a comparative study on classification of features by SVM and PSVM

Available online at www.sciencedirect.com Expert Systems with Applications Expert Systems with Applications 35 (2008) 1351–1366 www.elsevier.com/locate/eswa A comparative study on classification of features by SVM and PSVM extracted using Morlet wavelet for fault diagnosis of spur bevel gear box N Saravanan *, V.N.S Kumar Siddabattuni, K.I Ramachandran Department of Mechanical Engineering, Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India Abstract The condition of an inaccessible gear in an operating machine can be monitored using the vibration signal of the machine measured at some convenient location and further processed to unravel the significance of these signals This paper deals with the effectiveness of wavelet-based features for fault diagnosis using support vector machines (SVM) and proximal support vector machines (PSVM) The statistical feature vectors from Morlet wavelet coefficients are classified using J48 algorithm and the predominant features were fed as input for training and testing SVM and PSVM and their relative efficiency in classifying the faults in the bevel gear box was compared Ó 2007 Elsevier Ltd All rights reserved Keywords: Support vector machine; Proximal support vector machines; Bevel gear box; Morlet wavelet; Statistical features; Fault detection Introduction Fault diagnosis is an important process in preventive maintenance of gear box which avoids serious damage if defects occur to one of the gears during operation condition Early detection of the defects, therefore, is crucial to prevent the system from malfunction that could cause damage or entire system halt Diagnosing a gear system by examining vibration signals is the most commonly used method for detecting gear failures In the recent past reports of fault diagnosis of critical components using machine learning algorithms like SVM, PSVM are reported (Sugumaran, Muralidharan, & Ramachandran, 2006) The conventional methods for processing measured data contain the frequency domain technique, time domain technique, and time-frequency domain technique These methods have been widely employed to detect gear failures The use of vibration analysis for gear fault diagnosis and monitoring has been widely investigated and its application * Corresponding author Tel.: +91 4222656422; fax: +91 4222656274 E-mail address: n_saravanan@ettimadai.amrita.edu (N Saravanan) 0957-4174/$ - see front matter Ó 2007 Elsevier Ltd All rights reserved doi:10.1016/j.eswa.2007.08.026 in industry is well established (Cameron & Stuckey, 1994; Gadd & Mitchell, 1984; Leblanc, Dube, & Devereux, 1990) This is particularly reflected in the aviation industry where the helicopter engine, drive trains and rotor systems are fitted with vibration sensors for component health monitoring Support vector machine (SVM) is used in many applications of machine learning because of its high accuracy and good generalization capabilities SVM is based on statistical learning theory SVM classifies better than ANN because of the principle of risk minimization In artificial neural network (ANN) traditional empirical risk minimization (ERM) is used on training data set to minimize the error But in SVM, structural risk minimization (SRM) is used to minimize an upper bound on the expected risk SVM is modeled as an optimization problem and involves extensive computation, whereas, PSVM is modeled as a system of linear equations which involves less computation (Burgess, 1998) PSVM gives results very close to SVM Wavelet transform (WT) has attracted many researchers’ attention recently Wang and Mcfadden (1983) utilized the wavelet transform to represent all possible types of transients in vibration signals generated by faults in a 1352 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 gearbox Petrille, Paya, Esat, and Badi (1995) proposed the neural network to diagnose a simple gear system after the data have been pre-processed by the wavelet transform Boulahbal, Golnaraghi, and Ismail (1997) used the wavelet transform to analyze the vibration signal from the gear system with pitting on the gear The raw vibration signal in any mode from a single point on a machine is not a good indicator of the health or condition of a machine Vibration is a vectorial parameter with three dimensions and requires to be measured at several carefully selected points Vibration analysis can be carried out using Fourier transform techniques like Fourier series expansion (FSE), Fourier integral transform (FIT) and discrete Fourier transform (DFT) (Collacott) After the development of large-scale integration (LSI) and the associated microprocessor technology, fast Fourier transform (FFT) analyzers became cost effective for general applications The raw signatures acquired through a vibration sensor needed further processing and classification of the data for any meaningful surveillance of the condition of the system being monitored The flowchart in Fig explains the process of data acquisition