research article fault identification in industrial...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 516760 7 pageshttpdxdoiorg1011552013516760

Research ArticleFault Identification in Industrial Processes Using an IntegratedApproach of Neural Network and Analysis of Variance

Yuehjen E Shao and Chia-Ding Hou

Department of Statistics and Information Science Fu Jen Catholic University 510 Chungcheng Road Xinzhuang DistrictNew Taipei City 24205 Taiwan

Correspondence should be addressed to Chia-Ding Hou stat0002mailfjuedutw

Received 21 November 2012 Revised 28 April 2013 Accepted 14 May 2013

Academic Editor Jun Zhao

Copyright copy 2013 Y E Shao and C-D Hou 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

Due to its importance in process improvement the issue of determining exactly when faults occur has attracted considerableattention in recent years Most related studies have focused on the use of the maximum likelihood estimator (MLE) method todetermine the fault in univariate processes in which the underlying process distribution should be known in advance In additionmost studies have been devoted to identifying the faults of process mean shifts Different from most of the current researchthe present study proposes an effective approach to identify the faults of variance shifts in a multivariate process The proposedmechanism comprises the analysis of variance (ANOVA) approach a neural network (NN) classifier and an identification strategyTo demonstrate the effectiveness of our proposed approach a series of simulated experiments is conducted and the best resultsfrom our proposed approach are addressed

1 Introduction

Process personnel have always wanted to search for processfaults in real time to significantly improve the underly-ing process Statistical process control (SPC) charts havebeen successfully used to detect process faults for severaldecades Because technological progress allows more andmore advanced sensors to be used in a process it has becomepopular to monitor multiple quality characteristics during aprocess A multivariate process is simply defined as a processwith two quality characteristics or more to be monitoredDue to having multiple quality characteristics it is muchmore difficult to determine at what time a fault occurs ina multivariate process compared with that for a univariateprocess

Multivariate statistical process control (MSPC) chartshave been studied and developed [1ndash3] however their majorfunction is basically to generate an out-of-control signalwhenprocess faults occur It is extremely difficult to estimate oridentify the beginning time of a fault using onlyMSPC charts

In most situations the beginning time of a fault containsmost of the information behind the causes of the processfault Rapidly and accurately estimating the beginning timeof a fault would contribute much to the identification of theassociated root causes of the fault and would significantlyimprove the process

There have been many studies that investigated thebeginning time of a process fault howevermost of the studieshave focused on univariate processes [4ndash14] In additionmost related studies have focused on the use of themaximumlikelihood estimator (MLE) method [3ndash12] However theMLE method has a strict assumption the underlying processdistribution is known Because the real-world process distri-bution is typically unknown this strict assumption seriouslyrestricts the range of the applicability of the MLE methodBesides another problem that can be encountered is thatthere are a considerable number of explanatory variableswhen modeling a multivariate process with a considerablenumber of quality characteristics To overcome the limita-tions of the MLE method and the difficulties when there

2 Mathematical Problems in Engineering

are too many explanatory variables in a multivariate processthis study focuses on a multivariate process with ten qualitycharacteristics and considers process variance shifts as theunderlying process faults Additionally this study assumesthat the process covariance matrix has shifted from Σ

0to

Σ1when the process fault has occurred There are 56 input

variables considered in this study It is not practical to use all56 variables as inputs into the proposed neural network (NN)classifier Consequently this study uses a hybrid techniqueto select fewer but more significant explanatory variablesThis is the first stage of building the proposed scheme Thechosen significant variables are then used as inputs into theproposed NN models This modeling is the second stage ofcreating the scheme After conducting the NN classificationan identification strategy is combined with the scheme toestimate the beginning time of a process fault

The structure of this study is organized as followsSection 2 addresses the problems with previous studies Theresearch gaps and the proposed methodologies used arediscussed in Section 3 Section 4 discusses the experimentalsimulations where the results and analysis for the typicaland the proposed approaches are reported The final sectionconcludes this study

2 Problems and Process Models

In this section we discuss the difficulties that can beencountered in practice Several research studies that haveinvestigated determining the beginning times of faults will beaddressed In addition this section presents the models of ageneralized multivariate process and the process fault

21 Problems Statement In typical MSPC applications anout-of-control signal would indicate that a process faulthas occurred in the underlying process At that momentalthough we have evidence regarding the status of theunderlying process we would have difficulty determining thebeginning time of the fault In particular if the effects ofthe underlying process faults are minor the probability oftriggering a signal at the beginning time of the fault would beextremely lowAs a result it is almost impossible to determinethe beginning time of a process fault by only using the MSPCchart For example consider a multivariate process with tenquality characteristics monitored by |S| an MSPC chart Theprocess fault with a process variance shift has occurred attime 201 Due to the small magnitude of the fault it is notdetected until time 230 Observing Figure 1 it is apparentthat the beginning time of the fault is not equal to the MSPCsignalTheir difference gets larger when the magnitude of thefault gets smaller

Several studies have been conducted to address thedifficulty of determining the beginning time of a fault [13 14]Process faults have typically been divided into two typesprocess mean shifts and process variance shifts [15 16]Thesestudies propose the MLE approach to estimate the beginningtime of a fault when the process mean or variance has shiftedto a univariate process The MLE approach with the use ofEWMAcharts has been reported for a univariate process [17]

1 51 101 151 201 251Sample number

UCL

0E+00

5Eminus05

1Eminus04

15Eminus04

2Eminus04

25Eminus04

3Eminus04

35Eminus04

4Eminus04

45Eminus04

5Eminus04

MSPC signal (t = 230)

Starting time of a fault (t = 201)|S|

Figure 1 |S| an MSPC chart for the sample generalized variance

Whereas most of the existing MLE approaches have focusedon univariate processes the study in [3] derived anMLE for amultivariate process However the performance of this MLEwas not stable when the number of quality variables becamelarge

The MLE is criticized for its strict assumption that statesthat the underlying process distribution must be knownThis assumption is not feasible for practical processes As aresult machine learning (ML) methods have been used todetermine the beginning time of a fault [13 14] Howeverthe number of input variables from those studies is extremelysmall because of the simplicity of the process structureThereare few studies that have investigated how to identify thebeginning time of a fault when considerable input variablesare involved Too many input variables generally result ina time-consuming training stage with the ML approachHowever the study in [18] considered a large number ofinputs in their experiments and also assumed that all thequality variables were at faults in the process which is arare case in industrial applications Accordingly this studyproposes an effective hybrid scheme that integrates ANOVANN techniques and an identification strategy to overcomethe aforementioned difficulties

