

119865 flow measurement sensor

119863 density measurement sensor

119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column

Bottom product storage

Top product storage

Figure 8 A schematic diagram of the packed bed distillation column setup

Table 5 Steady state operating conditions of the packed bed distil-lation column

Process variable ValueFeed flow rate 40 kghrReflux flow rate 5 kghrFeed composition 03 mole fractionBottom level 400mm

stream More details about the process data collection andmodel estimation are presented next

631 Description of the Packed Bed Distillation Column Thepacked bed distillation column used in this experimentalmodeling example is a 6-inch diameter stainless steel columnconsisting of three packing sections (bottommiddle and topsection) rising to a height of 20 feet The column which isused to separate a methanol-water mixture has Koch-Sulzerstructured packing with liquid distributors above each pack-ing section An industrial quality Distributed Control System(DCS) is used to control the column A schematic diagram

of packed bed distillation column is shown in Figure 8 TenResistance Temperature Detector (RTD) sensors are fixedat various locations in the setup to monitor the columntemperature profile The flow rates and densities of variousstreams (eg feed reflux top product and bottom product)are also monitored In addition the setup includes fourpumps and five heat exchangers at different locations

The feed stream enters the column near its midpointThepart of the column above the feed constitutes the rectifyingsection and the part below (and including) the feed consti-tutes the stripping sectionThe feed flows down the strippingsection into the bottom of the column where a certain levelof liquid is maintained by a closed-loop controller A steam-heated reboiler is used to heat and vaporize part of the bottomstream which is then sent back to the column The vaporpasses up the entire column contacting descending liquid onits way down The bottom product is withdrawn from thebottom of the column and is then sent to a heat exchangerwhere it is used to heat the feed stream The vapors risingthrough the rectifying section are completely condensedin the condenser and the condensate is collected in thereflux drum in which a specified liquid level is maintained


0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)

Figure 9 Training and testing data used in the packed bed distillation column modeling example

A part of the condensate is sent back to the column using areflux pump The distillate not used as a reflux is cooled in aheat exchanger The cooled distillate and bottom streams arecollected in a feed tank where they are mixed and later sentas a feed to the column

632 Data Generation and Inferential Modeling A samplingtime of 4 s is chosen to collect the data used in this modelingproblem The data are generated by perturbing the flowrates of the feed and the reflux streams from their nominaloperating values which are shown in Table 5 First stepchanges of magnitudes plusmn50 in the feed flow rate around itsnominal value are introduced and in each case the processis allowed to settle to a new steady state After attaining thenominal conditions again similar step changes ofmagnitudesplusmn40 in the reflux flow rate around its nominal value areintroducedThese perturbations are used to generate trainingand testing data (each consisting of 4096 data samples) to be

used in developing the various models These perturbationsare shown in Figures 9(e) 9(f) 9(g) and 9(h) and the effectof these perturbations on the distillate and bottom streamcompositions are shown in Figures 9(a) 9(b) 9(c) and 9(d)

In this modeling problem the input variables consist ofsix temperatures at different positions in the column inaddition to the flow rates of the feed and reflux streams Theoutput variables on the other hand are the compositions ofthe light component (methane) in the distillate and bottomstreams (119909

119863and 119909

119861 resp) Because of the dynamic nature

of the column and the presence of a recycle stream thecolumn always runs under transient conditions These pro-cess dynamics can be accounted for in inferential models byincluding lagged inputs and outputs into the model [13 45ndash48] Therefore in this dynamic modeling problem laggedinputs and outputs are used in the LVR models to accountfor the dynamic behavior of the column Thus the modelinput matrix consists of 17 columns eight columns for theinputs (the six temperatures and the flow rates of the feed


0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR

Figure 10 Comparison of the model predictions using the variousmodeling methods for the experimental packed bed distillationcolumn example (solid blue line model prediction black dots plantdata)

and reflux streams) eight columns for the lagged inputs andone column for the lagged output To show the advantage ofthe IMSLVR algorithm its performance is compared to thoseof the conventional LVR models and the models estimatedusing multiscale prefiltered data and the results are shown inFigure 10The results clearly show that multiscale prefilteringprovides a significant improvement over the conventionalLVR (RCCA) method (which sought to overfit the measure-ments) and that the IMSLVR algorithm provides furtherimprovement in the smoothness and the prediction accuracyNote that Figure 10 shows only a part of the testing data forthe sake of clarity

7 Conclusions

Latent variable regression models are commonly used inpractice to estimate variables which are difficult to measurefrom other easier-to-measure variables This paper presentsa modeling technique to improve the prediction ability ofLVR models by integrating multiscale filtering and LVRmodel estimation which is called integrated multiscale LVR(IMSLVR)modelingThe idea behind the developed IMSLVRalgorithm is to filter the input and output data at differentscales construct different models using the filtered datafrom each scale and then select the model that providesthe minimum cross-validation MSE The performance of theIMSLVR modeling algorithm is compared to the conven-tional LVRmodeling methods as well as modeling prefiltereddata either using low pass filtering (such as mean filtering orEMWA filtering) or using multiscale filtering through threeexamples two simulated examples and one practical example

