Research ArticleBearing Fault Diagnosis Based on Domain Adaptation UsingTransferable Features under Different Working Conditions

Zhe Tong 1 Wei Li 1 Bo Zhang 2 andMeng Zhang 1

1School of Mechanical Engineering China University of Mining and Technology Xuzhou 221116 China2School of Computer Science and Technology China University of Mining and Technology Xuzhou 221116 China

Correspondence should be addressed to Bo Zhang zbcumt163com

Received 7 March 2018 Revised 16 May 2018 Accepted 31 May 2018 Published 28 June 2018

Academic Editor Paolo Pennacchi

Copyright copy 2018 Zhe Tong et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Bearing failure is the most common failure mode in rotating machinery and can result in large financial losses or even casualtiesHowever complex structures around bearing and actual variable working conditions can lead to large distribution differenceof vibration signal between a training set and a test set which causes the accuracy-dropping problem of fault diagnosis Thushow to improve efficiently the performance of bearing fault diagnosis under different working conditions is always a primarychallenge In this paper a novel bearing fault diagnosis under different working conditions method is proposed based on domainadaptation using transferable features(DATF) The datasets of normal bearing and faulty bearings are obtained through the fastFourier transformation (FFT) of raw vibration signals under different motor speeds and load conditionsThen we reduce marginaland conditional distributions simultaneously across domains based on maximum mean discrepancy (MMD) in feature space byrefining pseudo test labels which can be obtained by the nearest-neighbor (NN) classifier built on training data and then a robusttransferable feature representation for training and test domains is achieved after several iterationsWith the help of theNNclassifiertrained on transferable features bearing fault categories are identified accurately in final Extensive experiment results show thatthe proposed method under different working conditions can identify the bearing faults accurately and outperforms obviouslycompetitive approaches

1 Introduction

Bearings are the most critical components and widely usedin rotating machinery whose health conditions for examplethe fault degree in different places under different motorspeeds and loads may have a huge effect on the performancereliability and residual life of the equipment [1] or evencan lead to heavy casualties [2ndash4] Hence it is important todiagnose bearings under different working conditions

Cracks or spalls on the surfaces of the roller outer raceor inner race are commonly failure modes in bearings [5]Vibration signal is the most intuitive description for theoperating state of a bearing With the vibration signals underdifferent conditions being collected by sensors [6] manyintelligent fault diagnosis methods have already achieved sig-nificant success in the field of fault diagnosis In [7] a geneticalgorithm-based SVM (GA-SVM) model was presented and

it had high accuracy and generalization ability by optimizingparameters of SVM N Saravanan et al [8] proposed faultdiagnosis method based on DWT and ANN and it has beenproved such approach had the potential to diagnose variousfaults of the gearbox There are two key points for com-mon intelligent fault diagnosis technologies namely featureextraction and classification Raw vibration signal collectedby sensors is abound in redundant information Thus it isimportant for fault diagnosis to achieve effective features [9]Many signal processing approaches are applied to featureextraction from vibration signals Such as time-domainstatistical analysis frequency domain analysis [10] and time-frequency domain analysis [2]Then reducing the dimensionsis conducted for the sake of computational efficiency suchas principal component analysis (PCA) [11] locally linearembedding (LLE) [12] and linear discriminant analysis(LDA) [13] Finally with the help of a suitable classifier such

HindawiShock and VibrationVolume 2018 Article ID 6714520 12 pageshttpsdoiorg10115520186714520

2 Shock and Vibration

as nearest-neighbor (NN) support vector machine (SVM)or artificial neural networks (ANN) features acquired fromabove technological process are used for defect classifica-tion

To be true most of intelligent fault diagnosis methodswork well only under a general assumption the training andtest data are drawn from the same distribution Howeverin operation of rotating machinery because of complicatedworking conditions and complex sensor signals the distribu-tion of fault data is not consistent Vibration signals sampledunder different working conditions violate above assumptionand show large distribution differences between domains[9 14] which leads to drop dramatically of performanceMore specifically taking the roller bearing fault diagnosisproblem as an example classifier was trained under a veryconcrete type of data sampled under a certain motor speedand load however the actual application in fault diagnosis isto recognize test data collected under another motor speedand load Although the fault diameter and categories are notchanged the distribution differences between training data(training domain) and test data (test domain) changes withworking condition vary As a direct result the classifier canachieve high accuracy on training domain while performingpoorly on test domain [14] This is caused by distributiondifferences between two domains since features extractedfrom one domain can not represent for another domain Ofcourse we can spend lots of time and efforts to recollectdata to build a new classifier for effective fault diagnosis ontest domain However we can not always replace classifierby repetitively recollecting data Worse it is so expensive oreven impossible to rebuild the fault diagnosis model fromscratch using newly recollected training data for the actualtask Therefore there is still plenty of room for improve-ment

In order to avoid such recalibration effort we might wantto refine a fault diagnosis model trained in one condition(training domain) for a newworking condition (test domain)or to refine themodel trained on one rolling bearing (trainingdomain) for a new rolling bearing (test domain) This leadsto the research of domain adaptation (DA) [15 16] DAcan be considered as particular setting of transfer learning[17 18] which aims to leverage the knowledge learnt from atraining domain to use in a different but related test domainby reducing distribution differences [18 19] Maximummeandiscrepancy (MMD) [20ndash22] in the field of DA can be appliedto evaluate distribution divergences

In this paper considering actual fault diagnosis appli-cation we propose a novel bearing fault diagnosis underdifferent working conditions based on domain adaptationusing transferable features (DATF) Dataset of normal bear-ing and faulty bearings are achieved through the fast Fouriertransformation (FFT) of raw vibration signals under differentmotor speeds and load conditions Fault diagnosis model isbuilt by using nearest-neighbor (NN) classifier in trainingdomain and then we resort the pseudo outputs of NNclassifier in test domain to refine this model by reducingdistribution differences between domains constantly so thattransferable feature representation could be learnt fromtraining and test domains Finally NN classifier is built

with extracted transferable features and bearing faults areidentified accurately

The rest of this paper is organized as follows Section 2sketches out previous works and preliminaries includingdomain adaptation and maximum mean discrepancy Sec-tion 3 introduces fault diagnosis using transferable featuresincluding feature space generation and transferable featureextraction and diagnosis Section 4 presents the experimentalevaluations The conclusion is given in Section 5

2 Previous Works and Preliminaries

21 Domain Adaptation DA as one research of transferlearning is aimed at making full use of information comingfrom both training domain and test domain during thelearning process to adapt automatically [18 19 23] Generallydomain is considered as consisting of a feature space of inputsX and a probability distribution of inputs 119875(119883) where 119883 =1199091 119909119899 isin X is a series of learning samples Note thatdistributions of two domains are diverse when source domainand target domain are different that is119883119878 = 119883119879 and119875(119883119878) =119875(119883119879) [20 24]

In our work the objective of domain adaptation isto extract transferable features between two domains forrealizing successfully bearing fault diagnosis under differentworking conditions We denote the labeled training domain119883119905119903 = (1199091199051199031 1199101199051199031) (1199091199051199031198991 1199101199051199031198991 ) where 119909119905119903119894 isin X is theinput and 119910119905119903119894 isin Y is the related class label Similarly let theunlabeled test domain be 119883119905119890 = (1199091199051198901) (1199091199051198901198992 ) wherethe input 119909119905119890119894 isin X In the aspect of distribution let 119875(119883119905119903)and 119876(119883119905119890) be the marginal distributions of 119883119905119903 = 119909119905119903119894 and119883119905119890 = 119909119905119890119894 from the training and test domains respectivelySimilarly let 119875(119884119905119903|119883119905119903) and 119876(119884119905119890|119883119905119890) be the conditionaldistributions of119883119905119903 = 119909119905119903119894 and119883119905119890 = 119909119905119890119894 from the trainingdomain and test domain respectively [20 25 26]

In this literature we focus on the following settings(1) One training domain and one test domain share thesame fault types and feature space (2) Domain adaptationin our work is unsupervised and training domain 119883119905119903 isof labels while test domain 119883119905119890 is fully unlabeled (3) Themarginal distribution 119875(119883119905119903) = 119876(119883119905119890) and the conditionaldistribution 119875(119884119905119903|119883119905119903) = 119876(119884119905119890|119883119905119890) The above settings arewell suited to real-world variable working conditions faultdiagnosis Our task is to predict the fault types of bearingaccurately in the unlabeled test domainwith entirely differentdistribution by using the model built in training domain

22 Maximum Mean Discrepancy Typical procedure ofdomain adaptation is to reduce marginal distribution differ-ence across domains In our work domain adaptation is toreduce both marginal and conditional distribution differencesimultaneously by explicitly minimizing the empirical dis-tance measure which is more suitable for the situation ofbearing fault diagnosis under different working conditionsIn order to avoid expensive distribution calculation causedby the parametric criteria a nonparametric distance metricknown as MMD is employed for domain adaptation in ourwork Taking data from source domain119883119878 and target domain

Shock and Vibration 3

Labeled training data (Training domain)

PCA

Raw vibration signals Frequency domain

Trained model

Test data

iteration 1

Revise model

iteration T

pseudo-label

NO IF

BF OF

FFT

Data acquisition

Unlabeled test data (Test domain)

PCA

Raw vibration signals Frequency domain

FFT

Data acquisition

Test

Normal condition

Inner race fault

Ball fault

Outer race fault

NO IF

BF OF

Predict

Transferable feature extraction

middot middot middot

Figure 1 The framework of DATF for variable working condition fault diagnosis

119883119879 the MMD calculates the empirical estimate of distancesacross domains in the 119896-dimensional embedding [20 24]

119863119898 (119883119878 119883119879) =100381710038171003817100381710038171003817100381710038171003817100381710038171119899119904119899119904sum119894=1

119860119879119909119894 minus 1119899119905119899119904+119899119905sum119895=119899119904+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

(1)

where 119863119898 is the distance of marginal distributions acrossdomains119860 is the adaptationmatrix and 119899119904 and 119899119905 denote thenumber of source instances and target instances respectively

3 Fault Diagnosis Using Transferable Features

Asmentioned in Section 1 huge distribution difference acrosstraining domain and test domain under different workingconditions directly leads to poor performance of bearingfault diagnosis In order to solve this problem we needto learn the shift between two domains and extract morerobust transferable features for two domains In this sectionwe present our novel bearing fault diagnosis method undervariable working conditions The framework of our method

is illustrated in Figure 1 As shown in Figure 1 fault diagnosismodel built via labeled training data is iterated revisionaccording to pseudo-label and the final diagnostic results areobtained through the above revised model Details of eachpart are elaborated in the following subsections

31 Feature Space Generation Raw time series vibration sig-nals are readily available and abound in bearing informationOwning to the rotating nature of raw vibration signals froma defective bearing the periodic impulse would appear inobtained signals once a fault occursThus these fault impactscan be detected generally in frequency domain

In our work we directly catch FFT amplitudes fromthe raw time series vibration signals as samples where allsamples have the same dimension and these samples aregenerated under different motor speeds and load conditionsas described in Figure 2

They are divided into two parts labeled training data(119863119905119903) and unlabeled test data(119863119905119890) Then we use principalcomponent analysis (PCA) to generate feature space Themain steps of feature space generation are as follows

4 Shock and Vibration

Start

N = 12000NFFT = 2^nextpow2(N)f = fs2lowastlinspace(01NFFT2+1) fft_amplitude = abs(fft(xNFFT))N

Length of x(n) Next power of 2 from N Frequency resolution Fast Fourier transform of x(n)

The single-sided FFT spectrum amplitude is acquired through 2lowastfft_amplitude(1NFFT2+1) in Matlab

End

Vibration signal x(n) sampled with fs Hz

Figure 2 Flowchart of FFT spectrum amplitudes creation in MATLAB

Step 1 Catch FFT amplitudes from raw time series vibrationsignals collected under different working conditions as sam-ples119863119889119886119905119886Step 2 Take one of the conditions with different fault typesfrom 119863119889119886119905119886 as training samples 119883119905119903 isin 119877119899119905119903times119889 with label 119884119905119903 isin119877119899119905119903times1 and take another of the conditions with different faulttypes from119863119889119886119905119886 as unlabeled test samples119883119905119890 isin 119877119899119905119890times119889Step 3 Denote 119883119863 = 119883119905119903 119883119905119890 isin 119877119889times(119899119905119903+119899119905119890) and 119867 =119868 minus (1(119899119905119903 + 119899119905119890))119897119897119879 where 119868 denotes the identity matrix and119897 is considered as the ones vectors Then the 119896 dimensionalrepresentation is found by solving the following optimizationproblemmax119860119879119860=119868119905119903(119860119879119883119863119867119883119879119863119860) and then feature spaceis created by 119881 = 11986011987911988311986332 Transferable Feature Extraction and Diagnosis In orderto reduce the marginal distribution difference and extractrobust feature for two domains we resort MMD as thedistance measures between 119909119894119905119903 and 119909119895119905119890 to compare differentdistributions 10038171003817100381710038171003817100381710038171003817100381710038171003817

1119899119905119903119899119905119903sum119894=1

119860119879119909119894 minus 1119899119905119890119899119905119903+119899119905119890sum119895=119899119905119903+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

= 119905119903 (119860119879119883119863119872119898119883119879119863119860)(2)

where 119872119898 = [ (119872119898)119905119903119905119903 (119872119898)119905119903119905119890(119872119898)119905119890119905119903 (119872119898)119905119890119905119890] is the MMD matrix and is

computed as follows [24 26]

119872119898 =

1119899119905119903119899119905119903 119909119894 119909119895 isin 1198831199051199031119899119905119890119899119905119890 119909119894 119909119895 isin 119883119905119890minus1119899119905119903119899119905119890 119900119905ℎ119890119903119908119894119904119890

(3)

Themarginal distributions between training domain and testdomain are brought closer under the new representation119881 =119860119879119883119863 by minimizing (2)

In theory training and test data under different workingconditions collected from sensors should be of the samemarginal and conditional distributions while the reality isvery different For improving the performance of bearingfault diagnosis under different work conditions in our workthe differences of conditional distribution between domainsare also reduced bymining the class-conditional distributionFormally the class-conditional distributions can bemeasuredaccording to modified MMD

100381710038171003817100381710038171003817100381710038171003817100381710038171119899119905119903119899119905119903sum119894=1

119860T119909119894 minus 1119899119905119890119899119905119903+119899119905119890sum119895=119899119905119903+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

= 119905119903 (119860119879119883119863119872119888119883119879119863119860) (4)

where119872119888 = [ (119872119888)119905119903119905119903 (119872119888)119905119903119905119890(119872119888)119905119890119905119903 (119872119888)119905119890119905119890] is MMD coefficient matrix that

includes the class label 119888 and it can be calculated accordingto [24 26]

Shock and Vibration 5

119872119888 =

1119899119888119905119903119899119888119905119903 119909119894 119909119895 isin 1198831199051199031119899119888119905119890119899119888119905119890 119909119894 119909119895 isin 119883119905119890minus1119899119905119903119899119905119890

119909119894 isin 119883119888119905119903 119909119895 isin 119883119888119905119890119909119895 isin 119883119888119905119903 119909119894 isin 1198831198881199051198900 119900119905ℎ119890119903119908119894119904119890

(5)

The conditional distributions between training and testdomains are brought closer under the new representation119881 = 119860119879119883119863 by minimizing (4)

In order to obtain effective and robust transferable featurerepresentation and improve the quality of fault diagnosis ourwork aims to reduce the impact of discrepancies from boththe marginal and conditional distributions between trainingand test domains by resorting the pseudo labels of test data[26] on diagnosis and these pseudo labels can be obtainedfrom a base classifier (NN classifier) built on the labeledtraining data to predict the fully unlabeled test data Thusthe final optimization problem (6) in this paper comprised(2) and (4)

min119860119879119883119863119867119883

119879119863119860=119868

(1 minus 120582) 119862sum119888=0

119905119903 (119860119879119883119863119872119888119883119879119863119860) + 120582 1198602119865 (6)

where sdot 119865 is the Frobenius norm that guarantees the opti-mization problem to be well defined and 120582 is the regulariza-tion parameter [24] that trades off the impact of regulariza-tion term on the transformation matrix A The goal is to findthe latent feature space created by a transformation matrix119860 isin 119877119889times119896 where the discrepancies of both the marginal andconditional distributions between domains are significantlyreduced The Lagrange function for (7) is constructed whereΛ = diag(Λ 1 Λ 119896) isin 119877119896times119896 is the Lagrange multiplier