and further processing using PSVM This work deals with extraction of features from the vibration data of a bevel gear box system by Morlet wavelet and classification of Gear faults using support vector machine (SVM) and proximal support vector machine Fig Flow chart for bevel gear box condition diagnosis Fig Fault simulator setup Fig Inner view of the bevel gear box Fig Flowchart of fault diagnosis system (PSVM) The vibration signal from a piezoelectric transducer is captured for the following conditions: good bevel gear, bevel gear with tooth breakage (GTB), Bevel Gear N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 Table Details of faults under investigation Gears Fault description Dimension (mm) G1 G2 G3 G4 Good Gear tooth breakage (GTB) Gear with crack at root (GTC) Gear with face wear – 0.8 · 0.5 · 20 0.5 Table Gear wheel and pinion details Parameters Gear wheel Pinion wheel No of teeth Module Normal pressure angle Shaft angle Top clearance Addendum Whole depth Chordal tooth thickness Chordal tooth height Material 35 2.5 20° 90° 0.5 mm 2.5 mm 5.5 mm 3.93À0.150 mm 2.53 mm EN8 25 2.5 20° 90° 0.5 mm 2.5 mm 5.5 mm 3.92À0.110 mm 2.55 mm EN8 1353 vibration signals Continuous wavelet transform (CWT) could put the fine partition ability of wavelet transform to good use, and is quite suitable for the gear box fault diagnosis In this work, the coefficients of Morlet wavelet were used for feature extraction Even though different possible families of wavelets are available in wavelet application, but the Morlet wavelet has been used most commonly in the literature for the analysis of vibration signal from rotating machineries This is due to the fact that the Morlet wavelet is able to pickup impulses generated by rotating elements A group of statistical features like kurtosis, standard deviation, maximum value, etc form a set of features, which are widely used in fault diagnostics, are extracted from the wavelet coefficients of the time domain signals Selection of good features is an important phase in pattern recognition and requires detailed domain knowledge The Decision Tree using J48 algorithm was used for identifying the best features from a given set of samples The selected features were fed as input to SVM for classification 1.1 Different phases of present work with crack at root of the tooth (GTC), and bevel gear with face wear of the teeth (TFW) for various loading and lubrication conditions Wavelet transform is a time-frequency signal analysis method, which is widely used and well established It has the local characteristic of time domain as well as frequency domain In the processing of non-stationary signals, it presents better performance than the traditional Fourier analysis Hence, wavelet transform has got potential application in gear box fault diagnosis in which features are extracted from the wavelet transform coefficients of the The signals obtained are processed further for machine condition diagnosis as explained in the flow chart in Fig 2 Experimental studies The fault simulator with sensor is shown in Fig and the details of bevel gear box are shown in Fig A variable speed DC motor (0.5 hp) with speed up to 3000 rpm is the basic drive A short shaft of 30 mm diameter is attached to the shaft of the motor through a flexible coupling; this is to Fig (a) View of good pinion wheel; (b) view of pinion wheel with face wear (GFW); (c) view of pinion wheel with tooth breakage (GTB) 1354 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 minimize effects of misalignment and transmission of vibration from motor The shaft is supported at its ends through two roller bearings From this shaft the motion is transmitted to the bevel gear box by means of a belt drive The gear box is of dimension 150 mm · 170 mm · 120 mm and the full lubrication level is 110 mm and half lubrication level is 60 mm SAE 40 oil was used as a lubricant An electro- magnetic spring loaded disc brake was used to load the gear wheel A torque level of N-m was applied at the full load condition The various defects are created in the pinion wheels and the mating gear wheel is not disturbed With the sensor mounted on top of the gear box vibrations signals are obtained for various conditions The selected area is made flat and smooth to ensure effective coupling A piezoelectric accelerometer (Dytran model) Good-Dry-Unload Good-Dry-FullLoad 0.2 Amplitude Amplitude 0.2 -0.2 -0.4 2000 4000 6000 -0.2 -0.4 8000 2000 Sample No Good-HalfLub-Unload Amplitude Amplitude 0 2000 4000 6000 -0.2 8000 2000 Sample No Amplitude Amplitude -0.2 2000 4000 6000 -0.2 8000 2000 -0.2 2000 4000 6000 -0.2 -0.4 8000 2000 GTB-HalfLub-Unload 6000 8000 GTB-HalfLub-FullLoad Amplitude -0.2 4000 4000 Sample No 0.2 2000 8000 0.2 Sample No 6000 GTB-Dry-FullLoad Amplitude Amplitude GTB-Dry-Unload 0.2 4000 Sample No Sample No Amplitude 8000 0.2 6000 0.2 -0.2 -0.4 8000 2000 Sample No 4000 6000 8000 Sample No GTB-FullLub-Unload GTB-FullLub-FullLoad 0.2 Amplitude 0.2 Amplitude 6000 Good-Full-FullLoad Good-FullLub-Unload -0.2 -0.4 4000 Sample No 0.2 -0.4 8000 0.2 -0.2 -0.4 6000 Good-HalfLub-FullLoad 0.2 -0.4 4000 Sample No 2000 4000 Sample No 6000 8000 -0.2 -0.4 2000 4000 6000 8000 Sample No Fig (a) Vibration signal for good pinion wheel under different lubrication and loading conditions; (b) vibration signal for pinion wheels with teeth breakage under different lubrication and loading conditions; (c) vibration signal for pinion wheel with crack at root under different lubrication and loading conditions; (d) vibration signals for pinion wheel with teeth face wear under different lubrication and loading conditions N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 GTC-Dry-Unload GTC-Dry-fullLoad 0.1 Amplitude Amplitude 0.1 -0.1 2000 4000 6000 -0.1 8000 2000 Sample No Amplitude Amplitude -0.1 2000 4000 6000 -0.1 8000 2000 GTC-FullLub-Unload 6000 8000 GTC-FullLub-FullLoad 0.1 Amplitude Amplitude 4000 Sample No 0.1 -0.1 2000 4000 6000 -0.1 8000 2000 Sample No Amplitude -0.2 2000 4000 6000 -0.2 8000 2000 TFW-HalfLub-Unload 6000 8000 TFW-HalfLub-FullLoad Amplitude -0.2 4000 4000 Sample 0.2 2000 8000 0.2 Sample 6000 TFW-Dry-FullLoad 0.2 4000 Sample No TFW-Dry-Unload Amplitude 8000 0.1 Sample No Amplitude 6000 GTC-HalfLub-FullLoad 0.1 4000 Sample No GTC-HalfLub-Unload 6000 0.2 -0.2 8000 2000 Sample 4000 6000 8000 Sample TFW-FullLub-Unload TFW-FullLub-FullLoad 0.2 Amplitude 0.2 Amplitude 1355 -0.2 2000 4000 6000 -0.2 8000 Sample 2000 4000 6000 8000 Sample Fig (continued) is mounted on the flat surface using direct adhesive mounting technique The accelerometer is connected to the signal-conditioning unit (DACTRAN FFT analyzer), where the signal goes through the charge amplifier and an analogue-to-digital converter (ADC) The vibration signal in digital form is fed to the computer through a USB port The software RT Pro-series that accompanies the signal conditioning unit is used for recording the signals directly in the computer’s secondary memory The signal is then read from the memory and replayed and processed to extract different features 2.1 Experimental procedure In the present study, four pinion wheels whose details are as mentioned in Table were used One was a new wheel and was assumed to be free from defects In the other three pinion wheels, defects were created using EDM in 1356 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 order to keep the size of the defect under control The details of the various defects are depicted in Table and its views are shown in Fig The size of the defects is a little bigger than one can encounter in the practical situation; however, it is in-line with work reported in literature (Sugumaran et al., 2006) The vibration signal from the piezoelectric pickup mounted on the test bearing was taken, after allowing initial running of the bearing for sometime The sampling frequency was 12 000 Hz and sample length was 8192 for all speeds and all conditions The sample length was chosen arbitrarily; however, the following points were considered Statistical measures are more meaningful, when the number of samples is more On the other hand, as the number of samples increases the computational time increases To strike a balance, sample length of around 10 000 was chosen In some feature extraction techniques, which will be used with the same data, the number of samples is to be 2n The nearest 2n to 10 000 is 8192 and hence, it was taken as sample length Many trials were taken at the set speed and vibration signal was stored in the data The raw vibration signals acquired for various experimental conditions form the gear box using FFT are shown in Fig Feature extraction After acquiring the vibration signals in the time domain, it is processed to obtain feature vectors The continuous wavelet transform (CWT) is used for obtaining the wavelet coefficients of the signals The statistical parameters of the wavelet coefficients are extracted, which constitute the feature vectors The term wavelet means a small wave It is the representation of a signal in terms of finite length or fast decaying waveform known as mother wavelet This waveform is scaled and translated to match the input signal The continuous wavelet transform (Mallat, 1998) is defined as Z þ1 W s ðsÞ ¼ f ðtÞWs;j ðtÞdt; t À s Ws;j ðtÞ ¼ pffiffiffiffiffi W s jsj is a window function called the mother wavelet, s is a scale and s is a translation The term translation is related to the location of the window, as the window is shifted through the signal This corresponds to the time information in the transform domain But instead of a frequency parameter, we have a scale Scaling, as a mathematical operation, either dilates or compresses a signal Smaller scale corresponds to high frequency of signals and large scale corresponds to low frequency signals The wavelet series is simply a sampled version of the CWT, and the information it provides is highly redundant as far as the reconstruction of the signal is concerned This redundancy, on the other hand, requires a significant amount of computation time and resources 