Research ArticleRolling Bearing Degradation State Identification Based onLPP Optimized by GA
He Yu Hong-ru Li Zai-ke Tian and Wei-guo Wang
Mechanical Engineering College No 97 Heping West Road Shijiazhuang 050003 China
Correspondence should be addressed to He Yu 13832329446163com
Received 5 May 2016 Accepted 12 July 2016
Academic Editor Hyeong Joon Ahn
Copyright copy 2016 He Yu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In view of the problem that the actual degradation status of rolling bearing has a poor distinguishing characteristic and strongfuzziness a rolling bearing degradation state identification method based on multidomain feature fusion and dimension reductionof manifold learning combined with GG clustering is proposed Firstly the rolling bearing all-life data is preprocessed by localcharacteristic-scale decomposition (LCD) and six typical features including relative energy spectrum entropy (LREE) relativesingular spectrum entropy (LRSE) two-element multiscale entropy (TMSE) standard deviation (STD) RMS and root-squareamplitude (XR) are extracted and compose the original multidomain feature set And then locally preserving projection (LPP)is utilized to reduce dimension of original fusion feature set and genetic algorithm is applied to optimize the process of featurefusion Finally fuzzy recognition of rolling bearing degradation state is carried out by GG clustering and the principle of maximummembership degree and excellent performance of the proposed method is validated by comparing the recognition accuracy of LPPand GA-LPP
1 Introduction
Rolling element bearings are one of the most importantcomponents for carrying heavy loads and providing constantrotational speed in rotating machines [1] With continuousoperation of rotating machines for a long time rolling bear-ingsrsquo performance condition is changing all the time and thataffects performance stability of the whole machine directlyTherefore there is a practical significance for improvingrotating machinesrsquo service life by rolling bearing degradationstate identification in real time
The fault feature extracted from vibration signals isanalyzed to determine the bearing state [2] And fault featureextraction is the basis of realizing rolling bearing degradationstate recognition Scientific degradation features can charac-terize the degradation degree of rolling elements accuratelyand stably Degradation features are mainly selected formtime domain frequency domain time-frequency analysisand signal complexity aspects Considering that the actualvibration signal of rolling bearings is nonlinear and nonsta-tionary the ability of time and frequency domain statisticsto characterize different degradation states of the samebearings is relatively poor For instance kurtosis is insensitive
to initial damage [3] and it can hardly characterize theslight degradation state exactly These years the informationentropy theory is widely used in signal processing and faultdiagnosis and it develops into different forms of entropy withdifferent properties such as approximate entropy (ApEn)sample entropy (SampEn) multiscale entropy (MSE) spatialinformation entropy (SIE) and fuzzy entropy (FuzzyEn) [45] These entropy features apply nonlinear dynamics theorywhich is different from traditional time-domain indexes inhealth monitoring and fault identification and have madeachievements Compared with the preset fault pattern recog-nition rolling bearing degradation state recognition in itswhole life is more ambiguous and complex However a singlefeature of vibration signals can only reflect fault character-istics of rotating machines at a certain fault degree and thiscan result in problems such as recognition inaccuracy systeminstability and ambiguous recognition results [6] To addressthese problems multidomain feature fusion is widely used indegradation state recognition and fault prediction of rotatingmachines [7 8]
However high-dimensional feature vector composedby multidomain features inevitably has the problems ofinformation redundancy and characteristic conflict and the
Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2016 Article ID 9281098 10 pageshttpdxdoiorg10115520169281098
2 International Journal of Rotating Machinery
effective information is easy to be submerged by high-dimensional data [9] Moreover the use of high-dimensionaldata leads to a sharp increase in the amount of calculationthat is not conductive to the real-time identification of rollingbearing degradation state Manifold learning theory has theability to identify low-dimensional nonlinear structure whichis hidden in high-dimensional data and thus in recentyears those manifold learning algorithms including locallylinear embedding (LLE) locally preserving projection (LPP)isometric feature mapping (IsoMap) and Laplacian eigen-maps (LE) [10] By the neighbor graphs obtained from high-dimensional features LPP algorithm can gain its projectionin the low-dimensional space In this way fusion and reduc-tion of high-dimensional data are achieved Compared withIsoMap and LLE LPP has an advantage of simple calculationand fast processing speed The results of LPP are closelyrelated to nearest neighbor parameters that have no definitecriterion Therefore the optimized parameters are obtainedby repeated experiments Reference [11] proposed a modifiedkernel distance measure sensitivity factor to measure theability that fault features characterize different fault patternsIn view of this LPP algorithm can be optimized by taking thesensitivity factor as object function When the factor reachesits maximum the effect of LPP feature fusion is best
Considering that the actual rolling bearing degradationstates perform strong fuzziness and the boundaries of dif-ferent degradation states are difficult to determine FuzzyC-means (FCM) clustering [12] and Gustafson-Kessel (GK)clustering [13] are widely used in fault diagnosis And Gath-Geva (GG) clustering improves FCM and GK algorithm byfuzzymaximum likelihood estimation distance norm and theclustering effect is better [14]
Based on the above analysis a rolling bearing degradationstate identification method based on fusion and dimensionreduction of multidomain features and GG clustering is putforward in this paper Six features computed from informa-tion entropy and time domain are fused by LPP optimizedby genetic algorithm (GA-LPP) in order to separate thetraining points of different degradation degrees as clear aspossible Finally degradation state recognition is realized byGG clustering and the principle of maximummembership
2 Multidomain Feature Extraction
21 Time-Domain Features In order to fully characterizedifferent degradation states of rolling bearings multi-time-domain features are needed to analyze Common time-domain indexes include mean standard deviation (STD)root mean square (RMS) root- square amplitude skewnesspeak to peak waveform index pulse index margin indexpartial slope index and kurtosisThese features are examinedfrom three aspects of ability to follow degradation trendmonotonicity and data smoothing Three features includingSTD 119909std RMS 119909rms and root-square amplitude 119909
119903are
selected to compose a three-dimensional feature matrix asfollows
1198831= [119909std 119909rms 119909119903] (1)
22 Entropy Features Combined with LCD theory [4] rel-ative entropy theory and multivariate multiscale entropytheory the entropy features including LREE LRSE andTMSE are constructed below
221 LREE and LRSE (1) According to the LCDnoise reduc-tion criterion guided by the mutual correlation coefficient[15] the vibration signal is decomposed and reconstructedSuppose that 119872 samples in degradation state 119909
119894(119905) (119894 =
1 2 119872) and a single sample 119909119887(119905) in normal state are
acquired from the reconstructed signal(2) With the development of rolling bearing degradation
state the energy at characteristic frequency and its multipli-cationswill become larger in the frequency spectrumwhich isobtained by LCD andHilbert transform For rolling bearingsdifferent fault modes have different vibration characteris-tics For a certain rolling bearing fault mode just as innerring pitting its vibration characteristic frequency and itsfrequency multiplication can be calculated by the followingformula
222 TMSE Through LCD there is enough degradationstate information in the first two signal components whosecross-correlation coefficient is higher than others For thetwo components whose sequence length is 119873 after coarsegrain two-element embedding reconstruction compositedelay vectors and thresholds setting assuming that the twocomposite delay vectorsrsquo embedding dimensions are 119898 and119898 + 1 the conditional probabilities are respectively 119875119898(119903)and 119875
119898+1
(119903) when similar capacity limit is 119903 TMSE canbe expressed as the natural logarithm of the conditionalprobabilitiesrsquo ratio
TMSE (119872 120582 119903119873) = ln [
119875119898
(119903)
119875119898+1
(119903)
] (9)
where119872 is embedding vector and 120582 is delay vectorThe above three kinds of entropy features constitute
another three-dimensional feature matrix which can beexpressed as
1198832= [119909LREE 119909LRSE 119909TMSE] (10)
Above all entropy features and time-domain featuresconstitute a six-dimensional multidomain feature matrix
119883 = [1198831 1198832] (11)
3 Optimized LPP Based on GA
31 The Principle of LPP LPP algorithm can retain thenonlinear structure and local characteristics inside the datawhen it is applied for high-dimensional data reduction Thealgorithm principle can be shown as below [16]
For 119899 data samples with 119863 dimensional space 119883 =
1199091 1199092 119909
119899 the matrix 119885 = 119911
1 1199112 119911
119899 is its low-
dimensional space samples where 119911119894(119894 = 1 2 119889) is a 119889
dimensional vector (119889 ≪ 119863) The similarity matrix can bedefined by the following formula
119895are the nearest neighbor points and 119905 is a
constantLPP algorithm can be achieved by solving the following
optimization problem
argmin119882
sum
119894119895
(119882119879
119909119894minus 119882119879
119909119895)119876119894119895
= argmin119882
119882119879
119883119871119883119879
119882
(13)
which needs to satisfy 119882119879
119883119863119883119879
119882 = 1 and 119871 = 119863 minus
119876 is Laplace operator The matrix 119863119894119894
= sum119895119876119894119895reflects
the density of the data distribution Then the transformmatrix can be calculated by solving the generalized eigenvaluedecomposition problem
119883119871119883119879
119882 = 120582119883119863119883119879
119882 (14)
In the above formula the matrix 119883119863119883119879 is sometimes
a singular case For this problem the feature set is usuallyprojected onto a PCA subspace and in this way the singularitycan be eliminated And then the following linearmapping canbe obtained
119909 997888rarr 119911 = 119882119879
119909
119882 = 119882PCA119882LPP(15)
32 Kernel Space Measure Sensitivity Factor In order toevaluate the distinction effect of different degradation statesby training samples after fusion and dimension reductionZheyuan et al [17] propose that distance between differenttypes of samples in kernel space is taken as the basis of featureevaluation However in clustering analysis clustering centerselection not only depends on the distance but also dependson the degree of aggregation of the same type of pointsThere-fore reference [11] takes the ratio of different typesrsquo distanceand divergence of the same type as the measure factor inkernel space And this factor is regarded as the distinguishingcriterion for high accuracy The Gaussian radial basis kernelfunction is selected to calculate the distance between 119901
1and
1199012 The form is as follows
119870(119909 119910) = exp(minus
1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
21205902
) (16)
Then the distance between two points can be expressedas
119863119896(1199011 1199012) = radic2 minus 2119870 (119901
1 1199012) (17)
On this basis the average distance between training samplesof type 119886 and type 119887 can be calculated as follows
119863119886119887
119896=
1
119873119886119873119887
119873119886
sum
119894=1
119873119887
sum
119895=1
119863(119909119886
119894 119909119887
119895) (18)
where 119886 = 1 2 119862 119887 = 1 2 119862 119862 is the number ofsample categories 119873
119886and 119873
119887are the number of samples of
type 119886 and type 119887
4 International Journal of Rotating Machinery
The average distance between different sample categoriesis
119869119896=
1
119862 (119862 minus 1)
119862
sum
119886=1
119862
sum
119887=1
119863119886119887
119896 (19)
The divergence of the same sample category can beexpressed as
119878119896=
1
119862
119862
sum
119886=1
(
1
119873119886
119873119886
sum
119894=1
119863119896(119909119886
119894 119909119886)) (20)
where 119909119886is average of training samples of category 119886
According to the definition the kernel space measuresensitivity factor is
120576 =
119869119896
119878119896
(21)
33 Optimization Based on GA In order to make the fusionfeatures gained from LPP dimension reduction distinguishdifferent degradation states better genetic algorithm (GA) isapplied to optimize the kernel space where there are kindsof training samples GA is a newly developing algorithm tosearch an optimal solution The process of GA algorithmmainly includes population initialization crossover muta-tion fitness calculation (individual evaluation) and selection(population replacement)The kernel spacemeasure sensitiv-ity factor is taken as the fitness function for optimization andthe optimal individual is the case where the discrimination ofdifferent degradation states is highest