119871 = (1 minus 120582) 119905119903(119860119879(119883119863 119862sum119888=0

119872119888119883119879119863)119860) + 120582119905119903 (119860119879119860)+ 119905119903 ((119868 minus 119860119879119883119863119867119883119879119863119860)Λ)

(7)

According to 119889119871119889119860 = 0 the optimal solution of (6) can beacquired through the generalized eigen decomposition

((1 minus 120582)119883119863 119862sum119888=0

119872119888119883119879119863 + 120582119868)119860 = 119883119863119867119883119879119863119860Λ (8)

Finally the adaptation matrix A is obtained from solving (8)for 119896 smallest eigenvectors The procedure of fault diagnosisusing DAFT can be depicted as follows in detail

Step 1 For given training data 119883119905119903 isin 119877119899119905119903times119889 with label 119884119905119903 isin119877119899119905119903times1 andunlabeled test data119883119905119890 isin 119877119899119905119890times119889 in the feature spaceStep 2 Construct MMD matrix 119872119898 by (2) Adaptationmatrix 119860 generated by the 119896 smallest eigenvectors can beacquired by solving (8) through Lagrange multiplier Thenthe robust representation for two domains is obtained 119881 =119860119879119883119863

Step 3 Train the NN classifier on projected training data119860119879119883119905119903 119884119905119903 and then obtain pseudo test data labels 119884119905119890 thatdenote the conditional probability 119876(119884119905119890|119883119905119890) by using thetrained NN classifier

Step 4 Update MMD matrix 119872119888119862119888=1 by (5) according to119875(119884119905119903|119883119905119903) = 119876(119884119905119890|119883119905119890) and then obtain the updated adapta-tionmatrix119860 by solving (8) through LagrangemultiplierTheupdated robust representation for two domains is obtained119881 = 119860119879119883119863 and then jump to Step 3 until the end of theiteration

Step 5 Finally the test data labels 119884119905119890 are predicted accuratelyby the adaptive NN classifier

4 Experimental Evaluations

In order to demonstrate the effectiveness of the proposedfault diagnosis method the vast bearing vibration signalscollected from a bearing test rig are used Dataset is acquiredfrom the bearing data center of Case Western Reserve Uni-versity (CWRU) [27] DATF is compared with the baselineapproaches and several successful methods

(a) Baseline NN classifier with no projection and noadaptation is created That is original input is directly usedfor diagnosis

(b) NN NA NN classifier with no adaptation is createdSpecifically we use a new representation extracted fromoriginal input by PCA without domain adaptation

(c) NN SA NN classifier with projection and domainadaptation using subspace alignment that only reduces themarginal distribution [28]

(a) is a baseline method without projection and domainadaptation techniques which is widely used in the field offault diagnosis (b) is a classical method without domainadaptation which has achieved success inmany fault diagno-sis applications (c) is one of the novel and efficient approachin domain adaptation

41 Experimental Setup and Dataset Preparation The test-bed illustrated in Figure 3 consists of a driving motor a2 hp motor for loading a torque sensorencoder a powermeter accelerometers and electronic control unit [27 29]The test bearings locate in the motor shaft Subjected toelectrosparking inner-race faults (IF) outer-race faults (OF)and ball fault (BF) of different sizes (0007in 0014in and0021in) are introduced into the drive-end bearing of motor[30] The vibration signals are sampled with the help ofaccelerometers installed to the rack with magnetic bases

The working condition of the rotating machinery isusually complex in real-world For purpose of simulatingthe actual application and making the experimental resultsmore persuasive in our experiment dataset collected fromDrive-End Bearing Fault Data and sampled at a frequencyof 12kHz is obtained from different working conditionsDataset includes three kinds of fault degrees (0007in 0014inand 0021in) Each fault degree contains four fault types ofbearings NO IF OF and BF Each fault type of vibration datais collected from four kinds of working conditions ie L0

6 Shock and Vibration

Figure 3 Bearing test rig of Case Western Reserve University DataCenter

= 0 hp1797 rpm L1 = 1 hp1772 rpm L2 = 2 hp1750 rpmand L3 = 3 hp1730 rpm Each sample contains 2049 Fouriercoefficients transformed from the raw vibration signals usingFFT Each domain on dataset contains four fault types andeach fault type contains 200 samples Under our experimentalsetup it is impossible to find the optimal 119896 and 120582 via crossvalidation since labeled training data and unlabeled testdata are sampled from different working conditions Thusempirically searching the parameter space is used to findthe optimal parameter settings and details are described inSection 4 Finally 120582 = 01 and 119896 = 100 are used in our work

In order to verify the benefits of DATF contrast methodsof (a)-(c) are also carried out simultaneously The scenariosettings of all experiments are trained on labeled trainingdata under one single load (training domain) to diagnosethe unlabeled test data under another load (test domain)In all 48 different transferring tests are conducted and thedescription of experimental setup in detail is shown inTable 1

42 Diagnosis Results of the ProposedMethod Thediagnosticresults for fault size being 0007in 0014in and 0021in areshown in Figures 4 5 and 6 The average classificationaccuracies of four methods are described in Figure 7

Each figure is composed of four subfigures and testdomains in every figure are ordered clockwise from (a)L0 L1 L2 and L3 The left of the symbol rdquominus gtrdquo in everysubfigures represents the training domain and the rightrepresents the test domain For each set of bars in Figures 45 and 6 the performances indicate transferring from trainingdomain to test domain which simulates fault diagnosis underdifferent working conditions The load and speed betweendifferent domains have large discrepancies For example inFigure 4(a) the test domain is L0 (the motor load is 0hp andspeed is 1797rpm) the training domain is L1 (the motor loadis 1hp and speed is 1772rpm) L2 (the motor load is 2hp andspeed is 1750rpm) and L3 (the motor load is 3hp and speedis 1730rpm)

From the performances of bearing fault diagnosis inFigures 4 5 and 6 the highest accuracy rates can alwaysbe achieved when the training set of one domain is thesame with the testing set of one domain and this phe-nomenon is reasonable theoretically We can obviously findthat performances of the baseline method and NN NA areall very poor For example in Figures 6(a) 6(b) and 6(c)the accuracies are only about 75 when we transfer L3

Table 1 Description of the experimental setup

Task oftests

Diagnose unlabeled test samples in test domainLabeled training Unlabeled test Fault Fault(training domain) (test domain) type size

1 L0L1L2L3 L0 NOIF 0007inBFOF

2 L0L1L2L3 L1 NOIF 0007inBFOF

3 L0L1L2L3 L2 NOIF 0007inBFOF

4 L0L1L2L3 L3 NOIF 0007inBFOF

5 L0L1L2L3 L0 NOIF 0014inBFOF

6 L0L1L2L3 L1 NOIF 0014inBFOF

7 L0L1L2L3 L2 NOIF 0014inBFOF

8 L0L1L2L3 L3 NOIF 0014inBFOF

9 L0L1L2L3 L0 NOIF 0021inBFOF

10 L0L1L2L3 L1 NOIF 0021inBFOF

11 L0L1L2L3 L2 NOIF 0021inBFOF

12 L0L1L2L3 L3 NOIF 0021inBFOF

to L0 L1 and L2 respectively Especially in Figure 4 alot of accuracies of baseline method and NN NA can notreach 70 when we transfer L1 to L2 These results illustratetraditional methods without domain adaptation can not beapplied to fault diagnosis in variable working conditionsTheperformances of NN SA are better than the first two types ofmethods In Figures 5 and 6 the accuracies of NN NA forvariable working condition bearing fault diagnosis are veryhigh However in Figure 4(c) the performance transferringbetween L1 and L2 is only about 90 and the accuracy isabout 94 when we transfer L3 to L2 Similar phenomenaalso appear in Figure 4(a) These results mentioned aboveindicate that NN NA also can not be applied to complex andvariable working condition bearing fault diagnosis What isexciting is that the proposed method is evidently superior tothe other three compared methods in all cases whatever thetraining domain and test domain are Note that the accuraciesof DATF all can achieve 100 in Figures 4 5 and 6 Evenin Figure 4(a) DATF can still achieve a favorable accuracy(100) while baseline method and NN NA just reach about60 and NN SA only achieve 90 when transferring fromL1 to L2 Compared to the other three methods the averageclassification accuracy (100) of DATF has been markedlyimprovedThese results are all obtained from the benchmark

Shock and Vibration 7

100100

7763 7863

100100100 9975 100 1009575100

BaselineNN NA

NN SADATF

7788 8113

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

100 100

0

20

40

60

80

100Ac

cura

cy (

)

(a)

BaselineNN NA

NN SADATF

955100 100

96389912 100 93

79258237

100 100 100 100 100 100 100

L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL10

20

40

60

80

100

Accu

racy

()

(b)

BaselineNN NA

NN SADATF

8425 7959663

8425

100

9459025

1009425

100 100 100 100

63755737

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL2

985

0

20

40

60

80

100

Accu

racy

()

(c)

BaselineNN NA

NN SADATF

9875

91

100

7588

995

7687 100

100 100 100 100 9963 100 100 100

L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL30

20

40

60

80

100

Accu

racy

()

(d)

Figure 4 The results with fault size being 0007in

datasets of fault diagnosis research under a relatively fairexperiment condition Through the above analysis result wecan conclude that the proposed method is very potentialfor solving bearing fault diagnosis problems under differentworking conditions

To further illustrate the influence of extracted transferablefeatures on the results receiver operating characteristics(ROC) are applied for evaluation [32] An ROC curve isgenerated by plotting the false positive rate and true positiverate as the threshold level is varied In this paper ROCcurves are obtained from different models based on NNclassifier which are built on different extracted featuresand we only report ROC results on transferring test thattransfers L1 to L2 with fault size being 0007in in Figure 8and similar trends on all other tests Before the iterationbegins in Figure 8(a) performances of the model built onextracted features are unsatisfactory After iteration 1 time inFigure 8(b) performances of the model built on extractedtransferable features are improved dramatically and what isexciting is that performances based on extracted transferablefeatures achieve the perfect detection results ultimately

43 Parameter Sensitivity In this section we investigate theinfluence of the parameter 120582 which represents regularizationparameter during transferable feature extraction Theoreti-cally larger values of 120582 can make shrinkage regularization

more important in our work When 120582 rarr 0 and 120582 rarr1 the optimization problem is ill-defined Different 120582 hasdifferent effects on classification accuracy Figure 9 reportsthe results From Figure 9 it is obvious that different 120582 havea great influence on diagnostic results with fault size being0007in and performances with fault size being 0021in andit has little overall effect on results with fault size being0014in What is noticeable is that results are little affected byparameter 120582 when the training domain and test domain arethe same and 120582 isin [00505] can be optimal parameter valueswhich can indicate the proposed method can achieve stableand excellent performance under a wide range of parametervalues

44 Domain Discrepancy Effect of Empirical Analysis Inmany actual fault diagnosis and classification scenarios thedistribution of training data domain is different from thetesting data domain which leads to fault diagnostic accuracy-dropping In fact the data distribution differences betweendomains (training data domain and test data domain) reflectthe differences of the data structures that contain plenty offault messages It is a key point for fault diagnosis to extractfault features from data structures In order to profoundlyunderstand the effect of distribution differences between twodomains and explain why the proposed method works weresort the t-SNE technique [31] to visualize high dimensional

8 Shock and Vibration

1008875

90

100100

8775

9437100100100 100 100 100 100100 100

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(d)

Figure 5 The results with fault size being 0014in

representation of mentioned methods in our experiment in atwo-dimensional map

In all of the above-mentioned cases take the transferringtest that transfers L1 to L2 with fault size being 0007in as anexample in Figure 10

From Figure 10 it is clear that the distribution discrep-ancies of transferable features extracted via DATF betweentraining domain and test domain are much smaller thanthe compared methods and transferable features are muchmore divisible than othersThese results verify that DATF canfigure out a robust feature representation for training domainand test domain and test samples can be discriminatedsignificantly with NN classifier built in training domain byusing extracted transferable features

45 Discussion The proposed method provides a way ofdomain adaptation to extract robust fault features and clas-sify fault types under different working conditions Severalremarks still need to be described(1) This work presents a new point of view thatuses domain adaptation to realize bearing fault diagnosisunder different working conditions Li [30] utilized spec-trum images as features to conduct bearing fault diagnosiswhich applied two-dimensional principal component anal-ysis (2DPCA) into the dimension reduction of the spec-trum images of vibration signals and feature extraction andmost accuracies were very high Unfortunately there are

still several instances having lower accuracies To solve thisproblem we apply the domain adaptation into this field andtransferable features for training domain and test domainare extracted to classify fault types Finally the accuraciesall can reach 100 In this paper our work considers morebearing conditions (fault size being 0007in) Compared withthe method [30] in this situation advantages of our methodare highlighted(2) The vast results indicate that the proposed methodis suitable for effectively classifying mechanical health con-ditions under different working conditions In [9] DeepConvolutional Neural Networks with Wide First-Layer Ker-nel (WDCNN) and AdaBN are applied to diagnose threedatasets which contain 10 kinds of health conditions (BF IFOF with fault size being 0007 in 0014 in and 0021 in)under three load conditions (Load 1 Load 2 and Load 3)respectively which is similar to L1 L2 and L3 in this paperThe average accuracy of this method in [9] is 959 whereasaverage accuracy of DATF is 100 The main reason is thattransferable features extracted based on domain adaptationtake full advantage of structure information of trainingdomain and test domain and the distributions of transferablefeatures extracted from training domain and testing domainare very close after our methods as shown in Figure 10(3) It is noted that our method is unsupervised andfocuses on fault transfer diagnosis based on the same fault di-ameter under different working conditions In [14] a method

Shock and Vibration 9

100

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Figure 6 The results with fault size being 0021in

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80828486889092949698

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Figure 7 The average classification accuracies

based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

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Figure 8 ROC curves of faults detection based on DATF

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(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

Shock and Vibration 3

Labeled training data (Training domain)

PCA

Raw vibration signals Frequency domain

Trained model

Test data

iteration 1

Revise model

iteration T

pseudo-label

NO IF

BF OF

FFT

Data acquisition

Unlabeled test data (Test domain)

PCA

Raw vibration signals Frequency domain

FFT

Data acquisition

Test

Normal condition

Inner race fault

Ball fault

Outer race fault

NO IF

BF OF

Predict

Transferable feature extraction

middot middot middot

Figure 1 The framework of DATF for variable working condition fault diagnosis

119883119879 the MMD calculates the empirical estimate of distancesacross domains in the 119896-dimensional embedding [20 24]

119863119898 (119883119878 119883119879) =100381710038171003817100381710038171003817100381710038171003817100381710038171119899119904119899119904sum119894=1

119860119879119909119894 minus 1119899119905119899119904+119899119905sum119895=119899119904+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

(1)

where 119863119898 is the distance of marginal distributions acrossdomains119860 is the adaptationmatrix and 119899119904 and 119899119905 denote thenumber of source instances and target instances respectively

3 Fault Diagnosis Using Transferable Features

Asmentioned in Section 1 huge distribution difference acrosstraining domain and test domain under different workingconditions directly leads to poor performance of bearingfault diagnosis In order to solve this problem we needto learn the shift between two domains and extract morerobust transferable features for two domains In this sectionwe present our novel bearing fault diagnosis method undervariable working conditions The framework of our method

is illustrated in Figure 1 As shown in Figure 1 fault diagnosismodel built via labeled training data is iterated revisionaccording to pseudo-label and the final diagnostic results areobtained through the above revised model Details of eachpart are elaborated in the following subsections