The simulated examples use synthetic data and simulateddistillation column data while the practical example usesexperimental packed bed distillation columndataThe resultsof all examples show that data prefiltering (especially usingmultiscale filtering) provides a significant improvement overthe convectional LVR methods and that the IMSLVR algo-rithm provides a further improvement especially at highernoise levels The main reason for the advantages of theIMSLVR algorithm over modeling prefiltered data is that itintegratesmultiscale filtering and LVRmodeling which helpsretain themodel-relevant features in the data that can provideenhanced model predictions

Acknowledgment

This work was supported by the Qatar National ResearchFund (a member of the Qatar Foundation) under GrantNPRP 09ndash530-2-199

References

[1] B R kowalski and M B Seasholtz ldquoRecent developments inmultivariate calibrationrdquo Journal of Chemometrics vol 5 no 3pp 129ndash145 1991

[2] I Frank and J Friedman ldquoA statistical view of some chemomet-ric regression toolsrdquo Technometrics vol 35 no 2 pp 109ndash1481993

[3] M Stone and R J Brooks ldquoContinuum regression cross-validated sequentially constructed prediction embracing ordi-nary least squares partial least squares and principal compo-nents regressionrdquo Journal of the Royal Statistical Society SeriesB vol 52 no 2 pp 237ndash269 1990

[4] S Wold Soft Modeling The Basic Design and Some ExtensionsSystems under Indirect Observations Elsevier Amsterdam TheNetherlands 1982

[5] E C Malthouse A C Tamhane and R S H Mah ldquoNonlinearpartial least squaresrdquo Computers and Chemical Engineering vol21 no 8 pp 875ndash890 1997

[6] H Hotelling ldquoRelations between two sets of variablesrdquo Bio-metrika vol 28 pp 321ndash377 1936

[7] F R Bach and M I Jordan ldquoKernel independent componentanalysisrdquo Journal of Machine Learning Research vol 3 no 1 pp1ndash48 2003

[8] D R Hardoon S Szedmak and J Shawe-Taylor ldquoCanonicalcorrelation analysis an overview with application to learningmethodsrdquo Neural Computation vol 16 no 12 pp 2639ndash26642004

[9] M Borga T Landelius and H Knutsson ldquoA unified approachto pca pls mlr and cca technical reportrdquo Tech Rep LinkopingUniversity 1997

[10] J V Kresta T E Marlin and J F McGregor ldquodevelopment ofinferential process models using plsrdquo Computers amp ChemicalEngineering vol 18 pp 597ndash611 1994

[11] T Mejdell and S Skogestad ldquoEstimation of distillation compo-sitions frommultiple temperature measurements using partial-least squares regressionrdquo Industrial amp Engineering ChemistryResearch vol 30 pp 2543ndash2555 1991

[12] M Kano KMiyazaki S Hasebe and I Hashimoto ldquoInferentialcontrol system of distillation compositions using dynamic


partial least squares regressionrdquo Journal of Process Control vol10 no 2 pp 157ndash166 2000

[13] T Mejdell and S Skogestad ldquoComposition estimator in a pilot-plant distillation columnrdquo Industrial amp Engineering ChemistryResearch vol 30 pp 2555ndash2564 1991

[14] H YamamotoH Yamaji E Fukusaki HOhno andH FukudaldquoCanonical correlation analysis for multivariate regression andits application to metabolic fingerprintingrdquo Biochemical Engi-neering Journal vol 40 no 2 pp 199ndash204 2008

[15] B R Bakshi andG Stephanopoulos ldquoRepresentation of processtrends-IV Induction of real-time patterns from operating datafor diagnosis and supervisory controlrdquoComputers andChemicalEngineering vol 18 no 4 pp 303ndash332 1994

[16] B Bakshi ldquoMultiscale analysis and modeling using waveletsrdquoJournal of Chemometrics vol 13 no 3 pp 415ndash434 1999

[17] S Palavajjhala RMotrad and B Joseph ldquoProcess identificationusing discrete wavelet transform design of pre-filtersrdquo AIChEJournal vol 42 no 3 pp 777ndash790 1996

[18] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[19] A N Robertson K C Park and K F Alvin ldquoExtraction ofimpulse response data via wavelet transform for structural sys-tem identificationrdquo Journal of Vibration and Acoustics vol 120no 1 pp 252ndash260 1998

[20] M Nikolaou and P Vuthandam ldquoFIR model identificationparsimony through kernel compression with waveletsrdquo AIChEJournal vol 44 no 1 pp 141ndash150 1998

[21] M N Nounou and H N Nounou ldquoMultiscale fuzzy systemidentificationrdquo Journal of Process Control vol 15 no 7 pp 763ndash770 2005

