research article a framework for final drive simultaneous...

Research ArticleA Framework for Final Drive Simultaneous FailureDiagnosis Based on Fuzzy Entropy and Sparse BayesianExtreme Learning Machine

Qing Ye1 Hao Pan1 and Changhua Liu2

1School of Computer Science and Technology Wuhan University of Technology Wuhan 430000 China2Yangtze University College of Technology and Engineering Jingzhou 430023 China

Correspondence should be addressed to Qing Ye yq0712163com

Received 6 October 2014 Revised 4 January 2015 Accepted 19 January 2015

Academic Editor J Alfredo Hernandez

Copyright copy 2015 Qing Ye et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This research proposes a novel framework of final drive simultaneous failure diagnosis containing feature extraction training paireddiagnosticmodels generating decision threshold and recognizing simultaneous failuremodes In feature extractionmodule adoptwavelet package transform and fuzzy entropy to reduce noise interference and extract representative features of failure modeUse single failure sample to construct probability classifiers based on paired sparse Bayesian extreme learning machine whichis trained only by single failure modes and have high generalization and sparsity of sparse Bayesian learning approach To generateoptimal decision threshold which can convert probability output obtained from classifiers into final simultaneous failure modesthis research proposes using samples containing both single and simultaneous failure modes and Grid search method which issuperior to traditional techniques in global optimization Compared with other frequently used diagnostic approaches based onsupport vector machine and probability neural networks experiment results based on F

1-measure value verify that the diagnostic

accuracy and efficiency of the proposed framework which are crucial for simultaneous failure diagnosis are superior to the existingapproach

1 Introduction

With sustained increase of work condition complexity simul-taneous failures occur more frequently in final drive which isthe pivotal part of car and seriously affect running status andcomfort and safety of car Final drive is mainly consisting ofa pair of gears which are meshing together when car runsOwing to the complex structure a certain function disorderin final drive usually stems frommore than one single failureat the same time which is called simultaneous failure Tra-ditional manual technology cannot accomplish simultaneousfailure diagnosis (SFD) This paper is focusing on final drivesimultaneous failure diagnosis which is essential for automanufacturer and maintenance industry

Failure diagnosis by using vibration signal is almost themost frequently used approach because vibration signal isrelatively precise and accurate against diagnosis based onsound It can be divided into three main steps featureextraction training diagnostic models and failure mode

identification The vibration signal collected from final drivehas the characteristics of being nonlinear and nonstationaryand it is enclosed with a lot of uncorrelated and superfluousinformation It is impossible to extract valid failure modeinformation from original vibration signal because of thenoise and interference embedded in it The frequently usedpreprocessed methods include wavelet analysis [1 2] waveletpackage transform (WPT) [3 4] and empiricalmode decom-position (EMD) [5]Wavelet package transform is suitable fornonstationary vibration signal by decomposing original sig-nal into several subfrequency bands which contains differentfailure information and effectively reduces noise interference

Data contained in preprocessed signal is high-dimen-sional so that it cannot be directly inputted into diagnosticsystem Feature extraction has a deep effect on accuracyand reliability of failure diagnosis Recently researchershave introduced entropy into the field of feature extractionincluding approximate entropy sample entropy [6] andfuzzy entropy [7] Compared with approximation entropy

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2015 Article ID 427965 11 pageshttpdxdoiorg1011552015427965

2 Computational Intelligence and Neuroscience

and sample entropy which are based on Heaviside stepfunction which is mutational at the classification boundaryfuzzy entropy eliminates the influence of baseline drift ofdata and guarantees the entropy to vary smoothly andcontinuously with similarity tolerance [8] so that it isexcellent in measuring complexity and self-similarity of thepreprocessed vibration signal and fully reflecting changesof the vibration performance of mechanical equipment[9]

In recent years many machine learning methods areapplied in failure diagnosis including support vectormachine(SVM) [10] artificial neural networks (ANN) [11] extremelearningmachine (ELM) [12 13] and kernel extreme learningmachine (KELM) [14] ELM is single-hidden-layer feedfor-ward neural networks and without human intervention intuning parameters which differs from SVM and ANN andmakes it superior in high generalization and less learningtime KELM apply kernel function to ELM to improve gener-alization and nonlinear approximation ability [15] Howevercomputational cost and memory cost of KELM are high withregard to large scale problem Recently Bayesianmethods areemployed into ELM to learn the output weights by estimatingthe probability distribution of output with high generaliza-tion Soria-Olivas and Gomez-Sanchis proposed Bayesianextreme learning machine [16] for linear regression with-out solving classification problem Sparse Bayesian extremelearningmachine (SBELM) [17] is a novel method for findingthe sparse representatives of hidden layer output weights byimposing a hyperparameter on each weight During learningphase SBELM tunes some output weights into zero to obtaincompact model In summary SBELM has the advantagesof probability output high generalization sparsity and fasttraining speed To solve the problem of simultaneous failurediagnosis a proper classifier has to offer the probability of allpossible failures In this research the proposed frameworkconstructs classifiers based on paired SBELM in which eachclassifier based on SBELM is trained by a pair of singlefailure samples The paired SBELM effectively reflect theprobability distribution of failure modes In general onlysingle failure samples are used for constructing diagnosticmodels Since it is impossible to collect all combinationsof existing single failure modes for training the proposedframework can effectively solve the practical bottleneck insimultaneous failure diagnosis With the purpose of rec-ognizing simultaneous failure modes use both single andsimultaneous failure samples and Gird search method togenerate optimal decision threshold which could convertprobability result of classifier into final multiple failuremodes Considering that partial matching is valid and instru-mental in simultaneous failure diagnosis this research adoptsF1-measure to evaluate the performance of the proposedframework

This paper is organized as follows Section 2 presentsthe proposed framework Section 3 presents the experimentsetup data acquisition and preprocessing The results ofexperiment are discussed in Section 4 Finally a conclusionis given in Section 5

2 The Proposed Framework for Final DriveSimultaneous Failure Diagnosis

21 Feature Extraction Based on Wavelet PackageTransform and Fuzzy Entropy

211 Wavelet Package Transform In failure diagnosis one ofthe key points is extraction of features from original vibrationsignal which is nonstationary for mechanical equipmentWavelet package transform (WPT) is an extended form ofwavelet transform to analyse nonstationary and non-linearsignal and to supply better partition of frequency bandbecause the same frequency bandwidths can provide goodresolution regardless of high and low frequencies [18] As amultiresolution analysis method WPT can effectively pre-process nonstationary vibration signal in both time domainand frequency domain Two-scale equation ofWPT is shownbelow in which ℎ

0119896and ℎ1119896represent the filter coefficients

1199092119899(119905) = radic2sum

119896isin119885

ℎ0119896119909119899(2119905 minus 119896)

1199092119899+1

(119905) = radic2sum

119896isin119885

ℎ1119896119909119899(2119905 minus 119896)

(1)

The recursion formula of wavelet package coefficient is

119889119895+12119899

119896= sum

119897

ℎ0(2119897minus119896)

119889119895119899

119897 119889

119895+12119899+1

119896= sum

119897

ℎ1(2119897minus119896)

119889119895119899

119897 (2)

212 Fuzzy Entropy When failure occurred in final drive thecomplexity of oscillation feature will change hencewe shouldextract representative features containing in the signal Fuzzyentropy is an extension of Shannon entropy and fuzzy sets[19] The procedure of fuzzy entropy is described as follows

(1) Consider a time series with the length of119873 119909(119894) 1 le119894 le 119873 For given 119898 119899 and 119903 construct a vector set in theform of 119883119898

119894 119894 = 1 2 119873 minus 119898 + 1 in which each vector

contains119898 sequential elements starting from 119909(119894) shown as

119883119898

119894= 119909 (119894) 119909 (119894 + 1) 119909 (119909 + 119898 minus 1) minus 119909

0(119894)