31 Feature Space Generation Raw time series vibration sig-nals are readily available and abound in bearing informationOwning to the rotating nature of raw vibration signals froma defective bearing the periodic impulse would appear inobtained signals once a fault occursThus these fault impactscan be detected generally in frequency domain

In our work we directly catch FFT amplitudes fromthe raw time series vibration signals as samples where allsamples have the same dimension and these samples aregenerated under different motor speeds and load conditionsas described in Figure 2

They are divided into two parts labeled training data(119863119905119903) and unlabeled test data(119863119905119890) Then we use principalcomponent analysis (PCA) to generate feature space Themain steps of feature space generation are as follows

4 Shock and Vibration

Start

N = 12000NFFT = 2^nextpow2(N)f = fs2lowastlinspace(01NFFT2+1) fft_amplitude = abs(fft(xNFFT))N

Length of x(n) Next power of 2 from N Frequency resolution Fast Fourier transform of x(n)

The single-sided FFT spectrum amplitude is acquired through 2lowastfft_amplitude(1NFFT2+1) in Matlab

End

Vibration signal x(n) sampled with fs Hz

Figure 2 Flowchart of FFT spectrum amplitudes creation in MATLAB

Step 1 Catch FFT amplitudes from raw time series vibrationsignals collected under different working conditions as sam-ples119863119889119886119905119886Step 2 Take one of the conditions with different fault typesfrom 119863119889119886119905119886 as training samples 119883119905119903 isin 119877119899119905119903times119889 with label 119884119905119903 isin119877119899119905119903times1 and take another of the conditions with different faulttypes from119863119889119886119905119886 as unlabeled test samples119883119905119890 isin 119877119899119905119890times119889Step 3 Denote 119883119863 = 119883119905119903 119883119905119890 isin 119877119889times(119899119905119903+119899119905119890) and 119867 =119868 minus (1(119899119905119903 + 119899119905119890))119897119897119879 where 119868 denotes the identity matrix and119897 is considered as the ones vectors Then the 119896 dimensionalrepresentation is found by solving the following optimizationproblemmax119860119879119860=119868119905119903(119860119879119883119863119867119883119879119863119860) and then feature spaceis created by 119881 = 11986011987911988311986332 Transferable Feature Extraction and Diagnosis In orderto reduce the marginal distribution difference and extractrobust feature for two domains we resort MMD as thedistance measures between 119909119894119905119903 and 119909119895119905119890 to compare differentdistributions 10038171003817100381710038171003817100381710038171003817100381710038171003817

1119899119905119903119899119905119903sum119894=1

119860119879119909119894 minus 1119899119905119890119899119905119903+119899119905119890sum119895=119899119905119903+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

= 119905119903 (119860119879119883119863119872119898119883119879119863119860)(2)

where 119872119898 = [ (119872119898)119905119903119905119903 (119872119898)119905119903119905119890(119872119898)119905119890119905119903 (119872119898)119905119890119905119890] is the MMD matrix and is

computed as follows [24 26]

119872119898 =

1119899119905119903119899119905119903 119909119894 119909119895 isin 1198831199051199031119899119905119890119899119905119890 119909119894 119909119895 isin 119883119905119890minus1119899119905119903119899119905119890 119900119905ℎ119890119903119908119894119904119890

(3)

Themarginal distributions between training domain and testdomain are brought closer under the new representation119881 =119860119879119883119863 by minimizing (2)

In theory training and test data under different workingconditions collected from sensors should be of the samemarginal and conditional distributions while the reality isvery different For improving the performance of bearingfault diagnosis under different work conditions in our workthe differences of conditional distribution between domainsare also reduced bymining the class-conditional distributionFormally the class-conditional distributions can bemeasuredaccording to modified MMD

100381710038171003817100381710038171003817100381710038171003817100381710038171119899119905119903119899119905119903sum119894=1

119860T119909119894 minus 1119899119905119890119899119905119903+119899119905119890sum119895=119899119905119903+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

= 119905119903 (119860119879119883119863119872119888119883119879119863119860) (4)

where119872119888 = [ (119872119888)119905119903119905119903 (119872119888)119905119903119905119890(119872119888)119905119890119905119903 (119872119888)119905119890119905119890] is MMD coefficient matrix that

includes the class label 119888 and it can be calculated accordingto [24 26]

Shock and Vibration 5

119872119888 =

1119899119888119905119903119899119888119905119903 119909119894 119909119895 isin 1198831199051199031119899119888119905119890119899119888119905119890 119909119894 119909119895 isin 119883119905119890minus1119899119905119903119899119905119890

119909119894 isin 119883119888119905119903 119909119895 isin 119883119888119905119890119909119895 isin 119883119888119905119903 119909119894 isin 1198831198881199051198900 119900119905ℎ119890119903119908119894119904119890

(5)

The conditional distributions between training and testdomains are brought closer under the new representation119881 = 119860119879119883119863 by minimizing (4)

In order to obtain effective and robust transferable featurerepresentation and improve the quality of fault diagnosis ourwork aims to reduce the impact of discrepancies from boththe marginal and conditional distributions between trainingand test domains by resorting the pseudo labels of test data[26] on diagnosis and these pseudo labels can be obtainedfrom a base classifier (NN classifier) built on the labeledtraining data to predict the fully unlabeled test data Thusthe final optimization problem (6) in this paper comprised(2) and (4)

min119860119879119883119863119867119883

119879119863119860=119868

(1 minus 120582) 119862sum119888=0

119905119903 (119860119879119883119863119872119888119883119879119863119860) + 120582 1198602119865 (6)

where sdot 119865 is the Frobenius norm that guarantees the opti-mization problem to be well defined and 120582 is the regulariza-tion parameter [24] that trades off the impact of regulariza-tion term on the transformation matrix A The goal is to findthe latent feature space created by a transformation matrix119860 isin 119877119889times119896 where the discrepancies of both the marginal andconditional distributions between domains are significantlyreduced The Lagrange function for (7) is constructed whereΛ = diag(Λ 1 Λ 119896) isin 119877119896times119896 is the Lagrange multiplier

119871 = (1 minus 120582) 119905119903(119860119879(119883119863 119862sum119888=0

119872119888119883119879119863)119860) + 120582119905119903 (119860119879119860)+ 119905119903 ((119868 minus 119860119879119883119863119867119883119879119863119860)Λ)

(7)

According to 119889119871119889119860 = 0 the optimal solution of (6) can beacquired through the generalized eigen decomposition

((1 minus 120582)119883119863 119862sum119888=0

119872119888119883119879119863 + 120582119868)119860 = 119883119863119867119883119879119863119860Λ (8)

Finally the adaptation matrix A is obtained from solving (8)for 119896 smallest eigenvectors The procedure of fault diagnosisusing DAFT can be depicted as follows in detail

Step 1 For given training data 119883119905119903 isin 119877119899119905119903times119889 with label 119884119905119903 isin119877119899119905119903times1 andunlabeled test data119883119905119890 isin 119877119899119905119890times119889 in the feature spaceStep 2 Construct MMD matrix 119872119898 by (2) Adaptationmatrix 119860 generated by the 119896 smallest eigenvectors can beacquired by solving (8) through Lagrange multiplier Thenthe robust representation for two domains is obtained 119881 =119860119879119883119863

Step 3 Train the NN classifier on projected training data119860119879119883119905119903 119884119905119903 and then obtain pseudo test data labels 119884119905119890 thatdenote the conditional probability 119876(119884119905119890|119883119905119890) by using thetrained NN classifier

Step 4 Update MMD matrix 119872119888119862119888=1 by (5) according to119875(119884119905119903|119883119905119903) = 119876(119884119905119890|119883119905119890) and then obtain the updated adapta-tionmatrix119860 by solving (8) through LagrangemultiplierTheupdated robust representation for two domains is obtained119881 = 119860119879119883119863 and then jump to Step 3 until the end of theiteration

Step 5 Finally the test data labels 119884119905119890 are predicted accuratelyby the adaptive NN classifier

4 Experimental Evaluations

In order to demonstrate the effectiveness of the proposedfault diagnosis method the vast bearing vibration signalscollected from a bearing test rig are used Dataset is acquiredfrom the bearing data center of Case Western Reserve Uni-versity (CWRU) [27] DATF is compared with the baselineapproaches and several successful methods

(a) Baseline NN classifier with no projection and noadaptation is created That is original input is directly usedfor diagnosis

(b) NN NA NN classifier with no adaptation is createdSpecifically we use a new representation extracted fromoriginal input by PCA without domain adaptation

(c) NN SA NN classifier with projection and domainadaptation using subspace alignment that only reduces themarginal distribution [28]

(a) is a baseline method without projection and domainadaptation techniques which is widely used in the field offault diagnosis (b) is a classical method without domainadaptation which has achieved success inmany fault diagno-sis applications (c) is one of the novel and efficient approachin domain adaptation

41 Experimental Setup and Dataset Preparation The test-bed illustrated in Figure 3 consists of a driving motor a2 hp motor for loading a torque sensorencoder a powermeter accelerometers and electronic control unit [27 29]The test bearings locate in the motor shaft Subjected toelectrosparking inner-race faults (IF) outer-race faults (OF)and ball fault (BF) of different sizes (0007in 0014in and0021in) are introduced into the drive-end bearing of motor[30] The vibration signals are sampled with the help ofaccelerometers installed to the rack with magnetic bases

The working condition of the rotating machinery isusually complex in real-world For purpose of simulatingthe actual application and making the experimental resultsmore persuasive in our experiment dataset collected fromDrive-End Bearing Fault Data and sampled at a frequencyof 12kHz is obtained from different working conditionsDataset includes three kinds of fault degrees (0007in 0014inand 0021in) Each fault degree contains four fault types ofbearings NO IF OF and BF Each fault type of vibration datais collected from four kinds of working conditions ie L0

6 Shock and Vibration

Figure 3 Bearing test rig of Case Western Reserve University DataCenter

= 0 hp1797 rpm L1 = 1 hp1772 rpm L2 = 2 hp1750 rpmand L3 = 3 hp1730 rpm Each sample contains 2049 Fouriercoefficients transformed from the raw vibration signals usingFFT Each domain on dataset contains four fault types andeach fault type contains 200 samples Under our experimentalsetup it is impossible to find the optimal 119896 and 120582 via crossvalidation since labeled training data and unlabeled testdata are sampled from different working conditions Thusempirically searching the parameter space is used to findthe optimal parameter settings and details are described inSection 4 Finally 120582 = 01 and 119896 = 100 are used in our work

In order to verify the benefits of DATF contrast methodsof (a)-(c) are also carried out simultaneously The scenariosettings of all experiments are trained on labeled trainingdata under one single load (training domain) to diagnosethe unlabeled test data under another load (test domain)In all 48 different transferring tests are conducted and thedescription of experimental setup in detail is shown inTable 1

42 Diagnosis Results of the ProposedMethod Thediagnosticresults for fault size being 0007in 0014in and 0021in areshown in Figures 4 5 and 6 The average classificationaccuracies of four methods are described in Figure 7

Each figure is composed of four subfigures and testdomains in every figure are ordered clockwise from (a)L0 L1 L2 and L3 The left of the symbol rdquominus gtrdquo in everysubfigures represents the training domain and the rightrepresents the test domain For each set of bars in Figures 45 and 6 the performances indicate transferring from trainingdomain to test domain which simulates fault diagnosis underdifferent working conditions The load and speed betweendifferent domains have large discrepancies For example inFigure 4(a) the test domain is L0 (the motor load is 0hp andspeed is 1797rpm) the training domain is L1 (the motor loadis 1hp and speed is 1772rpm) L2 (the motor load is 2hp andspeed is 1750rpm) and L3 (the motor load is 3hp and speedis 1730rpm)

From the performances of bearing fault diagnosis inFigures 4 5 and 6 the highest accuracy rates can alwaysbe achieved when the training set of one domain is thesame with the testing set of one domain and this phe-nomenon is reasonable theoretically We can obviously findthat performances of the baseline method and NN NA areall very poor For example in Figures 6(a) 6(b) and 6(c)the accuracies are only about 75 when we transfer L3

Table 1 Description of the experimental setup

Task oftests

Diagnose unlabeled test samples in test domainLabeled training Unlabeled test Fault Fault(training domain) (test domain) type size

1 L0L1L2L3 L0 NOIF 0007inBFOF

2 L0L1L2L3 L1 NOIF 0007inBFOF

3 L0L1L2L3 L2 NOIF 0007inBFOF

4 L0L1L2L3 L3 NOIF 0007inBFOF

5 L0L1L2L3 L0 NOIF 0014inBFOF

6 L0L1L2L3 L1 NOIF 0014inBFOF

7 L0L1L2L3 L2 NOIF 0014inBFOF

8 L0L1L2L3 L3 NOIF 0014inBFOF

9 L0L1L2L3 L0 NOIF 0021inBFOF

10 L0L1L2L3 L1 NOIF 0021inBFOF

11 L0L1L2L3 L2 NOIF 0021inBFOF

12 L0L1L2L3 L3 NOIF 0021inBFOF

to L0 L1 and L2 respectively Especially in Figure 4 alot of accuracies of baseline method and NN NA can notreach 70 when we transfer L1 to L2 These results illustratetraditional methods without domain adaptation can not beapplied to fault diagnosis in variable working conditionsTheperformances of NN SA are better than the first two types ofmethods In Figures 5 and 6 the accuracies of NN NA forvariable working condition bearing fault diagnosis are veryhigh However in Figure 4(c) the performance transferringbetween L1 and L2 is only about 90 and the accuracy isabout 94 when we transfer L3 to L2 Similar phenomenaalso appear in Figure 4(a) These results mentioned aboveindicate that NN NA also can not be applied to complex andvariable working condition bearing fault diagnosis What isexciting is that the proposed method is evidently superior tothe other three compared methods in all cases whatever thetraining domain and test domain are Note that the accuraciesof DATF all can achieve 100 in Figures 4 5 and 6 Evenin Figure 4(a) DATF can still achieve a favorable accuracy(100) while baseline method and NN NA just reach about60 and NN SA only achieve 90 when transferring fromL1 to L2 Compared to the other three methods the averageclassification accuracy (100) of DATF has been markedlyimprovedThese results are all obtained from the benchmark

Shock and Vibration 7

100100

7763 7863

100100100 9975 100 1009575100

BaselineNN NA

NN SADATF

7788 8113

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

100 100

0

20

40

60

80

100Ac

cura

cy (

)

(a)

BaselineNN NA

NN SADATF

955100 100

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79258237

100 100 100 100 100 100 100

L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL10

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()

(b)

BaselineNN NA

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8425 7959663

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63755737

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL2

985

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BaselineNN NA

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L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL30

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Accu

racy

()

(d)

Figure 4 The results with fault size being 0007in

datasets of fault diagnosis research under a relatively fairexperiment condition Through the above analysis result wecan conclude that the proposed method is very potentialfor solving bearing fault diagnosis problems under differentworking conditions

To further illustrate the influence of extracted transferablefeatures on the results receiver operating characteristics(ROC) are applied for evaluation [32] An ROC curve isgenerated by plotting the false positive rate and true positiverate as the threshold level is varied In this paper ROCcurves are obtained from different models based on NNclassifier which are built on different extracted featuresand we only report ROC results on transferring test thattransfers L1 to L2 with fault size being 0007in in Figure 8and similar trends on all other tests Before the iterationbegins in Figure 8(a) performances of the model built onextracted features are unsatisfactory After iteration 1 time inFigure 8(b) performances of the model built on extractedtransferable features are improved dramatically and what isexciting is that performances based on extracted transferablefeatures achieve the perfect detection results ultimately