Studies have shown that the clustering effect of LPPfusion features will change along with the changing kernelspace In the interest of finding the optimal kernel spaceall training samples need to do affine transformation Take3D fusion features as an example one training point is setas 1198751(1199090 1199100 1199110) and affine transform angles are set as 120579
1isin
[0 2120587] and 1205792isin [0 2120587] So the affine transformation matrix
is
119860
=
[
[
[
[
[
[
sin 1205791cos 1205792sin 1205791sin 1205792cos 1205791
0
1 0 0 minus sin 1205791cos 1205792
0 1 0 minus sin 1205791sin 1205792
0 0 1 minus cos 1205791
]
]
]
]
]
]
(22)
The new sample feature points after kernel space transfor-mation can be computed by the following equation
119883 = 119860minus1
119887 (23)
The two affine transform angles are used as the trainingentity and the individuals are randomly generated to com-plete initialization By the optimization process of GA thetraining sample clustering effect is found to be the best
4 GG Clustering Algorithm
For the training sample set 119883 = 1198831 1198832 119883
119873 it is
assumed that each sample is made up of 119889 characteristics119883119896
= 1199091198961 1199091198962 119909
119896119889 After initialization all samples are
divided into 119862 categories namely the number of clusteringclassifications is 119862 (2 le 119862 le 119873) The clustering centers ofall categories are 119881
119894= V1 V2 V
119894 and the membership
matrix is 119880 = 119906119894119896119862times119873
The element 119906119894119896
isin [0 1] representsthe membership degree of the 119896 training sample to the 119894
degradation state (1 le 119894 le 119862) In GG algorithm the followingobjective function can reach the minimum value with theiterative adjustment of 119880 and 119881
119869119898(119880 119881) =
119862
sum
119894=1
119873
sum
119896=1
(120583119894119896)119898
1198632
119894119896 (24)
where119898 is the weighted index generally taken to 2Different from FCM clustering119863
119894119896indicates the distance
measure calculated by the covariance matrix in GG cluster-ing In that way the data samples of different directions andshapes can be reflected effectively
5 The Process of DegradationState Identification
The original vibration signal is preprocessed by LCD Thetime-domain features of STD RMS and root-square ampli-tude and the entropy features of LREE LRSE and TMSE areextracted from the selected signal components to composethe original characteristic set The degradation state recogni-tion processes are as shown in Figure 1
The degradation state recognition algorithmmainly con-tains the following key steps
(i) LCD PretreatmentAccording to the cross-correlationcoefficient between the LCD components and theoriginal signal the useful components can be chosenConsidering the amount of information existing incomponents and the time of computation the firsttwo components whose coefficient is higher thanothers are selected for further analysis after manytests
(ii) Feature Extraction and Fusion Six-dimensional mul-tiple domain features are fused by LPP algorithmand the intrinsic dimension is three according to themaximum likelihood estimationTherefore the targetdimension of feature fusion is set as three On thebasis of the maximum sensitive factor principle thefusion features are optimized by GA to find the bestkernel space for clustering analysis
(iii) The clustering centers are determined by GG algo-rithm and the rolling bearing degradation identi-fication is achieved by the principle of maximummembership degree
6 Instance Verification
61 Experimental Platform andData Preprocessing Thebear-ing full-life data used in this paper comes from Hangzhou
International Journal of Rotating Machinery 5
Training samples of all-
life data
Testing samples
LCD process
STD
RMS
XR
LREE
LRSE
TMSE
Eigenvector 1
Eigenvector 2
Eigenvector 3
GGclustering
Membership matrix
Normal state
Slight degradation
Severe degradation
Failure state
Component selection by LCD preprocessing Multidomain feature fusion based on GA-LPP Degradation state recognition based on GG
clustering
Degradation state
identification
Maximum membership
principle
1
2
3
4
Figure 1 Flow chart of the identification method
Table 1 The experimental parameters
Motor speed Samplinginterval Sampling time Sampling
frequency1500 rmin 10min 1 s 256 kHz
bearing test and research center [18] As is shown in Fig-ure 2(a) the test platform mainly consists of a ABLT-1Abearing test machine a signal acquisition module and statemonitoring equipment As Figure 2(b) shows four CA-YD-139 acceleration sensors are respectively fixed up on fourbearing test stations and connected to DH-5920 dynamicsignal acquisition instrument Four sets of rolling bearingscan be intensively tested andmultiple sets of full-life vibrationdata can be stored simultaneouslyWhat ismore four thermalresistors and a YD-1 acceleration sensor are connected with asignal amplifier to monitor the operating parameters Whenthe index exceeds the alarm threshold the test machine willstop working
Deep groove ball bearings are widely used in rotatingmachinery There is practical significance in engineeringtaking typical type of 6204 bearing as testing object The realbearing in normal state is shown in Figure 3(a) The specificparameters are set as shown in Table 1
When the test bench running time reaches 9600minutesthe machine is shut down Inner ring pitting occurs in thebearing at number 4 station and that result in bearing failure(as shown in Figure 3(b))
The collected 960 groups of vibration data record thewhole process of rolling bearing from normal state to failurestate Figure 4 shows the real-time monitoring curves ofaverage amplitude versus time which reflect different degra-dation states of rolling bearing clearly According to thechange of curve amplitude and curvature the rolling bearingperformance variation can be initially divided into four statesnormal state slight degradation severe degradation andfailure state The details are presented in Table 2
The original signal is preprocessed by LCD to get 10intrinsic scale components (ISCs) and the first 5 ISCs are
Table 2 The division of degradation state of rolling bearingperformance
shown in Figure 5 Further the cross-correlation coefficientbetween each component and the original signal is calculatedand the value relation is as follows
ISC1gt ISC
3gt ISC
2gt ISC
4gt ISC
5 (25)
What is more there are only the first and the third ISC whosecoefficient is more than 05 respectively 06487 and 05395Therefore the two components are taken as signal source fordegradation feature extraction
62 Degradation Feature Fusion and Optimization Accord-ing to the degradation state division in Table 2 100 groups ofnormal data 100 groups of slight degradation data 60 groupsof severe degradation data and 30 groups of failure dataare selected as training samples The characteristic indexesof different degradation states are extracted and normalizedrespectively The 3D time-domain feature points are shownin Figure 6 In the bearing degradation process from normalstate to failure state these three features are monotonicallyincreasing and the effect of failure state distinguishing isobvious However the points of the other three degradationstates aremixing severely and cannot be distinguished clearlyAlthough the time-domain features such as RMS are easy toget and have good stability to characterize degradation statesliterature [19] indicates that these time-domain features arenot sensitive to early bearing fault including slight degra-dation and severe degradation until bearing failure occursWhat is more reference [20] points out that rolling bearingsrsquovibration signals present nonlinear characteristics and thesethree traditional time-domain features are similar and can
6 International Journal of Rotating Machinery
ABLT-1A bearing
test machine
SensorsSignal
acquisition module
(a)
ThermistorsCA-YD-139acceleration
sensors
YD-1acceleration
sensors
No 1 No 2 No 3 No 4
(b)
Figure 2 Bearing life experiment layout (a) complete machine and (b) sensors
(a) (b)
Figure 3 6204 Bearing (a) normal state and (b) inner ring pitting
0 100 200 300 400 500 600 700 800 900 1000
008
009
01
011
012
013
014
015
016
017
Aver
age a
mpl
itude
Time (1 data point = 10 minutes)
Normal stateSlight degradation
Severe degradationFailure state
Figure 4 The preliminary division of degradation state based onaverage amplitude
hardly make an accurate evaluation of the early degradationstates of the bearings These arguments explain clearly whythe other three degradation states except for the failure one
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05minus02
minus02
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
001
Time (s)
ISC 1
minus02
minus02
ISC 2
ISC 3
ISC 4
ISC 5
minus01
Figure 5 The first five-order components of LCD
are mixed severely and cannot be distinguished by 3D time-domain features
Similarly the 3D complexity feature points made up ofentropy indexes of LREE LRSE and TMSE (scale factor is
International Journal of Rotating Machinery 7
002 04
0608
1
002
0406
081
0
02
04
06
08
1
STDRMS
XR
Normal stateSlight degradation
Severe degradationFailure state
Figure 6 Space distribution of time-domain features
002
0406
081
002
0406
081
0
02
04
06
08
1
LREELRSE
TMSE
Normal stateSlight degradation
Severe degradationFailure state
Figure 7 Space distribution of entropy features
15) are shown in Figure 7 The entropy vector can distinguishnormal state slight degradation and severe degradation onthe whole Nevertheless in the failure state the trainingsamplesrsquo clustering effect is unsatisfying Reference [21]demonstrates that entropy indexes are sole dependent on theprobability distribution of the event occurrence in bearingfault signals They are sensitive to the degradation statechanging but are more susceptible to spurious vibrationsWhen the bearing comes to failure state the violent conditionchanging will make the vibration signals mixed with a lot ofspurious components and the entropy features cannot stablycharacterize the failure state of bearings Therefore the 3Dentropy features at failure state show strong discreteness inFigure 7
In order to improve the discrimination effect of dif-ferent degradation states the above time-domain featuresand entropy features need to be fused Therefore the six-dimensional multidomain feature vectors are input to the
0020
05
001020304
Eigenvector XEigenvector Y
Eige
nvec
tor Z
minus01
minus02
minus03
minus05
minus1
minus02
minus04
minus06
Normal stateSlight degradation
Severe degradationFailure state
Figure 8 Space distribution of LPP fusion features
LPP for feature fusion and dimension reduction In orderto ensure the information exchanging among the neighbor-hoods the neighborhood number 119896 should not be too smallyet if 119896 is too large the local features can be incompleteGenerally analyzed the size of 119896 should be between 119889 and119873 where 119889 is the intrinsic dimension and119873 is the number oftraining samples in each category In this paper 119889 = 3 and119873 = 30 Thus 3 lt 119896 lt 30
The clustering effect is better when 119896 = 7 that is presentedin Figure 8 Compared with the time-domain features andthe entropy features the degradation state distinguishingability of the LPP fusion features is better and the clusteringeffects of normal state slight degradation and failure stateare satisfying But the robustness of fusion features in severedegradation state is relatively poor and this results in the factthat the same severe degradation state is divided into twosample partsMeanwhile the sample class spacing is relativelysmall and the clustering effect is not good So the process offeature fusion needs to be optimized
The kernel space measure sensitive factor is taken as theobjective function According to formula (22) and formula(23) the kernel space is optimized byGA so that the factor hasamaximumvalue In order to improve the convergence speedand ensure the search quality the population size is set as 119873= 20sim200 After several experiments 119873 = 30 The larger thecrossover probability is the higher the loss rate of excellentresults is But when the probability is too small the searchwill be blocked In general crossover probability 119875
119888= 06sim
10 and here it is 08 Mutation probability generally shouldnot be too large otherwise GA will become a random searchmethod and the precision and speed of convergence will beinfluenced Therefore the mutation probability 119875
119898= 003
As shown in Figure 9 after 26 iterations the kernelspace measure sensitivity factor tends to be stable andthe maximum value is achieved And the optimized affinetransformation angles are 120579
1= 14910 and 120579
2= 38532
Figure 10 presents the space distribution of optimized fusion
8 International Journal of Rotating Machinery
5 10 15 20 25 30 35 400
01
02
03
04
05
06
07
08
09
1
Iterations
Obj
ectiv
e fun
ctio
n va
lue
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
effective information is easy to be submerged by high-dimensional data [9] Moreover the use of high-dimensionaldata leads to a sharp increase in the amount of calculationthat is not conductive to the real-time identification of rollingbearing degradation state Manifold learning theory has theability to identify low-dimensional nonlinear structure whichis hidden in high-dimensional data and thus in recentyears those manifold learning algorithms including locallylinear embedding (LLE) locally preserving projection (LPP)isometric feature mapping (IsoMap) and Laplacian eigen-maps (LE) [10] By the neighbor graphs obtained from high-dimensional features LPP algorithm can gain its projectionin the low-dimensional space In this way fusion and reduc-tion of high-dimensional data are achieved Compared withIsoMap and LLE LPP has an advantage of simple calculationand fast processing speed The results of LPP are closelyrelated to nearest neighbor parameters that have no definitecriterion Therefore the optimized parameters are obtainedby repeated experiments Reference [11] proposed a modifiedkernel distance measure sensitivity factor to measure theability that fault features characterize different fault patternsIn view of this LPP algorithm can be optimized by taking thesensitivity factor as object function When the factor reachesits maximum the effect of LPP feature fusion is best