[22] M S Reis ldquoAmultiscale empirical modeling framework for sys-tem identificationrdquo Journal of Process Control vol 19 pp 1546ndash1557 2009

[23] M Nounou ldquoMultiscale finite impulse response modelingrdquoEngineering Applications of Artificial Intelligence vol 19 pp289ndash304 2006

[24] M N Nounou and H N Nounou ldquoImproving the predictionand parsimony of ARX models using multiscale estimationrdquoApplied Soft Computing Journal vol 7 no 3 pp 711ndash721 2007

[25] M N Nounou and H N Nounou ldquoMultiscale latent variableregressionrdquo International Journal of Chemical Engineering vol2010 Article ID 935315 5 pages 2010

[26] M N Nounou and H N Nounou ldquoReduced noise effect innonlinear model estimation using multiscale representationrdquoModelling and Simulation in Engineering vol 2010 Article ID217305 8 pages 2010

[27] J F Carrier and G Stephanopoulos ldquoWavelet-Based Modula-tion inControl-Relevant Process IdentificationrdquoAIChE Journalvol 44 no 2 pp 341ndash360 1998

[28] MMadakyaruMNounou andHNounou ldquoLinear inferentialmodeling theoretical perspectives extensions and compara-tive analysisrdquo Intelligent Control andAutomation vol 3 pp 376ndash389 2012

[29] R Rosipal and N Kramer ldquoOverview and recent advances inpartial least squaresrdquo in Subspace Latent Structure and Fea-ture Selection Lecture Notes in Computer Science pp 34ndash51Springer New York NY USA 2006

[30] P Geladi and B R Kowalski ldquoPartial least-squares regression atutorialrdquo Analytica Chimica Acta vol 185 no C pp 1ndash17 1986

[31] SWold ldquoCross-validatory estimation of the number of compo-nents in factor and principal components modelsrdquo Technomet-rics vol 20 no 4 p 397 1978

[32] R D Strum and D E Kirk First Principles of Discrete Systemsand Digital Signal Procesing Addison-Wesley Reading MassUSA 1989

[33] M N Nounou and B R Bakshi ldquoOn-line multiscale filtering ofrandom and gross errors without process modelsrdquo AIChE Jour-nal vol 45 no 5 pp 1041ndash1058 1999

[34] G Strang Introduction to Applied Mathematics Wellesley-Cambridge Press Wellesley Mass USA 1986

[35] G Strang ldquoWavelets and dilation equations a brief introduc-tionrdquo SIAM Review vol 31 no 4 pp 614ndash627 1989

[36] I Daubechies ldquoOrthonormal bases of compactly supportedwaveletsrdquo Communications on Pure and Applied Mathematicsvol 41 no 7 pp 909ndash996 1988

[37] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[38] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[39] D Donoho and I Johnstone ldquoIdeal de-noising in an orthonor-mal basis chosen from a library of basesrdquo Tech Rep Depart-ment of Statistics Stanford University 1994

[40] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B vol 57 no 2 pp 301ndash369 1995

[41] M Nounou and B R Bakshi ldquoMultiscale methods for de-noising and compresionrdquo in Wavelets in Analytical ChimistryB Walczak Ed pp 119ndash150 Elsevier AmsterdamThe Nether-lands 2000

[42] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[43] G P Nason ldquoWavelet shrinkage using cross-validationrdquo Journalof the Royal Statistical Society Series B vol 58 no 2 pp 463ndash479 1996

[44] M N Nounou ldquoDealing with collinearity in fir modelsusing bayesian shrinkagerdquo Indsutrial and Engineering ChemsitryResearch vol 45 pp 292ndash298 2006

[45] N L Ricker ldquoThe use of biased least-squares estimators forparameters in discrete-time pulse-response modelsrdquo Industrialand Engineering Chemistry Research vol 27 no 2 pp 343ndash3501988

[46] J F MacGregor and A K L Wong ldquoMultivariate model iden-tification and stochastic control of a chemical reactorrdquo Techno-metrics vol 22 no 4 pp 453ndash464 1980

[47] T Mejdell and S Skogestad ldquoEstimation of distillation compo-sitions frommultiple temperature measurements using partial-least-squares regressionrdquo Industrial amp Engineering ChemistryResearch vol 30 no 12 pp 2543ndash2555 1991

[48] T Mejdell and S Skogestad ldquoOutput estimation using multiplesecondarymeasurements high-purity distillationrdquoAIChE Jour-nal vol 39 no 10 pp 1641ndash1653 1993

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



by maximizing the covariance between the output and thetransformed input variablesThat is why PLS has been widelyutilized in practice such as in the chemical industry to esti-mate distillation column compositions [10ndash13] Other LVRmodel estimationmethods include regularized canonical cor-relation analysis (RCCA) RCCA is an extension of anotherestimation technique called canonical correlation analysis(CCA) which determines the transformed input variables bymaximizing the correlation between the transformed inputsand the output(s) [6 14] Thus CCA also takes the input-output relationship into account when transforming thevariables CCA however requires computing the inverses ofthe input covariance matrix Thus in the case of collinearityamong the variables or rank deficiency regularization ofthese matrices is performed to enhance the conditioning ofthe estimated model and thus is referred to as regularizedCCA (RCCA) Since the covariance and correlation of thetransformed variables are related RCCA reduces to PLSunder certain assumptions

