Computational Intelligence and Neuroscience

Volume 2015 (2015), Article ID 427965, 11 pages

http://dx.doi.org/10.1155/2015/427965

## A Framework for Final Drive Simultaneous Failure Diagnosis Based on Fuzzy Entropy and Sparse Bayesian Extreme Learning Machine

^{1}School of Computer Science and Technology, Wuhan University of Technology, Wuhan 430000, China^{2}Yangtze University College of Technology and Engineering, Jingzhou 430023, China

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

Academic Editor: J. Alfredo Hernandez

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

#### Abstract

This research proposes a novel framework of final drive simultaneous failure diagnosis containing feature extraction, training paired diagnostic models, generating decision threshold, and recognizing simultaneous failure modes. In feature extraction module, adopt wavelet package transform and fuzzy entropy to reduce noise interference and extract representative features of failure mode. Use single failure sample to construct probability classifiers based on paired sparse Bayesian extreme learning machine which is trained only by single failure modes and have high generalization and sparsity of sparse Bayesian learning approach. To generate optimal decision threshold which can convert probability output obtained from classifiers into final simultaneous failure modes, this research proposes using samples containing both single and simultaneous failure modes and Grid search method which is superior to traditional techniques in global optimization. Compared with other frequently used diagnostic approaches based on support vector machine and probability neural networks, experiment results based on -measure value verify that the diagnostic accuracy and efficiency of the proposed framework which are crucial for simultaneous failure diagnosis are superior to the existing approach.

#### 1. Introduction

With sustained increase of work condition complexity, simultaneous failures occur more frequently in final drive which is the pivotal part of car and seriously affect running status and comfort and safety of car. Final drive is mainly consisting of a pair of gears which are meshing together when car runs. Owing to the complex structure, a certain function disorder in final drive usually stems from more than one single failure at the same time which is called simultaneous failure. Traditional manual technology cannot accomplish simultaneous failure diagnosis (SFD). This paper is focusing on final drive simultaneous failure diagnosis which is essential for auto manufacturer and maintenance industry.

Failure diagnosis by using vibration signal is almost the most frequently used approach because vibration signal is relatively precise and accurate against diagnosis based on sound. It can be divided into three main steps: feature extraction, training diagnostic models, and failure mode identification. The vibration signal collected from final drive has the characteristics of being nonlinear and nonstationary, and it is enclosed with a lot of uncorrelated and superfluous information. It is impossible to extract valid failure mode information from original vibration signal because of the noise and interference embedded in it. The frequently used preprocessed methods include wavelet analysis [1, 2], wavelet package transform (WPT) [3, 4], and empirical mode decomposition (EMD) [5]. Wavelet package transform is suitable for nonstationary vibration signal by decomposing original signal into several subfrequency bands which contains different failure information and effectively reduces noise interference.

Data contained in preprocessed signal is high-dimensional so that it cannot be directly inputted into diagnostic system. Feature extraction has a deep effect on accuracy and reliability of failure diagnosis. Recently, researchers have introduced entropy into the field of feature extraction including approximate entropy, sample entropy [6], and fuzzy entropy [7]. Compared with approximation entropy and sample entropy which are based on Heaviside step function which is mutational at the classification boundary, fuzzy entropy eliminates the influence of baseline drift of data and guarantees the entropy to vary smoothly and continuously with similarity tolerance [8] so that it is excellent in measuring complexity and self-similarity of the preprocessed vibration signal and fully reflecting changes of the vibration performance of mechanical equipment [9].

In recent years, many machine learning methods are applied in failure diagnosis including support vector machine (SVM) [10], artificial neural networks (ANN) [11], extreme learning machine (ELM) [12, 13], and kernel extreme learning machine (KELM) [14]. ELM is single-hidden-layer feedforward neural networks and without human intervention in tuning parameters which differs from SVM and ANN and makes it superior in high generalization and less learning time. KELM apply kernel function to ELM to improve generalization and nonlinear approximation ability [15]. However, computational cost and memory cost of KELM are high with regard to large scale problem. Recently, Bayesian methods are employed into ELM to learn the output weights by estimating the probability distribution of output with high generalization. Soria-Olivas and Gómez-Sanchis proposed Bayesian extreme learning machine [16] for linear regression without solving classification problem. Sparse Bayesian extreme learning machine (SBELM) [17] is a novel method for finding the sparse representatives of hidden layer output weights by imposing a hyperparameter on each weight. During learning phase, SBELM tunes some output weights into zero to obtain compact model. In summary, SBELM has the advantages of probability output, high generalization, sparsity, and fast training speed. To solve the problem of simultaneous failure diagnosis, a proper classifier has to offer the probability of all possible failures. In this research, the proposed framework constructs classifiers based on paired SBELM in which each classifier based on SBELM is trained by a pair of single failure samples. The paired SBELM effectively reflect the probability distribution of failure modes. In general, only single failure samples are used for constructing diagnostic models. Since it is impossible to collect all combinations of existing single failure modes for training, the proposed framework can effectively solve the practical bottleneck in simultaneous failure diagnosis. With the purpose of recognizing simultaneous failure modes, use both single and simultaneous failure samples and Gird search method to generate optimal decision threshold which could convert probability result of classifier into final multiple failure modes. Considering that partial matching is valid and instrumental in simultaneous failure diagnosis, this research adopts -measure to evaluate the performance of the proposed framework.