Considering that the actual rolling bearing degradationstates perform strong fuzziness and the boundaries of dif-ferent degradation states are difficult to determine FuzzyC-means (FCM) clustering [12] and Gustafson-Kessel (GK)clustering [13] are widely used in fault diagnosis And Gath-Geva (GG) clustering improves FCM and GK algorithm byfuzzymaximum likelihood estimation distance norm and theclustering effect is better [14]
Based on the above analysis a rolling bearing degradationstate identification method based on fusion and dimensionreduction of multidomain features and GG clustering is putforward in this paper Six features computed from informa-tion entropy and time domain are fused by LPP optimizedby genetic algorithm (GA-LPP) in order to separate thetraining points of different degradation degrees as clear aspossible Finally degradation state recognition is realized byGG clustering and the principle of maximummembership
2 Multidomain Feature Extraction
21 Time-Domain Features In order to fully characterizedifferent degradation states of rolling bearings multi-time-domain features are needed to analyze Common time-domain indexes include mean standard deviation (STD)root mean square (RMS) root- square amplitude skewnesspeak to peak waveform index pulse index margin indexpartial slope index and kurtosisThese features are examinedfrom three aspects of ability to follow degradation trendmonotonicity and data smoothing Three features includingSTD 119909std RMS 119909rms and root-square amplitude 119909
119903are
selected to compose a three-dimensional feature matrix asfollows
1198831= [119909std 119909rms 119909119903] (1)
22 Entropy Features Combined with LCD theory [4] rel-ative entropy theory and multivariate multiscale entropytheory the entropy features including LREE LRSE andTMSE are constructed below
221 LREE and LRSE (1) According to the LCDnoise reduc-tion criterion guided by the mutual correlation coefficient[15] the vibration signal is decomposed and reconstructedSuppose that 119872 samples in degradation state 119909
119894(119905) (119894 =
1 2 119872) and a single sample 119909119887(119905) in normal state are
acquired from the reconstructed signal(2) With the development of rolling bearing degradation
state the energy at characteristic frequency and its multipli-cationswill become larger in the frequency spectrumwhich isobtained by LCD andHilbert transform For rolling bearingsdifferent fault modes have different vibration characteris-tics For a certain rolling bearing fault mode just as innerring pitting its vibration characteristic frequency and itsfrequency multiplication can be calculated by the followingformula
222 TMSE Through LCD there is enough degradationstate information in the first two signal components whosecross-correlation coefficient is higher than others For thetwo components whose sequence length is 119873 after coarsegrain two-element embedding reconstruction compositedelay vectors and thresholds setting assuming that the twocomposite delay vectorsrsquo embedding dimensions are 119898 and119898 + 1 the conditional probabilities are respectively 119875119898(119903)and 119875
119898+1
(119903) when similar capacity limit is 119903 TMSE canbe expressed as the natural logarithm of the conditionalprobabilitiesrsquo ratio
TMSE (119872 120582 119903119873) = ln [
119875119898
(119903)
119875119898+1
(119903)
] (9)
where119872 is embedding vector and 120582 is delay vectorThe above three kinds of entropy features constitute
another three-dimensional feature matrix which can beexpressed as
1198832= [119909LREE 119909LRSE 119909TMSE] (10)
Above all entropy features and time-domain featuresconstitute a six-dimensional multidomain feature matrix
119883 = [1198831 1198832] (11)
3 Optimized LPP Based on GA
31 The Principle of LPP LPP algorithm can retain thenonlinear structure and local characteristics inside the datawhen it is applied for high-dimensional data reduction Thealgorithm principle can be shown as below [16]
For 119899 data samples with 119863 dimensional space 119883 =
1199091 1199092 119909
119899 the matrix 119885 = 119911
1 1199112 119911
119899 is its low-
dimensional space samples where 119911119894(119894 = 1 2 119889) is a 119889
dimensional vector (119889 ≪ 119863) The similarity matrix can bedefined by the following formula
119895are the nearest neighbor points and 119905 is a
constantLPP algorithm can be achieved by solving the following
optimization problem
argmin119882
sum
119894119895
(119882119879
119909119894minus 119882119879
119909119895)119876119894119895
= argmin119882
119882119879
119883119871119883119879
119882
(13)
which needs to satisfy 119882119879
119883119863119883119879
119882 = 1 and 119871 = 119863 minus
119876 is Laplace operator The matrix 119863119894119894
= sum119895119876119894119895reflects
the density of the data distribution Then the transformmatrix can be calculated by solving the generalized eigenvaluedecomposition problem
119883119871119883119879
119882 = 120582119883119863119883119879
119882 (14)
In the above formula the matrix 119883119863119883119879 is sometimes
a singular case For this problem the feature set is usuallyprojected onto a PCA subspace and in this way the singularitycan be eliminated And then the following linearmapping canbe obtained
119909 997888rarr 119911 = 119882119879
119909
119882 = 119882PCA119882LPP(15)
32 Kernel Space Measure Sensitivity Factor In order toevaluate the distinction effect of different degradation statesby training samples after fusion and dimension reductionZheyuan et al [17] propose that distance between differenttypes of samples in kernel space is taken as the basis of featureevaluation However in clustering analysis clustering centerselection not only depends on the distance but also dependson the degree of aggregation of the same type of pointsThere-fore reference [11] takes the ratio of different typesrsquo distanceand divergence of the same type as the measure factor inkernel space And this factor is regarded as the distinguishingcriterion for high accuracy The Gaussian radial basis kernelfunction is selected to calculate the distance between 119901
1and
1199012 The form is as follows
119870(119909 119910) = exp(minus
1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
21205902
) (16)
Then the distance between two points can be expressedas
119863119896(1199011 1199012) = radic2 minus 2119870 (119901
1 1199012) (17)
On this basis the average distance between training samplesof type 119886 and type 119887 can be calculated as follows
119863119886119887
119896=
1
119873119886119873119887
119873119886
sum
119894=1
119873119887
sum
119895=1
119863(119909119886
119894 119909119887
119895) (18)
where 119886 = 1 2 119862 119887 = 1 2 119862 119862 is the number ofsample categories 119873
119886and 119873
119887are the number of samples of
type 119886 and type 119887
4 International Journal of Rotating Machinery
The average distance between different sample categoriesis
119869119896=
1
119862 (119862 minus 1)
119862
sum
119886=1
119862
sum
119887=1
119863119886119887
119896 (19)
The divergence of the same sample category can beexpressed as
119878119896=
1
119862
119862
sum
119886=1
(
1
119873119886
119873119886
sum
119894=1
119863119896(119909119886
119894 119909119886)) (20)
where 119909119886is average of training samples of category 119886
According to the definition the kernel space measuresensitivity factor is
120576 =
119869119896
119878119896
(21)
33 Optimization Based on GA In order to make the fusionfeatures gained from LPP dimension reduction distinguishdifferent degradation states better genetic algorithm (GA) isapplied to optimize the kernel space where there are kindsof training samples GA is a newly developing algorithm tosearch an optimal solution The process of GA algorithmmainly includes population initialization crossover muta-tion fitness calculation (individual evaluation) and selection(population replacement)The kernel spacemeasure sensitiv-ity factor is taken as the fitness function for optimization andthe optimal individual is the case where the discrimination ofdifferent degradation states is highest
Studies have shown that the clustering effect of LPPfusion features will change along with the changing kernelspace In the interest of finding the optimal kernel spaceall training samples need to do affine transformation Take3D fusion features as an example one training point is setas 1198751(1199090 1199100 1199110) and affine transform angles are set as 120579
1isin
[0 2120587] and 1205792isin [0 2120587] So the affine transformation matrix
is
119860
=
[
[
[
[
[
[
sin 1205791cos 1205792sin 1205791sin 1205792cos 1205791
0
1 0 0 minus sin 1205791cos 1205792
0 1 0 minus sin 1205791sin 1205792
0 0 1 minus cos 1205791
]
]
]
]
]
]
(22)
The new sample feature points after kernel space transfor-mation can be computed by the following equation
119883 = 119860minus1
119887 (23)
The two affine transform angles are used as the trainingentity and the individuals are randomly generated to com-plete initialization By the optimization process of GA thetraining sample clustering effect is found to be the best
4 GG Clustering Algorithm
For the training sample set 119883 = 1198831 1198832 119883
119873 it is
assumed that each sample is made up of 119889 characteristics119883119896
= 1199091198961 1199091198962 119909
119896119889 After initialization all samples are
divided into 119862 categories namely the number of clusteringclassifications is 119862 (2 le 119862 le 119873) The clustering centers ofall categories are 119881
119894= V1 V2 V
119894 and the membership
matrix is 119880 = 119906119894119896119862times119873
The element 119906119894119896
isin [0 1] representsthe membership degree of the 119896 training sample to the 119894
degradation state (1 le 119894 le 119862) In GG algorithm the followingobjective function can reach the minimum value with theiterative adjustment of 119880 and 119881
119869119898(119880 119881) =
119862
sum
119894=1
119873
sum
119896=1
(120583119894119896)119898
1198632
119894119896 (24)
where119898 is the weighted index generally taken to 2Different from FCM clustering119863
119894119896indicates the distance
measure calculated by the covariance matrix in GG cluster-ing In that way the data samples of different directions andshapes can be reflected effectively
5 The Process of DegradationState Identification
The original vibration signal is preprocessed by LCD Thetime-domain features of STD RMS and root-square ampli-tude and the entropy features of LREE LRSE and TMSE areextracted from the selected signal components to composethe original characteristic set The degradation state recogni-tion processes are as shown in Figure 1
The degradation state recognition algorithmmainly con-tains the following key steps
(i) LCD PretreatmentAccording to the cross-correlationcoefficient between the LCD components and theoriginal signal the useful components can be chosenConsidering the amount of information existing incomponents and the time of computation the firsttwo components whose coefficient is higher thanothers are selected for further analysis after manytests
(ii) Feature Extraction and Fusion Six-dimensional mul-tiple domain features are fused by LPP algorithmand the intrinsic dimension is three according to themaximum likelihood estimationTherefore the targetdimension of feature fusion is set as three On thebasis of the maximum sensitive factor principle thefusion features are optimized by GA to find the bestkernel space for clustering analysis
(iii) The clustering centers are determined by GG algo-rithm and the rolling bearing degradation identi-fication is achieved by the principle of maximummembership degree
6 Instance Verification
61 Experimental Platform andData Preprocessing Thebear-ing full-life data used in this paper comes from Hangzhou
International Journal of Rotating Machinery 5
Training samples of all-
life data
Testing samples
LCD process
STD
RMS
XR
LREE
LRSE
TMSE
Eigenvector 1
Eigenvector 2
Eigenvector 3
GGclustering
Membership matrix
Normal state
Slight degradation
Severe degradation
Failure state
Component selection by LCD preprocessing Multidomain feature fusion based on GA-LPP Degradation state recognition based on GG
clustering
Degradation state
identification
Maximum membership
principle
1
2
3
4
Figure 1 Flow chart of the identification method
Table 1 The experimental parameters
Motor speed Samplinginterval Sampling time Sampling
frequency1500 rmin 10min 1 s 256 kHz
bearing test and research center [18] As is shown in Fig-ure 2(a) the test platform mainly consists of a ABLT-1Abearing test machine a signal acquisition module and statemonitoring equipment As Figure 2(b) shows four CA-YD-139 acceleration sensors are respectively fixed up on fourbearing test stations and connected to DH-5920 dynamicsignal acquisition instrument Four sets of rolling bearingscan be intensively tested andmultiple sets of full-life vibrationdata can be stored simultaneouslyWhat ismore four thermalresistors and a YD-1 acceleration sensor are connected with asignal amplifier to monitor the operating parameters Whenthe index exceeds the alarm threshold the test machine willstop working