22 The Multivariate Process and the Fault Models In con-trast to the traditional multivariate normal distributionassumption for a process this study considers a multivariateprocess that follows an unknown multivariate distributionAssume that themultivariate process is initially in control andthe sample observations are from an unknown distribution119865(120583

Σ) with a known mean vector 120583

and covariance matrixΣ0 After an unknown time 120591 + 1 we assume that the process

covariance matrix changes from Σ0to Σ1 Let

119883

119894119895

= [1198831198941198951

1198831198941198952

119883119894119895119901

]1015840

(1)

Mathematical Problems in Engineering 3

119901 times 1 be a vector that represents the 119901 characteristics on the119895th observation in subgroup 119894with the unknown distributionfunction 119865(120583

Σ) Accordingly we have

119883

11

119883

1119899

iid119865(120583

Σ0)

119883

21

119883

2119899

iid119865(120583

Σ0)

119883

1205911

119883

120591119899

iid119865(120583

Σ0)

119883

120591+11

119883

120591+1119899

iid119865(120583

Σ1)

119883

1198791

119883

119879119899

iid119865(120583

Σ1)

(2)

where 119899 is the sample size 120591 + 1 is the change point 119879is the signal time in which a subgroup covariance matrixexceeds the limits of the control chart |S| ldquoiidsimrdquo meansldquoindependent and identically distributedrdquo and Σ

0is the in-

control covariance matrix which is defined as follows

Σ0=

[[[[[[[[[[[[

[

12059011

12059012

sdot sdot sdot 1205901119895


12059021

12059022

sdot sdot sdot sdot sdot sdot 120590

2119901

d

sdot sdot sdot

1205901198941

sdot sdot sdot 120590119894119895

sdot sdot sdot

d

1205901199011

1205901199012

sdot sdot sdot 120590119901119895

sdot sdot sdot 120590119901119901

]]]]]]]]]]]]

]119901times119901

(3)

Following the suggestion of [19] this study considers thefollowing variance shift as the process fault

Σ1=

[[[[[[[[[[[[

[

12059011

12059012

sdot sdot sdot 1205791205901119895

1205901119895+1


12059021

12059022

sdot sdot sdot 1205791205902119895

1205902119895+1


d

sdot sdot sdot

1205791205901198941

1205791205901198942

1205792120590119894119895

120579120590119894119895+1

sdot sdot sdot 120579120590119894119901

120590119894+11

120590119894+12

120579120590119894+1119895

120590119894+1119895+1

120590119894+1119901

d

1205901199011

1205901199012

sdot sdot sdot 120579120590119901119895

120590119901119895+1

sdot sdot sdot 120590119901119901

]]]]]]]]]]]]

]119901times119901

(4)

where 120579 is the inflated ratio Let the sample variance-covariance matrix in subgroup 119894 be defined as

S119894=

1

119899 minus 1

119899

sum

119895=1

(119883

119894119895

minus 119883

119894

)(119883

119894119895

minus 119883

119894

)

1015840

=

[[[[

[

11987811989411

11987811989412

sdot sdot sdot 1198781198941119901

11987811989421

11987811989422

sdot sdot sdot 1198781198942119901

1198781198941199011

1198781198941199012

sdot sdot sdot 119878119894119901119901

]]]]

]119901times119901

(5)

To monitor a multivariate process variance shift thesample generalized variances |S

119894| 119894 = 1 2 and the

following control limits are used [2]

UCL =1003816100381610038161003816Σ0

1003816100381610038161003816 (1198871 + 3radic1198872)

LCL = max(0 1003816100381610038161003816Σ01003816100381610038161003816 (1198871 minus 3radic119887

2))

(6)

where |Σ0| is the determinant of Σ

0and

1198871=

1

(119899 minus 1)119901

119901

prod

119894=1

(119899 minus 119894)

1198872=

1

(119899 minus 1)2119901

119901

prod

119894=1

(119899 minus 119894) (

119901

prod

119894=1

(119899 minus 119894 + 2) minus

119901

prod

119894=1

(119899 minus 119894))

(7)

3 The Proposed Scheme

In recent years intelligent approaches such as neural net-works and support vector machines have had an importantrole in the development of industrial technologies [20ndash22] Although acceptable results may be obtained usingtraditional intelligent approaches these approaches may notfulfill the particular needs of industrial applications Recentstudies have shown that hybrid intelligent approaches canhelp achieve a better performance for particular applications[18 19 23 24] In this study we develop a hybrid schemeto effectively determine the change point of a multivariateprocess The proposed scheme includes the ANOVA an NNand the identification strategyThe scheme can be used whenthe multivariate process distribution is unknown and whenthere are a large number of input variables The followingsections address these components

31 ANOVA The proposed hybrid two-stage method inte-grates the framework of ANOVAand anNN In stage I a one-way ANOVA test is applied to select important influentialvariables In stage II the selected significant variables aretaken as the input variables into the NN

The purpose of performing a one-way ANOVA in stageI is to determine whether data from the ldquoin-controlrdquo andldquoout-of-controlrdquo groups have a common mean that is todetermine whether the measured characteristics from theldquoin-controlrdquo and ldquoout-of-controlrdquo groups are actually dif-ferent Because matrix S

119894is symmetric only the elements


on and above the diagonal need to be examined by theone-way ANOVA To simplify the notation let 119884

1198941=

11987811989411

1198841198942

= 11987811989412

119884119894119901

= 1198781198941119901

119884119894119901+1

= 11987811989422

119884119894119901+2

=

11987811989423

119884119894119873minus1

= 119878119894119901119901

and 119884119894119873

= |S| where 119873 = 1 +

119901(119901 + 1)2 Let 119884119894119895119896119897

be the lth observation at the kth level ofthe factor (where level 1 represents an ldquoin-controlrdquo group andlevel 2 represents an ldquoout-of-controlrdquo group) for the variable119884119894119895mentioned above 119894 = 1 2 119879 119895 = 1 2 119873 119896 = 1 2