119894 = 1 2 119873 minus 119898 + 1

(3)

where 1199090(119894) is the average of vector119883119898

119894

(2) Define the distance 119889119898119894119895between 119883119898

119894and 119883119898

119895where

119894 119895 = 1 2 119873 minus 119898 119894 = 119895 as follows

119889119898

119894119895= max119896isin(0119898minus1)

1003816100381610038161003816[119909 (119894 + 119896) minus 1199090 (119894)] minus [119909 (119895 + 119896) minus 1199090 (119895)]

1003816100381610038161003816

(4)

(3) Calculate similarity between 119883119898119894and 119883119898

119895using fuzzy

function

119863119898

119894119895= 120583 (119889

119898

119894119895 119899 119903) = 119890

minus In 2(119889119898119894119895119903)119899

(5)

(4) Define function 120601119898 as follows

120601119898(119899 119903) =

1

119873 minus 119898

119873minus119898

sum

119894=1

(1

119873 minus 119898 + 1

119873minus119898

sum

119895=1119895 =119894

119863119898

119894119895) (6)

Computational Intelligence and Neuroscience 3

(5) Change119898 to119898 + 1 and repeat step (1) to (4)

120601119898+1(119899 119903) =

1

119873 minus 119898

119873minus119898

sum

119894=1

(1

119873 minus 119898 + 1

119873minus119898

sum

119895=1119895 =119894

119863119898+1

119894119895) (7)

(6) Fuzzy entropy of sequence 119909(119894) 1 le 119894 le 119873 isdefined as follows

FuzzyEn (119898 119899 119903) = lim119873rarrinfin

[In120601119898 (119899 119903) minus In120601119898+1 (119899 119903)] (8)

(7) If the length 119873 is finite FuzzyEn(119898 119899 119903) can bechanged as follows

FuzzyEn (119898 119899 119903) = In120601119898 (119899 119903) minus In120601119898+1 (119899 119903) (9)

22 Sparse Bayesian Extreme Learning Machine (SBELM)Given a preprocessed data set 119863 = (119909

119894 119905119894) 119894 = 1 119873

119909119894isin 119877119899 119905119894isin 119877119898 The output function of ELM with 119871 hidden

nodes is shown as follows

119891 (119909) =

119871

sum

119894=1

120573119894ℎ119894(119909) = 120573 sdot 119867 (119909) (10)

where 120573 = [1205731 120573

119871]119879 is output weight connecting hidden

nodes and output nodes 119867(119909) = [ℎ1(119909) ℎ

119871(119909)] is the

hidden layer output matrix for input 119909 in which ℎ119894(119909) is the

hidden output of the 119894th hidden node Equation (10) can bewritten as follows

119867120573 = 119879 (11)

where 119879 is the training data target matrix SBELM learnsoutput weight by using Bayesian method instead of bycalculatingMoore-Penrose generalized inverse of119867 [17]Thehidden layer output 119867 becomes the input of SBELM Treateach training sample as an independent Bernoulli event sothat probability119901(119905 | 119909) satisfies Bernoulli distribution Applysigmoid function to convert the predicted output Y(ℎ 120573) asfollows

120590 (Y (ℎ 120573)) =1

(1 + exp (minusY (ℎ 120573))) (12)

The likelihood function of sample set is expressed asfollows

119901 (119905 | 119867 120573) =

119873

prod

119895=1

120590 (Y (ℎ 120573))119905119895

[1 minus 120590 (Y(ℎ 120573))]1minus119905119895

(13)

where 119905119895is the target of training sample 119909

119895Y(ℎ 120573) = ℎ120573 and

119905119895isin 0 1 Conditioned on a hyperparameter 120572

119895 zero-mean

Gaussian prior distribution over 120573119894is as follows

119901 (120573 | 120572) =

119871

prod

119894=1

120572119894

radic2120587exp(minus

1205721198941205732

119894

2) (14)

The typical step of SBELM is to establish the distributionof marginal likelihood over 119905 conditioned on 120572 and 119867 and

determine 120572 by maximizing the marginal likelihood 119901(119905 | 119867120572) by Laplace approximation method

argmax ln 119901 (119905 | 120573119867) 119901 (120573 | 120572)

= argmax[

[

ln

119873

prod

119895=1

119910119905119895

119895(1 minus 119910

119895)1minus119905119895

+ ln119871

prod

119894=1

120572119894


1205721198941205732

119894

2)]

]

= argmax[

[

119873

sum

119895=1

119905119895ln119910119895+ (1 minus 119905

119895)

sdot ln (1 minus 119910119895) minus

120573119879119860120573

2+ const]

]

(15)

where 119910119895= 120590(Y(ℎ

119895 120573))119860 = diag(120572) and const =sum119871

119894=1ln120572119894minus

12 ln 2120587 Then make quadratic approximation for log ofposterior probability

nabla120573nabla120573 ln 119901 (119905 | 120573119867) 119901 (120573 | 120572) = minus (119867119879119870119867 + 119860) (16)

where 119870 is a diagonal matrix in which 119896119895= 119910119895(1 minus 119910

119895) with

119895 = 1 119873 Therefore the center and covariance matrix ofGauss distribution of 120573 expressed as 1205731015840 andΦ are obtained asfollows

1205731015840= Φ119867

1198791198701199051015840 Φ = (119867

119879119870119867 + 119860)

minus1

(17)

where 1199051015840 = 119867120573+119870minus1(119905minus119910) By obtainingGauss approximationof 120573 the log of marginal likelihood is represented as follows

L (120572) = ln119901 (119905 | 120572119867)

= minus1

2[119873 ln (2120587) + ln |119862| + (1199051015840)

119879

119862minus11199051015840]

(18)

where119862 = 119870+119867119860119867119879 By setting the differential ofL(120572)withrespect to 120572 as 0 update the hyperparameter 120572 as follows

120572119894=1 minus 120572119894Φ119894119894

(1205731198941015840)2 (19)

The main procedure of SBELM is described as follows

(1) Initialize 120572119894and 120573

119894randomly with 119894 = 1 119871

(2) By utilizing Laplace approximation approach obtainapproximated Gauss distribution of 120573 and update 1205731015840and Φ by using (17)

(3) Bymaximizing themarginal likelihood utilize (19) toupdate hyperparameter 120572 until reaching the termina-tion criteria

(4) By tuning some 120573119894into 0 obtain the sparse represen-

tation of hidden layer output weight(5) For an unknown sample 119909

119906 utilize (12) to predict

probability distribution 119901(119905 | 119909119906 1205731015840)


SBELM12

SBELMi1

SBELMm1

SBELM1m

SBELMim

SBELMm(mminus1)

PSBELM1

PSBELMi

PSBELMm

p1

pi

pm

middot middot middot



Figure 1 Structure of paired SBELM for simultaneous failure diag-nosis

23 Paired SBELM SBELM is excellent in solving binaryclassification by obtaining probability distribution of eachclass 119901(119905 | 119909 1205731015840) With the purpose of final drive simulta-neous failure diagnosis in which training samples of singlefailure are ample while training samples of simultaneousfailure are scarce this paper combines the state-of-the-artcoupling approach proposed in [20]with SBELM to constructa set of paired SBELM (PSBELM) classifiers expressed as[PSBELM

1 PSBELM

119898] for a 119898-label classification prob-

lem shown in Figure 1 and each paired classifier PSBELM119894=

[SBELM1198941 SBELM

119894119895 SBELM

119894119898] in which SBELM

119894119895is

trained by every pair of classes and its output is 119901119894119895(119905119894| 120573 119909)

for sample 119909 belonging to the 119894th against the 119895th class Thetotal number of classifiers is119898(119898 minus 1)2

In simultaneous failure diagnosis more than one failuremay occur at the same time that can infer the conceptsum119898

119894=1119901119894= 1 [21] By estimating each probability output of

binary classifiers SBELM119894119895to measure correlation between

various classes obtain the paired probability output 119901119894as

follows

119901119894=sum119889

119895=1119895 =119894119899119894119895119901119894119895

sum119889

119895=1119895 =119894119899119894119895

119894 = 1 119898 (20)

where 119899119894119895is the number of training sample belonging to the

119894th and the 119895th class

24 Optimization of DecisionThreshold for Simultaneous Fail-ure Mode Recognition For a 119898-class classification problemthe output of classifiers based on PSBELM is a probabilityvector 119875 = [119901