Deep groove ball bearings are widely used in rotatingmachinery There is practical significance in engineeringtaking typical type of 6204 bearing as testing object The realbearing in normal state is shown in Figure 3(a) The specificparameters are set as shown in Table 1
When the test bench running time reaches 9600minutesthe machine is shut down Inner ring pitting occurs in thebearing at number 4 station and that result in bearing failure(as shown in Figure 3(b))
The collected 960 groups of vibration data record thewhole process of rolling bearing from normal state to failurestate Figure 4 shows the real-time monitoring curves ofaverage amplitude versus time which reflect different degra-dation states of rolling bearing clearly According to thechange of curve amplitude and curvature the rolling bearingperformance variation can be initially divided into four statesnormal state slight degradation severe degradation andfailure state The details are presented in Table 2
The original signal is preprocessed by LCD to get 10intrinsic scale components (ISCs) and the first 5 ISCs are
Table 2 The division of degradation state of rolling bearingperformance
shown in Figure 5 Further the cross-correlation coefficientbetween each component and the original signal is calculatedand the value relation is as follows
ISC1gt ISC
3gt ISC
2gt ISC
4gt ISC
5 (25)
What is more there are only the first and the third ISC whosecoefficient is more than 05 respectively 06487 and 05395Therefore the two components are taken as signal source fordegradation feature extraction
62 Degradation Feature Fusion and Optimization Accord-ing to the degradation state division in Table 2 100 groups ofnormal data 100 groups of slight degradation data 60 groupsof severe degradation data and 30 groups of failure dataare selected as training samples The characteristic indexesof different degradation states are extracted and normalizedrespectively The 3D time-domain feature points are shownin Figure 6 In the bearing degradation process from normalstate to failure state these three features are monotonicallyincreasing and the effect of failure state distinguishing isobvious However the points of the other three degradationstates aremixing severely and cannot be distinguished clearlyAlthough the time-domain features such as RMS are easy toget and have good stability to characterize degradation statesliterature [19] indicates that these time-domain features arenot sensitive to early bearing fault including slight degra-dation and severe degradation until bearing failure occursWhat is more reference [20] points out that rolling bearingsrsquovibration signals present nonlinear characteristics and thesethree traditional time-domain features are similar and can
6 International Journal of Rotating Machinery
ABLT-1A bearing
test machine
SensorsSignal
acquisition module
(a)
ThermistorsCA-YD-139acceleration
sensors
YD-1acceleration
sensors
No 1 No 2 No 3 No 4
(b)
Figure 2 Bearing life experiment layout (a) complete machine and (b) sensors
(a) (b)
Figure 3 6204 Bearing (a) normal state and (b) inner ring pitting
0 100 200 300 400 500 600 700 800 900 1000
008
009
01
011
012
013
014
015
016
017
Aver
age a
mpl
itude
Time (1 data point = 10 minutes)
Normal stateSlight degradation
Severe degradationFailure state
Figure 4 The preliminary division of degradation state based onaverage amplitude
hardly make an accurate evaluation of the early degradationstates of the bearings These arguments explain clearly whythe other three degradation states except for the failure one
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05minus02
minus02
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
001
Time (s)
ISC 1
minus02
minus02
ISC 2
ISC 3
ISC 4
ISC 5
minus01
Figure 5 The first five-order components of LCD
are mixed severely and cannot be distinguished by 3D time-domain features
Similarly the 3D complexity feature points made up ofentropy indexes of LREE LRSE and TMSE (scale factor is
International Journal of Rotating Machinery 7
002 04
0608
1
002
0406
081
0
02
04
06
08
1
STDRMS
XR
Normal stateSlight degradation
Severe degradationFailure state
Figure 6 Space distribution of time-domain features
002
0406
081
002
0406
081
0
02
04
06
08
1
LREELRSE
TMSE
Normal stateSlight degradation
Severe degradationFailure state
Figure 7 Space distribution of entropy features
15) are shown in Figure 7 The entropy vector can distinguishnormal state slight degradation and severe degradation onthe whole Nevertheless in the failure state the trainingsamplesrsquo clustering effect is unsatisfying Reference [21]demonstrates that entropy indexes are sole dependent on theprobability distribution of the event occurrence in bearingfault signals They are sensitive to the degradation statechanging but are more susceptible to spurious vibrationsWhen the bearing comes to failure state the violent conditionchanging will make the vibration signals mixed with a lot ofspurious components and the entropy features cannot stablycharacterize the failure state of bearings Therefore the 3Dentropy features at failure state show strong discreteness inFigure 7
In order to improve the discrimination effect of dif-ferent degradation states the above time-domain featuresand entropy features need to be fused Therefore the six-dimensional multidomain feature vectors are input to the
0020
05
001020304
Eigenvector XEigenvector Y
Eige
nvec
tor Z
minus01
minus02
minus03
minus05
minus1
minus02
minus04
minus06
Normal stateSlight degradation
Severe degradationFailure state
Figure 8 Space distribution of LPP fusion features
LPP for feature fusion and dimension reduction In orderto ensure the information exchanging among the neighbor-hoods the neighborhood number 119896 should not be too smallyet if 119896 is too large the local features can be incompleteGenerally analyzed the size of 119896 should be between 119889 and119873 where 119889 is the intrinsic dimension and119873 is the number oftraining samples in each category In this paper 119889 = 3 and119873 = 30 Thus 3 lt 119896 lt 30
The clustering effect is better when 119896 = 7 that is presentedin Figure 8 Compared with the time-domain features andthe entropy features the degradation state distinguishingability of the LPP fusion features is better and the clusteringeffects of normal state slight degradation and failure stateare satisfying But the robustness of fusion features in severedegradation state is relatively poor and this results in the factthat the same severe degradation state is divided into twosample partsMeanwhile the sample class spacing is relativelysmall and the clustering effect is not good So the process offeature fusion needs to be optimized
The kernel space measure sensitive factor is taken as theobjective function According to formula (22) and formula(23) the kernel space is optimized byGA so that the factor hasamaximumvalue In order to improve the convergence speedand ensure the search quality the population size is set as 119873= 20sim200 After several experiments 119873 = 30 The larger thecrossover probability is the higher the loss rate of excellentresults is But when the probability is too small the searchwill be blocked In general crossover probability 119875
119888= 06sim
10 and here it is 08 Mutation probability generally shouldnot be too large otherwise GA will become a random searchmethod and the precision and speed of convergence will beinfluenced Therefore the mutation probability 119875
119898= 003
As shown in Figure 9 after 26 iterations the kernelspace measure sensitivity factor tends to be stable andthe maximum value is achieved And the optimized affinetransformation angles are 120579
1= 14910 and 120579
2= 38532
Figure 10 presents the space distribution of optimized fusion
8 International Journal of Rotating Machinery
5 10 15 20 25 30 35 400
01
02
03
04
05
06
07
08
09
1
Iterations
Obj
ectiv
e fun
ctio
n va
lue
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
222 TMSE Through LCD there is enough degradationstate information in the first two signal components whosecross-correlation coefficient is higher than others For thetwo components whose sequence length is 119873 after coarsegrain two-element embedding reconstruction compositedelay vectors and thresholds setting assuming that the twocomposite delay vectorsrsquo embedding dimensions are 119898 and119898 + 1 the conditional probabilities are respectively 119875119898(119903)and 119875
119898+1
(119903) when similar capacity limit is 119903 TMSE canbe expressed as the natural logarithm of the conditionalprobabilitiesrsquo ratio
TMSE (119872 120582 119903119873) = ln [
119875119898
(119903)
119875119898+1
(119903)
] (9)
where119872 is embedding vector and 120582 is delay vectorThe above three kinds of entropy features constitute
another three-dimensional feature matrix which can beexpressed as
1198832= [119909LREE 119909LRSE 119909TMSE] (10)
Above all entropy features and time-domain featuresconstitute a six-dimensional multidomain feature matrix
119883 = [1198831 1198832] (11)
3 Optimized LPP Based on GA
31 The Principle of LPP LPP algorithm can retain thenonlinear structure and local characteristics inside the datawhen it is applied for high-dimensional data reduction Thealgorithm principle can be shown as below [16]
For 119899 data samples with 119863 dimensional space 119883 =
1199091 1199092 119909
119899 the matrix 119885 = 119911
1 1199112 119911
119899 is its low-
dimensional space samples where 119911119894(119894 = 1 2 119889) is a 119889
dimensional vector (119889 ≪ 119863) The similarity matrix can bedefined by the following formula
119895are the nearest neighbor points and 119905 is a
constantLPP algorithm can be achieved by solving the following
optimization problem
argmin119882
sum
119894119895
(119882119879
119909119894minus 119882119879
119909119895)119876119894119895
= argmin119882
119882119879
119883119871119883119879
119882
(13)
which needs to satisfy 119882119879
119883119863119883119879
119882 = 1 and 119871 = 119863 minus
119876 is Laplace operator The matrix 119863119894119894
= sum119895119876119894119895reflects
the density of the data distribution Then the transformmatrix can be calculated by solving the generalized eigenvaluedecomposition problem
119883119871119883119879
119882 = 120582119883119863119883119879
119882 (14)
In the above formula the matrix 119883119863119883119879 is sometimes
a singular case For this problem the feature set is usuallyprojected onto a PCA subspace and in this way the singularitycan be eliminated And then the following linearmapping canbe obtained
119909 997888rarr 119911 = 119882119879
119909
119882 = 119882PCA119882LPP(15)
32 Kernel Space Measure Sensitivity Factor In order toevaluate the distinction effect of different degradation statesby training samples after fusion and dimension reductionZheyuan et al [17] propose that distance between differenttypes of samples in kernel space is taken as the basis of featureevaluation However in clustering analysis clustering centerselection not only depends on the distance but also dependson the degree of aggregation of the same type of pointsThere-fore reference [11] takes the ratio of different typesrsquo distanceand divergence of the same type as the measure factor inkernel space And this factor is regarded as the distinguishingcriterion for high accuracy The Gaussian radial basis kernelfunction is selected to calculate the distance between 119901
1and
1199012 The form is as follows
119870(119909 119910) = exp(minus
1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
21205902
) (16)
Then the distance between two points can be expressedas
119863119896(1199011 1199012) = radic2 minus 2119870 (119901
1 1199012) (17)
On this basis the average distance between training samplesof type 119886 and type 119887 can be calculated as follows
119863119886119887
119896=
1
119873119886119873119887
119873119886
sum
119894=1
119873119887
sum
119895=1
119863(119909119886
119894 119909119887
119895) (18)
where 119886 = 1 2 119862 119887 = 1 2 119862 119862 is the number ofsample categories 119873
119886and 119873
119887are the number of samples of
type 119886 and type 119887
4 International Journal of Rotating Machinery
The average distance between different sample categoriesis
119869119896=
1
119862 (119862 minus 1)
119862
sum
119886=1
119862
sum
119887=1
119863119886119887
119896 (19)
The divergence of the same sample category can beexpressed as
119878119896=
1
119862
119862
sum
119886=1
(
1
119873119886
119873119886
sum
119894=1
119863119896(119909119886
119894 119909119886)) (20)
where 119909119886is average of training samples of category 119886
According to the definition the kernel space measuresensitivity factor is
120576 =
119869119896
119878119896
(21)
33 Optimization Based on GA In order to make the fusionfeatures gained from LPP dimension reduction distinguishdifferent degradation states better genetic algorithm (GA) isapplied to optimize the kernel space where there are kindsof training samples GA is a newly developing algorithm tosearch an optimal solution The process of GA algorithmmainly includes population initialization crossover muta-tion fitness calculation (individual evaluation) and selection(population replacement)The kernel spacemeasure sensitiv-ity factor is taken as the fitness function for optimization andthe optimal individual is the case where the discrimination ofdifferent degradation states is highest