3.1 Wavelet-based feature extraction The multilevel 1D wavelet decomposition function, available in Matlab is chosen with the Morlet wavelets specified It returns the wavelet coefficients of signal X at scale N (Soman & Ramachandran, 2005) Fig shows Morlet wavelet Sixty-four scales are initially chosen to extract the Morlet wavelet coefficients of the signal data The efficiency of sixty-four scales of Morlet wavelets were obtained using WEKA data mining software and the coefficients of highest À1 Fig Morlet wavelet (Collacott) where Efficiency of Morlet Coefficients 97 % Efficiency 96 95 94 93 92 91 12 16 20 24 28 32 36 40 44 48 Morlet Wavelet Scale Fig % Efficiency of Morlet wavelet coefficients 52 56 60 64 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 1357 Fig (a) Good-Dry-No Load Vs GTB, GTC, TFW-Dry-No Load; (b) Good-Dry -Full Load Vs GTB, GTC, TFW-Dry -Full Load; (c) Good-Half LubNo Load Vs GTB, GTC, TFW-Half Lub-No Load; (d) Good-Half Lub-Full Load Vs GTB, GTC, TFW-Half Lub-Full Load; (e) Good-Full Lub-No Load Vs GTB, GTC, TFW-Full Lub-No Load; (f) Good-Full Lub-Full Load Vs GTB, GTC, TFW-Full Lub-Full Load 1358 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 Fig (continued) level are considered for classification Since the eighth level gave maximum efficiency of 96.5%, the statistical features corresponding to it were given as input for J48 algorithm to determine the predominant features to be given as an input for training and classification using SVM Fig gives the efficiencies of all scales Using J 48 algorithm in the present work A standard tree induced with c5.0 (or possibly ID3 or c4.5) consists of a number of branches, one root, a number of nodes and a number of leaves One branch is a chain of nodes from root to a leaf; and each node involves one attribute The occurrence of an attribute in a tree provides the information about the importance of the associated attribute as explained by Peng, Flach, Brazdil, and Soares (2002) A Decision Tree is a tree based knowledge representation methodology used to represent classification rules J48 algorithm (A WEKA implementation of c4.5 Algorithm) is a widely used one to construct Decision Trees as explained by Sugumaran et al (2006) The Decision Tree algorithm has been applied to the problem under discussion Input to the algorithm is set of statistical features of the eighth scale Morlet coefficients It is clear that the top node is the best node for classification The other features in the nodes of Decision Tree appear in descending order of importance It is to be stressed here that only features that contribute to the classification appear in the Decision Tree and others not Features, which have less discriminating capability, can be consciously discarded by deciding on the threshold This concept is made use for selecting good features The algorithm identifies the good features for the purpose of classification from the given training data set, and thus reduces the domain knowledge required to select good features for pattern classification problem The decision trees shown in Fig is for various lubrication and loading conditions of different faults compared with good conditions of the pinion gear wheel Based on above trees its clear that of all the statistical features, standard error, kurtosis, sample variance and minimum value play a dominant role in feature classification using Morlet coefficients These four predominant features are fed as an input to SVM for further classification The scatter plot showing the variation of the statistical parameters of Morlet coefficients are shown in Fig 10 These features were given as input for training and testing of classifying features using SVM Proximal support vector machine (PSVM) PSVM is a modified version of support vector machine (SVM) The SVM is a new generation learning system based on statistical learning theory SVM belongs to the class of supervised learning algorithms in which the learning machine is given a set of features (or inputs) with the associated labels (or output values) Each of these features can be looked upon as a dimension of a hyper-plane SVMs construct a hyper-plane that separates the data into two classes (this can be extended to multi-class problems) N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 While doing so, SVM algorithm tries to achieve maximum separation between the classes (see Fig 11) separating the classes with a large margin minimizes a bound on the expected generalization error By ‘minimum generalization error’, we mean that when a new set of features (that is data points with unknown class values) arrive for classification, the chance of making an error in the prediction (of the class to which it belongs) based on the learned classifier Good-Dry-Fullload Stat Parameters Stat Parameters Good-Dry-Noload -1 10 Sample No 15 -1 20 Good-HalfLub-Noload 10 Sample No 15 20 0.5 Stat Parameters Stat Parameters Good-HalfLub-Fullload 0.5 -0.5 1359 10 Sample No 15 20 -0.5 Good-FullLub-Noload 10 Sample No 15 20 