119897 = 1 2 119899119894119895119896

Let 120583119894119895

and 120591119894119895119896

be the correspondingoverall mean and treatment effect respectively Accordinglythe linear equation for the one-way ANOVA model is

119884119894119895119896119897

= 120583119894119895+ 120591119894119895119896

+ 120576119894119895119896119897

119896=1 2 119897=1 2 119899119894119895119896

119894=1 2 119879 119895=1 2 119873

(8)

To identify significant variables an F-test statistic is usedto test the differences between the in-control and out-ofcontrol groups Those significant variables selected in thisstage are then substituted into the NN to construct a two-stage model

32 Neural Network The purpose of using an NN is toclassify the process output as either an in-control or out-of-control processThe identification strategy uses this informa-tion to activate its function Accordingly the beginning timeof a process fault can be estimated in real time

The structure of the NN can be briefly described asfollows The NN nodes are divided into three layers whichinclude the input the output and the hidden layers Thenodes in the input layer receive input signals from an externalsource and the nodes in the output layer provide the targetoutput signalsThe output of each neuron in the input layer isthe same as the input to that neuron For each neuron 119895 in thehidden layer and neuron 119896 in the output layer the net inputsare given by

net119895= sum

119894

119908119895119894times 119900119894

net119896= sum

119895

119908119896119895times 119900119895

(9)

where 119894(119895) is a neuron in the previous layer 119900119894(119900119895) is the

output of node 119894(119895) and 119908119895119894(119908119896119895) is the connection weight

fromneuron 119894(119895) to neuron 119895(119896)The neuron outputs are givenas

119900119894= net119894

119900119894=

1

1 + expminus(net119894+120579119894)= 119891119894(net119894 120579119894)

119900119896=

1

1 + expminus(net119896+120579119896)= 119891119896(net119896 120579119896)

(10)

where net119895(net119896) is the input signal from the external source

to node 119895(119896) in the input layer and 120579119895(120579119896) is a bias The

transformation function shown in (10) is called the sigmoid

function and is the one most commonly used transformationfunction Accordingly this study uses the sigmoid function

The generalized delta rule is the conventional techniqueused to derive the connection weights of the feedforwardnetwork Firstly a set of random numbers is assigned to theconnection weights Then to obtain a pattern 119901 with targetoutput vector 119905

119901= [1199051199011 1199051199012 119905

119901119872]119879 the sum of the squared

error to be minimized is given as

119864119901=

1

2

119872

sum

119895=1

(119905119901119895minus 119900119901119895)2

(11)

where 119872 is the number of output nodes By minimizing theerror119864

119901using the gradient descent technique the connection

weights can be updated using the following equations

Δ119908119895119894(119901) = 120578120575

119901119895119900119901119895+ 120572Δ119908

119895119894(119901 minus 1) (12)

where for the output nodes

119878119901119895

= (119905119901119895minus 119900119901119895) 119900119901119895(1 minus 119900

119901119895) (13)

and for other nodes

120575119901119895

= (sum

119896

(120575119901119896

times 119908119896119895) 119900119901119895(1 minus 119900

119901119895)) (14)

33 An Identification Strategy This study uses an NN toclassify the status of a process at a certain time 119862 When theoutput of the NN is classified as ldquo0rdquo this indicates that theprocess fault has not occurred When the output of the NN isclassified as ldquo1rdquo this indicates that a process fault has intrudedinto the underlying process When an SPC chart is triggeredat time 119879 we know a fault has intruded into the underlyingprocess The identification component is then activated andthe NN begins to classify the status of the process from time119879 minus 1 to 1 in a backward sequence

If the NN output is ldquo1rdquo at time119879minus1 wemay conclude thatthe beginning time of the fault has been confirmed at time119879 minus 1 instead of time 119879 Then we can proceed to time 119879 minus 2If the NN output is ldquo1rdquo again at time 119879minus1 we could concludethat the beginning time of the fault has been confirmed attime119879minus2 instead of time119879minus1 However because all classifiersare not perfect we could obtainmisclassification resultsThatis we may encounter a problem in which the NN output is 0at time119879minus1 and the values of the outcome are all 1 s from time119879 minus 2 to 119879 minus 119862 (where 1 le 119862 le 119879 minus 1) one may ask what isthe subsequent decision The decision on the beginning timeof a fault is not definitively made by observing only a singleoutcome

In this study because the NN outputs are either 1 or 0we can consider them as the success or failure of a Bino-mial experiment respectively Accordingly we can use thecumulative probability distribution of a Binomial experimentto determine the beginning time of a fault If the NN hasa good classification capability we know that most of theoutput values from time 119879 to 119879 minus 120591 should be classifiedas 1 which implies that the cumulative probability of the


Binomial distribution is near 1 Due to there being no perfectclassifiers in reality several misclassifications of NN outputsmust be toleratedTherefore the cumulative probability of theBinomial distribution should be less than a certain thresholdvalue That is if the value of the cumulative probability isgreater than a threshold at a time 119879 minus 119894 we can concludethat the beginning time of a fault has occurred at time 119879 minus 119894However there is no theoretical threshold value Accordingto our experience and numerous simulationsrsquo results wetherefore estimate the threshold value as follows

(1) Determining theThreshold During the training and testingfor the NN modeling phase denoted previously as phase Iwe can obtain an accurate identification rate (AIR) for theclassification tasksTheAIR is equivalent to the probability ofa successful rate (119875

119878) from the Binomial experiments Because

the number of successes must be an integer the followingrelationship should be used

[119899119878] ge 119899 times 119875

119878 (15)

where 119899119878is the number of successes in 119899 Binomial exper-

iments and [119899119878] is the smallest integer that is greater than

or equal to the value of 119899 times 119875119878 The integer [119899

119878] is used as

a standard and the corresponding cumulative probability isconsidered to be the threshold As a result the threshold iscalculated as follows

119875 (119883119863le 119899119878) =

119899119878

sum

119909119863 =0

(119899

119909119863

)119901119909119863(1 minus 119901)

119899minus119909119863 (16)

where119883119863is the accumulation of the Binomial trial outputs

(2) Performing the Confirmation Test To perform the confir-mation data test the newprocess data vectors were generatedFor each confirmation data vector the phase I NNmodel thatclassifies the confirmation data was used This confirmationtest is referred to as phase II The accumulation of the NNoutputs in phase II is denoted as 119883NN The number ofsuccesses of the NN outputs in phase II is denoted as 119899