43 Parameter Sensitivity In this section we investigate theinfluence of the parameter 120582 which represents regularizationparameter during transferable feature extraction Theoreti-cally larger values of 120582 can make shrinkage regularization

more important in our work When 120582 rarr 0 and 120582 rarr1 the optimization problem is ill-defined Different 120582 hasdifferent effects on classification accuracy Figure 9 reportsthe results From Figure 9 it is obvious that different 120582 havea great influence on diagnostic results with fault size being0007in and performances with fault size being 0021in andit has little overall effect on results with fault size being0014in What is noticeable is that results are little affected byparameter 120582 when the training domain and test domain arethe same and 120582 isin [00505] can be optimal parameter valueswhich can indicate the proposed method can achieve stableand excellent performance under a wide range of parametervalues

44 Domain Discrepancy Effect of Empirical Analysis Inmany actual fault diagnosis and classification scenarios thedistribution of training data domain is different from thetesting data domain which leads to fault diagnostic accuracy-dropping In fact the data distribution differences betweendomains (training data domain and test data domain) reflectthe differences of the data structures that contain plenty offault messages It is a key point for fault diagnosis to extractfault features from data structures In order to profoundlyunderstand the effect of distribution differences between twodomains and explain why the proposed method works weresort the t-SNE technique [31] to visualize high dimensional

8 Shock and Vibration

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Figure 5 The results with fault size being 0014in

representation of mentioned methods in our experiment in atwo-dimensional map

In all of the above-mentioned cases take the transferringtest that transfers L1 to L2 with fault size being 0007in as anexample in Figure 10

From Figure 10 it is clear that the distribution discrep-ancies of transferable features extracted via DATF betweentraining domain and test domain are much smaller thanthe compared methods and transferable features are muchmore divisible than othersThese results verify that DATF canfigure out a robust feature representation for training domainand test domain and test samples can be discriminatedsignificantly with NN classifier built in training domain byusing extracted transferable features

45 Discussion The proposed method provides a way ofdomain adaptation to extract robust fault features and clas-sify fault types under different working conditions Severalremarks still need to be described(1) This work presents a new point of view thatuses domain adaptation to realize bearing fault diagnosisunder different working conditions Li [30] utilized spec-trum images as features to conduct bearing fault diagnosiswhich applied two-dimensional principal component anal-ysis (2DPCA) into the dimension reduction of the spec-trum images of vibration signals and feature extraction andmost accuracies were very high Unfortunately there are

still several instances having lower accuracies To solve thisproblem we apply the domain adaptation into this field andtransferable features for training domain and test domainare extracted to classify fault types Finally the accuraciesall can reach 100 In this paper our work considers morebearing conditions (fault size being 0007in) Compared withthe method [30] in this situation advantages of our methodare highlighted(2) The vast results indicate that the proposed methodis suitable for effectively classifying mechanical health con-ditions under different working conditions In [9] DeepConvolutional Neural Networks with Wide First-Layer Ker-nel (WDCNN) and AdaBN are applied to diagnose threedatasets which contain 10 kinds of health conditions (BF IFOF with fault size being 0007 in 0014 in and 0021 in)under three load conditions (Load 1 Load 2 and Load 3)respectively which is similar to L1 L2 and L3 in this paperThe average accuracy of this method in [9] is 959 whereasaverage accuracy of DATF is 100 The main reason is thattransferable features extracted based on domain adaptationtake full advantage of structure information of trainingdomain and test domain and the distributions of transferablefeatures extracted from training domain and testing domainare very close after our methods as shown in Figure 10(3) It is noted that our method is unsupervised andfocuses on fault transfer diagnosis based on the same fault di-ameter under different working conditions In [14] a method

Shock and Vibration 9

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Figure 6 The results with fault size being 0021in

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based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

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Figure 8 ROC curves of faults detection based on DATF

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Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

4 Shock and Vibration

Start

N = 12000NFFT = 2^nextpow2(N)f = fs2lowastlinspace(01NFFT2+1) fft_amplitude = abs(fft(xNFFT))N

Length of x(n) Next power of 2 from N Frequency resolution Fast Fourier transform of x(n)

The single-sided FFT spectrum amplitude is acquired through 2lowastfft_amplitude(1NFFT2+1) in Matlab

End

Vibration signal x(n) sampled with fs Hz

Figure 2 Flowchart of FFT spectrum amplitudes creation in MATLAB

Step 1 Catch FFT amplitudes from raw time series vibrationsignals collected under different working conditions as sam-ples119863119889119886119905119886Step 2 Take one of the conditions with different fault typesfrom 119863119889119886119905119886 as training samples 119883119905119903 isin 119877119899119905119903times119889 with label 119884119905119903 isin119877119899119905119903times1 and take another of the conditions with different faulttypes from119863119889119886119905119886 as unlabeled test samples119883119905119890 isin 119877119899119905119890times119889Step 3 Denote 119883119863 = 119883119905119903 119883119905119890 isin 119877119889times(119899119905119903+119899119905119890) and 119867 =119868 minus (1(119899119905119903 + 119899119905119890))119897119897119879 where 119868 denotes the identity matrix and119897 is considered as the ones vectors Then the 119896 dimensionalrepresentation is found by solving the following optimizationproblemmax119860119879119860=119868119905119903(119860119879119883119863119867119883119879119863119860) and then feature spaceis created by 119881 = 11986011987911988311986332 Transferable Feature Extraction and Diagnosis In orderto reduce the marginal distribution difference and extractrobust feature for two domains we resort MMD as thedistance measures between 119909119894119905119903 and 119909119895119905119890 to compare differentdistributions 10038171003817100381710038171003817100381710038171003817100381710038171003817

1119899119905119903119899119905119903sum119894=1

119860119879119909119894 minus 1119899119905119890119899119905119903+119899119905119890sum119895=119899119905119903+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

= 119905119903 (119860119879119883119863119872119898119883119879119863119860)(2)

where 119872119898 = [ (119872119898)119905119903119905119903 (119872119898)119905119903119905119890(119872119898)119905119890119905119903 (119872119898)119905119890119905119890] is the MMD matrix and is

computed as follows [24 26]

119872119898 =

1119899119905119903119899119905119903 119909119894 119909119895 isin 1198831199051199031119899119905119890119899119905119890 119909119894 119909119895 isin 119883119905119890minus1119899119905119903119899119905119890 119900119905ℎ119890119903119908119894119904119890

(3)

Themarginal distributions between training domain and testdomain are brought closer under the new representation119881 =119860119879119883119863 by minimizing (2)

In theory training and test data under different workingconditions collected from sensors should be of the samemarginal and conditional distributions while the reality isvery different For improving the performance of bearingfault diagnosis under different work conditions in our workthe differences of conditional distribution between domainsare also reduced bymining the class-conditional distributionFormally the class-conditional distributions can bemeasuredaccording to modified MMD

100381710038171003817100381710038171003817100381710038171003817100381710038171119899119905119903119899119905119903sum119894=1

119860T119909119894 minus 1119899119905119890119899119905119903+119899119905119890sum119895=119899119905119903+1

119860119879119909119895100381710038171003817100381710038171003817100381710038171003817100381710038172

= 119905119903 (119860119879119883119863119872119888119883119879119863119860) (4)

where119872119888 = [ (119872119888)119905119903119905119903 (119872119888)119905119903119905119890(119872119888)119905119890119905119903 (119872119888)119905119890119905119890] is MMD coefficient matrix that

includes the class label 119888 and it can be calculated accordingto [24 26]

Shock and Vibration 5

119872119888 =

1119899119888119905119903119899119888119905119903 119909119894 119909119895 isin 1198831199051199031119899119888119905119890119899119888119905119890 119909119894 119909119895 isin 119883119905119890minus1119899119905119903119899119905119890

119909119894 isin 119883119888119905119903 119909119895 isin 119883119888119905119890119909119895 isin 119883119888119905119903 119909119894 isin 1198831198881199051198900 119900119905ℎ119890119903119908119894119904119890

(5)

The conditional distributions between training and testdomains are brought closer under the new representation119881 = 119860119879119883119863 by minimizing (4)

In order to obtain effective and robust transferable featurerepresentation and improve the quality of fault diagnosis ourwork aims to reduce the impact of discrepancies from boththe marginal and conditional distributions between trainingand test domains by resorting the pseudo labels of test data[26] on diagnosis and these pseudo labels can be obtainedfrom a base classifier (NN classifier) built on the labeledtraining data to predict the fully unlabeled test data Thusthe final optimization problem (6) in this paper comprised(2) and (4)

min119860119879119883119863119867119883

119879119863119860=119868

(1 minus 120582) 119862sum119888=0

119905119903 (119860119879119883119863119872119888119883119879119863119860) + 120582 1198602119865 (6)

where sdot 119865 is the Frobenius norm that guarantees the opti-mization problem to be well defined and 120582 is the regulariza-tion parameter [24] that trades off the impact of regulariza-tion term on the transformation matrix A The goal is to findthe latent feature space created by a transformation matrix119860 isin 119877119889times119896 where the discrepancies of both the marginal andconditional distributions between domains are significantlyreduced The Lagrange function for (7) is constructed whereΛ = diag(Λ 1 Λ 119896) isin 119877119896times119896 is the Lagrange multiplier

119871 = (1 minus 120582) 119905119903(119860119879(119883119863 119862sum119888=0

119872119888119883119879119863)119860) + 120582119905119903 (119860119879119860)+ 119905119903 ((119868 minus 119860119879119883119863119867119883119879119863119860)Λ)

(7)

According to 119889119871119889119860 = 0 the optimal solution of (6) can beacquired through the generalized eigen decomposition

((1 minus 120582)119883119863 119862sum119888=0

119872119888119883119879119863 + 120582119868)119860 = 119883119863119867119883119879119863119860Λ (8)

Finally the adaptation matrix A is obtained from solving (8)for 119896 smallest eigenvectors The procedure of fault diagnosisusing DAFT can be depicted as follows in detail

Step 1 For given training data 119883119905119903 isin 119877119899119905119903times119889 with label 119884119905119903 isin119877119899119905119903times1 andunlabeled test data119883119905119890 isin 119877119899119905119890times119889 in the feature spaceStep 2 Construct MMD matrix 119872119898 by (2) Adaptationmatrix 119860 generated by the 119896 smallest eigenvectors can beacquired by solving (8) through Lagrange multiplier Thenthe robust representation for two domains is obtained 119881 =119860119879119883119863

Step 3 Train the NN classifier on projected training data119860119879119883119905119903 119884119905119903 and then obtain pseudo test data labels 119884119905119890 thatdenote the conditional probability 119876(119884119905119890|119883119905119890) by using thetrained NN classifier

Step 4 Update MMD matrix 119872119888119862119888=1 by (5) according to119875(119884119905119903|119883119905119903) = 119876(119884119905119890|119883119905119890) and then obtain the updated adapta-tionmatrix119860 by solving (8) through LagrangemultiplierTheupdated robust representation for two domains is obtained119881 = 119860119879119883119863 and then jump to Step 3 until the end of theiteration

Step 5 Finally the test data labels 119884119905119890 are predicted accuratelyby the adaptive NN classifier

4 Experimental Evaluations

In order to demonstrate the effectiveness of the proposedfault diagnosis method the vast bearing vibration signalscollected from a bearing test rig are used Dataset is acquiredfrom the bearing data center of Case Western Reserve Uni-versity (CWRU) [27] DATF is compared with the baselineapproaches and several successful methods

(a) Baseline NN classifier with no projection and noadaptation is created That is original input is directly usedfor diagnosis

(b) NN NA NN classifier with no adaptation is createdSpecifically we use a new representation extracted fromoriginal input by PCA without domain adaptation

(c) NN SA NN classifier with projection and domainadaptation using subspace alignment that only reduces themarginal distribution [28]

(a) is a baseline method without projection and domainadaptation techniques which is widely used in the field offault diagnosis (b) is a classical method without domainadaptation which has achieved success inmany fault diagno-sis applications (c) is one of the novel and efficient approachin domain adaptation

41 Experimental Setup and Dataset Preparation The test-bed illustrated in Figure 3 consists of a driving motor a2 hp motor for loading a torque sensorencoder a powermeter accelerometers and electronic control unit [27 29]The test bearings locate in the motor shaft Subjected toelectrosparking inner-race faults (IF) outer-race faults (OF)and ball fault (BF) of different sizes (0007in 0014in and0021in) are introduced into the drive-end bearing of motor[30] The vibration signals are sampled with the help ofaccelerometers installed to the rack with magnetic bases

The working condition of the rotating machinery isusually complex in real-world For purpose of simulatingthe actual application and making the experimental resultsmore persuasive in our experiment dataset collected fromDrive-End Bearing Fault Data and sampled at a frequencyof 12kHz is obtained from different working conditionsDataset includes three kinds of fault degrees (0007in 0014inand 0021in) Each fault degree contains four fault types ofbearings NO IF OF and BF Each fault type of vibration datais collected from four kinds of working conditions ie L0

6 Shock and Vibration

Figure 3 Bearing test rig of Case Western Reserve University DataCenter

= 0 hp1797 rpm L1 = 1 hp1772 rpm L2 = 2 hp1750 rpmand L3 = 3 hp1730 rpm Each sample contains 2049 Fouriercoefficients transformed from the raw vibration signals usingFFT Each domain on dataset contains four fault types andeach fault type contains 200 samples Under our experimentalsetup it is impossible to find the optimal 119896 and 120582 via crossvalidation since labeled training data and unlabeled testdata are sampled from different working conditions Thusempirically searching the parameter space is used to findthe optimal parameter settings and details are described inSection 4 Finally 120582 = 01 and 119896 = 100 are used in our work

In order to verify the benefits of DATF contrast methodsof (a)-(c) are also carried out simultaneously The scenariosettings of all experiments are trained on labeled trainingdata under one single load (training domain) to diagnosethe unlabeled test data under another load (test domain)In all 48 different transferring tests are conducted and thedescription of experimental setup in detail is shown inTable 1

42 Diagnosis Results of the ProposedMethod Thediagnosticresults for fault size being 0007in 0014in and 0021in areshown in Figures 4 5 and 6 The average classificationaccuracies of four methods are described in Figure 7

Each figure is composed of four subfigures and testdomains in every figure are ordered clockwise from (a)L0 L1 L2 and L3 The left of the symbol rdquominus gtrdquo in everysubfigures represents the training domain and the rightrepresents the test domain For each set of bars in Figures 45 and 6 the performances indicate transferring from trainingdomain to test domain which simulates fault diagnosis underdifferent working conditions The load and speed betweendifferent domains have large discrepancies For example inFigure 4(a) the test domain is L0 (the motor load is 0hp andspeed is 1797rpm) the training domain is L1 (the motor loadis 1hp and speed is 1772rpm) L2 (the motor load is 2hp andspeed is 1750rpm) and L3 (the motor load is 3hp and speedis 1730rpm)

From the performances of bearing fault diagnosis inFigures 4 5 and 6 the highest accuracy rates can alwaysbe achieved when the training set of one domain is thesame with the testing set of one domain and this phe-nomenon is reasonable theoretically We can obviously findthat performances of the baseline method and NN NA areall very poor For example in Figures 6(a) 6(b) and 6(c)the accuracies are only about 75 when we transfer L3

Table 1 Description of the experimental setup

Task oftests

Diagnose unlabeled test samples in test domainLabeled training Unlabeled test Fault Fault(training domain) (test domain) type size

