Fault diagnosis of spur gearbox based on random forest and wavelet packet decomposition

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/281863415 Fault diagnosis of spur gearbox based on random forest and wavelet packet decomposition ArticleinFrontiers of Mechanical Engineering · September 2015 DOI: 10.1007/s11465-015-0348-8 CITATIONS 25 READS 258 7 authors, including: Some of the authors of this publication are also working on these related projects: Fault Detection in Rotatory Machines based on Vibration Signal Analysis View project electromagnetic vibration energy harvester View project Diego Román Cabrera Universidad Politécnica Salesiana (UPS) 72 PUBLICATIONS1,395 CITATIONS SEE PROFILE René Vinicio Sánchez Universidad Politécnica Salesiana (UPS) 66 PUBLICATIONS1,497 CITATIONS SEE PROFILE Grover V. Zurita Universidad Privada Boliviana 44 PUBLICATIONS1,009 CITATIONS SEE PROFILE Mariela Cerrada Universidad Politécnica Salesiana (UPS) 141 PUBLICATIONS1,857 CITATIONS SEE PROFILE All content following this page was uploaded by Mariela Cerrada on 27 May 2016. The user has requested enhancement of the downloaded file. Diego CABRERA, Fernando SANCHO, René-Vinicio SÁNCHEZ, Grover ZURITA, Mariela CERRADA, Chuan LI, Rafael E. VÁSQUEZ Fault diagnosis of spur gearbox based on random forest and wavelet packet decomposition Abstract This paper addresses the development of a random forest classifier for the multi-class fault diagnosis in spur gearboxes. The vibration signal’s condition parameters are first extracted by applying the wavelet packet decomposition with multiple mother wavelets, and the coefficients’energy content for terminal nodes is used as the input feature for the classification problem. Then, a study through the parameters’space to find the best values for the number of trees and the number of random features is performed. In this way, the best set of mother wavelets for the application is identified and the best features are selected through the internal ranking of the random forest classifier. The results show that the proposed method reached 98.68%in classification accuracy, and high efficiency and robustness in the models. Keywords fault diagnosis, spur gearbox, wavelet packet decomposition, random forest 1 Introduction Power transmission mechanisms are fundamental components used in several types of machines, and spur gearboxes are one of the most used types in low and medium speed applications. As a fundamental tool in condition-based maintenance, fault diagnosis of power transmission devices has become an interesting tool in order to avoid unwanted downtime in industrial processes. Specialized literature shows that several works have been developed in dynamic analysis and fault diagnosis for gearboxes. Walha et al. [1] examined the effect of backlash in a two-stage gear system using standard methods for nonlinear systems. Abbes et al. [2] performed the dynamic analysis of gearboxes by using substructures, considering the static error as a vibratory excitation. Tian et al. [3] identified effective and sensitive health indicators in order to monitor the crack propagation in a one-stage gearbox using the dynamic model and the discrete wavelet transform. In order to avoid devastating consequences following to unexpected failures, several works addressing detection and diagnosis methods have been developed. Within such methods, one can mention: Lubricant monitoring using chemical analysis [4], analysis of motor current signals using spectral analysis based on the fast Fourier transform [5], and analysis of vibration signals [6], among others. One of the major drawbacks of all the mentioned methods is that they require the presence of a human expert to obtain results, which can produce biased conclusions that rely on the expert’s experience. Therefore, there is a need to automate the fault detection and diagnosis processes using more robust methods such as those coming from data mining and machine learning. Rafiee et al. [7] and Sanchez et al. [8] showed the use of multi- Diego CABRERA (✉), René-Vinicio SÁNCHEZ, Grover ZURITA, Mariela CERRADA, Chuan LI Departamento de Ingeniería Mecánica, Universidad Politécnica Salesiana, Cuenca, Ecuador E-mail: [email protected] Diego CABRERA, Fernando SANCHO Departamento de Ciencias de la Computación e Inteligencia Artificial, Universidad de Sevilla, España Grover ZURITA Departamento de Ingeniería Electro-Mecánica, Universidad Privada Boliviana, Cochabamba, Bolivia Mariela CERRADA Departamento de Sistemas de Control, Universidad de Los Andes, Mérida, Venezuela Chuan LI Research Center of System Health Maintenance, Chongqing