119860

At time 119905119894 the value of the cumulative probability can be

calculated as the following

119875 (119883NN le 119899119860) =

119899119860

sum

119883NN=0

(119899

119909NN)119901119909NN(1 minus 119901)

119899minus119909NN (17)

(3) Conducting the Decision Rule After performing steps (1)and (2) the decision rule can be set up as follows

If 119875 (119883NN le 119899119860) ge 119875 (119883

119863le 119899119878)

time 119905119894is the beginning time of a process fault

(18)

4 Simulated Examples

This study performs a series of simulations to compare theexisting single-stage NN method with the proposed hybridscheme proposed in Section 3The corresponding estimatorsof 120591 for these two methods are denoted as 120591ANN and 120591AArespectively

41 Assumptions Without loss of generality we assumethat each quality characteristic is sampled from a normaldistribution with zero mean and one standard deviation Inaddition we assume that we monitor ten quality characteris-tics simultaneously (ie p = 10) and the in-control covariancematrix is as follows

Σ0=

[[[[[[[

[

10 05 sdot sdot sdot sdot sdot sdot 05

05 10 05 sdot sdot sdot 05

05 10 05 05

05 d 05

05 sdot sdot sdot sdot sdot sdot 05 10

]]]]]]]

]10times10

(19)

For the out-of-control covariance structure without lossof generality we assume that a variance shift occurs at thefirst quality characteristic Consequently the following out-of-control covariance matrix is considered

Σ1=

[[[[

[

120579212059011

12057912059012

sdot sdot sdot 1205791205901119901

12057912059021

12059022


d

1205791205901199011

1205901199012

sdot sdot sdot 120590119901119901

]]]]

]119901times119901

(20)

In this study the training data sets include 1000 datavectors for every possible parameter settingWhereas the first500 data vectors are all from an in-control state the last 500data vectors are from an out-of-control state The structureof the testing data sets is the same as that of the training datasets that is the testing data sets involve 1000 data vectorsThefirst 500 data vectors are from an in-control state and the last500 data vectors are from an out-of-control state

This study considers four values of the inflated ratio 120579 1112 13 and 14 In our proposed two-stage model we have7 10 10 and 10 input nodes for the ANOVA-NN models for120579 = 11 120579 = 12 120579 = 13 and 120579 = 14 respectively For allthe models there is only one output node This output nodeindicates the classification results of the process status wherea value of 0 indicates that the process is in control and a valueof 1 implies that the process is out of control Furthermore thechange point of the process is assumed to be 201 (120591+1 = 201)For each data structure we use a sample size (n) of 12 andrepeat the simulation 5 times The average of the estimatesof each approach for the 5 simulation replicates was thenrecorded along with their standard errors

42 Modeling Results and Analysis In stage I we use asignificance level of 005 and apply a one-way ANOVA testto select the important influential variables The results aregiven in Table 1The significant variables selected in this stageare then used as the input variables into the NN In additionfrom Table 2 it can be seen that between the two methodsdiscussed above the two-stage ANOVA-NN scheme tendedto have a better performance than that of the existing single-stage NN method

To evaluate the performance of the two estimators dis-cussed above the bias and the mean squared error (MSE)were used in this study The bias of an estimator 120591 is


Table 1 Significant variables selected using the one-way ANOVA

120579 Significant explanatory variables11 Y 1 Y2 Y3 Y6 Y7 Y9 Y 10

12 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

13 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

14 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

Table 2 Average beginning time of a fault estimate and standarderror for two estimators

120579Method

120591ANN 120591AA

11 24250 (769) 23817 (1036)12 24817 (1314) 22350 (2032)13 24200 (1304) 22367 (1656)14 23050 (575) 20750 (720)

the distance between the expected value of the estimator andthe parameter being estimated It is used to indicate theaccuracy of the estimator and is defined as follows

bias (120591) = 119864 (120591) minus 120591 (21)

The MSE is the expected value of the squared errors and isdefined as follows

MSE (120591) = 119864(120591 minus 120591)2 (22)

It is used to indicate how far on average the collectionsof estimates are from the parameters being estimated Theeffects of the inflated ratio 120579 on the biases and the MSE ofthe two estimators are shown in Figures 2 and 3 respectivelyFrom Figure 2 it is found that the biases of the two estimatorsdecrease as 120579 increases and the bias of the two-stage schemeappears to be smaller than the one of the other method Onthe other side again Figure 3 shows that the mean squarederror of the two-stage scheme tends to be smaller than theone of single-stage NN method Consequently it seems thatthe proposed two-stageANOVA-NN scheme ismore efficientin detecting the actual change point than the existing single-stage NN method

5 Conclusions

The objective of this work is to develop an effective schemeto identify the beginning time of a fault specifically for aprocess variance shift in a multivariate process with a generaldistribution On the basis of our numerical study the two-stage procedure introduced here was generally more efficientin detecting the beginning time of a fault than that of thesingle-stage NNmethodThis work could be a useful guide toengineers attempting to search for the root cause of a processdisturbance

Based on our results further studies can be expanded Forexample extensions of the proposed two-stage procedure todiscrete multivariate processes or other statistical techniquesare possible Such work deserves further research and is ourfuture concern

Bias

504540353025201510

50

11 12 13 14120579

ANNAA

Figure 2 Biases of the two estimators

MSE

3000

2500

2000

1500

1000

500

011 12 13 14

120579

ANNAA

Figure 3 Mean squared errors of the two estimators

Acknowledgment

This work is partially supported by the National ScienceCouncil of China Grant no NSC 99-2221-E-030-014-MY3and Grant no NSC 100-2118-M-030-001

References

[1] H Hotelling ldquoMultivariate quality controlrdquo in Techniques ofStatistical Analysis C EisenhartMWHastay andWAWallisEds McGraw Hill New York NY USA 1947

[2] F B Alt ldquoMultivariate quality controlrdquo in Encyclopedia ofStatistical Sciences N L Johnson and S Kotz Eds vol 6 JohnWiley amp Sons New York NY USA 1985

[3] C D Hou Y E Shao and S Huang ldquoA combined MLE andgeneralized P chart approach to estimate the change point ofa multinomial processrdquo Applied Mathematics and InformationSciences vol 7 no 4 pp 1487ndash1493 2013