1 L0L1L2L3 L0 NOIF 0007inBFOF

2 L0L1L2L3 L1 NOIF 0007inBFOF

3 L0L1L2L3 L2 NOIF 0007inBFOF

4 L0L1L2L3 L3 NOIF 0007inBFOF

5 L0L1L2L3 L0 NOIF 0014inBFOF

6 L0L1L2L3 L1 NOIF 0014inBFOF

7 L0L1L2L3 L2 NOIF 0014inBFOF

8 L0L1L2L3 L3 NOIF 0014inBFOF

9 L0L1L2L3 L0 NOIF 0021inBFOF

10 L0L1L2L3 L1 NOIF 0021inBFOF

11 L0L1L2L3 L2 NOIF 0021inBFOF

12 L0L1L2L3 L3 NOIF 0021inBFOF

to L0 L1 and L2 respectively Especially in Figure 4 alot of accuracies of baseline method and NN NA can notreach 70 when we transfer L1 to L2 These results illustratetraditional methods without domain adaptation can not beapplied to fault diagnosis in variable working conditionsTheperformances of NN SA are better than the first two types ofmethods In Figures 5 and 6 the accuracies of NN NA forvariable working condition bearing fault diagnosis are veryhigh However in Figure 4(c) the performance transferringbetween L1 and L2 is only about 90 and the accuracy isabout 94 when we transfer L3 to L2 Similar phenomenaalso appear in Figure 4(a) These results mentioned aboveindicate that NN NA also can not be applied to complex andvariable working condition bearing fault diagnosis What isexciting is that the proposed method is evidently superior tothe other three compared methods in all cases whatever thetraining domain and test domain are Note that the accuraciesof DATF all can achieve 100 in Figures 4 5 and 6 Evenin Figure 4(a) DATF can still achieve a favorable accuracy(100) while baseline method and NN NA just reach about60 and NN SA only achieve 90 when transferring fromL1 to L2 Compared to the other three methods the averageclassification accuracy (100) of DATF has been markedlyimprovedThese results are all obtained from the benchmark

Shock and Vibration 7

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Figure 4 The results with fault size being 0007in

datasets of fault diagnosis research under a relatively fairexperiment condition Through the above analysis result wecan conclude that the proposed method is very potentialfor solving bearing fault diagnosis problems under differentworking conditions

To further illustrate the influence of extracted transferablefeatures on the results receiver operating characteristics(ROC) are applied for evaluation [32] An ROC curve isgenerated by plotting the false positive rate and true positiverate as the threshold level is varied In this paper ROCcurves are obtained from different models based on NNclassifier which are built on different extracted featuresand we only report ROC results on transferring test thattransfers L1 to L2 with fault size being 0007in in Figure 8and similar trends on all other tests Before the iterationbegins in Figure 8(a) performances of the model built onextracted features are unsatisfactory After iteration 1 time inFigure 8(b) performances of the model built on extractedtransferable features are improved dramatically and what isexciting is that performances based on extracted transferablefeatures achieve the perfect detection results ultimately

43 Parameter Sensitivity In this section we investigate theinfluence of the parameter 120582 which represents regularizationparameter during transferable feature extraction Theoreti-cally larger values of 120582 can make shrinkage regularization

more important in our work When 120582 rarr 0 and 120582 rarr1 the optimization problem is ill-defined Different 120582 hasdifferent effects on classification accuracy Figure 9 reportsthe results From Figure 9 it is obvious that different 120582 havea great influence on diagnostic results with fault size being0007in and performances with fault size being 0021in andit has little overall effect on results with fault size being0014in What is noticeable is that results are little affected byparameter 120582 when the training domain and test domain arethe same and 120582 isin [00505] can be optimal parameter valueswhich can indicate the proposed method can achieve stableand excellent performance under a wide range of parametervalues

44 Domain Discrepancy Effect of Empirical Analysis Inmany actual fault diagnosis and classification scenarios thedistribution of training data domain is different from thetesting data domain which leads to fault diagnostic accuracy-dropping In fact the data distribution differences betweendomains (training data domain and test data domain) reflectthe differences of the data structures that contain plenty offault messages It is a key point for fault diagnosis to extractfault features from data structures In order to profoundlyunderstand the effect of distribution differences between twodomains and explain why the proposed method works weresort the t-SNE technique [31] to visualize high dimensional

8 Shock and Vibration

1008875

90

100100

8775

9437100100100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL00

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cy (

)

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(a)

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racy

()

(d)

Figure 5 The results with fault size being 0014in

representation of mentioned methods in our experiment in atwo-dimensional map

In all of the above-mentioned cases take the transferringtest that transfers L1 to L2 with fault size being 0007in as anexample in Figure 10

From Figure 10 it is clear that the distribution discrep-ancies of transferable features extracted via DATF betweentraining domain and test domain are much smaller thanthe compared methods and transferable features are muchmore divisible than othersThese results verify that DATF canfigure out a robust feature representation for training domainand test domain and test samples can be discriminatedsignificantly with NN classifier built in training domain byusing extracted transferable features

45 Discussion The proposed method provides a way ofdomain adaptation to extract robust fault features and clas-sify fault types under different working conditions Severalremarks still need to be described(1) This work presents a new point of view thatuses domain adaptation to realize bearing fault diagnosisunder different working conditions Li [30] utilized spec-trum images as features to conduct bearing fault diagnosiswhich applied two-dimensional principal component anal-ysis (2DPCA) into the dimension reduction of the spec-trum images of vibration signals and feature extraction andmost accuracies were very high Unfortunately there are

still several instances having lower accuracies To solve thisproblem we apply the domain adaptation into this field andtransferable features for training domain and test domainare extracted to classify fault types Finally the accuraciesall can reach 100 In this paper our work considers morebearing conditions (fault size being 0007in) Compared withthe method [30] in this situation advantages of our methodare highlighted(2) The vast results indicate that the proposed methodis suitable for effectively classifying mechanical health con-ditions under different working conditions In [9] DeepConvolutional Neural Networks with Wide First-Layer Ker-nel (WDCNN) and AdaBN are applied to diagnose threedatasets which contain 10 kinds of health conditions (BF IFOF with fault size being 0007 in 0014 in and 0021 in)under three load conditions (Load 1 Load 2 and Load 3)respectively which is similar to L1 L2 and L3 in this paperThe average accuracy of this method in [9] is 959 whereasaverage accuracy of DATF is 100 The main reason is thattransferable features extracted based on domain adaptationtake full advantage of structure information of trainingdomain and test domain and the distributions of transferablefeatures extracted from training domain and testing domainare very close after our methods as shown in Figure 10(3) It is noted that our method is unsupervised andfocuses on fault transfer diagnosis based on the same fault di-ameter under different working conditions In [14] a method

Shock and Vibration 9

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Figure 6 The results with fault size being 0021in

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Figure 7 The average classification accuracies

based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

0

01

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ue P

ositi

ve R

ate

060402 08 10

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(c) Iteration 10 times

Figure 8 ROC curves of faults detection based on DATF

00001 0001 001 01 0995000

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(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

Shock and Vibration 5

119872119888 =

1119899119888119905119903119899119888119905119903 119909119894 119909119895 isin 1198831199051199031119899119888119905119890119899119888119905119890 119909119894 119909119895 isin 119883119905119890minus1119899119905119903119899119905119890

119909119894 isin 119883119888119905119903 119909119895 isin 119883119888119905119890119909119895 isin 119883119888119905119903 119909119894 isin 1198831198881199051198900 119900119905ℎ119890119903119908119894119904119890

(5)

The conditional distributions between training and testdomains are brought closer under the new representation119881 = 119860119879119883119863 by minimizing (4)

In order to obtain effective and robust transferable featurerepresentation and improve the quality of fault diagnosis ourwork aims to reduce the impact of discrepancies from boththe marginal and conditional distributions between trainingand test domains by resorting the pseudo labels of test data[26] on diagnosis and these pseudo labels can be obtainedfrom a base classifier (NN classifier) built on the labeledtraining data to predict the fully unlabeled test data Thusthe final optimization problem (6) in this paper comprised(2) and (4)

min119860119879119883119863119867119883

119879119863119860=119868

(1 minus 120582) 119862sum119888=0

119905119903 (119860119879119883119863119872119888119883119879119863119860) + 120582 1198602119865 (6)

where sdot 119865 is the Frobenius norm that guarantees the opti-mization problem to be well defined and 120582 is the regulariza-tion parameter [24] that trades off the impact of regulariza-tion term on the transformation matrix A The goal is to findthe latent feature space created by a transformation matrix119860 isin 119877119889times119896 where the discrepancies of both the marginal andconditional distributions between domains are significantlyreduced The Lagrange function for (7) is constructed whereΛ = diag(Λ 1 Λ 119896) isin 119877119896times119896 is the Lagrange multiplier

119871 = (1 minus 120582) 119905119903(119860119879(119883119863 119862sum119888=0

119872119888119883119879119863)119860) + 120582119905119903 (119860119879119860)+ 119905119903 ((119868 minus 119860119879119883119863119867119883119879119863119860)Λ)

(7)

According to 119889119871119889119860 = 0 the optimal solution of (6) can beacquired through the generalized eigen decomposition

((1 minus 120582)119883119863 119862sum119888=0

119872119888119883119879119863 + 120582119868)119860 = 119883119863119867119883119879119863119860Λ (8)

Finally the adaptation matrix A is obtained from solving (8)for 119896 smallest eigenvectors The procedure of fault diagnosisusing DAFT can be depicted as follows in detail

Step 1 For given training data 119883119905119903 isin 119877119899119905119903times119889 with label 119884119905119903 isin119877119899119905119903times1 andunlabeled test data119883119905119890 isin 119877119899119905119890times119889 in the feature spaceStep 2 Construct MMD matrix 119872119898 by (2) Adaptationmatrix 119860 generated by the 119896 smallest eigenvectors can beacquired by solving (8) through Lagrange multiplier Thenthe robust representation for two domains is obtained 119881 =119860119879119883119863

Step 3 Train the NN classifier on projected training data119860119879119883119905119903 119884119905119903 and then obtain pseudo test data labels 119884119905119890 thatdenote the conditional probability 119876(119884119905119890|119883119905119890) by using thetrained NN classifier

Step 4 Update MMD matrix 119872119888119862119888=1 by (5) according to119875(119884119905119903|119883119905119903) = 119876(119884119905119890|119883119905119890) and then obtain the updated adapta-tionmatrix119860 by solving (8) through LagrangemultiplierTheupdated robust representation for two domains is obtained119881 = 119860119879119883119863 and then jump to Step 3 until the end of theiteration

Step 5 Finally the test data labels 119884119905119890 are predicted accuratelyby the adaptive NN classifier

4 Experimental Evaluations

In order to demonstrate the effectiveness of the proposedfault diagnosis method the vast bearing vibration signalscollected from a bearing test rig are used Dataset is acquiredfrom the bearing data center of Case Western Reserve Uni-versity (CWRU) [27] DATF is compared with the baselineapproaches and several successful methods

(a) Baseline NN classifier with no projection and noadaptation is created That is original input is directly usedfor diagnosis

(b) NN NA NN classifier with no adaptation is createdSpecifically we use a new representation extracted fromoriginal input by PCA without domain adaptation

(c) NN SA NN classifier with projection and domainadaptation using subspace alignment that only reduces themarginal distribution [28]

(a) is a baseline method without projection and domainadaptation techniques which is widely used in the field offault diagnosis (b) is a classical method without domainadaptation which has achieved success inmany fault diagno-sis applications (c) is one of the novel and efficient approachin domain adaptation

41 Experimental Setup and Dataset Preparation The test-bed illustrated in Figure 3 consists of a driving motor a2 hp motor for loading a torque sensorencoder a powermeter accelerometers and electronic control unit [27 29]The test bearings locate in the motor shaft Subjected toelectrosparking inner-race faults (IF) outer-race faults (OF)and ball fault (BF) of different sizes (0007in 0014in and0021in) are introduced into the drive-end bearing of motor[30] The vibration signals are sampled with the help ofaccelerometers installed to the rack with magnetic bases

The working condition of the rotating machinery isusually complex in real-world For purpose of simulatingthe actual application and making the experimental resultsmore persuasive in our experiment dataset collected fromDrive-End Bearing Fault Data and sampled at a frequencyof 12kHz is obtained from different working conditionsDataset includes three kinds of fault degrees (0007in 0014inand 0021in) Each fault degree contains four fault types ofbearings NO IF OF and BF Each fault type of vibration datais collected from four kinds of working conditions ie L0

6 Shock and Vibration

Figure 3 Bearing test rig of Case Western Reserve University DataCenter

= 0 hp1797 rpm L1 = 1 hp1772 rpm L2 = 2 hp1750 rpmand L3 = 3 hp1730 rpm Each sample contains 2049 Fouriercoefficients transformed from the raw vibration signals usingFFT Each domain on dataset contains four fault types andeach fault type contains 200 samples Under our experimentalsetup it is impossible to find the optimal 119896 and 120582 via crossvalidation since labeled training data and unlabeled testdata are sampled from different working conditions Thusempirically searching the parameter space is used to findthe optimal parameter settings and details are described inSection 4 Finally 120582 = 01 and 119896 = 100 are used in our work

In order to verify the benefits of DATF contrast methodsof (a)-(c) are also carried out simultaneously The scenariosettings of all experiments are trained on labeled trainingdata under one single load (training domain) to diagnosethe unlabeled test data under another load (test domain)In all 48 different transferring tests are conducted and thedescription of experimental setup in detail is shown inTable 1

42 Diagnosis Results of the ProposedMethod Thediagnosticresults for fault size being 0007in 0014in and 0021in areshown in Figures 4 5 and 6 The average classificationaccuracies of four methods are described in Figure 7

Each figure is composed of four subfigures and testdomains in every figure are ordered clockwise from (a)L0 L1 L2 and L3 The left of the symbol rdquominus gtrdquo in everysubfigures represents the training domain and the rightrepresents the test domain For each set of bars in Figures 45 and 6 the performances indicate transferring from trainingdomain to test domain which simulates fault diagnosis underdifferent working conditions The load and speed betweendifferent domains have large discrepancies For example inFigure 4(a) the test domain is L0 (the motor load is 0hp andspeed is 1797rpm) the training domain is L1 (the motor loadis 1hp and speed is 1772rpm) L2 (the motor load is 2hp andspeed is 1750rpm) and L3 (the motor load is 3hp and speedis 1730rpm)

From the performances of bearing fault diagnosis inFigures 4 5 and 6 the highest accuracy rates can alwaysbe achieved when the training set of one domain is thesame with the testing set of one domain and this phe-nomenon is reasonable theoretically We can obviously findthat performances of the baseline method and NN NA areall very poor For example in Figures 6(a) 6(b) and 6(c)the accuracies are only about 75 when we transfer L3

Table 1 Description of the experimental setup

Task oftests

Diagnose unlabeled test samples in test domainLabeled training Unlabeled test Fault Fault(training domain) (test domain) type size

1 L0L1L2L3 L0 NOIF 0007inBFOF

2 L0L1L2L3 L1 NOIF 0007inBFOF

3 L0L1L2L3 L2 NOIF 0007inBFOF

4 L0L1L2L3 L3 NOIF 0007inBFOF

5 L0L1L2L3 L0 NOIF 0014inBFOF

6 L0L1L2L3 L1 NOIF 0014inBFOF

7 L0L1L2L3 L2 NOIF 0014inBFOF

8 L0L1L2L3 L3 NOIF 0014inBFOF

9 L0L1L2L3 L0 NOIF 0021inBFOF

10 L0L1L2L3 L1 NOIF 0021inBFOF

11 L0L1L2L3 L2 NOIF 0021inBFOF

12 L0L1L2L3 L3 NOIF 0021inBFOF

to L0 L1 and L2 respectively Especially in Figure 4 alot of accuracies of baseline method and NN NA can notreach 70 when we transfer L1 to L2 These results illustratetraditional methods without domain adaptation can not beapplied to fault diagnosis in variable working conditionsTheperformances of NN SA are better than the first two types ofmethods In Figures 5 and 6 the accuracies of NN NA forvariable working condition bearing fault diagnosis are veryhigh However in Figure 4(c) the performance transferringbetween L1 and L2 is only about 90 and the accuracy isabout 94 when we transfer L3 to L2 Similar phenomenaalso appear in Figure 4(a) These results mentioned aboveindicate that NN NA also can not be applied to complex andvariable working condition bearing fault diagnosis What isexciting is that the proposed method is evidently superior tothe other three compared methods in all cases whatever thetraining domain and test domain are Note that the accuraciesof DATF all can achieve 100 in Figures 4 5 and 6 Evenin Figure 4(a) DATF can still achieve a favorable accuracy(100) while baseline method and NN NA just reach about60 and NN SA only achieve 90 when transferring fromL1 to L2 Compared to the other three methods the averageclassification accuracy (100) of DATF has been markedlyimprovedThese results are all obtained from the benchmark