Good-FullLub-Fullload Stat Parameters Stat Parameters 0.5 0.5 -0.5 10 Sample No 15 -0.5 10 Sample No 20 GTB-Dry-Noload 20 GTB-Dry-Fullload 0.5 0.2 Stat Var Stat Var 15 0 -0.2 -0.4 -0.5 10 Sample 15 20 GTB-HalfLub-Noload 20 0.2 Stat Var Stat Var 15 GTB-HalfLub-Fullload 0.2 -0.2 -0.4 10 Sample 15 -0.2 -0.4 20 GTB-FullLub-Noload Stat Var -0.2 -0.4 10 Sample 15 15 20 -0.2 -0.4 5 10 Sample GTB-FullLub-Fullload 0.2 Stat Var 10 Sample 20 10 Sample 15 20 Fig 10 (a) Vibration signal for good pinion wheel under different lubrication and loading conditions; (b) vibration signal for pinion wheel with teeth breakage under different lubrication and loading conditions; (c) vibration signal for pinion wheel with crack at root under different lubrication and loading conditions; (d) vibration signals for pinion wheel with teeth face wear under different lubrication and loading conditions 1360 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 GTC-Dry-Noload GTC-Dry-Fullload Stat Var Stat Var 0.2 -0.2 10 Sample 15 20 10 Sample GTC-HalfLub-Noload Stat Var Stat Var 0.4 0.2 -0.2 0.2 -0.2 10 Sample 15 20 5 10 Sample 15 20 Sat Parameters Sat Parameters 0.5 10 Sample No 15 20 Sat Parameters 10 Sample No 15 20 -0.5 Sat Parameters 15 10 Sample No 15 20 0.5 -0.5 10 Sample No 15 20 TFW-FullLub-Fullload 10 Sample No 20 0.5 TFW-FullLub-Noload 0.5 15 TFW-HalfLub-Fullload 0.5 10 Sample TFW-Dry-Fullload TFW-HalfLub-Noload Sat Parameters 20 0.2 TFW-Dry-Noload Sat Parameters 15 -0.2 -0.5 10 Sample GTC-FullLub-Fullload Stat Var Stat Var GTC-FullLub-Noload 0.6 0.4 0.2 -0.2 -0.5 20 GTC-HalfLub-Fullload 0.4 -0.5 15 0.5 -0.5 20 10 Sample No 15 20 Fig 10 (continued) (hyper-plane) should be minimum Intuitively, such a classifier is one, which achieves maximum separation-margin between the classes The above process of maximizing separation leads to two hyper-planes parallel to the separating plane, on either side of it These two can have one or more points on them The planes are known as ‘bounding planes’ and the distance between them is called as ‘margin’ By SVM ‘learning’, we mean, finding a hyper-plane, which maximizes the margin The points lying beyond the bounding planes are called support vectors As for as data points belonging to AÀ are concerned P1, P2, P3, P4, and P5 are support vectors (see Fig 12), but P6, P7 are not support vectors Similar thing hold good for class A+ These points play a crucial role in the theory and hence the name N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 1361 Fig 13 Proximal support vector classification Fig 11 Flowchart of PSVM Fig 12 Standard SVM classifier support vector machines Here by ‘machine’, we mean an algorithm me0 y þ w0 w s:t: DðAw À ecÞ þ y P e ðw;c;yÞ2Rnþ1þm ð1Þ yP0 where A2R mÂn Vapnik has shown that if the training features are separated without errors by an optimal hyper-plane, the expected error rate on a test sample is bounded by the ratio of the expectation of the support vectors to the number of training vectors The smaller the size of the support vector set more general the above result Further the generalization is independent of the dimension of the problem In case such a hyper-plane is not possible the next best is to minimize the number of misclassifications whilst maximizing the margin with respect to the correctly classified features Recently a much simpler classifier, proximal support vector machine, was implemented wherein each class of points is assigned to the closer of two parallel planes (in input or feature space) that are pushed apart as far as possible This formulation leads to a fast and simple algorithm for generating a classifier – (linear or nonlinear) that is obtained by solving a single system of linear equations The point of departure from SVM is that, the optimization problem given by Eq (1) is replaced by the following problem: ðw;c;yÞ ; D fÀ1; þ1g mÂ1 ; e¼1 mÂ1 s:t: 1 m kyk þ ðwT w þ c2 Þ 2 DðAw À ecÞ þ y ¼ e Dry / Half / Full Lubrication No Load Condition Good Vs Gear Tooth Breakage (GTB) Good Vs Tooth Crack (GTC) Good Vs Face Wear (TFW) Full Load Condition Good Vs Gear Tooth Breakage (GTB) Good Vs Tooth Crack (GTC) Good Vs Face Wear (TFW) Fig 14 Methodology of classification using SVM and PSVM ð2Þ 1362 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 The geometrical interpretation of the above formulation is in Fig 12 Referring to Fig 13, y represents deviation (scaled by 1/ kwk) of the point from the plane passing through the centroid of the data cluster (A+ or AÀ) to which the point belongs Hence, there is no non-negativity constraint on y Further the 2-norm of the error vector y is minimized instead of the 1-norm; the margin between the bounding planes is maximized with respect to both orientation w and relative location c to the origin Extensive computational experience, as in Peng et al (2002) indicates that the formulation (Eq (2)) is almost as good as the classical formulation (Eq (1)) with some added advantages such as strong convexity of the objective function The key idea in this formulation is to make