The other challenge in constructing inferential models isthe presence of measurement noise in the data Measuredprocess data are usually contaminated by random and grosserrors due to normal fluctuations disturbances instrumentdegradation and human errors Such errors mask the impor-tant features in the data and degrade the prediction abilityof the estimated inferential model Therefore measurementnoise needs to be filtered for improved modelrsquos predic-tion Unfortunately measured data are usually multiscale innature whichmeans that they contain features and noisewithvarying contributions over both time and frequency [15] Forexample an abrupt change in the data spans a wide rangein the frequency domain and a small range in the timedomain while a slow change spans a wide range in the timedomain and a small range in the frequency domain Filteringsuch data using conventional low pass filters such as themean filter (MF) or exponentially weighted moving average(EWMA) filter does not usually provide a good noise-featureseparation because these filtering techniques classify noise ashigh frequency features and filter the data by removing allfeatures having frequencies higher than a defined thresholdThus modeling multiscale data requires developing multi-scalemodeling techniques that can take thismultiscale natureof the data into account

Many investigators have used multiscale techniques toimprove the accuracy of estimated empirical models [16ndash27] For example in [17] the authors used multiscale rep-resentation of data to design wavelet prefilters for modelingpurposes In [16] on the other hand the author discussedthe advantages of usingmultiscale representation in empiricalmodeling and in [18] he developed amultiscale PCAmodel-ing technique and used it in process monitoring Also in [1920 23] the authors used multiscale representation to reducecollinearity and shrink the large variations in FIR modelparameters Furthermore in [21 24] multiscale representa-tion was utilized to enhance the prediction and parsimonyof fuzzy and ARX models respectively In [22] the authorextends the classic single-scale system identification tools tothe description of multiscale systems In [25] the authorsdeveloped a multiscale latent variable regression (MSLVR)

modeling algorithm by decomposing the input-output dataat multiple scales using wavelet and scaling functions andthen constructing multiple latent variable regression modelsat multiple scales using the scaled signal approximations ofthe data Note that in this MSLVR approach [25] the LVRmodels are estimated using only the scaled signals and thusneglect the effect of any significant wavelet coefficients onthe model input-output relationship Later the same authorsextended the same principle to construct nonlinear modelsusingmultiscale representation [26] Finally in [27] waveletswere used asmodulating functions for control-related systemidentification Unfortunately the advantages of multiscalefiltering have not been fully utilized to enhance the predictionaccuracy of the general class of latent variable regression(LVR)models (eg PCR PLS and RCCA) which is the focusof this paper

The objective of this paper is to utilize wavelet-basedmul-tiscale filtering to enhance the prediction accuracy of LVRmodels by developing a modeling technique that integratesmultiscale filtering and LVR model estimation The soughttechnique should provide improvement over conventionalLVR methods as well as those obtained by prefiltering theprocess data (using low pass or multiscale filters)

The remainder of this paper is organized as followsIn Section 2 a statement of the problem addressed in thiswork is presented followed by descriptions of several com-monly used LVR model estimation techniques in Section 3In Section 4 brief descriptions of low pass and multiscalefiltering techniques are presented Then in Section 5 theadvantages of utilizing multiscale filtering in empirical mod-eling are discussed followed by a description of an algorithmcalled integrated multiscale LVR modeling (IMSLVR) thatintegrates multiscale filtering and LVR modeling Thenin Section 6 the performance of the developed IMSLVRmodeling technique is assessed through three examplestwo simulated examples using synthetic data and distillationcolumn data and one experimental example using practicalpacked bed distillation column data Finally concludingremarks are presented in Section 7

2 Problem Statement

This work addresses the problem of enhancing the predictionaccuracy of linear inferential models (that can be used toestimate or infer key process variables that are difficult orexpensive to measure from more easily measured ones)using multiscale filtering All variables inputs and outputsare assumed to be contaminated with additive zero-meanGaussian noise Also it is assumed that there exists a strongcollinearity among the variables Thus given noisy measure-ments of the input and output data it is desired to constructa linear model with enhanced prediction ability (comparedto existing LVR modeling methods) using multiscale datafiltering A general form of a linear inferential model can beexpressed as

y = Xb + 120598 (1)







C = 119864 (DD119879) = 119864 ([Xy]119879 [Xy])

= [

119864 (X119879X) 119864 (X119879y)119864 (y119879X) 119864 (y119879y)

] = [

CXX CXyCyX Cyy

]

(2)



z119894= Xa119894 (3)


119894




a119894= arg max

a119894var (z

119894) (119894 = 1 119898)

st a119879119894a119894= 1 z

119894= Xa119894

(4)


a119894= arg max

a119894a119879119894CXX a

119894(119894 = 1 119898)

st a119879119894a119894= 1

(5)