Shock and Vibration 7

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Figure 4 The results with fault size being 0007in

datasets of fault diagnosis research under a relatively fairexperiment condition Through the above analysis result wecan conclude that the proposed method is very potentialfor solving bearing fault diagnosis problems under differentworking conditions

To further illustrate the influence of extracted transferablefeatures on the results receiver operating characteristics(ROC) are applied for evaluation [32] An ROC curve isgenerated by plotting the false positive rate and true positiverate as the threshold level is varied In this paper ROCcurves are obtained from different models based on NNclassifier which are built on different extracted featuresand we only report ROC results on transferring test thattransfers L1 to L2 with fault size being 0007in in Figure 8and similar trends on all other tests Before the iterationbegins in Figure 8(a) performances of the model built onextracted features are unsatisfactory After iteration 1 time inFigure 8(b) performances of the model built on extractedtransferable features are improved dramatically and what isexciting is that performances based on extracted transferablefeatures achieve the perfect detection results ultimately

43 Parameter Sensitivity In this section we investigate theinfluence of the parameter 120582 which represents regularizationparameter during transferable feature extraction Theoreti-cally larger values of 120582 can make shrinkage regularization

more important in our work When 120582 rarr 0 and 120582 rarr1 the optimization problem is ill-defined Different 120582 hasdifferent effects on classification accuracy Figure 9 reportsthe results From Figure 9 it is obvious that different 120582 havea great influence on diagnostic results with fault size being0007in and performances with fault size being 0021in andit has little overall effect on results with fault size being0014in What is noticeable is that results are little affected byparameter 120582 when the training domain and test domain arethe same and 120582 isin [00505] can be optimal parameter valueswhich can indicate the proposed method can achieve stableand excellent performance under a wide range of parametervalues

44 Domain Discrepancy Effect of Empirical Analysis Inmany actual fault diagnosis and classification scenarios thedistribution of training data domain is different from thetesting data domain which leads to fault diagnostic accuracy-dropping In fact the data distribution differences betweendomains (training data domain and test data domain) reflectthe differences of the data structures that contain plenty offault messages It is a key point for fault diagnosis to extractfault features from data structures In order to profoundlyunderstand the effect of distribution differences between twodomains and explain why the proposed method works weresort the t-SNE technique [31] to visualize high dimensional

8 Shock and Vibration

1008875

90

100100

8775

9437100100100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL00

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)

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L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL3

BaselineNN NA

NN SADATF

80

100

Accu

racy

()

(d)

Figure 5 The results with fault size being 0014in

representation of mentioned methods in our experiment in atwo-dimensional map

In all of the above-mentioned cases take the transferringtest that transfers L1 to L2 with fault size being 0007in as anexample in Figure 10

From Figure 10 it is clear that the distribution discrep-ancies of transferable features extracted via DATF betweentraining domain and test domain are much smaller thanthe compared methods and transferable features are muchmore divisible than othersThese results verify that DATF canfigure out a robust feature representation for training domainand test domain and test samples can be discriminatedsignificantly with NN classifier built in training domain byusing extracted transferable features

45 Discussion The proposed method provides a way ofdomain adaptation to extract robust fault features and clas-sify fault types under different working conditions Severalremarks still need to be described(1) This work presents a new point of view thatuses domain adaptation to realize bearing fault diagnosisunder different working conditions Li [30] utilized spec-trum images as features to conduct bearing fault diagnosiswhich applied two-dimensional principal component anal-ysis (2DPCA) into the dimension reduction of the spec-trum images of vibration signals and feature extraction andmost accuracies were very high Unfortunately there are

still several instances having lower accuracies To solve thisproblem we apply the domain adaptation into this field andtransferable features for training domain and test domainare extracted to classify fault types Finally the accuraciesall can reach 100 In this paper our work considers morebearing conditions (fault size being 0007in) Compared withthe method [30] in this situation advantages of our methodare highlighted(2) The vast results indicate that the proposed methodis suitable for effectively classifying mechanical health con-ditions under different working conditions In [9] DeepConvolutional Neural Networks with Wide First-Layer Ker-nel (WDCNN) and AdaBN are applied to diagnose threedatasets which contain 10 kinds of health conditions (BF IFOF with fault size being 0007 in 0014 in and 0021 in)under three load conditions (Load 1 Load 2 and Load 3)respectively which is similar to L1 L2 and L3 in this paperThe average accuracy of this method in [9] is 959 whereasaverage accuracy of DATF is 100 The main reason is thattransferable features extracted based on domain adaptationtake full advantage of structure information of trainingdomain and test domain and the distributions of transferablefeatures extracted from training domain and testing domainare very close after our methods as shown in Figure 10(3) It is noted that our method is unsupervised andfocuses on fault transfer diagnosis based on the same fault di-ameter under different working conditions In [14] a method

Shock and Vibration 9

100

995 97785

100 9925 9762

79

100 100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

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Figure 6 The results with fault size being 0021in

97819863

100 9997 100986

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94198933

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NN SADATF

80828486889092949698

100

Accu

racy

()

100

0007 00210014

Figure 7 The average classification accuracies

based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

0

01

02

03

04

05

06

07

08

09

1Tr

ue P

ositi

ve R

ate

060402 08 10

False Positive Rate

(a) Before the iteration begins

NOIF

OFBF

NO AUC= 1IF AUC= 1

OF AUC= 093BF AUC= 079

True

Pos

itive

Rat

e

09

08

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04

03

02

01

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060402 08 10

False Positive Rate

(b) Iteration 1 time

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OFBF

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07

08

09

1

True

Pos

itive

Rat

e

NO AUC= 1IF AUC= 1

OF AUC= 1BF AUC= 1

060402 08 10

False Positive Rate

(c) Iteration 10 times

Figure 8 ROC curves of faults detection based on DATF

00001 0001 001 01 0995000

10000

9500

9000

8500

8000

7500

7000

6500

6000

5500

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(a) Effects on fault size being 0007in

00001 0001 001 01 0998000

10000

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racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(b) Effects on fault size being 0014in

00001 0001 001 01 0996500

7000

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Accu

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L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

6 Shock and Vibration

Figure 3 Bearing test rig of Case Western Reserve University DataCenter

= 0 hp1797 rpm L1 = 1 hp1772 rpm L2 = 2 hp1750 rpmand L3 = 3 hp1730 rpm Each sample contains 2049 Fouriercoefficients transformed from the raw vibration signals usingFFT Each domain on dataset contains four fault types andeach fault type contains 200 samples Under our experimentalsetup it is impossible to find the optimal 119896 and 120582 via crossvalidation since labeled training data and unlabeled testdata are sampled from different working conditions Thusempirically searching the parameter space is used to findthe optimal parameter settings and details are described inSection 4 Finally 120582 = 01 and 119896 = 100 are used in our work

In order to verify the benefits of DATF contrast methodsof (a)-(c) are also carried out simultaneously The scenariosettings of all experiments are trained on labeled trainingdata under one single load (training domain) to diagnosethe unlabeled test data under another load (test domain)In all 48 different transferring tests are conducted and thedescription of experimental setup in detail is shown inTable 1

42 Diagnosis Results of the ProposedMethod Thediagnosticresults for fault size being 0007in 0014in and 0021in areshown in Figures 4 5 and 6 The average classificationaccuracies of four methods are described in Figure 7

Each figure is composed of four subfigures and testdomains in every figure are ordered clockwise from (a)L0 L1 L2 and L3 The left of the symbol rdquominus gtrdquo in everysubfigures represents the training domain and the rightrepresents the test domain For each set of bars in Figures 45 and 6 the performances indicate transferring from trainingdomain to test domain which simulates fault diagnosis underdifferent working conditions The load and speed betweendifferent domains have large discrepancies For example inFigure 4(a) the test domain is L0 (the motor load is 0hp andspeed is 1797rpm) the training domain is L1 (the motor loadis 1hp and speed is 1772rpm) L2 (the motor load is 2hp andspeed is 1750rpm) and L3 (the motor load is 3hp and speedis 1730rpm)

From the performances of bearing fault diagnosis inFigures 4 5 and 6 the highest accuracy rates can alwaysbe achieved when the training set of one domain is thesame with the testing set of one domain and this phe-nomenon is reasonable theoretically We can obviously findthat performances of the baseline method and NN NA areall very poor For example in Figures 6(a) 6(b) and 6(c)the accuracies are only about 75 when we transfer L3

Table 1 Description of the experimental setup

Task oftests

Diagnose unlabeled test samples in test domainLabeled training Unlabeled test Fault Fault(training domain) (test domain) type size

1 L0L1L2L3 L0 NOIF 0007inBFOF

2 L0L1L2L3 L1 NOIF 0007inBFOF

3 L0L1L2L3 L2 NOIF 0007inBFOF

4 L0L1L2L3 L3 NOIF 0007inBFOF

5 L0L1L2L3 L0 NOIF 0014inBFOF

6 L0L1L2L3 L1 NOIF 0014inBFOF

7 L0L1L2L3 L2 NOIF 0014inBFOF

8 L0L1L2L3 L3 NOIF 0014inBFOF

9 L0L1L2L3 L0 NOIF 0021inBFOF

10 L0L1L2L3 L1 NOIF 0021inBFOF

11 L0L1L2L3 L2 NOIF 0021inBFOF

12 L0L1L2L3 L3 NOIF 0021inBFOF

to L0 L1 and L2 respectively Especially in Figure 4 alot of accuracies of baseline method and NN NA can notreach 70 when we transfer L1 to L2 These results illustratetraditional methods without domain adaptation can not beapplied to fault diagnosis in variable working conditionsTheperformances of NN SA are better than the first two types ofmethods In Figures 5 and 6 the accuracies of NN NA forvariable working condition bearing fault diagnosis are veryhigh However in Figure 4(c) the performance transferringbetween L1 and L2 is only about 90 and the accuracy isabout 94 when we transfer L3 to L2 Similar phenomenaalso appear in Figure 4(a) These results mentioned aboveindicate that NN NA also can not be applied to complex andvariable working condition bearing fault diagnosis What isexciting is that the proposed method is evidently superior tothe other three compared methods in all cases whatever thetraining domain and test domain are Note that the accuraciesof DATF all can achieve 100 in Figures 4 5 and 6 Evenin Figure 4(a) DATF can still achieve a favorable accuracy(100) while baseline method and NN NA just reach about60 and NN SA only achieve 90 when transferring fromL1 to L2 Compared to the other three methods the averageclassification accuracy (100) of DATF has been markedlyimprovedThese results are all obtained from the benchmark

Shock and Vibration 7

100100

7763 7863

100100100 9975 100 1009575100

BaselineNN NA

NN SADATF

7788 8113

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cura

cy (

)

(a)

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955100 100

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L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL10

20

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racy

()

(b)

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NN SADATF

8425 7959663

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63755737

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL2

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()

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NN SADATF

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L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL30

20

40

60

80

100

Accu

racy

()

(d)

Figure 4 The results with fault size being 0007in

datasets of fault diagnosis research under a relatively fairexperiment condition Through the above analysis result wecan conclude that the proposed method is very potentialfor solving bearing fault diagnosis problems under differentworking conditions

To further illustrate the influence of extracted transferablefeatures on the results receiver operating characteristics(ROC) are applied for evaluation [32] An ROC curve isgenerated by plotting the false positive rate and true positiverate as the threshold level is varied In this paper ROCcurves are obtained from different models based on NNclassifier which are built on different extracted featuresand we only report ROC results on transferring test thattransfers L1 to L2 with fault size being 0007in in Figure 8and similar trends on all other tests Before the iterationbegins in Figure 8(a) performances of the model built onextracted features are unsatisfactory After iteration 1 time inFigure 8(b) performances of the model built on extractedtransferable features are improved dramatically and what isexciting is that performances based on extracted transferablefeatures achieve the perfect detection results ultimately

43 Parameter Sensitivity In this section we investigate theinfluence of the parameter 120582 which represents regularizationparameter during transferable feature extraction Theoreti-cally larger values of 120582 can make shrinkage regularization

more important in our work When 120582 rarr 0 and 120582 rarr1 the optimization problem is ill-defined Different 120582 hasdifferent effects on classification accuracy Figure 9 reportsthe results From Figure 9 it is obvious that different 120582 havea great influence on diagnostic results with fault size being0007in and performances with fault size being 0021in andit has little overall effect on results with fault size being0014in What is noticeable is that results are little affected byparameter 120582 when the training domain and test domain arethe same and 120582 isin [00505] can be optimal parameter valueswhich can indicate the proposed method can achieve stableand excellent performance under a wide range of parametervalues

44 Domain Discrepancy Effect of Empirical Analysis Inmany actual fault diagnosis and classification scenarios thedistribution of training data domain is different from thetesting data domain which leads to fault diagnostic accuracy-dropping In fact the data distribution differences betweendomains (training data domain and test data domain) reflectthe differences of the data structures that contain plenty offault messages It is a key point for fault diagnosis to extractfault features from data structures In order to profoundlyunderstand the effect of distribution differences between twodomains and explain why the proposed method works weresort the t-SNE technique [31] to visualize high dimensional

8 Shock and Vibration

1008875

90

100100

8775

9437100100100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL00

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cura

cy (

)

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BaselineNN NA

NN SADATF

80

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Accu

racy

()

(d)

Figure 5 The results with fault size being 0014in

representation of mentioned methods in our experiment in atwo-dimensional map

In all of the above-mentioned cases take the transferringtest that transfers L1 to L2 with fault size being 0007in as anexample in Figure 10

From Figure 10 it is clear that the distribution discrep-ancies of transferable features extracted via DATF betweentraining domain and test domain are much smaller thanthe compared methods and transferable features are muchmore divisible than othersThese results verify that DATF canfigure out a robust feature representation for training domainand test domain and test samples can be discriminatedsignificantly with NN classifier built in training domain byusing extracted transferable features

45 Discussion The proposed method provides a way ofdomain adaptation to extract robust fault features and clas-sify fault types under different working conditions Severalremarks still need to be described(1) This work presents a new point of view thatuses domain adaptation to realize bearing fault diagnosisunder different working conditions Li [30] utilized spec-trum images as features to conduct bearing fault diagnosiswhich applied two-dimensional principal component anal-ysis (2DPCA) into the dimension reduction of the spec-trum images of vibration signals and feature extraction andmost accuracies were very high Unfortunately there are

still several instances having lower accuracies To solve thisproblem we apply the domain adaptation into this field andtransferable features for training domain and test domainare extracted to classify fault types Finally the accuraciesall can reach 100 In this paper our work considers morebearing conditions (fault size being 0007in) Compared withthe method [30] in this situation advantages of our methodare highlighted(2) The vast results indicate that the proposed methodis suitable for effectively classifying mechanical health con-ditions under different working conditions In [9] DeepConvolutional Neural Networks with Wide First-Layer Ker-nel (WDCNN) and AdaBN are applied to diagnose threedatasets which contain 10 kinds of health conditions (BF IFOF with fault size being 0007 in 0014 in and 0021 in)under three load conditions (Load 1 Load 2 and Load 3)respectively which is similar to L1 L2 and L3 in this paperThe average accuracy of this method in [9] is 959 whereasaverage accuracy of DATF is 100 The main reason is thattransferable features extracted based on domain adaptationtake full advantage of structure information of trainingdomain and test domain and the distributions of transferablefeatures extracted from training domain and testing domainare very close after our methods as shown in Figure 10(3) It is noted that our method is unsupervised andfocuses on fault transfer diagnosis based on the same fault di-ameter under different working conditions In [14] a method

Shock and Vibration 9

100

995 97785

100 9925 9762

79

100 100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

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cura

cy (

)

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975 100 100

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racy

()

(d)

Figure 6 The results with fault size being 0021in

97819863

100 9997 100986

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8874

94198933

Fault diameter (in)

BaselineNN NA

NN SADATF

80828486889092949698

100

Accu

racy

()