[4] T R Samuel J J Pignatiello and J A Calvin ldquoIdentifyingthe time of a step change with X control chartsrdquo QualityEngineering vol 10 no 3 pp 521ndash527 1998


[5] T R Samuel J J Pignatiello and J A Calvin ldquoIdentifying thetime of a step change in a normal process variancerdquo QualityEngineering vol 10 no 3 pp 529ndash538 1998

[6] J J Pignatiello and T R Samuel ldquoEstimation of the changepoint of a normal process mean in SPC applicationsrdquo Journalof Quality Technology vol 33 no 1 pp 82ndash95 2001

[7] M B Perry J J Pignatiello and J R Simpson ldquoEstimation ofthe change point of a Poisson rate parameter with a linear trenddisturbancerdquo Quality and Reliability Engineering Internationalvol 22 no 4 pp 371ndash384 2006

[8] Y E Shao and C D Hou ldquoEstimation of the starting time of astep change disturbance in a 120574 processrdquo Journal of the ChineseInstitute of Industrial Engineers vol 23 no 4 pp 319ndash327 2006

[9] Y E Shao and C D Hou ldquoEstimation of the change pointof a uniform process using the EWMA chart and MLErdquo ICICExpress Letters vol 3 no 3 pp 451ndash456 2009

[10] J J Pignatiello and T R Samuel ldquoIdentifying the time of astep-change in the process fraction nonconformingrdquo QualityEngineering vol 13 no 3 pp 357ndash365 2001

[11] T R Samuel and J J Pignatiello ldquoIdentifying the time of achange in a poisson rate parameterrdquo Quality Engineering vol10 no 4 pp 673ndash681 1998

[12] R Noorossana A Saghaei K Paynabar and S Abdi ldquoIdentify-ing the period of a step change in high-yield processesrdquoQualityand Reliability Engineering International vol 25 no 7 pp 875ndash883 2009

[13] Y E Shao H Y Huang C D Hou K S Lin and J E TsaildquoChange point determination for an attribute processrdquo ICICExpress Letters vol 5 pp 3117ndash3122 2011

[14] Y E Shao ldquoAn integrated neural networks and SPC approach toidentify the starting time of a process disturbancerdquo ICIC ExpressLetters vol 3 pp 319ndash324 2009

[15] Y E Shao C J Lu and C C Chiu ldquoA fault detectionsystem for an autocorrelated process using SPCEPCANN andSPCEPCSVM schemesrdquo International Journal of InnovativeComputing Information and Control vol 7 pp 5417ndash5428 2011

[16] Y E Shao and B S Hsu ldquoDetermining the contributors for amultivariate SPC chart signal using artificial neural networksand support vectormachinerdquo International Journal of InnovativeComputing Information and Control vol 5 no 12 pp 4899ndash4906 2009

[17] Y E Shao and C D Hou ldquoA combined MLE and EWMA chartapproach to estimate the change point of a gamma process withindividual observationsrdquo International Journal of InnovativeComputing Information and Control vol 7 no 5 pp 2109ndash21222011

[18] Y E Shao and C D Hou ldquoChange point determination for amultivariate process using a two-stage hybrid schemerdquo AppliedSoft Computing vol 13 no 3 pp 1520ndash1527 2013

[19] C S Cheng andH P Cheng ldquoIdentifying the source of varianceshifts in the multivariate process using neural networks andsupport vectormachinesrdquo Expert Systems with Applications vol35 no 1-2 pp 198ndash206 2008

[20] W Bischoff and F Miller ldquoA minimax two-stage procedure forcomparing treatments looking at a hybrid test and estimationproblemas awholerdquo Statistica Sinica vol 12 no 4 pp 1133ndash11442002

[21] C J Lu Y E Shao and P H Li ldquoMixture control chartpatterns recognition using independent component analysisand support vector machinerdquo Neurocomputing vol 74 no 11pp 1908ndash1914 2011

[22] W Dai Y E Shao and C J Lu ldquoIncorporating featureselection method into support vector regression for stock indexforecastingrdquo Neural Computing and Applications 2012

[23] Y E Shao C J Lu and Y C Wang ldquoA hybrid ICA-SVMapproach for determining the fault quality variables in amultivariate processrdquo Mathematical Problems in Engineeringvol 2012 Article ID 284910 12 pages 2012

[24] Y E Shao ldquoPrediction of currency volume issued in Taiwanusing a hybrid artificial neural network and multiple regressionapproachrdquo Mathematical Problems in Engineering vol 2013Article ID 676742 9 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


are too many explanatory variables in a multivariate processthis study focuses on a multivariate process with ten qualitycharacteristics and considers process variance shifts as theunderlying process faults Additionally this study assumesthat the process covariance matrix has shifted from Σ

0to

Σ1when the process fault has occurred There are 56 input

variables considered in this study It is not practical to use all56 variables as inputs into the proposed neural network (NN)classifier Consequently this study uses a hybrid techniqueto select fewer but more significant explanatory variablesThis is the first stage of building the proposed scheme Thechosen significant variables are then used as inputs into theproposed NN models This modeling is the second stage ofcreating the scheme After conducting the NN classificationan identification strategy is combined with the scheme toestimate the beginning time of a process fault

The structure of this study is organized as followsSection 2 addresses the problems with previous studies Theresearch gaps and the proposed methodologies used arediscussed in Section 3 Section 4 discusses the experimentalsimulations where the results and analysis for the typicaland the proposed approaches are reported The final sectionconcludes this study

2 Problems and Process Models

In this section we discuss the difficulties that can beencountered in practice Several research studies that haveinvestigated determining the beginning times of faults will beaddressed In addition this section presents the models of ageneralized multivariate process and the process fault

21 Problems Statement In typical MSPC applications anout-of-control signal would indicate that a process faulthas occurred in the underlying process At that momentalthough we have evidence regarding the status of theunderlying process we would have difficulty determining thebeginning time of the fault In particular if the effects ofthe underlying process faults are minor the probability oftriggering a signal at the beginning time of the fault would beextremely lowAs a result it is almost impossible to determinethe beginning time of a process fault by only using the MSPCchart For example consider a multivariate process with tenquality characteristics monitored by |S| an MSPC chart Theprocess fault with a process variance shift has occurred attime 201 Due to the small magnitude of the fault it is notdetected until time 230 Observing Figure 1 it is apparentthat the beginning time of the fault is not equal to the MSPCsignalTheir difference gets larger when the magnitude of thefault gets smaller