Technology and Business University, Chongqing 400067, China Rafael E. VÁSQUEZ Facultad de Ingeniería Mecánica, Universidad Pontificia Bolivariana, Medellín, Colombia layer perceptron neural networks to recognize fault conditions in gearboxes. Other methods, like adaptive neural networks, able to change its topology with the inclusion of new samples were introduced by Barakat et al. [9], while Yang et al. [10] introduced the use of clustering techniques based on self-organized Kohonen neural networks. The problem of techniques based on neural networks (NN) is that they are highly dependent on the number of features; therefore, if the problem is highly dimensional, it is likely that the NN will not converge. Additionally, falling into local minima is possible due to an iterativedeterministic search carried out by the training algorithm, preventing the convergence to the global optimum. New techniques have been developed in order to overcome such problems: For instance, support vector machines (SVMs) to identify the presence or absence of faults [11]; SVMs sets with multi-classification least squares support vector machines in fault diagnosis [12]; or Bayesian network [13], which in comparison with back-propagation neural network and probabilistic neural network have shown better results both in the error rate and computation time. Breiman et al. [14] introduced the random forest (RF) model, based on the grouping of trees for classification and regression (CART) [15]. The method is based on the use of a large number of CARTs, called weak learners, which are trained using the technique of bagging (random sampling with replacement). Subsequently, the results of classification of these trees are weighted to provide a single response. The split feature in each node of the weak learner is selected from a random population constructed from the initial population of features, following the Gini criterion as a measure of information gain. Both, measurement of random population of features and size of the forest (number of weak learners trees), are parameters selected by the user. These two parameters change the correlation among the trees, keeping enough diversity in the forest while avoiding over-training [16]. The resulting RF model is inherently multi-class and the decision-making process is done by counting each tree in the forest. RF offers a performance metric called out of bag error (oob-error) calculated as the average of the rate of error in each weak learner, validating with untrained samples remaining from the bagging process. From the previous properties, the characteristics of RF over other methods are summarized into an ability to perform classification with few training samples and lots of attributes, a reduction in bias for better generalization and a reduction in variance, which provides a robust RF classification model. This model has already been considered for fault diagnosis earlier. Han et al. [17] proposed to use it in fault diagnosis for bearings in electric motors with features obtained from the standard deviation of wavelet coefficients. Yang et al. [18] proposed features extraction from time and frequency domains, and parameters optimization through the use of genetic algorithms. Karabadji et al. [19] made a comparative study of different models based on trees using genetic search, showing that RF is among the classifiers that provide better performance. This paper addresses the development of a RF classifier for the multi-class fault diagnosis in spur gearboxes. The rest of the paper is structured as follows: Section 2 presents the background of decision trees and RF model; Section 3 shows the methodology used with RF for faults diagnosis and the configuration of the experimental system used to obtain signals of vibration with the extraction and selection of features for the classification; in Section 4, the results obtained with the methodology proposed by performing an analysis of the features selected by the model and the best parameters are shown; finally, some conclusions and future work are presented. 2 Decision trees and random forest model An instance of the problem can be defined as a vector of features: v¼ðx1,x2,:::,xdÞ2Rd, where each element x i represents a feature of the instance, and dis the problem’s dimension. Hence, the feature space has ddimensions. Since dcould be very large (even infinite), which would make the problem unsolvable, it is necessary to use a function to select a reduced set of “best” features, fðvÞ¼ðxf1,xf2,:::,xfdÞ2Rd, where each element f i belongs to [1,d]. In a binary tree-based model, each node jhas a decision function, hðv,θjÞ:RdP ↕ ↓f0,1g, that receives the vector of features and splits parameters, θj2P, associated to the node j, and decides if the vector goes down right or left in the tree; Pis the space of parameters. 