100

0007 00210014

Figure 7 The average classification accuracies

based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

0

01

02

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1Tr

ue P

ositi

ve R

ate

060402 08 10

False Positive Rate

(a) Before the iteration begins

NOIF

OFBF

NO AUC= 1IF AUC= 1

OF AUC= 093BF AUC= 079

True

Pos

itive

Rat

e

09

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060402 08 10

False Positive Rate

(b) Iteration 1 time

NOIF

OFBF

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09

1

True

Pos

itive

Rat

e

NO AUC= 1IF AUC= 1

OF AUC= 1BF AUC= 1

060402 08 10

False Positive Rate

(c) Iteration 10 times

Figure 8 ROC curves of faults detection based on DATF

00001 0001 001 01 0995000

10000

9500

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8500

8000

7500

7000

6500

6000

5500

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(a) Effects on fault size being 0007in

00001 0001 001 01 0998000

10000

9800

9600

9400

9200

9000

8800

8600

8400

8200

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(b) Effects on fault size being 0014in

00001 0001 001 01 0996500

7000

7500

8000

8500

9000

9500

10000

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

Shock and Vibration 7

100100

7763 7863

100100100 9975 100 1009575100

BaselineNN NA

NN SADATF

7788 8113

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

100 100

0

20

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60

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cura

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)

(a)

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79258237

100 100 100 100 100 100 100

L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL10

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(b)

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NN SADATF

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63755737

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL2

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racy

()

(c)

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L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL30

20

40

60

80

100

Accu

racy

()

(d)

Figure 4 The results with fault size being 0007in

datasets of fault diagnosis research under a relatively fairexperiment condition Through the above analysis result wecan conclude that the proposed method is very potentialfor solving bearing fault diagnosis problems under differentworking conditions

To further illustrate the influence of extracted transferablefeatures on the results receiver operating characteristics(ROC) are applied for evaluation [32] An ROC curve isgenerated by plotting the false positive rate and true positiverate as the threshold level is varied In this paper ROCcurves are obtained from different models based on NNclassifier which are built on different extracted featuresand we only report ROC results on transferring test thattransfers L1 to L2 with fault size being 0007in in Figure 8and similar trends on all other tests Before the iterationbegins in Figure 8(a) performances of the model built onextracted features are unsatisfactory After iteration 1 time inFigure 8(b) performances of the model built on extractedtransferable features are improved dramatically and what isexciting is that performances based on extracted transferablefeatures achieve the perfect detection results ultimately

43 Parameter Sensitivity In this section we investigate theinfluence of the parameter 120582 which represents regularizationparameter during transferable feature extraction Theoreti-cally larger values of 120582 can make shrinkage regularization

more important in our work When 120582 rarr 0 and 120582 rarr1 the optimization problem is ill-defined Different 120582 hasdifferent effects on classification accuracy Figure 9 reportsthe results From Figure 9 it is obvious that different 120582 havea great influence on diagnostic results with fault size being0007in and performances with fault size being 0021in andit has little overall effect on results with fault size being0014in What is noticeable is that results are little affected byparameter 120582 when the training domain and test domain arethe same and 120582 isin [00505] can be optimal parameter valueswhich can indicate the proposed method can achieve stableand excellent performance under a wide range of parametervalues

44 Domain Discrepancy Effect of Empirical Analysis Inmany actual fault diagnosis and classification scenarios thedistribution of training data domain is different from thetesting data domain which leads to fault diagnostic accuracy-dropping In fact the data distribution differences betweendomains (training data domain and test data domain) reflectthe differences of the data structures that contain plenty offault messages It is a key point for fault diagnosis to extractfault features from data structures In order to profoundlyunderstand the effect of distribution differences between twodomains and explain why the proposed method works weresort the t-SNE technique [31] to visualize high dimensional

8 Shock and Vibration

1008875

90

100100

8775

9437100100100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL00

20

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)

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L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL10

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(c)

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L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL3

BaselineNN NA

NN SADATF

80

100

Accu

racy

()

(d)

Figure 5 The results with fault size being 0014in

representation of mentioned methods in our experiment in atwo-dimensional map

In all of the above-mentioned cases take the transferringtest that transfers L1 to L2 with fault size being 0007in as anexample in Figure 10

From Figure 10 it is clear that the distribution discrep-ancies of transferable features extracted via DATF betweentraining domain and test domain are much smaller thanthe compared methods and transferable features are muchmore divisible than othersThese results verify that DATF canfigure out a robust feature representation for training domainand test domain and test samples can be discriminatedsignificantly with NN classifier built in training domain byusing extracted transferable features

45 Discussion The proposed method provides a way ofdomain adaptation to extract robust fault features and clas-sify fault types under different working conditions Severalremarks still need to be described(1) This work presents a new point of view thatuses domain adaptation to realize bearing fault diagnosisunder different working conditions Li [30] utilized spec-trum images as features to conduct bearing fault diagnosiswhich applied two-dimensional principal component anal-ysis (2DPCA) into the dimension reduction of the spec-trum images of vibration signals and feature extraction andmost accuracies were very high Unfortunately there are

still several instances having lower accuracies To solve thisproblem we apply the domain adaptation into this field andtransferable features for training domain and test domainare extracted to classify fault types Finally the accuraciesall can reach 100 In this paper our work considers morebearing conditions (fault size being 0007in) Compared withthe method [30] in this situation advantages of our methodare highlighted(2) The vast results indicate that the proposed methodis suitable for effectively classifying mechanical health con-ditions under different working conditions In [9] DeepConvolutional Neural Networks with Wide First-Layer Ker-nel (WDCNN) and AdaBN are applied to diagnose threedatasets which contain 10 kinds of health conditions (BF IFOF with fault size being 0007 in 0014 in and 0021 in)under three load conditions (Load 1 Load 2 and Load 3)respectively which is similar to L1 L2 and L3 in this paperThe average accuracy of this method in [9] is 959 whereasaverage accuracy of DATF is 100 The main reason is thattransferable features extracted based on domain adaptationtake full advantage of structure information of trainingdomain and test domain and the distributions of transferablefeatures extracted from training domain and testing domainare very close after our methods as shown in Figure 10(3) It is noted that our method is unsupervised andfocuses on fault transfer diagnosis based on the same fault di-ameter under different working conditions In [14] a method

Shock and Vibration 9

100

995 97785

100 9925 9762

79

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L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

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(b)

9513 1001009263

100 100100 100 100 100 100 100 100 100

7575

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL2

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(c)

9213

100100

100

8688

100 100 100

100 100 100 100 100 100 100 100

L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL3

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(d)

Figure 6 The results with fault size being 0021in

97819863

100 9997 100986

100

9379

8874

94198933

Fault diameter (in)

BaselineNN NA

NN SADATF

80828486889092949698

100

Accu

racy

()

100

0007 00210014

Figure 7 The average classification accuracies

based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

0

01

02

03

04

05

06

07

08

09

1Tr

ue P

ositi

ve R

ate

060402 08 10

False Positive Rate

(a) Before the iteration begins

NOIF

OFBF

NO AUC= 1IF AUC= 1

OF AUC= 093BF AUC= 079

True

Pos

itive

Rat

e

09

08

07

06

05

04

03

02

01

0

1

060402 08 10

False Positive Rate

(b) Iteration 1 time

NOIF

OFBF

0

01

02

03

04

05

06

07

08

09

1

True

Pos

itive

Rat

e

NO AUC= 1IF AUC= 1

OF AUC= 1BF AUC= 1

060402 08 10

False Positive Rate

(c) Iteration 10 times

Figure 8 ROC curves of faults detection based on DATF

00001 0001 001 01 0995000

10000

9500

9000

8500

8000

7500

7000

6500

6000

5500

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(a) Effects on fault size being 0007in

00001 0001 001 01 0998000

10000

9800

9600

9400

9200

9000

8800

8600

8400

8200

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(b) Effects on fault size being 0014in

00001 0001 001 01 0996500

7000

7500

8000

8500

9000

9500

10000

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

8 Shock and Vibration

1008875

90

100100

8775

9437100100100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL00

20

40

60

80

100Ac

cura

cy (

)

BaselineNN NA

NN SADATF

(a)

100 100

9487

100100 100 9975

100 100 100 100 100 100 100 100

L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL10

20

40

60

80

100

Accu

racy

()

BaselineNN NA

NN SADATF

100

(b)

100 985 100 100100 100 100 100 100 100 100 100 100100

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL20

20

40

60

80

100

Accu

racy

()

BaselineNN NA

NN SADATF

100 100

(c)

9912

967597

100

9962 98 9812 100

100 100 100 100 9963 100 100 100

L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL3

BaselineNN NA

NN SADATF

80

100

Accu

racy

()

(d)

Figure 5 The results with fault size being 0014in

representation of mentioned methods in our experiment in atwo-dimensional map

In all of the above-mentioned cases take the transferringtest that transfers L1 to L2 with fault size being 0007in as anexample in Figure 10

From Figure 10 it is clear that the distribution discrep-ancies of transferable features extracted via DATF betweentraining domain and test domain are much smaller thanthe compared methods and transferable features are muchmore divisible than othersThese results verify that DATF canfigure out a robust feature representation for training domainand test domain and test samples can be discriminatedsignificantly with NN classifier built in training domain byusing extracted transferable features

45 Discussion The proposed method provides a way ofdomain adaptation to extract robust fault features and clas-sify fault types under different working conditions Severalremarks still need to be described(1) This work presents a new point of view thatuses domain adaptation to realize bearing fault diagnosisunder different working conditions Li [30] utilized spec-trum images as features to conduct bearing fault diagnosiswhich applied two-dimensional principal component anal-ysis (2DPCA) into the dimension reduction of the spec-trum images of vibration signals and feature extraction andmost accuracies were very high Unfortunately there are

still several instances having lower accuracies To solve thisproblem we apply the domain adaptation into this field andtransferable features for training domain and test domainare extracted to classify fault types Finally the accuraciesall can reach 100 In this paper our work considers morebearing conditions (fault size being 0007in) Compared withthe method [30] in this situation advantages of our methodare highlighted(2) The vast results indicate that the proposed methodis suitable for effectively classifying mechanical health con-ditions under different working conditions In [9] DeepConvolutional Neural Networks with Wide First-Layer Ker-nel (WDCNN) and AdaBN are applied to diagnose threedatasets which contain 10 kinds of health conditions (BF IFOF with fault size being 0007 in 0014 in and 0021 in)under three load conditions (Load 1 Load 2 and Load 3)respectively which is similar to L1 L2 and L3 in this paperThe average accuracy of this method in [9] is 959 whereasaverage accuracy of DATF is 100 The main reason is thattransferable features extracted based on domain adaptationtake full advantage of structure information of trainingdomain and test domain and the distributions of transferablefeatures extracted from training domain and testing domainare very close after our methods as shown in Figure 10(3) It is noted that our method is unsupervised andfocuses on fault transfer diagnosis based on the same fault di-ameter under different working conditions In [14] a method

Shock and Vibration 9

100

995 97785

100 9925 9762

79

100 100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

BaselineNN NA

NN SADATF

0

20

40

60

80

100Ac

cura

cy (

)

(a)

975 100 100

96259838 100 9587

76127612

100 100 100 100 100 100 100

L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL1

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(b)

9513 1001009263

100 100100 100 100 100 100 100 100 100

7575

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL2

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(c)

9213

100100

100

8688

100 100 100

100 100 100 100 100 100 100 100

L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL3

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(d)

Figure 6 The results with fault size being 0021in

97819863

100 9997 100986

100

9379

8874

94198933

Fault diameter (in)

BaselineNN NA

NN SADATF

80828486889092949698

100

Accu

racy

()

100

0007 00210014

Figure 7 The average classification accuracies

based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

0

01

02

03

04

05

06

07

08

09

1Tr

ue P

ositi

ve R

ate

060402 08 10

False Positive Rate

(a) Before the iteration begins

NOIF

OFBF

NO AUC= 1IF AUC= 1

OF AUC= 093BF AUC= 079

True

Pos

itive

Rat

e

09

08

07

06

05

04

03

02

01

0

1

060402 08 10

False Positive Rate

(b) Iteration 1 time

NOIF

OFBF

0

01

02

03

04

05

06

07

08

09

1

True

Pos

itive

Rat

e

NO AUC= 1IF AUC= 1

OF AUC= 1BF AUC= 1

060402 08 10

False Positive Rate

(c) Iteration 10 times

Figure 8 ROC curves of faults detection based on DATF

00001 0001 001 01 0995000

10000

9500

9000

8500

8000

7500

7000

6500

6000

5500

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(a) Effects on fault size being 0007in

00001 0001 001 01 0998000

10000

9800

9600

9400

9200

9000

8800

8600

8400

8200

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(b) Effects on fault size being 0014in

00001 0001 001 01 0996500

7000

7500

8000

8500

9000

9500

10000

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

Shock and Vibration 9

100

995 97785

100 9925 9762

79

100 100 100 100 100 100100 100

L0-gtL0 L1-gtL0 L2-gtL0 L3-gtL0

BaselineNN NA

NN SADATF

0

20

40

60

80

100Ac

cura

cy (

)

(a)

975 100 100

96259838 100 9587

76127612

100 100 100 100 100 100 100

L0-gtL1 L1-gtL1 L2-gtL1 L3-gtL1

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(b)

9513 1001009263

100 100100 100 100 100 100 100 100 100

7575

L0-gtL2 L1-gtL2 L2-gtL2 L3-gtL2

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(c)

9213

100100

100

8688

100 100 100

100 100 100 100 100 100 100 100

L0-gtL3 L1-gtL3 L2-gtL3 L3-gtL3

BaselineNN NA

NN SADATF

0

20

40

60

80

100

Accu

racy

()

(d)

Figure 6 The results with fault size being 0021in

97819863

100 9997 100986

100

9379

8874

94198933

Fault diameter (in)

BaselineNN NA

NN SADATF

80828486889092949698

100

Accu

racy

()

100

0007 00210014

Figure 7 The average classification accuracies

based on neural network by using transferring parameters isproposed and success for diagnosing two datasets including6 kinds of health conditions sampled from different faultdiameters (BF IF OF with fault size being 0007 in and 0021in) with the same motor load and speed (L0) and it focuseson fault diagnosis between two kinds of fault diameters underthe sameworking conditions In addition unlike ourmethodit should be noted that a small amount of labeled data in test

domain is needed when training modified neural networkswhile our method does not need labeled test data during thetraining

5 Conclusion

This paper presents a new way for solving bearing faultdiagnosis under different working conditions Although

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

0

01

02

03

04

05

06

07

08

09

1Tr

ue P

ositi

ve R

ate

060402 08 10

False Positive Rate

(a) Before the iteration begins

NOIF

OFBF

NO AUC= 1IF AUC= 1

OF AUC= 093BF AUC= 079

True

Pos

itive

Rat

e

09

08

07

06

05

04

03

02

01

0

1

060402 08 10

False Positive Rate

(b) Iteration 1 time

NOIF

OFBF

0

01

02

03

04

05

06

07

08

09

1

True

Pos

itive

Rat

e

NO AUC= 1IF AUC= 1

OF AUC= 1BF AUC= 1

060402 08 10

False Positive Rate

(c) Iteration 10 times

Figure 8 ROC curves of faults detection based on DATF

00001 0001 001 01 0995000

10000

9500

9000

8500

8000

7500

7000

6500

6000

5500

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(a) Effects on fault size being 0007in

00001 0001 001 01 0998000

10000

9800

9600

9400

9200

9000

8800

8600

8400

8200

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(b) Effects on fault size being 0014in