Studies have shown that the clustering effect of LPPfusion features will change along with the changing kernelspace In the interest of finding the optimal kernel spaceall training samples need to do affine transformation Take3D fusion features as an example one training point is setas 1198751(1199090 1199100 1199110) and affine transform angles are set as 120579
1isin
[0 2120587] and 1205792isin [0 2120587] So the affine transformation matrix
is
119860
=
[
[
[
[
[
[
sin 1205791cos 1205792sin 1205791sin 1205792cos 1205791
0
1 0 0 minus sin 1205791cos 1205792
0 1 0 minus sin 1205791sin 1205792
0 0 1 minus cos 1205791
]
]
]
]
]
]
(22)
The new sample feature points after kernel space transfor-mation can be computed by the following equation
119883 = 119860minus1
119887 (23)
The two affine transform angles are used as the trainingentity and the individuals are randomly generated to com-plete initialization By the optimization process of GA thetraining sample clustering effect is found to be the best
4 GG Clustering Algorithm
For the training sample set 119883 = 1198831 1198832 119883
119873 it is
assumed that each sample is made up of 119889 characteristics119883119896
= 1199091198961 1199091198962 119909
119896119889 After initialization all samples are
divided into 119862 categories namely the number of clusteringclassifications is 119862 (2 le 119862 le 119873) The clustering centers ofall categories are 119881
119894= V1 V2 V
119894 and the membership
matrix is 119880 = 119906119894119896119862times119873
The element 119906119894119896
isin [0 1] representsthe membership degree of the 119896 training sample to the 119894
degradation state (1 le 119894 le 119862) In GG algorithm the followingobjective function can reach the minimum value with theiterative adjustment of 119880 and 119881
119869119898(119880 119881) =
119862
sum
119894=1
119873
sum
119896=1
(120583119894119896)119898
1198632
119894119896 (24)
where119898 is the weighted index generally taken to 2Different from FCM clustering119863
119894119896indicates the distance
measure calculated by the covariance matrix in GG cluster-ing In that way the data samples of different directions andshapes can be reflected effectively
5 The Process of DegradationState Identification
The original vibration signal is preprocessed by LCD Thetime-domain features of STD RMS and root-square ampli-tude and the entropy features of LREE LRSE and TMSE areextracted from the selected signal components to composethe original characteristic set The degradation state recogni-tion processes are as shown in Figure 1
The degradation state recognition algorithmmainly con-tains the following key steps
(i) LCD PretreatmentAccording to the cross-correlationcoefficient between the LCD components and theoriginal signal the useful components can be chosenConsidering the amount of information existing incomponents and the time of computation the firsttwo components whose coefficient is higher thanothers are selected for further analysis after manytests
(ii) Feature Extraction and Fusion Six-dimensional mul-tiple domain features are fused by LPP algorithmand the intrinsic dimension is three according to themaximum likelihood estimationTherefore the targetdimension of feature fusion is set as three On thebasis of the maximum sensitive factor principle thefusion features are optimized by GA to find the bestkernel space for clustering analysis
(iii) The clustering centers are determined by GG algo-rithm and the rolling bearing degradation identi-fication is achieved by the principle of maximummembership degree
6 Instance Verification
61 Experimental Platform andData Preprocessing Thebear-ing full-life data used in this paper comes from Hangzhou
International Journal of Rotating Machinery 5
Training samples of all-
life data
Testing samples
LCD process
STD
RMS
XR
LREE
LRSE
TMSE
Eigenvector 1
Eigenvector 2
Eigenvector 3
GGclustering
Membership matrix
Normal state
Slight degradation
Severe degradation
Failure state
Component selection by LCD preprocessing Multidomain feature fusion based on GA-LPP Degradation state recognition based on GG
clustering
Degradation state
identification
Maximum membership
principle
1
2
3
4
Figure 1 Flow chart of the identification method
Table 1 The experimental parameters
Motor speed Samplinginterval Sampling time Sampling
frequency1500 rmin 10min 1 s 256 kHz
bearing test and research center [18] As is shown in Fig-ure 2(a) the test platform mainly consists of a ABLT-1Abearing test machine a signal acquisition module and statemonitoring equipment As Figure 2(b) shows four CA-YD-139 acceleration sensors are respectively fixed up on fourbearing test stations and connected to DH-5920 dynamicsignal acquisition instrument Four sets of rolling bearingscan be intensively tested andmultiple sets of full-life vibrationdata can be stored simultaneouslyWhat ismore four thermalresistors and a YD-1 acceleration sensor are connected with asignal amplifier to monitor the operating parameters Whenthe index exceeds the alarm threshold the test machine willstop working
Deep groove ball bearings are widely used in rotatingmachinery There is practical significance in engineeringtaking typical type of 6204 bearing as testing object The realbearing in normal state is shown in Figure 3(a) The specificparameters are set as shown in Table 1
When the test bench running time reaches 9600minutesthe machine is shut down Inner ring pitting occurs in thebearing at number 4 station and that result in bearing failure(as shown in Figure 3(b))
The collected 960 groups of vibration data record thewhole process of rolling bearing from normal state to failurestate Figure 4 shows the real-time monitoring curves ofaverage amplitude versus time which reflect different degra-dation states of rolling bearing clearly According to thechange of curve amplitude and curvature the rolling bearingperformance variation can be initially divided into four statesnormal state slight degradation severe degradation andfailure state The details are presented in Table 2
The original signal is preprocessed by LCD to get 10intrinsic scale components (ISCs) and the first 5 ISCs are
Table 2 The division of degradation state of rolling bearingperformance
shown in Figure 5 Further the cross-correlation coefficientbetween each component and the original signal is calculatedand the value relation is as follows
ISC1gt ISC
3gt ISC
2gt ISC
4gt ISC
5 (25)
What is more there are only the first and the third ISC whosecoefficient is more than 05 respectively 06487 and 05395Therefore the two components are taken as signal source fordegradation feature extraction
62 Degradation Feature Fusion and Optimization Accord-ing to the degradation state division in Table 2 100 groups ofnormal data 100 groups of slight degradation data 60 groupsof severe degradation data and 30 groups of failure dataare selected as training samples The characteristic indexesof different degradation states are extracted and normalizedrespectively The 3D time-domain feature points are shownin Figure 6 In the bearing degradation process from normalstate to failure state these three features are monotonicallyincreasing and the effect of failure state distinguishing isobvious However the points of the other three degradationstates aremixing severely and cannot be distinguished clearlyAlthough the time-domain features such as RMS are easy toget and have good stability to characterize degradation statesliterature [19] indicates that these time-domain features arenot sensitive to early bearing fault including slight degra-dation and severe degradation until bearing failure occursWhat is more reference [20] points out that rolling bearingsrsquovibration signals present nonlinear characteristics and thesethree traditional time-domain features are similar and can
6 International Journal of Rotating Machinery
ABLT-1A bearing
test machine
SensorsSignal
acquisition module
(a)
ThermistorsCA-YD-139acceleration
sensors
YD-1acceleration
sensors
No 1 No 2 No 3 No 4
(b)
Figure 2 Bearing life experiment layout (a) complete machine and (b) sensors
(a) (b)
Figure 3 6204 Bearing (a) normal state and (b) inner ring pitting
0 100 200 300 400 500 600 700 800 900 1000
008
009
01
011
012
013
014
015
016
017
Aver
age a
mpl
itude
Time (1 data point = 10 minutes)
Normal stateSlight degradation
Severe degradationFailure state
Figure 4 The preliminary division of degradation state based onaverage amplitude
hardly make an accurate evaluation of the early degradationstates of the bearings These arguments explain clearly whythe other three degradation states except for the failure one
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05minus02
minus02
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
001
Time (s)
ISC 1
minus02
minus02
ISC 2
ISC 3
ISC 4
ISC 5
minus01
Figure 5 The first five-order components of LCD
are mixed severely and cannot be distinguished by 3D time-domain features
Similarly the 3D complexity feature points made up ofentropy indexes of LREE LRSE and TMSE (scale factor is
International Journal of Rotating Machinery 7
002 04
0608
1
002
0406
081
0
02
04
06
08
1
STDRMS
XR
Normal stateSlight degradation
Severe degradationFailure state
Figure 6 Space distribution of time-domain features
002
0406
081
002
0406
081
0
02
04
06
08
1
LREELRSE
TMSE
Normal stateSlight degradation
Severe degradationFailure state
Figure 7 Space distribution of entropy features
15) are shown in Figure 7 The entropy vector can distinguishnormal state slight degradation and severe degradation onthe whole Nevertheless in the failure state the trainingsamplesrsquo clustering effect is unsatisfying Reference [21]demonstrates that entropy indexes are sole dependent on theprobability distribution of the event occurrence in bearingfault signals They are sensitive to the degradation statechanging but are more susceptible to spurious vibrationsWhen the bearing comes to failure state the violent conditionchanging will make the vibration signals mixed with a lot ofspurious components and the entropy features cannot stablycharacterize the failure state of bearings Therefore the 3Dentropy features at failure state show strong discreteness inFigure 7
In order to improve the discrimination effect of dif-ferent degradation states the above time-domain featuresand entropy features need to be fused Therefore the six-dimensional multidomain feature vectors are input to the
0020
05
001020304
Eigenvector XEigenvector Y
Eige
nvec
tor Z
minus01
minus02
minus03
minus05
minus1
minus02
minus04
minus06
Normal stateSlight degradation
Severe degradationFailure state
Figure 8 Space distribution of LPP fusion features
LPP for feature fusion and dimension reduction In orderto ensure the information exchanging among the neighbor-hoods the neighborhood number 119896 should not be too smallyet if 119896 is too large the local features can be incompleteGenerally analyzed the size of 119896 should be between 119889 and119873 where 119889 is the intrinsic dimension and119873 is the number oftraining samples in each category In this paper 119889 = 3 and119873 = 30 Thus 3 lt 119896 lt 30
The clustering effect is better when 119896 = 7 that is presentedin Figure 8 Compared with the time-domain features andthe entropy features the degradation state distinguishingability of the LPP fusion features is better and the clusteringeffects of normal state slight degradation and failure stateare satisfying But the robustness of fusion features in severedegradation state is relatively poor and this results in the factthat the same severe degradation state is divided into twosample partsMeanwhile the sample class spacing is relativelysmall and the clustering effect is not good So the process offeature fusion needs to be optimized
The kernel space measure sensitive factor is taken as theobjective function According to formula (22) and formula(23) the kernel space is optimized byGA so that the factor hasamaximumvalue In order to improve the convergence speedand ensure the search quality the population size is set as 119873= 20sim200 After several experiments 119873 = 30 The larger thecrossover probability is the higher the loss rate of excellentresults is But when the probability is too small the searchwill be blocked In general crossover probability 119875
119888= 06sim
10 and here it is 08 Mutation probability generally shouldnot be too large otherwise GA will become a random searchmethod and the precision and speed of convergence will beinfluenced Therefore the mutation probability 119875
119898= 003
As shown in Figure 9 after 26 iterations the kernelspace measure sensitivity factor tends to be stable andthe maximum value is achieved And the optimized affinetransformation angles are 120579
1= 14910 and 120579
2= 38532
Figure 10 presents the space distribution of optimized fusion
8 International Journal of Rotating Machinery
5 10 15 20 25 30 35 400
01
02
03
04
05
06
07
08
09
1
Iterations
Obj
ectiv
e fun
ctio
n va
lue
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
The average distance between different sample categoriesis
119869119896=
1
119862 (119862 minus 1)
119862
sum
119886=1
119862
sum
119887=1
119863119886119887
119896 (19)
The divergence of the same sample category can beexpressed as
119878119896=
1
119862
119862
sum
119886=1
(
1
119873119886
119873119886
sum
119894=1
119863119896(119909119886
119894 119909119886)) (20)
where 119909119886is average of training samples of category 119886
According to the definition the kernel space measuresensitivity factor is
120576 =
119869119896
119878119896
(21)
33 Optimization Based on GA In order to make the fusionfeatures gained from LPP dimension reduction distinguishdifferent degradation states better genetic algorithm (GA) isapplied to optimize the kernel space where there are kindsof training samples GA is a newly developing algorithm tosearch an optimal solution The process of GA algorithmmainly includes population initialization crossover muta-tion fitness calculation (individual evaluation) and selection(population replacement)The kernel spacemeasure sensitiv-ity factor is taken as the fitness function for optimization andthe optimal individual is the case where the discrimination ofdifferent degradation states is highest
