Elsevier

Knowledge-Based Systems

Volume 129, 1 August 2017, Pages 39-60
Knowledge-Based Systems

A Bayesian approach to consequent parameter estimation in probabilistic fuzzy systems and its application to bearing fault classification

https://doi.org/10.1016/j.knosys.2017.05.007Get rights and content

Abstract

A bearing is an essential component in rotating machinery, one of its principal cause of failure, and its health condition is directly related to the safety and effective operation of such machinery. To the best of our knowledge, it is the first time that a probabilistic fuzzy system is applied to bearing fault classification. The type of probabilistic fuzzy classifier considered is a parsimonious fuzzy rule based model where each rule can diagnose a set of faults each one with its probability. For this kind of real world application, it is desirable to develop interpretable and accurate MIMO fuzzy systems, able to deal with the dimensionality and uncertainty present in data (vibration signals). For parameter estimation we adopt a two steps sequential state-of-the-art data-driven method. First, the antecedents of the rules are estimated using an iterative supervised clustering algorithm. Based on the antecedents the consequent parameters are then estimated. For this, a new method for consequent estimation is proposed. This is based on the observation that for defining a rule consequent not all training data within the region of applicability of that rule are equality relevant. Two criteria for selecting a relevant region in the feature space for consequent parameter estimation are proposed. Results show a statistically significant improvement on the performance of probabilistic fuzzy diagnosers trained with the proposed method, independently of the criterion used for defining the relevant region, when compared with the above mentioned state-of-the-art method. Moreover, the proposed consequent parameter estimation method practically has no overhead on the overall training of the diagnoser. Results show that an equilibrium can be found between the model level of detail and its accuracy. However, when accuracy is the sole comparison criterion, the proposed probabilistic fuzzy systems systematically matches the performance of other data-driven models like distance based methods (K-nearest neighbors), connectionists (probabilistic neural networks), or maximum margin classifiers (support vector machines).

Introduction

A bearing is a mechanical component used to reduce friction between other mechanical moving parts. Bearings are one of the most common components in mechanical equipment and one of the principal cause of its malfunction [63]. For example, in induction motors metal bearing faults account up to 40% of the faults [44]. This makes bearing fault diagnosis (i.e., detection, classification and prognosis) an economically very relevant topic. Moreover, the literature on bearing fault diagnosis is extensive which indicates that the topic is also scientific and technically challenging.

Rolling element bearings, such as ball bearings, consist of an inner, an outer race or ring, inside which there is a cage of holding the rolling elements, see Fig. 1. All these elements are prone to faults due to excessive load, lubricant failure, fatigue, corrosion, or other causes. The bearing health condition is directly related to the safety and effective operation of mechanical systems [32]. For instance, should the metal engine bearings supporting a crankshaft fail the whole engine can disintegrate.

A fault can be classified according to its location and to its type. For instance, it can be a single point fault in the outer race as illustrated in Fig. 2. Often, general failures originate from incipient single point faults such as this. Thus, an early diagnosis of incipient faults is deemed necessary and this is the main motivation to focus this study on single point faults. Different faults can and do occur simultaneously and this possibility is also considered in this study. Bearing fault diagnosis involves at least the following stages: data acquisition and conditioning, feature extraction and selection, and finally classification. Fuzzy formalisms have been used in all of these stages as a framework for dealing with the inherent uncertainty of the feature space. Actually, i) the number of faulty samples is much smaller than healthy samples; ii) there is no guarantee that all relevant features are fully observable; iii) the interference between different faults is not easily identified; iv) and measurement is often noisy. Therefore, the knowledge that a fault diagnoser holds about the system is necessarily incomplete and uncertain. However, another important advantage of fuzzy models when compared with other nonlinear modeling and detection techniques such as artificial neural networks, is that fuzzy models provide an insight on the linguistic relationship between the variables of the system [50]; an issue that is often forgotten with some notable exceptions though [71], [72].

A diversity of fuzzy formalisms has been applied to the bearing fault diagnosis, including the neuro-fuzzy approaches, e.g., [22], [39], [48], clustering, e.g., [31], [54], [57], application of fuzzy measures, and in particular of fuzzy entropy, e.g., [35], [70], fuzzy support vector machines, e.g., [68], possibility and Dempster-Shafer evidence theory, e.g., [55], [59], fuzzy relations and fuzzy relation equations, e.g., [64], fuzzy fusion of multiple criteria, e.g., [11], [35], fuzzy numbers, e.g., [21], rough sets and fuzzy rough sets, e.g., [47], [69], semi-supervised approaches, e.g., [20], ensembles of fuzzy classifiers, e.g., [61], fuzzy grey-optimization methods for short-term fault prediction [67], fuzzy similarity operators to compare two time-domain phase trajectories [45], fuzzy lattice based diagnoser [27], and health degree evaluation index based on fuzzy sets [5], [62].