1 119901

119898] in which 119901

119895represents occur-

ring possibility of the 119895th failure In order to obtain finalsimultaneous failuremodes an appropriate threshold value isindispensable In general researchers usually use 05 to be thethreshold value [21] which is of generality but not suitable forspecific application This paper utilizes Grid Search methodand an independent sample set containing both single failureand simultaneous failure to generate an optimal decisionthreshold 120576lowast between 0 and 1 which can convert probabilityoutput vector into result vector 119865 = [119891

1 119891

119894 119891

119898]

effectively

119891119894= 1 119901119894ge 120576lowast

0 119901119894lt 120576lowast119894 = 1 119898 120576

lowastisin (0 1) (21)

The simultaneous failure modes are those single failuresthat their corresponding 119891

119894is equal to 1 Since the range of

searching is limited the time-consuming characteristic ofGrid Search method can not weaken its advantages of globaloptimization compared with GA and PSO [18]

25 Evaluation of Performance Based on F1-Measure Con-sidering that partial matching is valid and significant insimultaneous failure diagnosis [22] utilize an independenttesting set and F1-measure [23] which is commonly usedfor evaluation of information retrieval systems to evaluatediagnostic accuracy for the proposed simultaneous failurediagnostic framework Given a data set 119863 = (119909

119894 119905119894) 119894 =

1 119873 119909119894isin 119877119899 119905119894isin 119877119898 119905119894119895isin 0 1 119895 = 1 119898 define

two variables namely precision (119875) and recall (119877) amongwhich 119875 represents the ratio between correct identified singlefailure modes and the actual simultaneous failure modes and119877 represents the ratio between correct identified single failuremodes and the predicted simultaneous failure modes

119875 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119905119894119895

119877 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119891lowast119894119895

(22)

where 119891lowast119894= [119891

lowast

1198941 119891

lowast

119894119898] is the predicted simultaneous

failure modes by using the proposed framework and 119905119894=

[1199051198941 119905

119894119898] is the actual simultaneous failure modes of 119909

119894

The F1-measure value can be obtained as follows

119864 =2 lowast 119875 lowast 119877

119875 + 119877=

2sum119898

119895=1sum119873

119894=1119891lowast

119894119895119910119894119895

sum119898

119895=1sum119873

119894=1119910119894119895+ sum119898

119895=1sum119873

119894=1119891lowast119894119895

(23)

26 The Proposed Framework for Final Drive SimultaneousFailure Diagnosis The structure of the proposed frameworkis shown in Figure 2 and the procedure of the proposedframework is described as follows

(1) Divide sample set into four parts 119863training1 119863training2119863threshold and 119863testing All the sample should bepreprocessed by WPT and utilize fuzzy entropy tomeasure the feature of oscillatory

(2) Utilize 119863training1 containing only single failure modesto optimize parameters of WPT including number oflayer 119871 and mother wavelet in preprocessing by usingfailure diagnostic model of SBELM

(3) By using optimal parameters of WPT obtained fromstep 2 preprocess 119863training2 containing only singlefailure modes and train classifiers based on pairedSBELM

(4) The optimumdiagnostic model of PSBELMgeneratesa probability output vector [119901

1 119901

119898] in 119898-label

classification problem and uses 119863threshold containingboth single and simultaneous failure modes and GridSearch to confirm the decision threshold value 120576lowastwhich is used to obtain simultaneous failure modes

(5) Use119863testing and F1-measure to evaluate the diagnosticaccuracy of the proposed framework


Opt

imal

dia

gnos

tic m

odel

Evaluation of theproposed framework

Dtesting

Optimal WPT andfuzzy entropy

Optimal diagnosticmodel based onpaired SBELM

Decision threshold

Result vector

Evaluation



Optimizingdecision threshold

Confirmation of thethreshold value

Optimalparametersof WPT


Train diagnosticmodel based on paired SBELM

Training diagnosticmodel

Dtraining1

Determiningparameters of WPT

Standard SBELMdiagnostic model

Optimization of WPT

Dthreshold

Dtraining2

value 120576lowast

value 120576lowast

Optimal 120576lowast

Figure 2 The structure of the proposed framework

3 Experiment Setup and Preprocessing

31 Experiment Setup In order to obtain sample data withrepresentativeness for constructing diagnostic model andverify the efficiency of the proposed diagnostic platformimplement the experiments on a test bed containing a PC twosensors a signal amplifier and a simulation turntablewith thecomposition as shown in Figure 3 in quiet room to collectenough original vibration signal of final drive Two sensorsare laid on the final drive in horizontal and vertical directionas shown in Figure 4 to collect vibration signal when it is setinto running state Most failures of final drive such as gearerror gear hard point and tooth broken occurred in gear pairwhich is the hard core of final drive and consisted of a driving-gear and a driven-gear In this research simulate 9 commonfailures including 6 single failures and 3 simultaneous failureswhich are described in detail in Table 1 under the rotatingspeed of 1200 rm for driving motor As shown in Figure 5amplitudes of simultaneous failure are obviously greater thansingle failure because when simultaneous failure occurs thesesingle failures are coupled together severelyThewave profiles

between single and simultaneous failure patterns are similarso that it is difficult to distinguish them manually but thecharacteristics embedded in each vibration signal can beextracted and identified by using methods afore mentioned

Considering the universality of vibration signals whichare used to construct diagnostic models repeat simulatingeach failure mode for 100 times and record the most stable 2seconds in each time with the sampling rate of 12 kHz whichshould be higher than the gear meshing frequency so thateffective failure information may not be discarded duringthe sampling Eventually 1000 sample data are obtainedand prepared to be preprocessed All the simulations areimplemented in MATLAB 70 which is running in a PC withCPU of 34GHZ and RAM of 40GB

32 Feature Extraction Based on WPT and Fuzzy EntropyIn this paper fuzzy entropy is used to reflect the changeof complexity By calculating the average fuzzy entropy ofvibration signal corresponding to each failure mode which isshown in Table 2 we find out that the values of fuzzy entropyof these 10 failure modes are approximate so that it can not


Table 1 Simple and simultaneous failure modes description

Failurelabel Failure mode description

C1 Single failures Normal statusC2

Single failures

Gear errorC3 Gear burrC4 Gear hard pointC5 MisalignmentC6 Gear tooth brokenC7 Gear crackC8

Simultaneousfailures

Gear tooth broken and gear crack

C9 Gear tooth broken and gear hardpoint

C10 Misalignment and gear crack

Turntable part

Drive part

Figure 3 The test bed

correctly distinguish failures of final drive The reason is thatinformation supplied by fuzzy entropy of original signal islimited and unable to reflect the deep-seated information ofeach failure situation Therefore we employ the frequentlyused mother wavelet function called Daubechies waveletwhich has orthogonal characteristic and effectiveness infiltering signal of vibrating machinery to implement waveletpackage transform and decompose original vibration signalinto several subfrequency bands By calculating fuzzy entropyof each frequency band obtained from 119871-level wavelet pack-age transform decomposition construct a feature vectorwith the dimension of 2119871 which can effectively reflect thecomplexity and self-similarity of oscillation characteristic offailure modes occurring in final drive

Feature = [FuzzyEn1 FuzzyEn2 FuzzyEn2119871] (24)

33 Distribution Plan of Samples There are 1000 sample datacontaining 100 normal samples 600 single failure samples

Horizontaldirection

Verticaldirection

Figure 4 Two sensors on final drive

and 300 simultaneous failure samples In each trial ran-domly divide the whole sample set into four parts 119863training1119863training2 119863threshold and 119863testing 119863training1 and 119863training2 whichare consisting of only single failure modes are used for opti-mizing parameters of WPT and training optimal diagnosticmodel based on paired SBELM119863threshold which contains bothsingle and simultaneous failure modes is used to generateoptimal decision threshold which convert probability resultof diagnostic model based on paired SBELM into final simul-taneous failure modes119863testing is used to test and evaluate theproposed framework by using F1-measure Ensure that thewhole sample set be preprocessed and the size of trainingsamples should be more than testing samples to ensure thegeneralization of the proposed framework The distributionplan is shown in Table 3