computation simple, by replacing the inequality constraint by equality The modification, even though simple, changes the nature of optimization problem significantly An explicit exact solution can be written to the problem in terms of the problem data It is impossible to that in the previous formulations because of their combinatorial nature Geometrically the formulation obtained by (Eq (2)) can be interpreted as follows The planes XTW À c = ±1 are not bounding planes anymore, but can be thought of as ‘‘proximal’’ planes, around which the points of each class are clustered and which are pushed as far apart as possible by the term WTW + c2 in the objective function; in fact this term is the reciprocal of the 2-norm distance squared between the two planes in the (w, c) space The interpretation, however, is not based on the idea of maximizing the margin, the distance between the bounding parallel planes, which is a key feature of support vector machines After training, for any the new set of features prediction of its class is possible using the decision function as given below which is a function of ‘w’ and ‘c’ It is called testing f ðxÞ ¼ signðwT x À cÞ ð3Þ If the value of f(x) is positive then new set of features belongs to class A+ otherwise it belongs to class AÀ Classifying multiple classes is commonly performed by combining several binary SVM classifiers in a tournament manner, either one-against-all or one-against-one, the latter approach required substantially more computational effort (Petrille et al., 1995) Table SVM results for statistical features from Morlet coefficients Sl no.a W1 W2 W3 W4 Gamma Efficiency (%) 10 11 12 13 14 15 16 17 18 0.4968457 0.5108036 0.5003421 À0.4650681 0.1224614 À304.27068 0.0983717 0.1167805 0.0164757 À267.87401 0.1108383 À484.58268 À0.3098304 0.1429575 À3.0033355 À0.101011 0.16201 À8.4899387 0.484703 0.5520768 0.5309669 À4.2096613 0.208264 À10183.429 0.0902571 0.1679909 À0.3951278 À3195.119 0.1417259 À3675.914 À2.501089 0.2654004 À18.407903 À1.473269 0.345397 À50.618626 0.979318252 0.114672115 0.502348489 2.693242987 À0.25844467 3.603153258 0.14774685 0.038789747 65.72850705 0.216319558 À0.46081525 0.257437445 3.642236896 0.002027233 12.6936568 5.163634397 À0.64252924 16.28268003 1.37492531 À9.6845415 À1.1059302 228.538653 À22.62677 21.3864755 0.47803585 À14.921874 1166.77874 4.51519148 À12.983546 5.17913819 146.172721 À38.671339 509.45638 102.151553 À35.71901 323.252975 8.853585 2.720055 0.271705 À38.0839 1.638668 8.846569 0.583319 2.721864 288.999 À3.07958 À1.87604 À3.36361 À1.55253 4.675529 À2.94136 20.3311 À0.85299 66.02637 100 95 95 95 100 95 100 95 95 95 95 100 95 100 95 100 95 100 a The details of comparison of various conditions are given in Table Table Various conditions of comparison Good-dry-No Load Vs GTB-dry-No-Load Good-dry-No Load Vs GTC-dry-No Load Good-dry-No Load Vs TFW-dry-No Load Good-dry-Full Load Vs GTB-dry-Full Load Good-dry-Full Load Vs GTC-dry-Full Load Good-dry-Full Load Vs TFW-dry-Full Load Good-Half-No Load Vs GTB-Half-No Load Good-Half-No Load Vs GTC-Half-No Load Good-Half-No Load Vs TFW-Half-No Load 10 11 12 13 14 15 16 17 18 Good-Half-Full Load Vs GTB-Half-Full Load Good Half-Full Load Vs GTC-Half-Full Load Good Half-Full Load Vs TFW-Half-Full Load Good-Full-No Load Vs GTB-Full-No Load Good-Full-No Load Vs GTC-Full-No Load Good-Full-No Load Vs TFW-Full-No Load Good-Full-Full Load Vs GTB-Full-Full Load Good-Full-Full Load Vs GTC-Full-Full Load Good-Full-Full Load Vs TFW-Full-Full Load N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 Application of SVM and PSVM for problem at hand and results Fig 14 shows the methodology adopted for classification of various conditions of gear box using SVM and PSVM For each condition feature vectors consisting of 100 feature value sets were collected from the experiment at 2000 rpm Seventy samples in each class were used for training and 30 samples are reserved for testing SVM and PSVM Training was done and the values of ‘w’ (weights), ‘c’ (gamma) and efficiency obtained for statistical features of Morlet coefficients using SVM and PSVM are tabulated Tables and 5, respectively The relative efficiency of above said two types of classification are shown in Fig 15 The feature vectors corresponding to good gear were compared with those of the faulty gears at different loading and lubrication levels, by taking each fault at a time 1363 Discussion (1) The use of Morlet Wavelet and extraction of statistical features from the wavelet coefficients was found to be very efficient for classification using SVM and PSVM (2) Decision tree is a good tool in selecting the best features among the extracted feature vectors Standard error, sample varience, kurtosis and minimum value were found to be the most contributing features (3) In SVM, the parameters w (weight) and c (gamma) define the separating plane These can be used for testing a new set of data and classifying the faults accordingly The efficiency for various conditions using SVM for Morlet coefficients are shown in Table and that of PSVM in Table 4, respectively (4) From Tables and we can see that PSVM has slightly better classification capability than SVM Table 5a PSVM results for statistical features from Morlet coefficients Sl no.