CXX a119894= 120582119894a119894 (6)





120573 = arg min

120573

(

1003817100381710038171003817

Z120573 minus y1003817100381710038171003817

2

2) (7)


120573 = (Z119879Z)

minus1

Z119879y (8)





[14 29]

a119894= arg max

a119894cov (z

119894 y)

st a119879119894a119894= 1 z

119894= Xa119894

(9)



119894and the data


a119894= arg max

a119894a119879119894CXy

st a119879119894a119894= 1

(10)


CXy CyX a119894 = 120582

2

119894a119894

(11)




a119894= arg max

a119894corr (z

119894 y)

st z119894= Xa119894

(12)




119894= 1 and


a119894= arg max

a119894cov (z

119894 y)

st z119894= Xa119894 a119879119894CXX a

119894= 1

(13)


a119894= arg max

a119894a119879119894CXy

st a119879119894CXX a

119894= 1

(14)



2

119894a119894 (15)





a119894= arg max

a119894a119879119894CXy

st a119879119894((1 minus 120591

119886)CXX + 120591

119886I) a119894= 1

(16)


[(1 minus 120591119886)CXX + 120591


119894= 120582

2

119894a119894 (17)


119886)CXX + 120591



119886)CXX+120591119886I] instead


119886



119886= 1 the



119886



119886between


119886

119886= arg min

120591119886









4 Data Filtering



119896=

119873minus1

sum

119894=0

119908119894119910119896minus119894

(19)



119894= 1119873 while


119896= 120572119910119896+ (1 minus 120572)

119896minus1 (20)

where 119910119896and






1 ℎ2 ℎ



120601119895119896

(119905) =radic2

minus119895120601 (2

minus119895

119905 minus 119896) (21)


120595119895119896

(119905) =radic2

minus119895120595 (2

minus119895

119905 minus 119896) (22)


1 1198922 119892

119903] that


119909 (119905) =

1198992minus119869

sum

119896=1

a119869119896120601119869119896

(119905) +

119869

sum

119895=1

1198992minus119895

sum

119896=1

d119895119896120595119895119896

(119905) (23)






119895andG

119895 respectively

that isa119895= H119895a119895minus1

d119895= G119895a119895minus1

(24)where

H119895=

[

[

[

[

ℎ1

sdot ℎ119903

sdot sdot

0 ℎ1

sdot ℎ119903

0

0 0 sdot sdot sdot

0 0 ℎ1

sdot ℎ119903

]

]

]

]1198992119895times1198992119895

G119895=

[

[

[

[

1198921

119892119903

0 1198921

119892119903

0

0 0

0 0 1198921

119892119903

]

]

]

]1198992119895times1198992119895




Second scaledsignal

Third scaledsignal

(c)

(e)

(g)

Originaldata

(a)

First scaledsignal

(b)

(d)

(f)




H

H

H

G

G

G








119905119895= 120590119895radic2 log 119899 (26)



120590119895=

1

06745

median

10038161003816100381610038161003816

119889119895119896

10038161003816100381610038161003816

(27)










119895and y

119895 then


y119895= X119895b119895+ 120598119895 (28)










C119895= 119864 (D

119895D119879119895) = 119864 ([X

119895y119895]

119879

[X119895y119895]) = [

CX119895X119895 CX119895y119895Cy119895X119895 Cy119895y119895

]

(29)


z119894119895

= X119895a119894119895 (30)

where z119894119895




a119894119895

= arg maxa119894119895

a119879119894119895CX119895X119895a119894119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(31)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(32)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895

((1 minus 120591119886119895

)CX119895X119895 + 120591119886119895I) a119894119895

= 1

(33)







Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data








MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)





x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)


y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894




0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















C = 119864 (DD119879) = 119864 ([Xy]119879 [Xy])

= [

119864 (X119879X) 119864 (X119879y)119864 (y119879X) 119864 (y119879y)

] = [

CXX CXyCyX Cyy

]

(2)



z119894= Xa119894 (3)


119894




a119894= arg max

a119894var (z

119894) (119894 = 1 119898)

st a119879119894a119894= 1 z

119894= Xa119894

(4)


a119894= arg max

a119894a119879119894CXX a

119894(119894 = 1 119898)

st a119879119894a119894= 1

(5)


CXX a119894= 120582119894a119894 (6)





120573 = arg min

120573

(

1003817100381710038171003817

Z120573 minus y1003817100381710038171003817

2

2) (7)


120573 = (Z119879Z)

minus1

Z119879y (8)





[14 29]

a119894= arg max

a119894cov (z

119894 y)

st a119879119894a119894= 1 z

119894= Xa119894

(9)



119894and the data


a119894= arg max

a119894a119879119894CXy

st a119879119894a119894= 1

(10)


CXy CyX a119894 = 120582

2

119894a119894

(11)




a119894= arg max

a119894corr (z

119894 y)

st z119894= Xa119894

(12)