4 Result and Discussion

41 Optimization of Preprocessing and Feature Extraction Indata preprocessing optimum combination of decompositionlevel and mother wavelet of WPT and parameters of fuzzyentropy can achieve better performance in classification Weuse119863training1 containing random 250 single samples to obtainthe optimal combination of level number andmother waveletwhich is suitable for preprocessing samples collected fromfinal drive In order to simplify experiment and on the basisof previous research result focus on three wavelets Db3 Db4and Db5 and three decomposition levels from 3 to 5 Threeparameters of fuzzy entropy including119898 119899 and 119903 are definedempirically in advance Parameter 119898 is usually set to be 2Related to the boundary of fuzzy function parameters 119903 and119899 are setting as 02 and 2 STD where STD is the standarddeviation of original data [8]

By using single failure samples contained in 119863training1and standard diagnostic model based on SBELM with failureparameters find out appropriate parameters of WPT toachieve best performance of preprocessing The standarddiagnostic model based on SBELM is only used for selectingoptimal parameters of WPT that exist in the best failureidentification model in which the accuracy of classificationis highest The comparison result is shown in Figure 6 whichindicates that classification accuracy of the preprocessing by


Table 2 Average fuzzy entropy of 10 failure modes

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10Average fuzzy entropy 17761 18695 18855 17904 18734 19018 18080 18653 18107 19268

0 500 1000 1500 2000 2500minus02

0

02 C1

0 500 1000 1500 2000 2500

C2

minus02

0

02

C4

minus02

0

02

0 500 1000 1500 2000 2500

C6

minus02

0

02

0 500 1000 1500 2000 2500

C8

minus05

0

05

0 500 1000 1500 2000 2500

C10

minus05

0

05

0 500 1000 1500 2000 2500

C3

minus02

0

02

0 500 1000 1500 2000 2500

C5

minus02

0

02

0 500 1000 1500 2000 2500

C7

minus02

0

02

0 500 1000 1500 2000 2500

C9

minus05

0

05

0 500 1000 1500 2000 2500

Figure 5 Vibration waveforms of 9 failure modes and normal status

Table 3 Division of the whole sample dataset

Single failure Simultaneous failure Total number119863training1 250 250119863training2 250 250119863threshold 100 200 300119863testing 100 100 200Total 700 300 1000

using 3 level decomposition and Db4 as mother wavelet andstandard diagnostic model based on SBELM is highest withthe accuracy of 952 This parameter combination of WPTis suitable for preprocessing the dataset in this application

After decomposing vibration signal by using three-levelwavelet package decomposition calculate the correspondingvalue of fuzzy entropy as shown in Figure 7 In Figure 7horizontal ordinate represents eight subfrequency bands ofthree-level wavelet package decomposition and longitudinalcoordinate represents the fuzzy entropy value The FuzzyEnof the oscillation from final drive with simultaneous failuresis larger than that of single failures and normal status Whensimultaneous failures occur under rotation of gear pairdifferent failure points are coupling together to make the

9270

9520

9240

94809450

9250

93909420

92

9000

9100

9200

9300

9400

9500

9600

9700

L3D

b3

()

L3D

b4

L3D

b5

L4D

b3

L4D

b4

L4D

b5

L5D

b3

L5D

b4

L5D

b5

Diagnostic accuracy

Combination of parameters

Figure 6 The diagnostic accuracy of different parameters

oscillation complex and stronger Furthermore the values offuzzy entropy vary from one failure pattern to another Thischaracteristic denotes fuzzy entropy can be used as feature offailure diagnosis

By calculating fuzzy entropy of each frequency bandobtained from three-levelwavelet package decompositionwe


05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910

Number of feature vectors

Figure 7 Mean value of fuzzy entropy for failure modes

construct a feature vector with the dimension of 8 which caneffectively reflect the failure modes of final drive


42 Effectiveness of Optimal Decision Threshold After con-structing optimal diagnostic model based on paired SBELMwith optimal parameters of WPT in preprocessing by usingonly single failure modes generation of optimal decisionthreshold is the pivotal point which affects final diagnosticaccuracy of simultaneous failure Traditional machine learn-ingmethods usually adopt 05 as general threshold value (GT)[24]This research uses119863threshold containing 100 single failuremodes and 200 simultaneous failure modes and Grid Searchmethodwith interval of 001 to search final decision threshold120576lowast in range of 0 to 1 AlthoughGrid Search is time consumingit can obtain global optimum

With the purpose of verifying effectiveness of optimaldecision threshold utilize 5-fold cross validation method toimplement a set of experiments by using 119863threshold for bothsingle and simultaneous failure modes recognition Resultsare shown in Figure 8

After optimizing threshold the accuracy of diagnosticmodel improves by an average of 6 FixedGeneral thresholdis generated by experience so that it has generalization butwithout optimization [25] Even using the same diagnosticmodel to diagnose different sample set would require differ-ent threshold Therefore this research uses an independentsample set to generate optimal decision threshold

43 Sensitivity Analysis of SBELM For diagnosis based onELM diagnostic accuracy and training speed are sensitiveto the initial number of hidden nodes To analyze thesensitivity of SBELM on the number of hidden nodes in thisapplication use 500 single failure samples in 119863training1 and119863training2 to train classifier based on SBELM and the bestaverage accuracy along with the increase of hidden nodesis shown in Figure 9 As shown in Figure 9 the averageaccuracies of ELM with increment of hidden nodes are inlarger variation The reason for this fluctuation is that ELM

9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()

Simultaneous fault Total fault

General thresholdOptimal decision threshold

Figure 8 Diagnostic accuracies of models with general thresholdand optimal decision threshold

80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM

Figure 9 Variation of accuracy of ELM and SBELM

is in poor generalization because of data overfitting [17]However the average accuracies are stable and are obviouslyhigher than ELMThe result verifies that SBELM is relativelyinsensitive to the initial number of hidden nodes MoreoverSBELM can obtain an excellent accuracy with a small hiddenlayer which reduces the computational cost effectively

44 Evaluation of the Proposed Framework In order to effec-tively confirm the availability of the proposed simultaneousfailure diagnosis framework we use 119863testing containing 100single failure modes and 100 simultaneous failure modes andF1-measuremethod tomeasure performance of the proposedframework and diagnostic model based on PNN and SVM indiagnostic accuracy and diagnostic speed Firstly use sampleset which are consisting of119863training1 and119863training2 to constructand tune parameters of diagnostic model based on PNN andSVM separately and then use 119863threshold to generate optimalthreshold value Since SVM is essentially used for binary-classclassification [26] with the purpose of simultaneous failurediagnosis we combine SVM with multiclass classificationstrategy to construct a set of classifiers in which each classifier


Table 4 Comparison of paired strategy and one-to-all strategy

Classifier Decision threshold isin [0 1] Multiclass classification strategy Accuracy ()Single failures Simultaneous failures Entire sample

PNN 069 One-to-all 9154 (plusmn102) 8822 (plusmn155) 8942 (plusmn167)paired strategy 9321 (plusmn125) 8914 (plusmn176) 9233 (plusmn205)

SVM 069 One-to-all 9202 (plusmn235) 8090 (plusmn162) 8470 (plusmn187)paired strategy 9484 (plusmn175) 8432 (plusmn135) 8892 (plusmn214)