Several studies have been conducted to address thedifficulty of determining the beginning time of a fault [13 14]Process faults have typically been divided into two typesprocess mean shifts and process variance shifts [15 16]Thesestudies propose the MLE approach to estimate the beginningtime of a fault when the process mean or variance has shiftedto a univariate process The MLE approach with the use ofEWMAcharts has been reported for a univariate process [17]

1 51 101 151 201 251Sample number

UCL

0E+00

5Eminus05

1Eminus04

15Eminus04

2Eminus04

25Eminus04

3Eminus04

35Eminus04

4Eminus04

45Eminus04

5Eminus04

MSPC signal (t = 230)

Starting time of a fault (t = 201)|S|

Figure 1 |S| an MSPC chart for the sample generalized variance

Whereas most of the existing MLE approaches have focusedon univariate processes the study in [3] derived anMLE for amultivariate process However the performance of this MLEwas not stable when the number of quality variables becamelarge

The MLE is criticized for its strict assumption that statesthat the underlying process distribution must be knownThis assumption is not feasible for practical processes As aresult machine learning (ML) methods have been used todetermine the beginning time of a fault [13 14] Howeverthe number of input variables from those studies is extremelysmall because of the simplicity of the process structureThereare few studies that have investigated how to identify thebeginning time of a fault when considerable input variablesare involved Too many input variables generally result ina time-consuming training stage with the ML approachHowever the study in [18] considered a large number ofinputs in their experiments and also assumed that all thequality variables were at faults in the process which is arare case in industrial applications Accordingly this studyproposes an effective hybrid scheme that integrates ANOVANN techniques and an identification strategy to overcomethe aforementioned difficulties

22 The Multivariate Process and the Fault Models In con-trast to the traditional multivariate normal distributionassumption for a process this study considers a multivariateprocess that follows an unknown multivariate distributionAssume that themultivariate process is initially in control andthe sample observations are from an unknown distribution119865(120583

Σ) with a known mean vector 120583

and covariance matrixΣ0 After an unknown time 120591 + 1 we assume that the process

covariance matrix changes from Σ0to Σ1 Let

119883

119894119895

= [1198831198941198951

1198831198941198952

119883119894119895119901

]1015840

(1)




119883

11

119883

1119899

iid119865(120583

Σ0)

119883

21

119883

2119899

iid119865(120583

Σ0)

119883

1205911

119883

120591119899

iid119865(120583

Σ0)

119883

120591+11

119883

120591+1119899

iid119865(120583

Σ1)

119883

1198791

119883

119879119899

iid119865(120583

Σ1)

(2)


0is the in-


Σ0=

[[[[[[[[[[[[

[

12059011

12059012



12059021

12059022


2119901

d

sdot sdot sdot

1205901198941

sdot sdot sdot 120590119894119895

sdot sdot sdot

d

1205901199011

1205901199012

sdot sdot sdot 120590119901119895

sdot sdot sdot 120590119901119901

]]]]]]]]]]]]

]119901times119901

(3)


Σ1=

[[[[[[[[[[[[

[

12059011

12059012

sdot sdot sdot 1205791205901119895

1205901119895+1


12059021

12059022

sdot sdot sdot 1205791205902119895

1205902119895+1


d

sdot sdot sdot

1205791205901198941

1205791205901198942

1205792120590119894119895

120579120590119894119895+1

sdot sdot sdot 120579120590119894119901

120590119894+11

120590119894+12

120579120590119894+1119895

120590119894+1119895+1

120590119894+1119901

d

1205901199011

1205901199012

sdot sdot sdot 120579120590119901119895

120590119901119895+1

sdot sdot sdot 120590119901119901

]]]]]]]]]]]]

]119901times119901

(4)


S119894=

1

119899 minus 1

119899

sum

119895=1

(119883

119894119895

minus 119883

119894

)(119883

119894119895

minus 119883

119894

)

1015840

=

[[[[

[

11987811989411

11987811989412

sdot sdot sdot 1198781198941119901

11987811989421

11987811989422

sdot sdot sdot 1198781198942119901

1198781198941199011

1198781198941199012

sdot sdot sdot 119878119894119901119901

]]]]

]119901times119901

(5)


119894| 119894 = 1 2 and the


UCL =1003816100381610038161003816Σ0

1003816100381610038161003816 (1198871 + 3radic1198872)


2))

(6)


0and

1198871=

1

(119899 minus 1)119901

119901

prod

119894=1

(119899 minus 119894)

1198872=

1

(119899 minus 1)2119901

119901

prod

119894=1

(119899 minus 119894) (

119901

prod

119894=1

(119899 minus 119894 + 2) minus

119901

prod

119894=1

(119899 minus 119894))

(7)








1198941=

11987811989411

1198841198942

= 11987811989412

119884119894119901

= 1198781198941119901

119884119894119901+1

= 11987811989422

119884119894119901+2

=

11987811989423

119884119894119873minus1

= 119878119894119901119901

and 119884119894119873

= |S| where 119873 = 1 +

119901(119901 + 1)2 Let 119884119894119895119896119897


119897 = 1 2 119899119894119895119896

Let 120583119894119895

and 120591119894119895119896


119884119894119895119896119897

= 120583119894119895+ 120591119894119895119896

+ 120576119894119895119896119897

119896=1 2 119897=1 2 119899119894119895119896

119894=1 2 119879 119895=1 2 119873

(8)




net119895= sum

119894

119908119895119894times 119900119894

net119896= sum

119895

119908119896119895times 119900119895

(9)




119900119894= net119894

119900119894=

1

1 + expminus(net119894+120579119894)= 119891119894(net119894 120579119894)

119900119896=

1

1 + expminus(net119896+120579119896)= 119891119896(net119896 120579119896)

(10)






119901= [1199051199011 1199051199012 119905



119864119901=

1

2

119872

sum

119895=1

(119905119901119895minus 119900119901119895)2

(11)




Δ119908119895119894(119901) = 120578120575

119901119895119900119901119895+ 120572Δ119908

119895119894(119901 minus 1) (12)