Studies have shown that the clustering effect of LPPfusion features will change along with the changing kernelspace In the interest of finding the optimal kernel spaceall training samples need to do affine transformation Take3D fusion features as an example one training point is setas 1198751(1199090 1199100 1199110) and affine transform angles are set as 120579
1isin
[0 2120587] and 1205792isin [0 2120587] So the affine transformation matrix
is
119860
=
[
[
[
[
[
[
sin 1205791cos 1205792sin 1205791sin 1205792cos 1205791
0
1 0 0 minus sin 1205791cos 1205792
0 1 0 minus sin 1205791sin 1205792
0 0 1 minus cos 1205791
]
]
]
]
]
]
(22)
The new sample feature points after kernel space transfor-mation can be computed by the following equation
119883 = 119860minus1
119887 (23)
The two affine transform angles are used as the trainingentity and the individuals are randomly generated to com-plete initialization By the optimization process of GA thetraining sample clustering effect is found to be the best
4 GG Clustering Algorithm
For the training sample set 119883 = 1198831 1198832 119883
119873 it is
assumed that each sample is made up of 119889 characteristics119883119896
= 1199091198961 1199091198962 119909
119896119889 After initialization all samples are
divided into 119862 categories namely the number of clusteringclassifications is 119862 (2 le 119862 le 119873) The clustering centers ofall categories are 119881
119894= V1 V2 V
119894 and the membership
matrix is 119880 = 119906119894119896119862times119873
The element 119906119894119896
isin [0 1] representsthe membership degree of the 119896 training sample to the 119894
degradation state (1 le 119894 le 119862) In GG algorithm the followingobjective function can reach the minimum value with theiterative adjustment of 119880 and 119881
119869119898(119880 119881) =
119862
sum
119894=1
119873
sum
119896=1
(120583119894119896)119898
1198632
119894119896 (24)
where119898 is the weighted index generally taken to 2Different from FCM clustering119863
119894119896indicates the distance
measure calculated by the covariance matrix in GG cluster-ing In that way the data samples of different directions andshapes can be reflected effectively
5 The Process of DegradationState Identification
The original vibration signal is preprocessed by LCD Thetime-domain features of STD RMS and root-square ampli-tude and the entropy features of LREE LRSE and TMSE areextracted from the selected signal components to composethe original characteristic set The degradation state recogni-tion processes are as shown in Figure 1
The degradation state recognition algorithmmainly con-tains the following key steps
(i) LCD PretreatmentAccording to the cross-correlationcoefficient between the LCD components and theoriginal signal the useful components can be chosenConsidering the amount of information existing incomponents and the time of computation the firsttwo components whose coefficient is higher thanothers are selected for further analysis after manytests
(ii) Feature Extraction and Fusion Six-dimensional mul-tiple domain features are fused by LPP algorithmand the intrinsic dimension is three according to themaximum likelihood estimationTherefore the targetdimension of feature fusion is set as three On thebasis of the maximum sensitive factor principle thefusion features are optimized by GA to find the bestkernel space for clustering analysis
(iii) The clustering centers are determined by GG algo-rithm and the rolling bearing degradation identi-fication is achieved by the principle of maximummembership degree
6 Instance Verification
61 Experimental Platform andData Preprocessing Thebear-ing full-life data used in this paper comes from Hangzhou
International Journal of Rotating Machinery 5
Training samples of all-
life data
Testing samples
LCD process
STD
RMS
XR
LREE
LRSE
TMSE
Eigenvector 1
Eigenvector 2
Eigenvector 3
GGclustering
Membership matrix
Normal state
Slight degradation
Severe degradation
Failure state
Component selection by LCD preprocessing Multidomain feature fusion based on GA-LPP Degradation state recognition based on GG
clustering
Degradation state
identification
Maximum membership
principle
1
2
3
4
Figure 1 Flow chart of the identification method
Table 1 The experimental parameters
Motor speed Samplinginterval Sampling time Sampling
frequency1500 rmin 10min 1 s 256 kHz
bearing test and research center [18] As is shown in Fig-ure 2(a) the test platform mainly consists of a ABLT-1Abearing test machine a signal acquisition module and statemonitoring equipment As Figure 2(b) shows four CA-YD-139 acceleration sensors are respectively fixed up on fourbearing test stations and connected to DH-5920 dynamicsignal acquisition instrument Four sets of rolling bearingscan be intensively tested andmultiple sets of full-life vibrationdata can be stored simultaneouslyWhat ismore four thermalresistors and a YD-1 acceleration sensor are connected with asignal amplifier to monitor the operating parameters Whenthe index exceeds the alarm threshold the test machine willstop working
Deep groove ball bearings are widely used in rotatingmachinery There is practical significance in engineeringtaking typical type of 6204 bearing as testing object The realbearing in normal state is shown in Figure 3(a) The specificparameters are set as shown in Table 1
When the test bench running time reaches 9600minutesthe machine is shut down Inner ring pitting occurs in thebearing at number 4 station and that result in bearing failure(as shown in Figure 3(b))
The collected 960 groups of vibration data record thewhole process of rolling bearing from normal state to failurestate Figure 4 shows the real-time monitoring curves ofaverage amplitude versus time which reflect different degra-dation states of rolling bearing clearly According to thechange of curve amplitude and curvature the rolling bearingperformance variation can be initially divided into four statesnormal state slight degradation severe degradation andfailure state The details are presented in Table 2
The original signal is preprocessed by LCD to get 10intrinsic scale components (ISCs) and the first 5 ISCs are
Table 2 The division of degradation state of rolling bearingperformance
shown in Figure 5 Further the cross-correlation coefficientbetween each component and the original signal is calculatedand the value relation is as follows
ISC1gt ISC
3gt ISC
2gt ISC
4gt ISC
5 (25)
What is more there are only the first and the third ISC whosecoefficient is more than 05 respectively 06487 and 05395Therefore the two components are taken as signal source fordegradation feature extraction
62 Degradation Feature Fusion and Optimization Accord-ing to the degradation state division in Table 2 100 groups ofnormal data 100 groups of slight degradation data 60 groupsof severe degradation data and 30 groups of failure dataare selected as training samples The characteristic indexesof different degradation states are extracted and normalizedrespectively The 3D time-domain feature points are shownin Figure 6 In the bearing degradation process from normalstate to failure state these three features are monotonicallyincreasing and the effect of failure state distinguishing isobvious However the points of the other three degradationstates aremixing severely and cannot be distinguished clearlyAlthough the time-domain features such as RMS are easy toget and have good stability to characterize degradation statesliterature [19] indicates that these time-domain features arenot sensitive to early bearing fault including slight degra-dation and severe degradation until bearing failure occursWhat is more reference [20] points out that rolling bearingsrsquovibration signals present nonlinear characteristics and thesethree traditional time-domain features are similar and can
6 International Journal of Rotating Machinery
ABLT-1A bearing
test machine
SensorsSignal
acquisition module
(a)
ThermistorsCA-YD-139acceleration
sensors
YD-1acceleration
sensors
No 1 No 2 No 3 No 4
(b)
Figure 2 Bearing life experiment layout (a) complete machine and (b) sensors
(a) (b)
Figure 3 6204 Bearing (a) normal state and (b) inner ring pitting
0 100 200 300 400 500 600 700 800 900 1000
008
009
01
011
012
013
014
015
016
017
Aver
age a
mpl
itude
Time (1 data point = 10 minutes)
Normal stateSlight degradation
Severe degradationFailure state
Figure 4 The preliminary division of degradation state based onaverage amplitude
hardly make an accurate evaluation of the early degradationstates of the bearings These arguments explain clearly whythe other three degradation states except for the failure one
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05minus02
minus02
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
001
Time (s)
ISC 1
minus02
minus02
ISC 2
ISC 3
ISC 4
ISC 5
minus01
Figure 5 The first five-order components of LCD
are mixed severely and cannot be distinguished by 3D time-domain features
Similarly the 3D complexity feature points made up ofentropy indexes of LREE LRSE and TMSE (scale factor is
International Journal of Rotating Machinery 7
002 04
0608
1
002
0406
081
0
02
04
06
08
1
STDRMS
XR
Normal stateSlight degradation
Severe degradationFailure state
Figure 6 Space distribution of time-domain features
002
0406
081
002
0406
081
0
02
04
06
08
1
LREELRSE
TMSE
Normal stateSlight degradation
Severe degradationFailure state
Figure 7 Space distribution of entropy features
15) are shown in Figure 7 The entropy vector can distinguishnormal state slight degradation and severe degradation onthe whole Nevertheless in the failure state the trainingsamplesrsquo clustering effect is unsatisfying Reference [21]demonstrates that entropy indexes are sole dependent on theprobability distribution of the event occurrence in bearingfault signals They are sensitive to the degradation statechanging but are more susceptible to spurious vibrationsWhen the bearing comes to failure state the violent conditionchanging will make the vibration signals mixed with a lot ofspurious components and the entropy features cannot stablycharacterize the failure state of bearings Therefore the 3Dentropy features at failure state show strong discreteness inFigure 7
In order to improve the discrimination effect of dif-ferent degradation states the above time-domain featuresand entropy features need to be fused Therefore the six-dimensional multidomain feature vectors are input to the
0020
05
001020304
Eigenvector XEigenvector Y
Eige
nvec
tor Z
minus01
minus02
minus03
minus05
minus1
minus02
minus04
minus06
Normal stateSlight degradation
Severe degradationFailure state
Figure 8 Space distribution of LPP fusion features
LPP for feature fusion and dimension reduction In orderto ensure the information exchanging among the neighbor-hoods the neighborhood number 119896 should not be too smallyet if 119896 is too large the local features can be incompleteGenerally analyzed the size of 119896 should be between 119889 and119873 where 119889 is the intrinsic dimension and119873 is the number oftraining samples in each category In this paper 119889 = 3 and119873 = 30 Thus 3 lt 119896 lt 30
The clustering effect is better when 119896 = 7 that is presentedin Figure 8 Compared with the time-domain features andthe entropy features the degradation state distinguishingability of the LPP fusion features is better and the clusteringeffects of normal state slight degradation and failure stateare satisfying But the robustness of fusion features in severedegradation state is relatively poor and this results in the factthat the same severe degradation state is divided into twosample partsMeanwhile the sample class spacing is relativelysmall and the clustering effect is not good So the process offeature fusion needs to be optimized
The kernel space measure sensitive factor is taken as theobjective function According to formula (22) and formula(23) the kernel space is optimized byGA so that the factor hasamaximumvalue In order to improve the convergence speedand ensure the search quality the population size is set as 119873= 20sim200 After several experiments 119873 = 30 The larger thecrossover probability is the higher the loss rate of excellentresults is But when the probability is too small the searchwill be blocked In general crossover probability 119875
119888= 06sim
10 and here it is 08 Mutation probability generally shouldnot be too large otherwise GA will become a random searchmethod and the precision and speed of convergence will beinfluenced Therefore the mutation probability 119875
119898= 003
As shown in Figure 9 after 26 iterations the kernelspace measure sensitivity factor tends to be stable andthe maximum value is achieved And the optimized affinetransformation angles are 120579
1= 14910 and 120579
2= 38532
Figure 10 presents the space distribution of optimized fusion
8 International Journal of Rotating Machinery
5 10 15 20 25 30 35 400
01
02
03
04
05
06
07
08
09
1
Iterations
Obj
ectiv
e fun
ctio
n va
lue
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
Component selection by LCD preprocessing Multidomain feature fusion based on GA-LPP Degradation state recognition based on GG
clustering
Degradation state
identification
Maximum membership
principle
1
2
3
4
Figure 1 Flow chart of the identification method
Table 1 The experimental parameters
Motor speed Samplinginterval Sampling time Sampling
frequency1500 rmin 10min 1 s 256 kHz