* Nu 0.1 À0.55 À0.5 À0.55 À0.1 0.1 À0.1 À0.5 À0.55 * W À0.0011 À0.0053 0.2067 0.5357 À0.0054 À0.0257 0.999 2.6152 À0.0058 À0.0275 1.0658 2.8042 0.004 0.0249 0.9361 À2.8311 À0.0022 À0.013 À0.4191 1.4922 0.0009 0.0052 0.1796 À0.5934 0.0024 0.0079 À0.6125 À1.0344 À0.0035 À0.0119 0.9269 1.4925 À0.0033 À0.0114 0.879 1.4136 The different conditions as per Table G % Sl no.* Nu W G À0.55 0.0051 0.0389 À0.095 À2.8081 0.0009 0.0066 À0.0158 À0.4703 À0.0008 À0.0064 0.0119 0.4546 À0.0509 30 0.0034 40 1.709 98 8.82 98 À0.1 9.415 99 0.1 9.400 99 À4.458 0.1 1.696 100 0.2 À5.777 98 8.350 99 0.1 7.910 100 0.2 À0.1 À0.1 % À0.007 92.5 À0.002 À0.0095 À0.0212 1.2507 0.0016 0.008 0.0152 À1.0367 0.003 0.0147 0.024 À1.9117 À0.3965 0.2928 99 0.5129 100 0.0007 0.0032 À0.154 À0.3604 À0.0006 À0.0026 0.1311 0.299 À0.001 À0.0048 0.2305 0.5551 À1.2589 22.5 0.9364 90 1.6806 97.5 1364 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 Table 5b PSVM results for statistical features from Morlet coefficients Sl no Nu 0.1 0.15 0.17 À0.1 0.05 0.1 À0.3 0.1 0.5 W G % Sl no Nu W G % 0.835 80 10 0.2 À0.7277 77.5 1.177 87 0.5 À1.4014 80 1.302 95 0.6 À0.0016 À0.0107 À0.0874 0.2779 À0.0041 À0.0266 À0.1606 0.6525 À0.0049 À0.0318 À0.1772 0.7704 À1.561 95.5 À0.0009 À0.0042 À0.0513 0.8685 0.0004 0.0019 0.0231 À0.3941 0.0008 0.0037 0.0407 À0.7655 À0.612 20 À1.3327 92.5 0.246 95 0.3 À1.7426 95 0.460 95 0.4 À2.0527 97.5 À0.0192 À0.0826 3.906 5.9826 À0.0012 À0.0055 0.2019 0.4111 À0.0019 À0.0091 0.3062 0.6847 30.82 À0.0012 À0.0063 0.1074 0.2307 À0.0017 À0.0092 0.1493 0.3351 À0.0019 À0.0103 0.1647 0.3752 2.5 11 12 0.2 0.1 1.458 95 0.3 2.260 95 0.35 0.0019 0.0073 À0.1922 À1.4833 0.0027 0.0101 À0.2512 À2.0828 0.0033 0.0126 À0.2972 À2.6164 À0.001 À0.0055 0.0933 0.3571 À0.0028 À0.0152 0.2136 1.0138 À0.0032 À0.0175 0.2361 1.1689 0.5804 80 1.3742 87.5 1.5126 95 Table 5c PSVM results for statistical features from Morlet coefficients Sl no Nu W 13 À0.55 0.0074 0.0463 À0.1052 À4.4774 À0.0021 À0.0133 0.026 1.2515 À0.0008 À0.0048 0.0085 0.4563 0.2 0.07 14 0.3 0.4 0.0034 0.0128 À0.1414 À1.6839 0.0044 0.0166 À0.1773 À2.1749 G % Sl no Nu W 16 0.01 À0.0001 À0.0007 0.0049 0.0614 À0.0002 À0.0014 0.0094 0.1224 À0.0004 À0.0021 0.0139 0.183 0.06 2.5 À0.0271 97.5 0.02 À0.0092 95 0.03 À0.9302 90 À1.1684 92.5 17 0.1 0.15 0.0012 0.0047 À0.0662 À0.6439 0.0018 0.0069 À0.0911 À0.9438 G % 0.0245 90 0.0488 95 0.073 100 À0.4424 67.5 À0.6178 87.5 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 1365 Table 5c (continued) Sl no 15 Nu W G % Sl no Nu W G 0.5 0.0053 0.0201 À0.2095 À2.6373 À1.3786 95 0.16 0.0019 0.0073 À0.0956 À1.0023 À0.6499 95 0.1 À0.001 À0.005 0.105 0.5005 À0.0018 À0.0095 0.1905 0.9449 À0.0026 À0.0135 0.264 1.344 0.6836 65 18 0.09 1.2619 90 0.1 1.7558 95 0.13 À0.0008 À0.0042 0.0713 0.421 À0.0009 À0.0047 0.0786 0.4661 À0.0011 À0.006 0.1 0.5999 0.2 0.3 % 0.4674 92.5 0.5169 97.5 0.6625 100 Comparision of PSVM and SVM Results 100 98 Efficiency 96 94 92 90 17 18 P 16 15 14 12 PSVM 13 11 Condition 10 88 SVM Fig 15 Comparison of SVM and PSVM results Conclusion Fault diagnosis of gear box is one of the core research areas in the field of condition monitoring of rotating machines A comparative study of different classifying techniques and the ability of morlet wavelet in feature extraction for bevel gear box fault detection is carried out It was found that PSVM has an edge over SVM in classification of features References Boulahbal, D., Golnaraghi, M F & Ismail, F., (1997) In Proceedings of DETC’97, 1997, ASME design engineering technical conference DETC97/VIB-4009 Burgess, C J C (1998) A tutorial on support vector machines for pattern recognition Data Mining and Knowledge Discovery, 2, 955–974 Cameron, B G & Stuckey, M J., (1994) A review of transmission vibration monitoring at Westland Helicopter Ltd In Proceedings of the 20th European rotorcraft forum, (pp 16/1–116/16) Paper 116 Collacott, R A Mechanical fault diagnosis and condition monitoring Chapman & Hall Gadd, P & Mitchell, P J (1984) Condition monitoring of helicopter gearboxes using automatic vibration analysis techniques, AGARD CP 369 gears, and power transmission system for helicopter turboprops (pp 29/1–29/10) Leblanc, J F A., Dube, J R F & Devereux, B., (1990) Helicopter gearbox vibration analysis in the Canadian forces – applications and lessons In Proceedings of the first international conference, gearbox noise and vibration, (pp 173–177) IMechE, Cambridge, UK, C404/023 Mallat (1998) A wavelet tour of signal processing Academic Press 1366 N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 Peng, Y H., Flach, P A., Brazdil, P & Soares, C., (2002) Decision treebased data characterization for meta-learning, In ECML/PKDD-2002 Workshop IDDM-2002 Helsinki, Finland Petrille, O., Paya, B., Esat, I I., & Badi, M N M (1995) Proceedings of the energy-sources technology conference and exhibition: structural dynamics and vibration, PD-Vol 70, 97 Soman, K P., & Ramachandran, K I (2005) Insight into wavelets from theory to practice Prentice-Hall of India Private Limited Sugumaran, V., Muralidharan, V., & Ramachandran, K I (2006) Feature selection using decision tree and classification through proximal support vector machine for fault diagnostics of roller bearing Mechanical Systems and Signal Processing, 21, 930–942 Wang, W J., & Mcfadden, P D (1983) Early detection of gear failure by vibration analysis II: interpretation of the time-frequency distribution using image processing techniques Mechanical Systems and Signal Processing, 7(3), 205–215 [...]