119878119901119895

= (119905119901119895minus 119900119901119895) 119900119901119895(1 minus 119900

119901119895) (13)

and for other nodes

120575119901119895

= (sum

119896

(120575119901119896

times 119908119896119895) 119900119901119895(1 minus 119900

119901119895)) (14)









[119899119878] ge 119899 times 119875

119878 (15)




119878] is used as


119875 (119883119863le 119899119878) =

119899119878

sum

119909119863 =0

(119899

119909119863

)119901119909119863(1 minus 119901)

119899minus119909119863 (16)



119860



119875 (119883NN le 119899119860) =

119899119860

sum

119883NN=0

(119899

119909NN)119901119909NN(1 minus 119901)

119899minus119909NN (17)


If 119875 (119883NN le 119899119860) ge 119875 (119883

119863le 119899119878)


(18)




Σ0=

[[[[[[[

[



05 10 05 05

05 d 05


]]]]]]]

]10times10

(19)


Σ1=

[[[[

[

120579212059011

12057912059012

sdot sdot sdot 1205791205901119901

12057912059021

12059022


d

1205791205901199011

1205901199012

sdot sdot sdot 120590119901119901

]]]]

]119901times119901

(20)








12 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

13 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

14 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10


120579Method

120591ANN 120591AA

11 24250 (769) 23817 (1036)12 24817 (1314) 22350 (2032)13 24200 (1304) 22367 (1656)14 23050 (575) 20750 (720)


bias (120591) = 119864 (120591) minus 120591 (21)


MSE (120591) = 119864(120591 minus 120591)2 (22)


5 Conclusions



Bias

504540353025201510

50

11 12 13 14120579

ANNAA


MSE

3000

2500

2000

1500

1000

500

011 12 13 14

120579

ANNAA


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119883

11

119883

1119899

iid119865(120583

Σ0)

119883

21

119883

2119899

iid119865(120583

Σ0)

119883

1205911

119883

120591119899

iid119865(120583

Σ0)

119883

120591+11

119883

120591+1119899

iid119865(120583

Σ1)

119883

1198791

119883

119879119899

iid119865(120583

Σ1)

(2)


0is the in-


Σ0=

[[[[[[[[[[[[

[

12059011

12059012



12059021

12059022


2119901

d

sdot sdot sdot

1205901198941

sdot sdot sdot 120590119894119895

sdot sdot sdot

d

1205901199011

1205901199012

sdot sdot sdot 120590119901119895

sdot sdot sdot 120590119901119901

]]]]]]]]]]]]

]119901times119901

(3)


Σ1=

[[[[[[[[[[[[

[

12059011

12059012

sdot sdot sdot 1205791205901119895

1205901119895+1


12059021

12059022

sdot sdot sdot 1205791205902119895

1205902119895+1


d

sdot sdot sdot

1205791205901198941

1205791205901198942

1205792120590119894119895

120579120590119894119895+1

sdot sdot sdot 120579120590119894119901

120590119894+11

120590119894+12

120579120590119894+1119895

120590119894+1119895+1

120590119894+1119901

d

1205901199011

1205901199012

sdot sdot sdot 120579120590119901119895

120590119901119895+1

sdot sdot sdot 120590119901119901

]]]]]]]]]]]]

]119901times119901

(4)


S119894=

1

119899 minus 1

119899

sum

119895=1

(119883

119894119895

minus 119883

119894

)(119883

119894119895

minus 119883

119894

)

1015840

=

[[[[

[

11987811989411

11987811989412

sdot sdot sdot 1198781198941119901

11987811989421

11987811989422

sdot sdot sdot 1198781198942119901

1198781198941199011

1198781198941199012

sdot sdot sdot 119878119894119901119901

]]]]

]119901times119901

(5)


119894| 119894 = 1 2 and the


UCL =1003816100381610038161003816Σ0

1003816100381610038161003816 (1198871 + 3radic1198872)


2))

(6)


0and

1198871=

1

(119899 minus 1)119901

119901

prod

119894=1

(119899 minus 119894)

1198872=

1

(119899 minus 1)2119901

119901

prod

119894=1

(119899 minus 119894) (

119901

prod

119894=1

(119899 minus 119894 + 2) minus

119901

prod

119894=1

(119899 minus 119894))

(7)








1198941=

11987811989411

1198841198942

= 11987811989412

119884119894119901

= 1198781198941119901

119884119894119901+1

= 11987811989422

119884119894119901+2

=

11987811989423

119884119894119873minus1

= 119878119894119901119901

and 119884119894119873

= |S| where 119873 = 1 +

119901(119901 + 1)2 Let 119884119894119895119896119897


119897 = 1 2 119899119894119895119896

Let 120583119894119895

and 120591119894119895119896


119884119894119895119896119897

= 120583119894119895+ 120591119894119895119896

+ 120576119894119895119896119897

119896=1 2 119897=1 2 119899119894119895119896

119894=1 2 119879 119895=1 2 119873

(8)




net119895= sum

119894

119908119895119894times 119900119894

net119896= sum

119895

119908119896119895times 119900119895

(9)




119900119894= net119894

119900119894=

1

1 + expminus(net119894+120579119894)= 119891119894(net119894 120579119894)

119900119896=

1

1 + expminus(net119896+120579119896)= 119891119896(net119896 120579119896)

(10)






119901= [1199051199011 1199051199012 119905



119864119901=

1

2

119872

sum

119895=1

(119905119901119895minus 119900119901119895)2

(11)




Δ119908119895119894(119901) = 120578120575

119901119895119900119901119895+ 120572Δ119908

119895119894(119901 minus 1) (12)


119878119901119895

= (119905119901119895minus 119900119901119895) 119900119901119895(1 minus 119900

119901119895) (13)

and for other nodes

120575119901119895

= (sum

119896

(120575119901119896

times 119908119896119895) 119900119901119895(1 minus 119900

119901119895)) (14)









[119899119878] ge 119899 times 119875

119878 (15)




119878] is used as


119875 (119883119863le 119899119878) =

119899119878

sum

119909119863 =0

(119899

119909119863

)119901119909119863(1 minus 119901)

119899minus119909119863 (16)



119860



119875 (119883NN le 119899119860) =

119899119860

sum

119883NN=0

(119899

119909NN)119901119909NN(1 minus 119901)