00001 0001 001 01 0996500

7000

7500

8000

8500

9000

9500

10000

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

10 Shock and Vibration

NOIF

OFBF

NO AUC= 1IF AUC = 058

OF AUC = 054BF AUC = 091

0

01

02

03

04

05

06

07

08

09

1Tr

ue P

ositi

ve R

ate

060402 08 10

False Positive Rate

(a) Before the iteration begins

NOIF

OFBF

NO AUC= 1IF AUC= 1

OF AUC= 093BF AUC= 079

True

Pos

itive

Rat

e

09

08

07

06

05

04

03

02

01

0

1

060402 08 10

False Positive Rate

(b) Iteration 1 time

NOIF

OFBF

0

01

02

03

04

05

06

07

08

09

1

True

Pos

itive

Rat

e

NO AUC= 1IF AUC= 1

OF AUC= 1BF AUC= 1

060402 08 10

False Positive Rate

(c) Iteration 10 times

Figure 8 ROC curves of faults detection based on DATF

00001 0001 001 01 0995000

10000

9500

9000

8500

8000

7500

7000

6500

6000

5500

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(a) Effects on fault size being 0007in

00001 0001 001 01 0998000

10000

9800

9600

9400

9200

9000

8800

8600

8400

8200

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(b) Effects on fault size being 0014in

00001 0001 001 01 0996500

7000

7500

8000

8500

9000

9500

10000

Accu

racy

L0-gtL0L0-gtL1L0-gtL2L0-gtL3L1-gtL0L1-gtL1L1-gtL2L1-gtL3

L2-gtL0L2-gtL1L2-gtL2L2-gtL3L3-gtL0L3-gtL1L3-gtL2L3-gtL3

(c) Effects on fault size being 0021in

Figure 9 Accuracy () on different 120582

baseline approaches and several successful methods are allcapable of detecting the bearing defects distributional differ-ence of datasets sampled from different working conditionshas a huge impact on these methods and their shallowrepresentations are insensitive to distinguish different pat-terns under different working conditions To tackle thisproblem DATF extracts transferable feature representationfor training and test domain by reducing the discrepancybetween domains and strengthen the recognizable informa-tion in raw vibration signal To evaluate the proposed DATFmethod bearing fault diagnosis experiments were carriedout Extensive experiment results show that DATF is capableof improving the performance of bearing fault diagnosisunder different working conditions comparing with the peermethods

Data Availability

Data used in this paper is acquired from the bearing datacenter of CaseWestern Reserve University (CWRU) and webpage httpcsegroupscaseedubearingdatacenterhome (ac-cessed October 2015)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Key RampD Pro-gram of China (2016YFC0802900) National Natural Science

Shock and Vibration 11

minus60 minus40 minus20 0 20 40 60 80

minus20

minus40

minus60

0

20

40

60

NOIFBFOF

NOIFBFOF

(a) Baseline method

minus60

minus20

minus40

minus60

minus80minus80 minus40 minus20 0 20 40 60 80

0

20

40

60

NOIFBFOF

NOIFBFOF

(b) NN NA

minus80 minus60 minus40 minus20 0 20 40 60 80

minus20

minus40

minus60

minus80

0

20

40

60

80

NOIFBFOF

NOIFBFOF

(c) NN SA

minus60

minus20

minus40

minus60

minus80minus80 minus40 minus20 0 20 40 60 80

0

20

40

60

80

NOIFBFOF

NOIFBFOF

(d) DATF

Figure 10 Feature visualization via t-SNE [31] over a fault diagnosis task from training domain L1 (blue) to test domain L2 (red) underdifferent working conditions

Foundation of China (no 51475455) and the Natural ScienceFoundation of Jiangsu Province (no BK20160276)

References

[1] W Jacobs B Van Hooreweder R Boonen P Sas and DMoens ldquoThe influence of external dynamic loads on the life-time of rolling element bearings Experimental analysis of the

lubricant film and surface wearrdquoMechanical Systems and SignalProcessing vol 74 no 1 pp 144ndash164 2016

[2] G Li G L McDonald and Q Zhao ldquoSinusoidal synthesisbased adaptive tracking for rotating machinery fault detectionrdquoMechanical Systems and Signal Processing vol 83 pp 356ndash3702017

[3] K A Loparo M L Adams W Lin M Farouk Abdel-Magiedand N Afshari ldquoFault detection and diagnosis of rotating

12 Shock and Vibration

machineryrdquo IEEE Transactions on Industrial Electronics vol 47no 5 pp 1005ndash1014 2000

[4] A K Jalan and A R Mohanty ldquoModel based fault diagnosis ofa rotor-bearing system for misalignment and unbalance understeady-state conditionrdquo Journal of Sound and Vibration vol 327no 3-5 pp 604ndash622 2009

[5] B Qiao X Zhang J Gao and X Chen ldquoImpact-force sparsereconstruction from highly incomplete and inaccurate mea-surementsrdquo Journal of Sound and Vibration vol 376 Supple-ment C pp 72ndash94 2016

[6] Y B Li M Q Xu Y Wei and W H Huang ldquoA new roll-ing bearing fault diagnosis method based on multiscale per-mutation entropy and improved support vector machine basedbinary treerdquo Measurement vol 77 Supplement C pp 80ndash942016

[7] J Huang X Hu and F Yang ldquoSupport vector machine withgenetic algorithm for machinery fault diagnosis of high voltagecircuit breakerrdquoMeasurement vol 44 no 6 pp 1018ndash1027 2011

[8] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[9] W Zhang G Peng C Li Y Chen and Z Zhang ldquoA new deeplearning model for fault diagnosis with good anti-noise anddomain adaptation ability on raw vibration signalsrdquo Sensors vol17 no 2 2017

[10] C Joannin B Chouvion F Thouverez J-P Ousty and MMbaye ldquoA nonlinear component mode synthesis method forthe computation of steady-state vibrations in non-conservativesystemsrdquoMechanical Systems and Signal Processing vol 83 pp75ndash92 2017

[11] MMisraHH Yue S J Qin andC Ling ldquoMultivariate processmonitoring and fault diagnosis bymulti-scale PCArdquoComputersamp Chemical Engineering vol 26 no 9 pp 1281ndash1293 2002

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

[13] X Jin M Zhao T W S Chow and M Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2441ndash2451 2014

[14] R Zhang H Tao L Wu and Y Guan ldquoTransfer learning withneural networks for bearing fault diagnosis in changingworkingconditionsrdquo IEEE Access vol 5 pp 14347ndash14357 2017

[15] H Shimodaira ldquoImproving predictive inference under covari-ate shift by weighting the log-likelihood functionrdquo Journal ofStatistical Planning and Inference vol 90 no 2 pp 227ndash2442000

[16] D Tuia C Persello and L Bruzzone ldquoDomain adaptation forthe classification of remote sensing data An overview of recentadvancesrdquo IEEE Geoscience and Remote Sensing Magazine vol4 no 2 pp 41ndash57 2016

[17] S J Pan I W Tsang J T Kwok and Q Yang ldquoDomain adapta-tion via transfer component analysisrdquo IEEE Transactions onNeural Networks and Learning Systems vol 22 no 2 pp 199ndash210 2011

[18] S J Pan and Q Yang ldquoA survey on transfer learningrdquo IEEETransactions on Knowledge and Data Engineering vol 22 no10 pp 1345ndash1359 2010

[19] M Long J Wang G Ding S J Pan and P S Yu ldquoAdaptationregularizationA general framework for transfer learningrdquo IEEETransactions on Knowledge and Data Engineering vol 26 no 5pp 1076ndash1089 2014

[20] VM Patel R Gopalan R Li and R Chellappa ldquoVisual domainadaptation A survey of recent advancesrdquo IEEE Signal ProcessingMagazine vol 32 no 3 pp 53ndash69 2015

[21] W Lu B Liang Y Cheng D Meng J Yang and T ZhangldquoDeep model based domain adaptation for fault diagnosisrdquoIEEE Transactions on Industrial Electronics vol 64 no 3 pp2296ndash2305 2017

[22] K Nigam A K Mccallum S Thrun and T Mitchell ldquoTextclassification from labeled andunlabeled documents using EMrdquoMachine Learning vol 39 no 2 pp 103ndash134 2000

[23] R Gopalan R Li and R Chellappa ldquoDomain adaptation forobject recognition An unsupervised approachrdquo in Proceedingsof the 2011 IEEE International Conference on Computer VisionICCV 2011 pp 999ndash1006 Barcelona Spain November 2011

[24] J Tahmoresnezhad and S Hashemi ldquoVisual domain adaptationvia transfer feature learningrdquo Knowledge and Information Sys-tems vol 50 no 2 pp 585ndash605 2017

[25] G Csurka ldquoDomain adaptation for visual applications a com-prehensive surveyrdquo CoRR abs170205374 2017

[26] M Long JWang G Ding J Sun and P S Yu ldquoTransfer featurelearningwith joint distribution adaptationrdquo inProceedings of theIEEE International Conference on Computer Vision ICCV 2013pp 2200ndash2207 December 2013

[27] ldquoCase western reserve university bearings vibration datasetavailablerdquo 2015 httpcsegroupscaseedubearingdatacenterhome

[28] B Fernando A Habrard M Sebban and T Tuytelaars ldquoUnsu-pervised visual domain adaptation using subspace alignmentrdquoin Proceedings of the 2013 14th IEEE International Conferenceon Computer Vision ICCV 2013 pp 2960ndash2967 AustraliaDecember 2013

[29] H Al-Bugharbee and I Trendafilova ldquoA fault diagnosismethodology for rolling element bearings based on advancedsignal pretreatment and autoregressive modellingrdquo Journal ofSound and Vibration vol 369 pp 246ndash265 2016

[30] W Li M Qiu Z Zhu B Wu and G Zhou ldquoBearing faultdiagnosis based on spectrum images of vibration signalsrdquoMeasurement Science and Technology vol 27 no 3 p 0350052016

[31] LMaaten andGHinton ldquoVisualizing data using t-snerdquo Journalof Machine Learning Research vol 9 pp 2579ndash2605 2008

[32] S K Lee and P R White ldquoHigher-order time-frequency analy-sis and its application to fault detection in rotating machineryrdquoMechanical Systems and Signal Processing vol 11 no 4 pp 637ndash650 1997

12 Shock and Vibration

machineryrdquo IEEE Transactions on Industrial Electronics vol 47no 5 pp 1005ndash1014 2000

[4] A K Jalan and A R Mohanty ldquoModel based fault diagnosis ofa rotor-bearing system for misalignment and unbalance understeady-state conditionrdquo Journal of Sound and Vibration vol 327no 3-5 pp 604ndash622 2009

[5] B Qiao X Zhang J Gao and X Chen ldquoImpact-force sparsereconstruction from highly incomplete and inaccurate mea-surementsrdquo Journal of Sound and Vibration vol 376 Supple-ment C pp 72ndash94 2016

[6] Y B Li M Q Xu Y Wei and W H Huang ldquoA new roll-ing bearing fault diagnosis method based on multiscale per-mutation entropy and improved support vector machine basedbinary treerdquo Measurement vol 77 Supplement C pp 80ndash942016

[7] J Huang X Hu and F Yang ldquoSupport vector machine withgenetic algorithm for machinery fault diagnosis of high voltagecircuit breakerrdquoMeasurement vol 44 no 6 pp 1018ndash1027 2011

[8] N Saravanan and K I Ramachandran ldquoIncipient gear box faultdiagnosis using discrete wavelet transform (DWT) for featureextraction and classification using artificial neural network(ANN)rdquo Expert Systems with Applications vol 37 no 6 pp4168ndash4181 2010

[9] W Zhang G Peng C Li Y Chen and Z Zhang ldquoA new deeplearning model for fault diagnosis with good anti-noise anddomain adaptation ability on raw vibration signalsrdquo Sensors vol17 no 2 2017

[10] C Joannin B Chouvion F Thouverez J-P Ousty and MMbaye ldquoA nonlinear component mode synthesis method forthe computation of steady-state vibrations in non-conservativesystemsrdquoMechanical Systems and Signal Processing vol 83 pp75ndash92 2017

[11] MMisraHH Yue S J Qin andC Ling ldquoMultivariate processmonitoring and fault diagnosis bymulti-scale PCArdquoComputersamp Chemical Engineering vol 26 no 9 pp 1281ndash1293 2002

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

[13] X Jin M Zhao T W S Chow and M Pecht ldquoMotor bearingfault diagnosis using trace ratio linear discriminant analysisrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2441ndash2451 2014

[14] R Zhang H Tao L Wu and Y Guan ldquoTransfer learning withneural networks for bearing fault diagnosis in changingworkingconditionsrdquo IEEE Access vol 5 pp 14347ndash14357 2017

[15] H Shimodaira ldquoImproving predictive inference under covari-ate shift by weighting the log-likelihood functionrdquo Journal ofStatistical Planning and Inference vol 90 no 2 pp 227ndash2442000

[16] D Tuia C Persello and L Bruzzone ldquoDomain adaptation forthe classification of remote sensing data An overview of recentadvancesrdquo IEEE Geoscience and Remote Sensing Magazine vol4 no 2 pp 41ndash57 2016

[17] S J Pan I W Tsang J T Kwok and Q Yang ldquoDomain adapta-tion via transfer component analysisrdquo IEEE Transactions onNeural Networks and Learning Systems vol 22 no 2 pp 199ndash210 2011

[18] S J Pan and Q Yang ldquoA survey on transfer learningrdquo IEEETransactions on Knowledge and Data Engineering vol 22 no10 pp 1345ndash1359 2010

[19] M Long J Wang G Ding S J Pan and P S Yu ldquoAdaptationregularizationA general framework for transfer learningrdquo IEEETransactions on Knowledge and Data Engineering vol 26 no 5pp 1076ndash1089 2014

[20] VM Patel R Gopalan R Li and R Chellappa ldquoVisual domainadaptation A survey of recent advancesrdquo IEEE Signal ProcessingMagazine vol 32 no 3 pp 53ndash69 2015

[21] W Lu B Liang Y Cheng D Meng J Yang and T ZhangldquoDeep model based domain adaptation for fault diagnosisrdquoIEEE Transactions on Industrial Electronics vol 64 no 3 pp2296ndash2305 2017

[22] K Nigam A K Mccallum S Thrun and T Mitchell ldquoTextclassification from labeled andunlabeled documents using EMrdquoMachine Learning vol 39 no 2 pp 103ndash134 2000

[23] R Gopalan R Li and R Chellappa ldquoDomain adaptation forobject recognition An unsupervised approachrdquo in Proceedingsof the 2011 IEEE International Conference on Computer VisionICCV 2011 pp 999ndash1006 Barcelona Spain November 2011

[24] J Tahmoresnezhad and S Hashemi ldquoVisual domain adaptationvia transfer feature learningrdquo Knowledge and Information Sys-tems vol 50 no 2 pp 585ndash605 2017

[25] G Csurka ldquoDomain adaptation for visual applications a com-prehensive surveyrdquo CoRR abs170205374 2017

[26] M Long JWang G Ding J Sun and P S Yu ldquoTransfer featurelearningwith joint distribution adaptationrdquo inProceedings of theIEEE International Conference on Computer Vision ICCV 2013pp 2200ndash2207 December 2013

[27] ldquoCase western reserve university bearings vibration datasetavailablerdquo 2015 httpcsegroupscaseedubearingdatacenterhome

[28] B Fernando A Habrard M Sebban and T Tuytelaars ldquoUnsu-pervised visual domain adaptation using subspace alignmentrdquoin Proceedings of the 2013 14th IEEE International Conferenceon Computer Vision ICCV 2013 pp 2960ndash2967 AustraliaDecember 2013

[29] H Al-Bugharbee and I Trendafilova ldquoA fault diagnosismethodology for rolling element bearings based on advancedsignal pretreatment and autoregressive modellingrdquo Journal ofSound and Vibration vol 369 pp 246ndash265 2016

[30] W Li M Qiu Z Zhu B Wu and G Zhou ldquoBearing faultdiagnosis based on spectrum images of vibration signalsrdquoMeasurement Science and Technology vol 27 no 3 p 0350052016

[31] LMaaten andGHinton ldquoVisualizing data using t-snerdquo Journalof Machine Learning Research vol 9 pp 2579ndash2605 2008

[32] S K Lee and P R White ldquoHigher-order time-frequency analy-sis and its application to fault detection in rotating machineryrdquoMechanical Systems and Signal Processing vol 11 no 4 pp 637ndash650 1997

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