... samples in each class were used for training and 30 samples are reserved for testing SVM and PSVM Training was done and the values of ‘w’ (weights), ‘c’ (gamma) and efficiency obtained for statistical features of Morlet coefficients using SVM and PSVM are tabulated Tables 3 and 5, respectively The relative efficiency of above said two types of classification are shown in Fig 15 The feature vectors corresponding... Comparision of PSVM and SVM Results 100 98 Efficiency 96 94 92 90 17 18 P 16 15 14 12 9 PSVM 13 11 Condition 10 7 8 6 4 5 3 1 2 88 SVM Fig 15 Comparison of SVM and PSVM results 8 Conclusion Fault diagnosis of gear box is one of the core research areas in the field of condition monitoring of rotating machines A comparative study of different classifying techniques and the ability of morlet wavelet in feature... GTC-Full-Full Load Good-Full-Full Load Vs TFW-Full-Full Load N Saravanan et al / Expert Systems with Applications 35 (2008) 1351–1366 6 Application of SVM and PSVM for problem at hand and results Fig 14 shows the methodology adopted for classification of various conditions of gear box using SVM and PSVM For each condition feature vectors consisting of 100 feature value sets were collected from the experiment at... good gear were compared with those of the faulty gears at different loading and lubrication levels, by taking each fault at a time 1363 7 Discussion (1) The use of Morlet Wavelet and extraction of statistical features from the wavelet coefficients was found to be very efficient for classification using SVM and PSVM (2) Decision tree is a good tool in selecting the best features among the extracted feature... vectors Standard error, sample varience, kurtosis and minimum value were found to be the most contributing features (3) In SVM, the parameters w (weight) and c (gamma) define the separating plane These can be used for testing a new set of data and classifying the faults accordingly The efficiency for various conditions using SVM for Morlet coefficients are shown in Table 3 and that of PSVM in Table 4, respectively... 955–974 Cameron, B G & Stuckey, M J., (1994) A review of transmission vibration monitoring at Westland Helicopter Ltd In Proceedings of the 20th European rotorcraft forum, (pp 16/1–116/16) Paper 116 Collacott, R A Mechanical fault diagnosis and condition monitoring Chapman & Hall Gadd, P & Mitchell, P J (1984) Condition monitoring of helicopter gearboxes using automatic vibration analysis techniques, AGARD... as in Peng et al (2002) indicates that the formulation (Eq (2)) is almost as good as the classical formulation (Eq (1)) with some added advantages such as strong convexity of the objective function The key idea in this formulation is to make computation simple, by replacing the inequality constraint by equality The modification, even though simple, changes the nature of optimization problem significantly... vibration, PD-Vol 70, 97 Soman, K P., & Ramachandran, K I (2005) Insight into wavelets from theory to practice Prentice-Hall of India Private Limited Sugumaran, V., Muralidharan, V., & Ramachandran, K I (2006) Feature selection using decision tree and classification through proximal support vector machine for fault diagnostics of roller bearing Mechanical Systems and Signal Processing, 21, 930–942 Wang,... number of misclassifications whilst maximizing the margin with respect to the correctly classified features Recently a much simpler classifier, proximal support vector machine, was implemented wherein each class of points is assigned to the closer of two parallel planes (in input or feature space) that are pushed apart as far as possible This formulation leads to a fast and simple algorithm for generating a. .. gears, and power transmission system for helicopter turboprops (pp 29/1–29/10) Leblanc, J F A. , Dube, J R F & Devereux, B., (1990) Helicopter gearbox vibration analysis in the Canadian forces – applications and lessons In Proceedings of the first international conference, gearbox noise and vibration, (pp 173–177) IMechE, Cambridge, UK, C404/023 Mallat (1998) A wavelet tour of signal processing Academic ... bounding planes are called support vectors As for as data points belonging to AÀ are concerned P1, P2, P3, P4, and P5 are support vectors (see Fig 12), but P6, P7 are not support vectors Similar thing