119894= 1 and


a119894= arg max

a119894cov (z

119894 y)

st z119894= Xa119894 a119879119894CXX a

119894= 1

(13)


a119894= arg max

a119894a119879119894CXy

st a119879119894CXX a

119894= 1

(14)



2

119894a119894 (15)





a119894= arg max

a119894a119879119894CXy

st a119879119894((1 minus 120591

119886)CXX + 120591

119886I) a119894= 1

(16)


[(1 minus 120591119886)CXX + 120591


119894= 120582

2

119894a119894 (17)


119886)CXX + 120591



119886)CXX+120591119886I] instead


119886



119886= 1 the



119886



119886between


119886

119886= arg min

120591119886









4 Data Filtering



119896=

119873minus1

sum

119894=0

119908119894119910119896minus119894

(19)



119894= 1119873 while


119896= 120572119910119896+ (1 minus 120572)

119896minus1 (20)

where 119910119896and






1 ℎ2 ℎ



120601119895119896

(119905) =radic2

minus119895120601 (2

minus119895

119905 minus 119896) (21)


120595119895119896

(119905) =radic2

minus119895120595 (2

minus119895

119905 minus 119896) (22)


1 1198922 119892

119903] that


119909 (119905) =

1198992minus119869

sum

119896=1

a119869119896120601119869119896

(119905) +

119869

sum

119895=1

1198992minus119895

sum

119896=1

d119895119896120595119895119896

(119905) (23)






119895andG

119895 respectively

that isa119895= H119895a119895minus1

d119895= G119895a119895minus1

(24)where

H119895=

[

[

[

[

ℎ1

sdot ℎ119903

sdot sdot

0 ℎ1

sdot ℎ119903

0

0 0 sdot sdot sdot

0 0 ℎ1

sdot ℎ119903

]

]

]

]1198992119895times1198992119895

G119895=

[

[

[

[

1198921

119892119903

0 1198921

119892119903

0

0 0

0 0 1198921

119892119903

]

]

]

]1198992119895times1198992119895




Second scaledsignal

Third scaledsignal

(c)

(e)

(g)

Originaldata

(a)

First scaledsignal

(b)

(d)

(f)




H

H

H

G

G

G








119905119895= 120590119895radic2 log 119899 (26)



120590119895=

1

06745

median

10038161003816100381610038161003816

119889119895119896

10038161003816100381610038161003816

(27)










119895and y

119895 then


y119895= X119895b119895+ 120598119895 (28)










C119895= 119864 (D

119895D119879119895) = 119864 ([X

119895y119895]

119879

[X119895y119895]) = [

CX119895X119895 CX119895y119895Cy119895X119895 Cy119895y119895

]

(29)


z119894119895

= X119895a119894119895 (30)

where z119894119895




a119894119895

= arg maxa119894119895

a119879119894119895CX119895X119895a119894119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(31)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(32)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895

((1 minus 120591119886119895

)CX119895X119895 + 120591119886119895I) a119894119895

= 1

(33)







Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data








MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)





x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)


y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894




0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894and the data


a119894= arg max

a119894a119879119894CXy

st a119879119894a119894= 1

(10)


CXy CyX a119894 = 120582

2

119894a119894

(11)




a119894= arg max

a119894corr (z

119894 y)

st z119894= Xa119894

(12)




119894= 1 and


a119894= arg max

a119894cov (z

119894 y)

st z119894= Xa119894 a119879119894CXX a

119894= 1

(13)


a119894= arg max

a119894a119879119894CXy

st a119879119894CXX a

119894= 1

(14)



2

119894a119894 (15)





a119894= arg max

a119894a119879119894CXy

st a119879119894((1 minus 120591

119886)CXX + 120591

119886I) a119894= 1

(16)


[(1 minus 120591119886)CXX + 120591


119894= 120582

2

119894a119894 (17)


119886)CXX + 120591



119886)CXX+120591119886I] instead


119886



119886= 1 the



119886



119886between


119886

119886= arg min

120591119886









4 Data Filtering



119896=

119873minus1

sum

119894=0

119908119894119910119896minus119894

(19)



119894= 1119873 while


119896= 120572119910119896+ (1 minus 120572)

119896minus1 (20)

where 119910119896and






1 ℎ2 ℎ



120601119895119896

(119905) =radic2

minus119895120601 (2

minus119895

119905 minus 119896) (21)


120595119895119896

(119905) =radic2

minus119895120595 (2

minus119895

119905 minus 119896) (22)


1 1198922 119892

119903] that


119909 (119905) =

1198992minus119869

sum

119896=1

a119869119896120601119869119896

(119905) +

119869

sum

119895=1

1198992minus119895

sum

119896=1

d119895119896120595119895119896

(119905) (23)