119899minus119909NN (17)


If 119875 (119883NN le 119899119860) ge 119875 (119883

119863le 119899119878)


(18)




Σ0=

[[[[[[[

[



05 10 05 05

05 d 05


]]]]]]]

]10times10

(19)


Σ1=

[[[[

[

120579212059011

12057912059012

sdot sdot sdot 1205791205901119901

12057912059021

12059022


d

1205791205901199011

1205901199012

sdot sdot sdot 120590119901119901

]]]]

]119901times119901

(20)








12 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

13 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

14 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10


120579Method

120591ANN 120591AA

11 24250 (769) 23817 (1036)12 24817 (1314) 22350 (2032)13 24200 (1304) 22367 (1656)14 23050 (575) 20750 (720)


bias (120591) = 119864 (120591) minus 120591 (21)


MSE (120591) = 119864(120591 minus 120591)2 (22)


5 Conclusions



Bias

504540353025201510

50

11 12 13 14120579

ANNAA


MSE

3000

2500

2000

1500

1000

500

011 12 13 14

120579

ANNAA


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198941=

11987811989411

1198841198942

= 11987811989412

119884119894119901

= 1198781198941119901

119884119894119901+1

= 11987811989422

119884119894119901+2

=

11987811989423

119884119894119873minus1

= 119878119894119901119901

and 119884119894119873

= |S| where 119873 = 1 +

119901(119901 + 1)2 Let 119884119894119895119896119897


119897 = 1 2 119899119894119895119896

Let 120583119894119895

and 120591119894119895119896


119884119894119895119896119897

= 120583119894119895+ 120591119894119895119896

+ 120576119894119895119896119897

119896=1 2 119897=1 2 119899119894119895119896

119894=1 2 119879 119895=1 2 119873

(8)




net119895= sum

119894

119908119895119894times 119900119894

net119896= sum

119895

119908119896119895times 119900119895

(9)




119900119894= net119894

119900119894=

1

1 + expminus(net119894+120579119894)= 119891119894(net119894 120579119894)

119900119896=

1

1 + expminus(net119896+120579119896)= 119891119896(net119896 120579119896)

(10)






119901= [1199051199011 1199051199012 119905



119864119901=

1

2

119872

sum

119895=1

(119905119901119895minus 119900119901119895)2

(11)




Δ119908119895119894(119901) = 120578120575

119901119895119900119901119895+ 120572Δ119908

119895119894(119901 minus 1) (12)


119878119901119895

= (119905119901119895minus 119900119901119895) 119900119901119895(1 minus 119900

119901119895) (13)

and for other nodes

120575119901119895

= (sum

119896

(120575119901119896

times 119908119896119895) 119900119901119895(1 minus 119900

119901119895)) (14)









[119899119878] ge 119899 times 119875

119878 (15)




119878] is used as


119875 (119883119863le 119899119878) =

119899119878

sum

119909119863 =0

(119899

119909119863

)119901119909119863(1 minus 119901)

119899minus119909119863 (16)



119860



119875 (119883NN le 119899119860) =

119899119860

sum

119883NN=0

(119899

119909NN)119901119909NN(1 minus 119901)

119899minus119909NN (17)


If 119875 (119883NN le 119899119860) ge 119875 (119883

119863le 119899119878)


(18)




Σ0=

[[[[[[[

[



05 10 05 05

05 d 05


]]]]]]]

]10times10

(19)


Σ1=

[[[[

[

120579212059011

12057912059012

sdot sdot sdot 1205791205901119901

12057912059021

12059022


d

1205791205901199011

1205901199012

sdot sdot sdot 120590119901119901

]]]]

]119901times119901

(20)








12 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

13 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

14 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10


120579Method

120591ANN 120591AA

11 24250 (769) 23817 (1036)12 24817 (1314) 22350 (2032)13 24200 (1304) 22367 (1656)14 23050 (575) 20750 (720)


bias (120591) = 119864 (120591) minus 120591 (21)


MSE (120591) = 119864(120591 minus 120591)2 (22)


5 Conclusions



Bias

504540353025201510

50

11 12 13 14120579

ANNAA


MSE

3000

2500

2000

1500

1000

500

011 12 13 14

120579

ANNAA


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














[119899119878] ge 119899 times 119875

119878 (15)




119878] is used as


119875 (119883119863le 119899119878) =

119899119878

sum

119909119863 =0

(119899

119909119863

)119901119909119863(1 minus 119901)

119899minus119909119863 (16)



119860



119875 (119883NN le 119899119860) =

119899119860

sum

119883NN=0

(119899

119909NN)119901119909NN(1 minus 119901)

119899minus119909NN (17)


If 119875 (119883NN le 119899119860) ge 119875 (119883

119863le 119899119878)


(18)




Σ0=

[[[[[[[

[



05 10 05 05

05 d 05


]]]]]]]

]10times10

(19)


Σ1=

[[[[

[

120579212059011

12057912059012

sdot sdot sdot 1205791205901119901

12057912059021

12059022


d

1205791205901199011

1205901199012

sdot sdot sdot 120590119901119901

]]]]

]119901times119901

(20)








12 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

13 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

14 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10


120579Method

120591ANN 120591AA

11 24250 (769) 23817 (1036)12 24817 (1314) 22350 (2032)13 24200 (1304) 22367 (1656)14 23050 (575) 20750 (720)


bias (120591) = 119864 (120591) minus 120591 (21)


MSE (120591) = 119864(120591 minus 120591)2 (22)


5 Conclusions



Bias

504540353025201510

50

11 12 13 14120579

ANNAA


MSE

3000

2500

2000

1500

1000

500

011 12 13 14

120579

ANNAA


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












12 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

13 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10

14 Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y 10


120579Method

120591ANN 120591AA

11 24250 (769) 23817 (1036)12 24817 (1314) 22350 (2032)13 24200 (1304) 22367 (1656)14 23050 (575) 20750 (720)


bias (120591) = 119864 (120591) minus 120591 (21)


MSE (120591) = 119864(120591 minus 120591)2 (22)


5 Conclusions



Bias

504540353025201510

50

11 12 13 14120579

ANNAA


MSE

3000

2500

2000

1500

1000

500

011 12 13 14

120579

ANNAA


Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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