2.1 Training the model Training a tree-based classifier [15] is similar to building the tree in a recursive manner. For each node j, a set of split parameters is selected resolving the optimization problem: θj¼argmaxIðSj,θÞ θ2P , where Iis a fitness function, some gain information criterion, and S j is a subset of training set belonging to node j. The tree is built to achieve the maximum purity in set S j , and some stop criterion can be used to finish the construction, for example a defined depth. An RF classifier [14] is formed from a group of decision trees trained with injected randomness following two processes (see Fig. 1): 1) Bagging. The training set Sis sub-sampled with replacement to build Tnew subsets in order to train Ttrees. 2) Randomized node. In each node of each tree the optimization problem is modified to: θj¼argmaxIðSj,θÞ θ2Pj , where P j is a randomly selected subset of P,Pj   << P jj . We will use Pj   , rho and ρindistinctively. 2.2 Testing the model After building the group of trees, new samples can be classified. For that, the sample is classified for every tree, and from the set of results we obtain a probability distribution by averaging different results (Fig. 2). 3 Application of random forest for spur gearbox fault diagnosis From vibratory signals, a set of features based on wavelet packet decomposition (WPD) is obtained. RF is then used in feature ranker mode to select the most important features; the parameters of the model are optimized with a greedy algorithm. 3.1 Measurement set up The measurements were carried out at the Mechanical Engineering Division (Vibration Laboratory) from the Salesian Polytechnic University of Cuenca (Ecuador). Figure 3 shows the measurement set up. The vibration signatures are obtained from a uni-axial accelerometer. Seven classes are used to evaluate the classifier: Normal operation (Class 1), and the following 6 types for failures: 10%broken tooth (Class 2); Pinion pitting (Class 3); Wear of 0.5 mm in the face of the pinion (Class 4); Misaligned gear (Class 5); 50%broken tooth (Class 6); 100%broken tooth (Class 7). 3.2 Extraction of features Preprocessing raw data in order to obtain the features is done by using WPD: Firstly, the raw signal is decomposed using a mother wavelet in approximation signal and detail signal. After that, approximation and detail signals are decomposed again. This process continues until a level 6 of decomposes signals, as shown in Fig. 4(a). Then, for each signal obtained in level 6, the energy is calculated building a vector of 64 real values; this process is repeated using 5 different mother wavelets: Daubechies 7 (db7), symlet 3 (sym3), coiflet 4 (coif4), biorthogonal 6.8 (bior6.8) and reverse biorthogonal 6.8 (rbior6.8). All these vectors of energy are put together resulting in a features vector with 320 real values, as shown in Fig. 4(b). The purpose of using these families of wavelets is to verify that the diagnostic information specifically supplied by these wavelets is sufficient to achieve an acceptable ranking in the classification task. Fig. 1 Training process for the RF classifier Fig. 3 Configuration of system for failure’s simulation Fig. 2 Test process for the RF classifier 3.3 Features selection The features selection is performed in two stages: 1) In the first stage, the best set of wavelet mother is selected by incrementally adding each subset of features to the data, and training a RF model modifying iteratively the number of trees and the number of random features selected in each node. The oob-error is used as a performance measure to choose the best set of wavelets and the best parameters. Table 1 shows a summary of results of this process. It is identified that the combination db7 +sym3 +coif4 +bior6.8 presents the minimum ooberror with a total of 17 random variables (rho) and 727 planted trees (see Fig. 5 for a comparison of different parameters’sets). 2) The second stage uses the ranking assigned by RF to each feature in order to construct the diagram of importance. Figure 6 shows the importance of features diagram before making the selection of features (Fig. 6(a)), and after the selection where 185 features are selected from the original set of 256 (Fig. 6(b)). Features ranked lower than 0.2 are removed. 