SBELM 072 One-to-all 9513 (plusmn122) 9054 (plusmn205) 9294 (plusmn173)paired strategy 9842 (plusmn141) 9281 (plusmn235) 9623 (plusmn159)

is only focusing on two failure modes Trying to ensurethe excellent performance of classifiers based on SVM setthe value of regularization parameter 119862 of SVM to be 10120572where 120572 is between 0 and 2 Radial basis kernel functionis employed in SVM with 119862 = 10 and 119903 = 2 whichshow the best accuracy of classification As a probabilityclassifier the crucial hyperparameter of PNN is spread 119904 Inthis research the value of s is chosen from 1 to 3 with intervalof 05 according to conclusion of references Finally the besthyperparameters 119904 and threshold value 120576 for PNN are 1 and069

To verify the effectiveness of the paired strategy inthe proposed framework implement a set of experimentswith one-to-all strategy The experimental results are shownin Table 4 Comparing different classifiers with one-to-allstrategy and paired strategy the accuracies of classifiers withpaired strategy are generally 2 to 4 higher than that ofclassifiers with one-to-all strategyThe primary reason is thatpaired strategy which is used in the proposed frameworkfully considers the correlation between each single failureHowever one-to-all strategy may cause some indecisionregions between different classes The indecision region isprone to sinking into misclassification

To verify the performance of the proposed frameworkimplement a set of experiments about different classifierswith the same testing set and best parameters The decisionthreshold values training time testing time and testingaccuracy of diagnosticmodels based on paired SBELM SVMand PNN are shown in Table 5 The diagnostic accuracy ofpaired SBELM for single failure simultaneous failure andentire sample is 984 928 and 962 which are higherthan that of the SVM and PNN The reason is that SBELMestimates the probability distribution of output values insteadof fitting data to improve generalization [17] Moreover thetraining time and testing time of paired SBELM are 1454msand 487ms that are much fewer than SVMs The reason forthis disparity is that even though paired SBELM builds aset of binary classifiers the sparse characteristic of SBELMreduces the computational cost Consequently the disparitywill become more obvious if the size of sample is big

In practical application of auto manufacturer repre-sentative and valid samples are continuously collected andadded to the training sample database to improve trainingaccuracy Based on this learning speed becomes a crucialfactor for evaluating the efficiency of diagnostic platform Ingeneral considering both diagnostic accuracy and diagnosticefficiency the proposed platform is superior in simultaneous

Table 5 Performance of three classifiers

PNN SVM SBELMDecision threshold[0 1] 069 069 076

Accuracy of singlefailure () 9324 (plusmn186) 9481 (plusmn219) 9842 (plusmn155)

Accuracy ofsimultaneousfailure ()

8912 (plusmn241) 8734 (plusmn196) 9281 (plusmn185)

Accuracy of entiresample () 9230 (plusmn255) 9143 (plusmn232) 9623 (plusmn206)

Training time(ms) 2682 493 1454

Testing time(ms) 865 1194 487

9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials

Figure 10 Testing result of 100 trials

failure diagnosis and it is not only suitable for final drive ofcar but also it can be porting to other research fields

In order to verify the stability of the proposed diagnosticframework based on paired SBELM implement 100 trials andin each trial thewhole sample data is reshuffled and randomlydistributed into119863testing afresh andmake sure there are enoughsingle failure samples and simultaneous failure samples in119863testing The testing result is shown in Figure 10 in which thetesting accuracy is stable in the range between 95 and 97and there is no dramatic variation in 100 simulation trials

5 Conclusion

This paper proposes a novel framework based on SBELMand fuzzy entropy for simultaneous failure diagnosis offinal drive which is hardcore to affect the performance


and safety of car The proposed framework contains foursections preprocessing and feature extraction based onWPT and fuzzy entropy construction of diagnostic modelbased on paired SBELM generation of decision thresholdvalue and recognition of simultaneous failure modes Byusing single failure samples obtain optimal parameters ofWPT which are perfectly adequate for the data in thisapplication Diagnostic model based on paired SBELM inwhich each binary classifier is trained by only single failuresamplesWith an independent sample subset containing bothsingle and simultaneous failure samples use Grid Searchmethod to generate optimal decision threshold by whichprobability result obtained from diagnostic model can beconverted into final result of simultaneous failure modesCompared with frequently used diagnostic model based onSVM and PNN there are three superiorities of the proposedframework (1) The proposed framework based on SBELMinherits the advantages of ELM (efficient approximation andlearning speed) and sparse Bayesian learning (high sparsityand generalization) (2) Fully considering the difficulty andimpossibility of assembling all possible simultaneous failuremodes the proposed framework trains paired classifiersbased on SBELM by using only single failure samples andmoreover the paired strategy can effectively avoid indecisionregions between different classes which can result in misclas-sification (3) With the average testing accuracy of 962 andtesting time of 487ms the proposed framework outperformsother diagnostic models in diagnostic accuracy and learningspeedThe proposed framework is general and transplantablefor simultaneous failure diagnosis so it can be applied toother applications in industrial area in which accuracy andtime cost of failure identification are key factors

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 70701013 the Scien-tific Research and Technology Development Plan Project ofGuangxi Province under Grant no 2013F020202 and theResearch Project of Liuzhou GM-Wuling Limited LiabilityCompany under Grant no 20132h0261 The authors alsogratefully acknowledge the helpful comments and sugges-tions of the reviewers which have improved the paper

References

[1] X Chiementin B Kilundu L Rasolofondraibe S Crequyand B Pottier ldquoPerformance of wavelet denoising in vibrationanalysis highlightingrdquo Journal of Vibration and Control vol 18no 6 pp 850ndash858 2012

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

[3] J-DWu and J-J Chan ldquoFaulted gear identification of a rotatingmachinery based on wavelet Transform and artificial neuralnetworkrdquo Expert Systems with Applications vol 36 no 5 pp8862ndash8875 2009

[4] M Vannucci and V Colla ldquoNovel classificationmethod for sen-sitive problems and uneven datasets based on neural networksand fuzzy logicrdquo Applied Soft Computing Journal vol 11 no 2pp 2383ndash2390 2011

[5] A Janusauskas V Marozas and A Lukosevicius ldquoEnsembleempirical mode decomposition based feature enhancement ofcardio signalsrdquo Medical Engineering and Physics vol 35 no 8pp 1059ndash1069 2013

[6] W T Chen Z ZWang H B Xie andW Yu ldquoCharacterizationof surface EMG signal based on fuzzy entropyrdquo IEEE Transac-tions on Neural Systems and Rehabilitation Engineering vol 15no 2 pp 266ndash272 2007

[7] J S Richman and J R Moorman ldquoPhysiological time-seriesanalysis using approximate and sample entropyrdquoThe AmericanJournal of PhysiologymdashHeart and Circulatory Physiology vol278 no 6 pp H2039ndashH2049 2000

[8] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013

[9] G L Xiong L Zhang H S Liu H J Zou and W-ZGuo ldquoA comparative study on ApEn SampEn and their fuzzycounterparts in a multiscale framework for feature extractionrdquoJournal of Zhejiang University Science A vol 11 no 4 pp 270ndash279 2010

[10] S Deng S Y Lin and W L Chang ldquoApplication of multiclasssupport vector machines for fault diagnosis of field air defensegunrdquo Expert Systems with Applications vol 38 no 5 pp 6007ndash6013 2011

[11] W Jatmiko W P Nulad M I Elly I M A Setiawan and PMursanto ldquoHeart beat classification using wavelet feature basedon neural networkrdquoWSEASTransactions on Systems vol 10 no1 pp 17ndash26 2011

[12] G-B Huang Q-Y Zhu and C-K Siew ldquoExtreme learningmachine theory and applicationsrdquoNeurocomputing vol 70 no1ndash3 pp 489ndash501 2006

[13] E Cambria G-B Huang L L C Kasun et al ldquoExtremelearning machinesrdquo IEEE Intelligent Systems vol 28 no 6 pp30ndash59 2013

[14] Q YuanWZhou S Li andDCai ldquoEpileptic EEG classificationbased on extreme learning machine and nonlinear featuresrdquoEpilepsy Research vol 96 no 1-2 pp 29ndash38 2011

[15] G-B Huang H Zhou X Ding and R Zhang ldquoExtremelearning machine for regression and multiclass classificationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 513ndash529 2012