119895andG

119895 respectively

that isa119895= H119895a119895minus1

d119895= G119895a119895minus1

(24)where

H119895=

[

[

[

[

ℎ1

sdot ℎ119903

sdot sdot

0 ℎ1

sdot ℎ119903

0

0 0 sdot sdot sdot

0 0 ℎ1

sdot ℎ119903

]

]

]

]1198992119895times1198992119895

G119895=

[

[

[

[

1198921

119892119903

0 1198921

119892119903

0

0 0

0 0 1198921

119892119903

]

]

]

]1198992119895times1198992119895




Second scaledsignal

Third scaledsignal

(c)

(e)

(g)

Originaldata

(a)

First scaledsignal

(b)

(d)

(f)




H

H

H

G

G

G








119905119895= 120590119895radic2 log 119899 (26)



120590119895=

1

06745

median

10038161003816100381610038161003816

119889119895119896

10038161003816100381610038161003816

(27)










119895and y

119895 then


y119895= X119895b119895+ 120598119895 (28)










C119895= 119864 (D

119895D119879119895) = 119864 ([X

119895y119895]

119879

[X119895y119895]) = [

CX119895X119895 CX119895y119895Cy119895X119895 Cy119895y119895

]

(29)


z119894119895

= X119895a119894119895 (30)

where z119894119895




a119894119895

= arg maxa119894119895

a119879119894119895CX119895X119895a119894119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(31)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(32)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895

((1 minus 120591119886119895

)CX119895X119895 + 120591119886119895I) a119894119895

= 1

(33)







Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data








MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)





x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)


y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894




0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











4 Data Filtering



119896=

119873minus1

sum

119894=0

119908119894119910119896minus119894

(19)



119894= 1119873 while


119896= 120572119910119896+ (1 minus 120572)

119896minus1 (20)

where 119910119896and






1 ℎ2 ℎ



120601119895119896

(119905) =radic2

minus119895120601 (2

minus119895

119905 minus 119896) (21)


120595119895119896

(119905) =radic2

minus119895120595 (2

minus119895

119905 minus 119896) (22)


1 1198922 119892

119903] that


119909 (119905) =

1198992minus119869

sum

119896=1

a119869119896120601119869119896

(119905) +

119869

sum

119895=1

1198992minus119895

sum

119896=1

d119895119896120595119895119896

(119905) (23)






119895andG

119895 respectively

that isa119895= H119895a119895minus1

d119895= G119895a119895minus1

(24)where

H119895=

[

[

[

[

ℎ1

sdot ℎ119903

sdot sdot

0 ℎ1

sdot ℎ119903

0

0 0 sdot sdot sdot

0 0 ℎ1

sdot ℎ119903

]

]

]

]1198992119895times1198992119895

G119895=

[

[

[

[

1198921

119892119903

0 1198921

119892119903

0

0 0

0 0 1198921

119892119903

]

]

]

]1198992119895times1198992119895




Second scaledsignal

Third scaledsignal

(c)

(e)

(g)

Originaldata

(a)

First scaledsignal

(b)

(d)

(f)




H

H

H

G

G

G








119905119895= 120590119895radic2 log 119899 (26)



120590119895=

1

06745

median

10038161003816100381610038161003816

119889119895119896

10038161003816100381610038161003816

(27)










119895and y

119895 then


y119895= X119895b119895+ 120598119895 (28)










C119895= 119864 (D

119895D119879119895) = 119864 ([X

119895y119895]

119879

[X119895y119895]) = [

CX119895X119895 CX119895y119895Cy119895X119895 Cy119895y119895

]

(29)


z119894119895

= X119895a119894119895 (30)

where z119894119895




a119894119895

= arg maxa119894119895

a119879119894119895CX119895X119895a119894119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(31)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(32)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895

((1 minus 120591119886119895

)CX119895X119895 + 120591119886119895I) a119894119895

= 1

(33)







Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data








MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)





x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)


y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894




0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Second scaledsignal

Third scaledsignal

(c)

(e)

(g)

Originaldata

(a)

First scaledsignal

(b)

(d)

(f)




H

H

H

G

G

G








119905119895= 120590119895radic2 log 119899 (26)



120590119895=

1

06745

median

10038161003816100381610038161003816

119889119895119896

10038161003816100381610038161003816

(27)










119895and y

119895 then


y119895= X119895b119895+ 120598119895 (28)










C119895= 119864 (D

119895D119879119895) = 119864 ([X

119895y119895]

119879

[X119895y119895]) = [

CX119895X119895 CX119895y119895Cy119895X119895 Cy119895y119895

]

(29)


z119894119895

= X119895a119894119895 (30)

where z119894119895




a119894119895

= arg maxa119894119895

a119879119894119895CX119895X119895a119894119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(31)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(32)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895

((1 minus 120591119886119895

)CX119895X119895 + 120591119886119895I) a119894119895

= 1

(33)







Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data








MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)





x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)


y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894




0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119895and y

119895 then


y119895= X119895b119895+ 120598119895 (28)










C119895= 119864 (D

119895D119879119895) = 119864 ([X

119895y119895]