bearing test and research center [18] As is shown in Fig-ure 2(a) the test platform mainly consists of a ABLT-1Abearing test machine a signal acquisition module and statemonitoring equipment As Figure 2(b) shows four CA-YD-139 acceleration sensors are respectively fixed up on fourbearing test stations and connected to DH-5920 dynamicsignal acquisition instrument Four sets of rolling bearingscan be intensively tested andmultiple sets of full-life vibrationdata can be stored simultaneouslyWhat ismore four thermalresistors and a YD-1 acceleration sensor are connected with asignal amplifier to monitor the operating parameters Whenthe index exceeds the alarm threshold the test machine willstop working
Deep groove ball bearings are widely used in rotatingmachinery There is practical significance in engineeringtaking typical type of 6204 bearing as testing object The realbearing in normal state is shown in Figure 3(a) The specificparameters are set as shown in Table 1
When the test bench running time reaches 9600minutesthe machine is shut down Inner ring pitting occurs in thebearing at number 4 station and that result in bearing failure(as shown in Figure 3(b))
The collected 960 groups of vibration data record thewhole process of rolling bearing from normal state to failurestate Figure 4 shows the real-time monitoring curves ofaverage amplitude versus time which reflect different degra-dation states of rolling bearing clearly According to thechange of curve amplitude and curvature the rolling bearingperformance variation can be initially divided into four statesnormal state slight degradation severe degradation andfailure state The details are presented in Table 2
The original signal is preprocessed by LCD to get 10intrinsic scale components (ISCs) and the first 5 ISCs are
Table 2 The division of degradation state of rolling bearingperformance
shown in Figure 5 Further the cross-correlation coefficientbetween each component and the original signal is calculatedand the value relation is as follows
ISC1gt ISC
3gt ISC
2gt ISC
4gt ISC
5 (25)
What is more there are only the first and the third ISC whosecoefficient is more than 05 respectively 06487 and 05395Therefore the two components are taken as signal source fordegradation feature extraction
62 Degradation Feature Fusion and Optimization Accord-ing to the degradation state division in Table 2 100 groups ofnormal data 100 groups of slight degradation data 60 groupsof severe degradation data and 30 groups of failure dataare selected as training samples The characteristic indexesof different degradation states are extracted and normalizedrespectively The 3D time-domain feature points are shownin Figure 6 In the bearing degradation process from normalstate to failure state these three features are monotonicallyincreasing and the effect of failure state distinguishing isobvious However the points of the other three degradationstates aremixing severely and cannot be distinguished clearlyAlthough the time-domain features such as RMS are easy toget and have good stability to characterize degradation statesliterature [19] indicates that these time-domain features arenot sensitive to early bearing fault including slight degra-dation and severe degradation until bearing failure occursWhat is more reference [20] points out that rolling bearingsrsquovibration signals present nonlinear characteristics and thesethree traditional time-domain features are similar and can
6 International Journal of Rotating Machinery
ABLT-1A bearing
test machine
SensorsSignal
acquisition module
(a)
ThermistorsCA-YD-139acceleration
sensors
YD-1acceleration
sensors
No 1 No 2 No 3 No 4
(b)
Figure 2 Bearing life experiment layout (a) complete machine and (b) sensors
(a) (b)
Figure 3 6204 Bearing (a) normal state and (b) inner ring pitting
0 100 200 300 400 500 600 700 800 900 1000
008
009
01
011
012
013
014
015
016
017
Aver
age a
mpl
itude
Time (1 data point = 10 minutes)
Normal stateSlight degradation
Severe degradationFailure state
Figure 4 The preliminary division of degradation state based onaverage amplitude
hardly make an accurate evaluation of the early degradationstates of the bearings These arguments explain clearly whythe other three degradation states except for the failure one
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05minus02
minus02
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
001
Time (s)
ISC 1
minus02
minus02
ISC 2
ISC 3
ISC 4
ISC 5
minus01
Figure 5 The first five-order components of LCD
are mixed severely and cannot be distinguished by 3D time-domain features
Similarly the 3D complexity feature points made up ofentropy indexes of LREE LRSE and TMSE (scale factor is
International Journal of Rotating Machinery 7
002 04
0608
1
002
0406
081
0
02
04
06
08
1
STDRMS
XR
Normal stateSlight degradation
Severe degradationFailure state
Figure 6 Space distribution of time-domain features
002
0406
081
002
0406
081
0
02
04
06
08
1
LREELRSE
TMSE
Normal stateSlight degradation
Severe degradationFailure state
Figure 7 Space distribution of entropy features
15) are shown in Figure 7 The entropy vector can distinguishnormal state slight degradation and severe degradation onthe whole Nevertheless in the failure state the trainingsamplesrsquo clustering effect is unsatisfying Reference [21]demonstrates that entropy indexes are sole dependent on theprobability distribution of the event occurrence in bearingfault signals They are sensitive to the degradation statechanging but are more susceptible to spurious vibrationsWhen the bearing comes to failure state the violent conditionchanging will make the vibration signals mixed with a lot ofspurious components and the entropy features cannot stablycharacterize the failure state of bearings Therefore the 3Dentropy features at failure state show strong discreteness inFigure 7
In order to improve the discrimination effect of dif-ferent degradation states the above time-domain featuresand entropy features need to be fused Therefore the six-dimensional multidomain feature vectors are input to the
0020
05
001020304
Eigenvector XEigenvector Y
Eige
nvec
tor Z
minus01
minus02
minus03
minus05
minus1
minus02
minus04
minus06
Normal stateSlight degradation
Severe degradationFailure state
Figure 8 Space distribution of LPP fusion features
LPP for feature fusion and dimension reduction In orderto ensure the information exchanging among the neighbor-hoods the neighborhood number 119896 should not be too smallyet if 119896 is too large the local features can be incompleteGenerally analyzed the size of 119896 should be between 119889 and119873 where 119889 is the intrinsic dimension and119873 is the number oftraining samples in each category In this paper 119889 = 3 and119873 = 30 Thus 3 lt 119896 lt 30
The clustering effect is better when 119896 = 7 that is presentedin Figure 8 Compared with the time-domain features andthe entropy features the degradation state distinguishingability of the LPP fusion features is better and the clusteringeffects of normal state slight degradation and failure stateare satisfying But the robustness of fusion features in severedegradation state is relatively poor and this results in the factthat the same severe degradation state is divided into twosample partsMeanwhile the sample class spacing is relativelysmall and the clustering effect is not good So the process offeature fusion needs to be optimized
The kernel space measure sensitive factor is taken as theobjective function According to formula (22) and formula(23) the kernel space is optimized byGA so that the factor hasamaximumvalue In order to improve the convergence speedand ensure the search quality the population size is set as 119873= 20sim200 After several experiments 119873 = 30 The larger thecrossover probability is the higher the loss rate of excellentresults is But when the probability is too small the searchwill be blocked In general crossover probability 119875
119888= 06sim
10 and here it is 08 Mutation probability generally shouldnot be too large otherwise GA will become a random searchmethod and the precision and speed of convergence will beinfluenced Therefore the mutation probability 119875
119898= 003
As shown in Figure 9 after 26 iterations the kernelspace measure sensitivity factor tends to be stable andthe maximum value is achieved And the optimized affinetransformation angles are 120579
1= 14910 and 120579
2= 38532
Figure 10 presents the space distribution of optimized fusion
8 International Journal of Rotating Machinery
5 10 15 20 25 30 35 400
01
02
03
04
05
06
07
08
09
1
Iterations
Obj
ectiv
e fun
ctio
n va
lue
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
Figure 2 Bearing life experiment layout (a) complete machine and (b) sensors
(a) (b)
Figure 3 6204 Bearing (a) normal state and (b) inner ring pitting
0 100 200 300 400 500 600 700 800 900 1000
008
009
01
011
012
013
014
015
016
017
Aver
age a
mpl
itude
Time (1 data point = 10 minutes)
Normal stateSlight degradation
Severe degradationFailure state
Figure 4 The preliminary division of degradation state based onaverage amplitude
hardly make an accurate evaluation of the early degradationstates of the bearings These arguments explain clearly whythe other three degradation states except for the failure one
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05minus02
minus02
002
0 005 01 015 02 025 03 035 04 045 05
002
0 005 01 015 02 025 03 035 04 045 05
001
Time (s)
ISC 1
minus02
minus02
ISC 2
ISC 3
ISC 4
ISC 5
minus01
Figure 5 The first five-order components of LCD
are mixed severely and cannot be distinguished by 3D time-domain features
Similarly the 3D complexity feature points made up ofentropy indexes of LREE LRSE and TMSE (scale factor is
International Journal of Rotating Machinery 7
002 04
0608
1
002
0406
081
0
02
04
06
08
1
STDRMS
XR
Normal stateSlight degradation
Severe degradationFailure state
Figure 6 Space distribution of time-domain features
002
0406
081
002
0406
081
0
02
04
06
08
1
LREELRSE
TMSE
Normal stateSlight degradation
Severe degradationFailure state
Figure 7 Space distribution of entropy features
15) are shown in Figure 7 The entropy vector can distinguishnormal state slight degradation and severe degradation onthe whole Nevertheless in the failure state the trainingsamplesrsquo clustering effect is unsatisfying Reference [21]demonstrates that entropy indexes are sole dependent on theprobability distribution of the event occurrence in bearingfault signals They are sensitive to the degradation statechanging but are more susceptible to spurious vibrationsWhen the bearing comes to failure state the violent conditionchanging will make the vibration signals mixed with a lot ofspurious components and the entropy features cannot stablycharacterize the failure state of bearings Therefore the 3Dentropy features at failure state show strong discreteness inFigure 7
In order to improve the discrimination effect of dif-ferent degradation states the above time-domain featuresand entropy features need to be fused Therefore the six-dimensional multidomain feature vectors are input to the
0020
05
001020304
Eigenvector XEigenvector Y
Eige
nvec
tor Z
minus01
minus02
minus03
minus05
minus1
minus02
minus04
minus06
Normal stateSlight degradation
Severe degradationFailure state
Figure 8 Space distribution of LPP fusion features
LPP for feature fusion and dimension reduction In orderto ensure the information exchanging among the neighbor-hoods the neighborhood number 119896 should not be too smallyet if 119896 is too large the local features can be incompleteGenerally analyzed the size of 119896 should be between 119889 and119873 where 119889 is the intrinsic dimension and119873 is the number oftraining samples in each category In this paper 119889 = 3 and119873 = 30 Thus 3 lt 119896 lt 30
The clustering effect is better when 119896 = 7 that is presentedin Figure 8 Compared with the time-domain features andthe entropy features the degradation state distinguishingability of the LPP fusion features is better and the clusteringeffects of normal state slight degradation and failure stateare satisfying But the robustness of fusion features in severedegradation state is relatively poor and this results in the factthat the same severe degradation state is divided into twosample partsMeanwhile the sample class spacing is relativelysmall and the clustering effect is not good So the process offeature fusion needs to be optimized
The kernel space measure sensitive factor is taken as theobjective function According to formula (22) and formula(23) the kernel space is optimized byGA so that the factor hasamaximumvalue In order to improve the convergence speedand ensure the search quality the population size is set as 119873= 20sim200 After several experiments 119873 = 30 The larger thecrossover probability is the higher the loss rate of excellentresults is But when the probability is too small the searchwill be blocked In general crossover probability 119875
119888= 06sim
10 and here it is 08 Mutation probability generally shouldnot be too large otherwise GA will become a random searchmethod and the precision and speed of convergence will beinfluenced Therefore the mutation probability 119875
119898= 003
As shown in Figure 9 after 26 iterations the kernelspace measure sensitivity factor tends to be stable andthe maximum value is achieved And the optimized affinetransformation angles are 120579
1= 14910 and 120579
2= 38532
Figure 10 presents the space distribution of optimized fusion
8 International Journal of Rotating Machinery
5 10 15 20 25 30 35 400
01
02
03
04
05
06
07
08
09
1
Iterations
Obj
ectiv
e fun
ctio
n va
lue
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
Figure 6 Space distribution of time-domain features
002
0406
081
002
0406
081
0
02
04
06
08
1
LREELRSE