[16] E Soria-Olivas and J Gomez-Sanchis ldquoBELM bayesianextreme learning machinerdquo IEEE Transaction on Neural Net-works vol 22 no 3 pp 505ndash509 2011

[17] J Luo C-M Vong and P-K Wong ldquoSparse bayesian extremelearningmachine formulti-classificationrdquo IEEETransactions onNeural Networks and Learning Systems vol 25 no 4 pp 836ndash843 2014

[18] Z Yang P K Wong C M Vong J Zhong and J LiangldquoSimultaneous-fault diagnosis of gas turbine generator systemsusing a pairwise-coupled probabilistic classifierrdquoMathematicalProblems in Engineering vol 2013 Article ID 827128 14 pages2013


[19] D Karaboga and B Basturk ldquoA powerful and efficientalgorithm for numerical function optimization artificial beecolony(ABC)algorithmrdquo Journal of Global Optimization vol 39no 3 pp 459ndash471 2007

[20] T-F Wu C-J Lin and R C Weng ldquoProbability estimatesfor multi-class classification by pairwise couplingrdquo Journal ofMachine Learning Research vol 5 pp 975ndash1005 2004

[21] F Schwenker ldquoHierarchical support vector machines for multi-class pattern recognitionrdquo in Proceedings of the 4th Interna-tional Conference on Knowledge-Based Intellingent EngineeringSystems amp Allied Technologies pp 561ndash565 Brighton UKSeptember 2000

[22] T Yingthawornsuk ldquoClassification of cardiac arrhythmia viaSVMrdquo in Proceedings of the 2nd International Conference onBiomedical Engineering and Technology vol 34 IPCBEE 2012

[23] R Baeza-Yates and B Ribeiro-Neto Modern InformationRetrieval ACMPress Addision-WesleyWokingham UK 1999

[24] J Cheng K Zhang and Y Yang ldquoAn order tracking techniquefor the gear fault diagnosis using local mean decompositionmethodrdquo Mechanism and Machine Theory vol 55 pp 67ndash762012

[25] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo Modeling Decisions for Artificial Intelligence vol4617 pp 318ndash329 2007

[26] C M Vong P K Wong and W F Ip ldquoA new frameworkof simultaneous-fault diagnosis using pairwise probabilisticmulti-label classification for time-dependent patternsrdquo IEEETransactions on Industrial Electronics vol 60 no 8 pp 3372ndash3385 2013

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



and sample entropy which are based on Heaviside stepfunction which is mutational at the classification boundaryfuzzy entropy eliminates the influence of baseline drift ofdata and guarantees the entropy to vary smoothly andcontinuously with similarity tolerance [8] so that it isexcellent in measuring complexity and self-similarity of thepreprocessed vibration signal and fully reflecting changesof the vibration performance of mechanical equipment[9]

In recent years many machine learning methods areapplied in failure diagnosis including support vectormachine(SVM) [10] artificial neural networks (ANN) [11] extremelearningmachine (ELM) [12 13] and kernel extreme learningmachine (KELM) [14] ELM is single-hidden-layer feedfor-ward neural networks and without human intervention intuning parameters which differs from SVM and ANN andmakes it superior in high generalization and less learningtime KELM apply kernel function to ELM to improve gener-alization and nonlinear approximation ability [15] Howevercomputational cost and memory cost of KELM are high withregard to large scale problem Recently Bayesianmethods areemployed into ELM to learn the output weights by estimatingthe probability distribution of output with high generaliza-tion Soria-Olivas and Gomez-Sanchis proposed Bayesianextreme learning machine [16] for linear regression with-out solving classification problem Sparse Bayesian extremelearningmachine (SBELM) [17] is a novel method for findingthe sparse representatives of hidden layer output weights byimposing a hyperparameter on each weight During learningphase SBELM tunes some output weights into zero to obtaincompact model In summary SBELM has the advantagesof probability output high generalization sparsity and fasttraining speed To solve the problem of simultaneous failurediagnosis a proper classifier has to offer the probability of allpossible failures In this research the proposed frameworkconstructs classifiers based on paired SBELM in which eachclassifier based on SBELM is trained by a pair of singlefailure samples The paired SBELM effectively reflect theprobability distribution of failure modes In general onlysingle failure samples are used for constructing diagnosticmodels Since it is impossible to collect all combinationsof existing single failure modes for training the proposedframework can effectively solve the practical bottleneck insimultaneous failure diagnosis With the purpose of rec-ognizing simultaneous failure modes use both single andsimultaneous failure samples and Gird search method togenerate optimal decision threshold which could convertprobability result of classifier into final multiple failuremodes Considering that partial matching is valid and instru-mental in simultaneous failure diagnosis this research adoptsF1-measure to evaluate the performance of the proposedframework

This paper is organized as follows Section 2 presentsthe proposed framework Section 3 presents the experimentsetup data acquisition and preprocessing The results ofexperiment are discussed in Section 4 Finally a conclusionis given in Section 5

2 The Proposed Framework for Final DriveSimultaneous Failure Diagnosis

21 Feature Extraction Based on Wavelet PackageTransform and Fuzzy Entropy

211 Wavelet Package Transform In failure diagnosis one ofthe key points is extraction of features from original vibrationsignal which is nonstationary for mechanical equipmentWavelet package transform (WPT) is an extended form ofwavelet transform to analyse nonstationary and non-linearsignal and to supply better partition of frequency bandbecause the same frequency bandwidths can provide goodresolution regardless of high and low frequencies [18] As amultiresolution analysis method WPT can effectively pre-process nonstationary vibration signal in both time domainand frequency domain Two-scale equation ofWPT is shownbelow in which ℎ

0119896and ℎ1119896represent the filter coefficients

1199092119899(119905) = radic2sum

119896isin119885

ℎ0119896119909119899(2119905 minus 119896)

1199092119899+1

(119905) = radic2sum

119896isin119885

ℎ1119896119909119899(2119905 minus 119896)

(1)

The recursion formula of wavelet package coefficient is

119889119895+12119899

119896= sum

119897

ℎ0(2119897minus119896)

119889119895119899

119897 119889

119895+12119899+1

119896= sum

119897

ℎ1(2119897minus119896)

119889119895119899

119897 (2)

212 Fuzzy Entropy When failure occurred in final drive thecomplexity of oscillation feature will change hencewe shouldextract representative features containing in the signal Fuzzyentropy is an extension of Shannon entropy and fuzzy sets[19] The procedure of fuzzy entropy is described as follows

(1) Consider a time series with the length of119873 119909(119894) 1 le119894 le 119873 For given 119898 119899 and 119903 construct a vector set in theform of 119883119898

119894 119894 = 1 2 119873 minus 119898 + 1 in which each vector

contains119898 sequential elements starting from 119909(119894) shown as

119883119898

119894= 119909 (119894) 119909 (119894 + 1) 119909 (119909 + 119898 minus 1) minus 119909

0(119894)

119894 = 1 2 119873 minus 119898 + 1

(3)

where 1199090(119894) is the average of vector119883119898

119894

(2) Define the distance 119889119898119894119895between 119883119898

119894and 119883119898

119895where

119894 119895 = 1 2 119873 minus 119898 119894 = 119895 as follows

119889119898

119894119895= max119896isin(0119898minus1)

1003816100381610038161003816[119909 (119894 + 119896) minus 1199090 (119894)] minus [119909 (119895 + 119896) minus 1199090 (119895)]

1003816100381610038161003816

(4)

(3) Calculate similarity between 119883119898119894and 119883119898

119895using fuzzy

function

119863119898

119894119895= 120583 (119889

119898

119894119895 119899 119903) = 119890

minus In 2(119889119898119894119895119903)119899

(5)

(4) Define function 120601119898 as follows

120601119898(119899 119903) =

1

119873 minus 119898

119873minus119898

sum

119894=1

(1

119873 minus 119898 + 1

119873minus119898

sum

119895=1119895 =119894

119863119898

119894119895) (6)