This paper is organized as follows: Section 2 presents the proposed framework. Section 3 presents the experiment setup, data acquisition, and preprocessing. The results of experiment are discussed in Section 4. Finally, a conclusion is given in Section 5.

#### 2. The Proposed Framework for Final Drive Simultaneous Failure Diagnosis

##### 2.1. Feature Extraction Based on Wavelet Package Transform and Fuzzy Entropy

###### 2.1.1. Wavelet Package Transform

In failure diagnosis, one of the key points is extraction of features from original vibration signal which is nonstationary for mechanical equipment. Wavelet package transform (WPT) is an extended form of wavelet transform to analyse nonstationary and non-linear signal and to supply better partition of frequency band because the same frequency bandwidths can provide good resolution regardless of high and low frequencies [18]. As a multiresolution analysis method, WPT can effectively preprocess nonstationary vibration signal in both time domain and frequency domain. Two-scale equation of WPT is shown below, in which and represent the filter coefficients:

The recursion formula of wavelet package coefficient is

###### 2.1.2. Fuzzy Entropy

When failure occurred in final drive, the complexity of oscillation feature will change; hence we should extract representative features containing in the signal. Fuzzy entropy is an extension of Shannon entropy and fuzzy sets [19]. The procedure of fuzzy entropy is described as follows.

(1) Consider a time series with the length of . For given , , and , construct a vector set in the form of in which each vector contains sequential elements starting from shown as where is the average of vector .

(2) Define the distance between and where , as follows:

(3) Calculate similarity between and using fuzzy function:

(4) Define function as follows:

(5) Change to and repeat step (1) to (4):

(6) Fuzzy entropy of sequence is defined as follows:

(7) If the length is finite, can be changed as follows:

##### 2.2. Sparse Bayesian Extreme Learning Machine (SBELM)

Given a preprocessed data set , , , . The output function of ELM with hidden nodes is shown as follows: where is output weight connecting hidden nodes and output nodes; is the hidden layer output matrix for input in which is the hidden output of the th hidden node. Equation (10) can be written as follows: where is the training data target matrix. SBELM learns output weight by using Bayesian method instead of by calculating Moore-Penrose generalized inverse of [17]. The hidden layer output becomes the input of SBELM. Treat each training sample as an independent Bernoulli event so that probability satisfies Bernoulli distribution. Apply sigmoid function to convert the predicted output as follows:

The likelihood function of sample set is expressed as follows: where is the target of training sample , , and . Conditioned on a hyperparameter , zero-mean Gaussian prior distribution over is as follows:

The typical step of SBELM is to establish the distribution of marginal likelihood over conditioned on and and determine by maximizing the marginal likelihood by Laplace approximation method: where , , and const = . Then, make quadratic approximation for log of posterior probability: where is a diagonal matrix in which with . Therefore, the center and covariance matrix of Gauss distribution of expressed as and are obtained as follows: where . By obtaining Gauss approximation of , the log of marginal likelihood is represented as follows: where . By setting the differential of with respect to as 0, update the hyperparameter as follows:

The main procedure of SBELM is described as follows.(1)Initialize and randomly with .(2)By utilizing Laplace approximation approach, obtain approximated Gauss distribution of and update and by using (17).(3)By maximizing the marginal likelihood, utilize (19) to update hyperparameter until reaching the termination criteria.(4)By tuning some into 0, obtain the sparse representation of hidden layer output weight.(5)For an unknown sample , utilize (12) to predict probability distribution .

##### 2.3. Paired SBELM

SBELM is excellent in solving binary classification by obtaining probability distribution of each class . With the purpose of final drive simultaneous failure diagnosis in which training samples of single failure are ample while training samples of simultaneous failure are scarce, this paper combines the state-of-the-art coupling approach proposed in [20] with SBELM to construct a set of paired SBELM (PSBELM) classifiers expressed as for a -label classification problem shown in Figure 1 and each paired classifier in which is trained by every pair of classes and its output is for sample belonging to the th against the th class. The total number of classifiers is .