119879

[X119895y119895]) = [

CX119895X119895 CX119895y119895Cy119895X119895 Cy119895y119895

]

(29)


z119894119895

= X119895a119894119895 (30)

where z119894119895




a119894119895

= arg maxa119894119895

a119879119894119895CX119895X119895a119894119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(31)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895a119894119895

= 1

(32)


a119894119895

= arg maxa119894119895

a119879119894119895CX119895y119895 (119894 = 1 119898 119895 = 0 119869)

st a119879119894119895

((1 minus 120591119886119895

)CX119895X119895 + 120591119886119895I) a119894119895

= 1

(33)







Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data








MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)





x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)


y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894




0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Multiscalefiltering

LVRmodeling

LVR 1

LVR 2

LVR 119895

Scale 119869 LVR 119869

Scale 2

Scale 1

Scaledata

Raw input-output

data








MSE =

1

119873

119899

sum

119896=1

(119910 (119896) minus (119896))

2

(34)





x3= x1+ x2

x4= 03x

1+ 07x

2

x5= 03x

3+ 02x

4

x6= 22x

1minus 17x

3

x7= 21x

6+ 12x

5

x8= 14x

2minus 12x

7

x9= 13x

2+ 21x

1

x10

= 13x6minus 23x

9

(35)


y =

10

sum

119894=1

119887119894x119894 (36)

where 119887119894




0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 50 100 150 200 250 300 350 400 450 500

0

5

10

15

20

Out

put

Samples

minus5

minus10

minus15

minus20

minus25




119900) and even (y



119890119894= (12)(y

119900119894+ y119900119894+1


CVMSEy119890

=

1198732

sum

119894=1

(y119890119894

minus y119890119894)

2

(37)



0 50 100 150 200 250

0

10IMSLVR

Samples

minus10

minus20

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MSF + LVR

119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

EWMA + LVR119910

0 50 100 150 200 250

0

10

Samples

minus10

minus20

MF + LVR

119910

0 50 100 150 200 250

0

10

Samples

LVR

minus10

minus20

119910







IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














IMSLVR

055

06

065

07

RCCA

MSF + LVR

IMSLVR

06

065

07

075

PLS

MSF + LVR

db3db2Haar

IMSLVR

06

065

07

075

PCR

MSF + LVR




0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 100 200 300 400 500094

096

098

Training data

Samples

119909119863

(a)

0 100 200 300 400 500094

096

098

Testing data

Samples

119909119863

(b)

Training data

0 100 200 300 400 500

002

004

Samples

119909119861

(c)

Testing data

0 100 200 300 400 500

001

002

003

Samples

119909119861

(d)

Training data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(e)

Testing data

0 100 200 300 400 500

098

1

102

Feed

flow

Samples

(f)

Training data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(g)

Testing data

0 100 200 300 400 500

62

64

Reflu

x flo

w

Samples

(h)







119863and





RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












RCCA 08971 09616 14573 15973 36553PLS 09512 10852 14562 16106 36568PCR 09586 10675 14504 16101 36904

SNR = 10

RCCA 05719 06281 09184 10119 18694PLS 05930 06964 09325 10239 18733PCR 06019 06823 09211 10240 18876

SNR = 20

RCCA 03816 04100 05676 06497 09395PLS 03928 04507 05994 06733 09423PCR 03946 04443 05872 06670 09508




6 PaT 322 K 119909

1198630979

P 17225 times 10





119861001








shown in Figure 7








50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















50 100 150 200 250095

096

097

098

IMSLVR

Samples

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MSF + LVR119909119863

0 50 100 150 200 250095

096

097

098

Samples

EWMA + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

MF + LVR

119909119863

0 50 100 150 200 250095

096

097

098

Samples

LVR

119909119863




Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Reflux drum

Condenser




119879 119865119863

119879 119865119863

119879 119865119863

119879 119865119863

Feed tank

Reboiler

119879 119865

119879

119879119879

119879

119879

119879

Distillation column


Top product storage









0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1000 2000 3000 4000085

09

095

Training data

Samples

119909119863

(a)

0 1000 2000 3000 4000085

09

095

Testing data

Samples

119909119863

(b)

Training data

0 1000 2000 3000 4000

00501

015

Samples

119909119861

(c)

0 1000 2000 3000 4000

00501

015

Samples

119909119861

Testing data

(d)

Training data

0 1000 2000 3000 400020

40

60

Feed

flow

Samples

(e)

0 1000 2000 3000 400020

40

60Fe

ed fl

ow

Samples

Testing data

(f)

Training data

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

(g)

0 1000 2000 3000 4000

4

6

Reflu

x flo

w

Samples

Testing data

(h)






119863and 119909




0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

IMSLVR

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

0 20 40 60 80 100 120 140 160 180 200088

0885089

0895

Samples

Samples

Samples

LVR

119909119863

119909119863

119909119863

MSF + LVR



7 Conclusions



Acknowledgment


References





















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article integrated multiscale latent variable...

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