120601119898+1(119899 119903) =

1

119873 minus 119898

119873minus119898

sum

119894=1

(1

119873 minus 119898 + 1

119873minus119898

sum

119895=1119895 =119894

119863119898+1

119894119895) (7)



[In120601119898 (119899 119903) minus In120601119898+1 (119899 119903)] (8)




119894 119905119894) 119894 = 1 119873



119891 (119909) =

119871

sum

119894=1

120573119894ℎ119894(119909) = 120573 sdot 119867 (119909) (10)

where 120573 = [1205731 120573



119871(119909)] is the



119867120573 = 119879 (11)


120590 (Y (ℎ 120573)) =1

(1 + exp (minusY (ℎ 120573))) (12)


119901 (119905 | 119867 120573) =

119873

prod

119895=1

120590 (Y (ℎ 120573))119905119895

[1 minus 120590 (Y(ℎ 120573))]1minus119905119895

(13)


119895Y(ℎ 120573) = ℎ120573 and


119895 zero-mean


119901 (120573 | 120572) =

119871

prod

119894=1

120572119894


1205721198941205732

119894

2) (14)



argmax ln 119901 (119905 | 120573119867) 119901 (120573 | 120572)

= argmax[

[

ln

119873

prod

119895=1

119910119905119895

119895(1 minus 119910

119895)1minus119905119895

+ ln119871

prod

119894=1

120572119894


1205721198941205732

119894

2)]

]

= argmax[

[

119873

sum

119895=1

119905119895ln119910119895+ (1 minus 119905

119895)


120573119879119860120573

2+ const]

]

(15)

where 119910119895= 120590(Y(ℎ


119894=1ln120572119894minus




119895) with


1205731015840= Φ119867

1198791198701199051015840 Φ = (119867

119879119870119867 + 119860)

minus1

(17)


L (120572) = ln119901 (119905 | 120572119867)

= minus1

2[119873 ln (2120587) + ln |119862| + (1199051015840)

119879

119862minus11199051015840]

(18)


120572119894=1 minus 120572119894Φ119894119894

(1205731198941015840)2 (19)


(1) Initialize 120572119894and 120573

119894randomly with 119894 = 1 119871








SBELM12

SBELMi1

SBELMm1

SBELM1m

SBELMim

SBELMm(mminus1)

PSBELM1

PSBELMi

PSBELMm

p1

pi

pm






1 PSBELM



[SBELM1198941 SBELM

119894119895 SBELM


119894119895is







follows

119901119894=sum119889

119895=1119895 =119894119899119894119895119901119894119895

sum119889

119895=1119895 =119894119899119894119895

119894 = 1 119898 (20)




1 119901

119898] in which 119901



1 119891

119894 119891

119898]

effectively

119891119894= 1 119901119894ge 120576lowast

0 119901119894lt 120576lowast119894 = 1 119898 120576






119894 119905119894) 119894 =



119875 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119905119894119895

119877 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119891lowast119894119895

(22)

where 119891lowast119894= [119891

lowast

1198941 119891

lowast



[1199051198941 119905


119894


119864 =2 lowast 119875 lowast 119877

119875 + 119877=

2sum119898

119895=1sum119873

119894=1119891lowast

119894119895119910119894119895

sum119898

119895=1sum119873

119894=1119910119894119895+ sum119898

119895=1sum119873

119894=1119891lowast119894119895

(23)






1 119901

119898] in 119898-label




Opt

imal

dia

gnos

tic m

odel


Dtesting



Decision threshold

Result vector

Evaluation









Dtraining1



Optimization of WPT

Dthreshold

Dtraining2

value 120576lowast

value 120576lowast












Single failures






Turntable part

Drive part





Horizontaldirection

Verticaldirection









0 500 1000 1500 2000 2500minus02

0

02 C1

0 500 1000 1500 2000 2500

C2

minus02

0

02

C4

minus02

0

02

0 500 1000 1500 2000 2500

C6

minus02

0

02

0 500 1000 1500 2000 2500

C8

minus05

0

05

0 500 1000 1500 2000 2500

C10

minus05

0

05

0 500 1000 1500 2000 2500

C3

minus02

0

02

0 500 1000 1500 2000 2500

C5

minus02

0

02

0 500 1000 1500 2000 2500

C7

minus02

0

02

0 500 1000 1500 2000 2500

C9

minus05

0

05

0 500 1000 1500 2000 2500






9270

9520

9240

94809450

9250

93909420

92

9000

9100

9200

9300

9400

9500

9600

9700

L3D

b3

()

L3D

b4

L3D

b5

L4D

b3

L4D

b4

L4D

b5

L5D

b3

L5D

b4

L5D

b5

Diagnostic accuracy






05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910









9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()




80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM






















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120601119898+1(119899 119903) =

1

119873 minus 119898

119873minus119898

sum

119894=1

(1

119873 minus 119898 + 1

119873minus119898

sum

119895=1119895 =119894

119863119898+1

119894119895) (7)



[In120601119898 (119899 119903) minus In120601119898+1 (119899 119903)] (8)




119894 119905119894) 119894 = 1 119873



119891 (119909) =

119871

sum

119894=1

120573119894ℎ119894(119909) = 120573 sdot 119867 (119909) (10)

where 120573 = [1205731 120573



119871(119909)] is the



119867120573 = 119879 (11)


120590 (Y (ℎ 120573)) =1

(1 + exp (minusY (ℎ 120573))) (12)


119901 (119905 | 119867 120573) =

119873

prod

119895=1

120590 (Y (ℎ 120573))119905119895

[1 minus 120590 (Y(ℎ 120573))]1minus119905119895

(13)


119895Y(ℎ 120573) = ℎ120573 and


119895 zero-mean


119901 (120573 | 120572) =

119871

prod

119894=1

120572119894


1205721198941205732

119894

2) (14)



argmax ln 119901 (119905 | 120573119867) 119901 (120573 | 120572)

= argmax[

[

ln

119873

prod

119895=1

119910119905119895

119895(1 minus 119910

119895)1minus119905119895

+ ln119871

prod

119894=1

120572119894


1205721198941205732

119894

2)]

]

= argmax[

[

119873

sum

119895=1

119905119895ln119910119895+ (1 minus 119905

119895)


120573119879119860120573

2+ const]

]

(15)

where 119910119895= 120590(Y(ℎ


119894=1ln120572119894minus




119895) with


1205731015840= Φ119867

1198791198701199051015840 Φ = (119867

119879119870119867 + 119860)

minus1

(17)


L (120572) = ln119901 (119905 | 120572119867)

= minus1

2[119873 ln (2120587) + ln |119862| + (1199051015840)

119879

119862minus11199051015840]

(18)


120572119894=1 minus 120572119894Φ119894119894

(1205731198941015840)2 (19)


(1) Initialize 120572119894and 120573

119894randomly with 119894 = 1 119871








SBELM12

SBELMi1

SBELMm1

SBELM1m

SBELMim

SBELMm(mminus1)

PSBELM1

PSBELMi

PSBELMm

p1

pi

pm






1 PSBELM



[SBELM1198941 SBELM

119894119895 SBELM


119894119895is







follows

119901119894=sum119889

119895=1119895 =119894119899119894119895119901119894119895

sum119889

119895=1119895 =119894119899119894119895

119894 = 1 119898 (20)




1 119901

119898] in which 119901



1 119891

119894 119891

119898]

effectively

119891119894= 1 119901119894ge 120576lowast

0 119901119894lt 120576lowast119894 = 1 119898 120576






119894 119905119894) 119894 =



119875 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119905119894119895

119877 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119891lowast119894119895

(22)

where 119891lowast119894= [119891

lowast

1198941 119891

lowast



[1199051198941 119905


119894


119864 =2 lowast 119875 lowast 119877

119875 + 119877=

2sum119898

119895=1sum119873

119894=1119891lowast

119894119895119910119894119895

sum119898

119895=1sum119873

119894=1119910119894119895+ sum119898

119895=1sum119873

119894=1119891lowast119894119895

(23)