TMSE
Normal stateSlight degradation
Severe degradationFailure state
Figure 7 Space distribution of entropy features
15) are shown in Figure 7 The entropy vector can distinguishnormal state slight degradation and severe degradation onthe whole Nevertheless in the failure state the trainingsamplesrsquo clustering effect is unsatisfying Reference [21]demonstrates that entropy indexes are sole dependent on theprobability distribution of the event occurrence in bearingfault signals They are sensitive to the degradation statechanging but are more susceptible to spurious vibrationsWhen the bearing comes to failure state the violent conditionchanging will make the vibration signals mixed with a lot ofspurious components and the entropy features cannot stablycharacterize the failure state of bearings Therefore the 3Dentropy features at failure state show strong discreteness inFigure 7
In order to improve the discrimination effect of dif-ferent degradation states the above time-domain featuresand entropy features need to be fused Therefore the six-dimensional multidomain feature vectors are input to the
0020
05
001020304
Eigenvector XEigenvector Y
Eige
nvec
tor Z
minus01
minus02
minus03
minus05
minus1
minus02
minus04
minus06
Normal stateSlight degradation
Severe degradationFailure state
Figure 8 Space distribution of LPP fusion features
LPP for feature fusion and dimension reduction In orderto ensure the information exchanging among the neighbor-hoods the neighborhood number 119896 should not be too smallyet if 119896 is too large the local features can be incompleteGenerally analyzed the size of 119896 should be between 119889 and119873 where 119889 is the intrinsic dimension and119873 is the number oftraining samples in each category In this paper 119889 = 3 and119873 = 30 Thus 3 lt 119896 lt 30
The clustering effect is better when 119896 = 7 that is presentedin Figure 8 Compared with the time-domain features andthe entropy features the degradation state distinguishingability of the LPP fusion features is better and the clusteringeffects of normal state slight degradation and failure stateare satisfying But the robustness of fusion features in severedegradation state is relatively poor and this results in the factthat the same severe degradation state is divided into twosample partsMeanwhile the sample class spacing is relativelysmall and the clustering effect is not good So the process offeature fusion needs to be optimized
The kernel space measure sensitive factor is taken as theobjective function According to formula (22) and formula(23) the kernel space is optimized byGA so that the factor hasamaximumvalue In order to improve the convergence speedand ensure the search quality the population size is set as 119873= 20sim200 After several experiments 119873 = 30 The larger thecrossover probability is the higher the loss rate of excellentresults is But when the probability is too small the searchwill be blocked In general crossover probability 119875
119888= 06sim
10 and here it is 08 Mutation probability generally shouldnot be too large otherwise GA will become a random searchmethod and the precision and speed of convergence will beinfluenced Therefore the mutation probability 119875
119898= 003
As shown in Figure 9 after 26 iterations the kernelspace measure sensitivity factor tends to be stable andthe maximum value is achieved And the optimized affinetransformation angles are 120579
1= 14910 and 120579
2= 38532
Figure 10 presents the space distribution of optimized fusion
8 International Journal of Rotating Machinery
5 10 15 20 25 30 35 400
01
02
03
04
05
06
07
08
09
1
Iterations
Obj
ectiv
e fun
ctio
n va
lue
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
Figure 9 The curve of objective function with iterations
0 02005
minus01
0
01
02
03
04
05
minus02
minus02minus05
minus1 minus04minus06
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
Normal stateSlight degradation
Severe degradationFailure state
Figure 10 Space distribution of GA-LPP fusion features
feature points In comparison with Figure 8 the optimizedfusion features distinguish different degradation states betterthan the original features and especially the clustering effectof training samples in severe degradation state improves alot What is more the different class distinctions are furtherwidening Thus the optimization effect is obvious
In order to furtherly illustrate the excellent performanceof the proposed method the sensitivity factors of time-domain features entropy features LPP fusion features andGA-LPP fusion features are calculated respectively and theresult is just as Figure 11 shows The kernel space measuresensitivity factor of GA-LPP fusion features is the maximumone and it indicates that the fusion features have a strongability to characterize different bearing degradation statesafter GA optimization
63 Degradation State Recognition Based on GG MethodAccording to the number of bearing degradation states thenumber of clustering centers is determined as 119888 = 4 The
Time-domain Entropy LPP GA-LPP0
100
200
300
400
500
600
700
800
900
1000
Sens
itivi
ty fa
ctor
Figure 11 Clustering effects of different combinations of features
weighted factor is119898 = 2 and the iterative stopping thresholdvalue is 10minus5The 290 times 3matrix composed byGA-LPP fusionfeatures is computed by GG clustering and the clusteringcenter matrix is
119881 =
[
[
[
[
[
[
minus02388 00078 00097
minus00076 01597 minus01367
00237 minus04320 03697
00595 minus00016 minus00027
]
]
]
]
]
]
(26)
In accordance with Table 1 every 5 groups of data arechosen randomly as testing samples from each degradationstate The selected 20 groups of datarsquos multidomain featuresare optimized by GA-LPP at the same affine transformationangles The fusion feature space distribution is shown inFigure 12 where the testing feature points are well distributedaround the clustering centers and the testing sample spacingis large enough This method can effectively avoid identifica-tion misjudgment and improve the recognition accuracy
The membership matrix 119880 is established based on greycorrelation analysis Based on this bearing degradation staterecognition is realized guided by the principle of maximummembership value Table 3 is themembershipmatrix betweenthe testing samples and each standard degradation state Bycomparing the membership value of the same sample pointand different degradation states the recognition result isthe degradation state whose membership value is maximumHere are two LPP results before and after GA optimizationWithout GA optimization LPP fusion features judge slightdegradation state as normal state and severe degradationstate is mistaken as failure state The accuracy of degradationstate recognition is only 85 In comparison GA-LPP fusionfeatures have a better distinguishing ability 20 groups ofidentification results are in complete agreement with thereal degradation states and the excellent performance of theproposed method is verified
International Journal of Rotating Machinery 9
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
Table 3 Result of the rolling bearing degradation state identification
Real state Sample number Normal state Slight degradation Severe degradation Failure state Recognition resultLPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP LPP GA-LPP
Note ldquoradicrdquo represents the right recognition result and ldquotimesrdquo represents the wrong one
0 01005
0
01
02
03
04
05
Eige
nvec
torZ
1
Eigenvector Y1 Eigenvector X1
minus01
minus02
minus05
minus01minus02
minus03minus04
Testing samplesClustering samples
V1
V2
V3
V4
Figure 12 Space distribution of test samples and clustering centers
7 Conclusion
In order to improve degradation state recognition accuracyin rolling bearing all-life cycle this paper proposes a newdegradation state identification method based on GA-LPPand GG clustering Through the actual signal processing andanalysis the following conclusions can be obtained
(1) Compared with preset fault degrees it is difficult todistinguish different degradation states in the bearingcycle life Single domain features usually measure
degradation states from only one perspective sothe ability of single domain features to characterizecomplex and fuzzy degradation states can be insuf-ficient In manifold learning theory LPP algorithmcan fuse multidomain features and reduce dimensionto improve distinguishing effects of different degrada-tion states
(2) The kernel space measure sensitivity factor is takenas the optimization criterion GA algorithm based onkernel space transformation is applied to optimizethe LPP feature fusion process which can separatedifferent degradation samples better In this way theclustering effect of the same degradation state is moresatisfactory and the accuracy is higher
(3) There is some engineering value combining GA-LPPmultidomain feature fusion and GG clustering in thefield of degradation state recognition
(4) The GA parameter setting has a certain effect onthe convergence speed and the calculation precisionEven if the parameters are the same for repeatedexperiments the results can fluctuate Therefore thefollowing work is to improve the proposed methodapplicability by parameter optimization and enhanc-ing GA searching stability
Competing Interests
The authors declare that they have no competing interests
10 International Journal of Rotating Machinery
Acknowledgments
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
This project is supported by National Natural Science Foun-dation of China (Grant no 51541506)
References
[1] P Nguyen M Kang J-M Kim B-H Ahn J-M Ha andB-K Choi ldquoRobust condition monitoring of rolling elementbearings using de-noising and envelope analysis with signaldecomposition techniquesrdquo Expert Systems with Applicationsvol 42 no 22 pp 9024ndash9032 2015
[2] S Zhang Y Zhang and J Zhu ldquoRolling element-bearing featureextraction based on combined wavelets and quantum-behavedparticle swarm optimizationrdquo Indian Journal of Thoracic andCardiovascular Surgery vol 31 no 1 pp 605ndash610 2015
[3] T Xiao B-P Tang Y Qin and C Chen ldquoDegradation trendprediction of rolling bearing based on manifold learning andleast squares support vector machinerdquo Journal of Vibration andShock vol 34 no 9 pp 149ndash153 2015
[4] J Zheng J Cheng and Y Yang ldquoA rolling bearing fault diagno-sis approach based on LCD and fuzzy entropyrdquoMechanism andMachine Theory vol 70 pp 441ndash453 2013
[5] Y-K Wang H-R Li B Wang and B-H Xu ldquoSpatial infor-mation entropy and its application in the degradation stateidentification of hydraulic pumprdquo Mathematical Problems inEngineering vol 2015 Article ID 532684 7 pages 2015
[6] L Yun Z Lifeng and Z Shujun ldquoA hand gesture recognitionmethod based on multi-feature fusion and template matchingrdquoProcedia Engineering vol 29 pp 1678ndash1684 2012
[7] X Li P Li L Jiang and Y Cao ldquoFault diagnosis methodof asynchronous motor based on heterogeneous informationfeature fusionrdquo Chinese Journal of Scientific Instrument vol 34no 1 pp 227ndash233 2013
[8] X Ding Q He and N Luo ldquoA fusion feature and itsimprovement based on locality preserving projections forrolling element bearing fault classificationrdquo Journal of Soundand Vibration vol 335 pp 367ndash383 2015
[9] H-Y Pan Y Yang Y-G Li and J Cheng ldquoThe rollingbearings fault diagnosis method based on manifold learningand improved VPMCDrdquo Journal of Vibration Engineering vol27 no 6 pp 934ndash941 2014
[10] Q He ldquoVibration signal classification by wavelet packet energyflow manifold learningrdquo Journal of Sound and Vibration vol332 no 7 pp 1881ndash1894 2013
[11] Z-Q Su B-P Tang and J-B Yao ldquoFault diagnosis methodbased on sensitive feature selection and manifold learningdimension reductionrdquo Journal of Vibration and Shock vol 33no 3 pp 70ndash75 2014
[12] M Zarinbal M H Fazel Zarandi and I B Turksen ldquoRelativeentropy fuzzy c-means clusteringrdquo Information Sciences vol260 pp 74ndash97 2014
[13] Y-I KimD-WKimD Lee andKH Lee ldquoA cluster validationindex for GK cluster analysis based on relative degree ofsharingrdquo Information Sciences vol 168 no 1ndash4 pp 225ndash2422004
[14] L Zhang P Li M Li S Zhang and Z Zhang ldquoFault diagnosisof rolling bearing based on ITD fuzzy entropy and GG cluster-ingrdquo Chinese Journal of Scientific Instrument vol 35 no 11 pp2624ndash2632 2014
[15] Y Yang H Pan and J Cheng ldquoA rolling bearing fault diag-nosis method based on LCD de-noising and VPMCDrdquo ChinaMechanical Engineering vol 24 no 24 pp 3338ndash3344 2013
[16] X-X Ding and Q-B He ldquoMachine fault diagnosis based onWPD and LPPrdquo Journal of Vibration and Shock vol 33 no 3pp 89ndash93 2014
[17] C Zheyuan Y Jianguo L Xianpeng et al ldquoFeature selectionalgorithm based on kernel distance measurerdquo PR amp AI vol 23no 2 pp 235ndash240 2010
[18] B Wang H-R Li Q-H Chen and B-H Xu ldquoRolling bearingperformance degradative state recognition based onmathemat-ical morphological fractal dimension and fuzzy center meansrdquoActa Armamentarii vol 36 no 10 pp 1982ndash1990 2015
[19] Y N Pan J Chen and X L Li ldquoSpectral entropy a comple-mentary index for rolling element bearing performance degra-dation assessmentrdquo Proceedings of the Institution of MechanicalEngineers Part C vol 223 no 5 pp 1223ndash1231 2009
[20] K Zhu X Song and D Xue ldquoA roller bearing fault diagnosismethod based on hierarchical entropy and support vectormachine with particle swarm optimization algorithmrdquo Mea-surement vol 47 no 1 pp 669ndash675 2014
[21] B Tao L Zhu H Ding and Y Xiong ldquoAn alternative time-domain index for condition monitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering andSystem Safety vol 92 no 5 pp 660ndash670 2007