3.4 Validation process The database is divided into a 75%‒25%for the training- Fig. 4 Feature extraction process. (a) WPD; (b) energy extraction and features vector building Table 1 The best selected wavelets and the best model parameters Mother wavelets oob-error Feature’s number Tree’s number db7 0.0590 12 1901 db7 +sym3 0.0419 8 1713 db7 +sym3 +coif4 0.0410 7 1671 db7 + sym3 + coif4 + bior6.8 0.0390 17 727 db7 +sym3 +coif4 +bior6.8 +rbior6.8 0.0438 10 1191 test sets. Table 2 shows the confusion matrix resulting from the process of validation. The largest error can be found in pinion pitting fault which shows a big confusion with a 10%broken tooth. It is important to note that in Classes 3 and 4 the classifier makes mistakes with a specific fault, allowing focus on improving the accuracy without changing the classifier completely. The general performance is shown in Table 3. The accuracy analysis leads to the conclusion that the classifier has a great success separating different faults. Through sensibility analysis we can focus specifically on the ability of the classifier to identify samples belonging to a group. In our case, F score weights the relationship between accuracy and sensibility metrics. We have assigned the same importance to both values, showing a more realistic metric without bias introduced for percentages of both, true positives and true negatives. Another measure of assessment is the area under the Fig. 5 Curves of training: (a) oob-error for each set of wavelets with variable number of random features; (b) oob-error for the set of the best wavelets for maximum randomness (1), maximum correlation (256) and perfect randomness (17) with a variable number of trees Fig. 6 Selection of features: (a) Importance of features without selection; (b) importance of features with selection; (c) curve of further training to the selection of features curve (AUC) of each class against each of the remaining classes, as shown in Table 4, and obtained from the relative operating characteristic curve. From this measure we obtain the degree of classification percentage between each of the classes. Figure 7 shows the separation of data and the importance of each coordinate after the application of the multidimensional scaling (MDS) technique to the proximity matrix. 4 Analysis and results Figure 5(a) shows that the set of features with the best group of mother wavelets gets always an error lower than the other groups of wavelets, regardless of the parameters of the RF model. Among the possible sets of random features, it is observed that with a specific set of 17 of them we get the minimum oob-error regardless of the number of trees. The ranking obtained from the importance of features, as shown in Fig. 6(a), shows that the Feature 33, belonging to the wavelet db7, influences the model negatively. After applying the threshold to reduce the number of features according to their importance, an increment of 0.1%of the oob-error is observed (from 3.9%to 4%), a value that is irrelevant for the total classification. The results shown in Tables 3 and 4 are justified by observing the eigen-values in Fig. 7(b), which shows a high degree of importance up to the sixth coordinate of the scale space; hence, the first 3 dimensions shown in are not enough for an adequate graphic representation of the separation between the data. However, it is easy to see that the classes most clearly separable are normal, tooth broken 100%, tooth broken 10%and gear misalignment. Nevertheless, the joint information provided by MDS and by the eigen-values helps to understand the distribution of data in a high-dimensional space as the one we are considering, so MDS can be seen as an additional and valuable tool in combination with classification models that work on high dimensional structures. 5 Conclusions and future work The results obtained with this classifier show a lower error rate and over 96%of classification accuracy for the gear faults classes. It must be stressed out the efficiency in the use of wavelet energy coefficients as features predictors. The lowest AUC is located between classes 2 (10% broken tooth) and 3 (pinion pitting), concluding that the energy of these two faults are most similar and can produce more errors for a correct classification. From a computational point of view, the training of the RF model does not present high complexity in time, and it is easy to do a Table 2 Confusion matrix Class 1 2 3 4567 1 37000000 2 03700000 3 03341000 4 00235100 500003800 6 0 0 0 0 0 36 1 7 00000037 Table 3 Classifier’s performance measures Metric Value/% Accuracy 96.950 Sensibility 97.000 F score 96.975 Table 4 AUC individual unit: % Class 2 3 4 5 6 7 1 100 100 100 100 100 100 2 96.05 100 100 100 100 3 96.30 100 100 100 4 98.68 100 100 5 100 100 6 98.65