1 119901

119898] in 119898-label




Opt

imal

dia

gnos

tic m

odel


Dtesting



Decision threshold

Result vector

Evaluation









Dtraining1



Optimization of WPT

Dthreshold

Dtraining2

value 120576lowast

value 120576lowast












Single failures






Turntable part

Drive part





Horizontaldirection

Verticaldirection









0 500 1000 1500 2000 2500minus02

0

02 C1

0 500 1000 1500 2000 2500

C2

minus02

0

02

C4

minus02

0

02

0 500 1000 1500 2000 2500

C6

minus02

0

02

0 500 1000 1500 2000 2500

C8

minus05

0

05

0 500 1000 1500 2000 2500

C10

minus05

0

05

0 500 1000 1500 2000 2500

C3

minus02

0

02

0 500 1000 1500 2000 2500

C5

minus02

0

02

0 500 1000 1500 2000 2500

C7

minus02

0

02

0 500 1000 1500 2000 2500

C9

minus05

0

05

0 500 1000 1500 2000 2500






9270

9520

9240

94809450

9250

93909420

92

9000

9100

9200

9300

9400

9500

9600

9700

L3D

b3

()

L3D

b4

L3D

b5

L4D

b3

L4D

b4

L4D

b5

L5D

b3

L5D

b4

L5D

b5

Diagnostic accuracy






05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910









9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()




80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM






















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




SBELM12

SBELMi1

SBELMm1

SBELM1m

SBELMim

SBELMm(mminus1)

PSBELM1

PSBELMi

PSBELMm

p1

pi

pm






1 PSBELM



[SBELM1198941 SBELM

119894119895 SBELM


119894119895is







follows

119901119894=sum119889

119895=1119895 =119894119899119894119895119901119894119895

sum119889

119895=1119895 =119894119899119894119895

119894 = 1 119898 (20)




1 119901

119898] in which 119901



1 119891

119894 119891

119898]

effectively

119891119894= 1 119901119894ge 120576lowast

0 119901119894lt 120576lowast119894 = 1 119898 120576






119894 119905119894) 119894 =



119875 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119905119894119895

119877 =sum119898

119895=1sum119873

119894=1119891lowast

119894119895119905119894119895

sum119898

119895=1sum119873

119894=1119891lowast119894119895

(22)

where 119891lowast119894= [119891

lowast

1198941 119891

lowast



[1199051198941 119905


119894


119864 =2 lowast 119875 lowast 119877

119875 + 119877=

2sum119898

119895=1sum119873

119894=1119891lowast

119894119895119910119894119895

sum119898

119895=1sum119873

119894=1119910119894119895+ sum119898

119895=1sum119873

119894=1119891lowast119894119895

(23)






1 119901

119898] in 119898-label




Opt

imal

dia

gnos

tic m

odel


Dtesting



Decision threshold

Result vector

Evaluation









Dtraining1



Optimization of WPT

Dthreshold

Dtraining2

value 120576lowast

value 120576lowast












Single failures






Turntable part

Drive part





Horizontaldirection

Verticaldirection









0 500 1000 1500 2000 2500minus02

0

02 C1

0 500 1000 1500 2000 2500

C2

minus02

0

02

C4

minus02

0

02

0 500 1000 1500 2000 2500

C6

minus02

0

02

0 500 1000 1500 2000 2500

C8

minus05

0

05

0 500 1000 1500 2000 2500

C10

minus05

0

05

0 500 1000 1500 2000 2500

C3

minus02

0

02

0 500 1000 1500 2000 2500

C5

minus02

0

02

0 500 1000 1500 2000 2500

C7

minus02

0

02

0 500 1000 1500 2000 2500

C9

minus05

0

05

0 500 1000 1500 2000 2500






9270

9520

9240

94809450

9250

93909420

92

9000

9100

9200

9300

9400

9500

9600

9700

L3D

b3

()

L3D

b4

L3D

b5

L4D

b3

L4D

b4

L4D

b5

L5D

b3

L5D

b4

L5D

b5

Diagnostic accuracy






05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910









9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()




80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM






















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Opt

imal

dia

gnos

tic m

odel


Dtesting



Decision threshold

Result vector

Evaluation









Dtraining1



Optimization of WPT

Dthreshold

Dtraining2

value 120576lowast

value 120576lowast












Single failures






Turntable part

Drive part





Horizontaldirection

Verticaldirection









0 500 1000 1500 2000 2500minus02

0

02 C1

0 500 1000 1500 2000 2500

C2

minus02

0

02

C4

minus02

0

02

0 500 1000 1500 2000 2500

C6

minus02

0

02

0 500 1000 1500 2000 2500

C8

minus05

0

05

0 500 1000 1500 2000 2500

C10

minus05

0

05

0 500 1000 1500 2000 2500

C3

minus02

0

02

0 500 1000 1500 2000 2500

C5

minus02

0

02

0 500 1000 1500 2000 2500

C7

minus02

0

02

0 500 1000 1500 2000 2500

C9

minus05

0

05

0 500 1000 1500 2000 2500






9270

9520

9240

94809450

9250

93909420

92

9000

9100

9200

9300

9400

9500

9600

9700

L3D

b3

()

L3D

b4

L3D

b5

L4D

b3

L4D

b4

L4D

b5

L5D

b3

L5D

b4

L5D

b5

Diagnostic accuracy






05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910









9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()




80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM






















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Single failures






Turntable part

Drive part





Horizontaldirection

Verticaldirection









0 500 1000 1500 2000 2500minus02

0

02 C1

0 500 1000 1500 2000 2500

C2

minus02

0

02

C4

minus02

0

02

0 500 1000 1500 2000 2500

C6

minus02

0

02

0 500 1000 1500 2000 2500

C8

minus05

0

05

0 500 1000 1500 2000 2500

C10

minus05

0

05

0 500 1000 1500 2000 2500

C3

minus02

0

02

0 500 1000 1500 2000 2500

C5

minus02

0

02

0 500 1000 1500 2000 2500

C7

minus02

0

02

0 500 1000 1500 2000 2500

C9

minus05

0

05

0 500 1000 1500 2000 2500






9270

9520

9240

94809450

9250

93909420

92

9000

9100

9200

9300

9400

9500

9600

9700

L3D

b3

()

L3D

b4

L3D

b5

L4D

b3

L4D

b4

L4D

b5

L5D

b3

L5D

b4

L5D

b5

Diagnostic accuracy






05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910









9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()




80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM






















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






0 500 1000 1500 2000 2500minus02

0

02 C1

0 500 1000 1500 2000 2500

C2

minus02

0

02

C4

minus02

0

02

0 500 1000 1500 2000 2500

C6

minus02

0

02

0 500 1000 1500 2000 2500

C8

minus05

0

05

0 500 1000 1500 2000 2500

C10

minus05

0

05

0 500 1000 1500 2000 2500

C3

minus02

0

02

0 500 1000 1500 2000 2500

C5

minus02

0

02

0 500 1000 1500 2000 2500

C7

minus02

0

02

0 500 1000 1500 2000 2500

C9

minus05

0

05

0 500 1000 1500 2000 2500






9270

9520

9240

94809450

9250

93909420

92

9000

9100

9200

9300

9400

9500

9600

9700

L3D

b3

()

L3D

b4

L3D

b5

L4D

b3

L4D

b4

L4D

b5

L5D

b3

L5D

b4

L5D

b5

Diagnostic accuracy






05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910









9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()




80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM






















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




05075

1125

15175

2225

1 2 3 4 5 6 7 8

12345

678910









9020

8450

8860

9780

91309450

7000

7500

8000

8500

9000

9500

10000

Single fault

()




80

85

90

95

100

10 20 30 40 50 60 70 80 90 100

Accu

racy

()

ELMSBELM






















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





















9300

()

9400

9500

9600

9700

9800

10 20 30 40 50 60 70 80 90 100Number of trials




5 Conclusion






Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Acknowledgments


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article a framework for final drive simultaneous...

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