In rule based bearing fault diagnosis either Mamdami or Sugeno models are used, see e.g., [6], [33], [43]. In this work we propose an innovative approach to bearing fault classification, i.e., we propose a parsimonious type of fuzzy rule based model where each rule can diagnose a set of faults each one with an associated probability. This type of model is known as a probabilistic fuzzy system [8], [40] and is usually composed by a set of rules combining linguistic information in the antecedents with probabilities in the consequents. Each rule can be viewed as describing a fuzzy region in the feature space where the consequent probability distribution over predicted classes is valid. That is, the j-th rule, r(j), can be informally viewed as: r(j):=iftheinputbelongstofuzzyregion(j)theny^(j)=c1withprob.p1,,y^(j)=cCwithprob.pC[w(j)]where y^(j) is the rule output, c1,,cC are class labels, and w(j) is a certainty factor representing a belief in the accuracy of the rule. Although we will be using this type of probabilistic fuzzy system, it should be clear that other types are available, e.g., [3], [36].

In real-world applications like bearing fault diagnosis there are various types of uncertainty to be handled. Uncertainties can result from partially observable dynamics, insufficient data, or coarse or noisy measurements, for instance [65]. While stochastic modeling methods can tackle stochastic uncertainty, fuzzy systems are useful to handle incomplete and vague information. Therefore, probabilistic fuzzy systems seem particularly suitable to cope with complex, stochastic, and vague environments [8], [10], [40]. Actually, probabilistic fuzzy systems have been successfully applied to real-world problems, e.g., in financial market analysis [4], [10], robotics [36], process modeling and control [65], or more recently to predict the mortality of septic shock patients [16]. However, to the best of our knowledge, it is the first time that a probabilistic fuzzy system is applied to any type of bearing fault diagnosis, i.e., detection, classification or prognosis of faults. This is unfortunate as i) the problem requires a multi-input multi-output (MIMO) model with typically one output for each fault, ii) the usual Mamdami or Sugeno rule based models do not scale sufficiently well in the MIMO case, iii) the larger the model the more difficult is to interpret it, and iv) the more prone is to overfitting. The main contributions of the work can be stated as follows:

  • The application of a parsimonious and accurate probabilistic fuzzy system to the fault classification in bearing diagnostics. The employment of a fault diagnoser exhibiting reduced computational complexity is particularly relevant in this application as both the number of input variables and output classes can be large.

  • A new parameter estimation method for the rule consequents. This is based on the observation that for defining a rule consequent not all training data points within its activation region are equality relevant. Criteria for selecting relevant data points are proposed revealing superior performance relatively to the currently used methods.

The remaining of the paper is organized as follows. Section 2 describes both the experimental apparatus and the theoretical background required in this study. Section 3 presents the proposed fuzzy probabilistic diagnoser. Section 4 presents the proposed method for consequent estimation. Results and discussion are presented in Section 5. Conclusions end the paper.

Section snippets

Material and methods

This section briefly describes the experimental apparatus used for data acquisition, the signal processing used for feature extraction, and the employed method of feature selection. Data considered in this work consist of vibration signals from which time, frequency, and time-frequency domain features are computed. A total of 1634 features are computed; see Section 2.1 for further details. For feature selection a decision tree-like entropy based criterion is used (Section 2.3).

A probabilistic fuzzy system as fault diagnoser

As previously discussed, we propose the application of a probabilistic fuzzy system, motivated by the need for using an easy to interpret and accurate MIMO fuzzy system, able to deal with the type of collected data.

This section presents the proposed fuzzy probabilistic diagnoser, its inference, and the proposed method for parameter estimation.

To the best of our knowledge, it is the first time that a probabilistic fuzzy system is applied to bearing fault diagnosis. The type of probabilistic

The proposed method for consequent estimation

Within the feature space XRd, the jth rule has a region of influence or activation region defined by the support of the fuzzy relation R(j)(x)={(x,A(j)(x))|xX}where A(j)(x) is given by (3) and the support by supp(R(j))={x|xXA(j)(x)>0}.

Let X={x1,,xk,,xN} be a finite (training) set of feature vectors in the Rd space. Associated with each feature vector xk there is a label yk{c1,,cC} representing the fault type (or class) of the kth input. Up to now, to estimate p(ci|r(j)) the

Results and discussion

This section presents some experimental results and a corresponding brief discussion. Before reporting the application of the proposed methodology to the more realistic experimental apparatus of Section 2.1 where fault interferences can be studied, we report the application to a simpler but widely used benchmark dataset relative to the 6203-2RS JEM SKF deep groove ball bearing from the Case Western Reserve University (CWRU) Bearing Data Centre [37]. This allows the comparison of performance

Conclusions

Bearing fault diagnosis is both an economically relevant and a scientific and technologic challenging topic. This paper has presented a first-time application of fuzzy probabilistic classifiers to bearing fault diagnosis. These are rule-based systems where each rule can diagnose a set of faults each one of them with an associated probability. Each rule can be viewed as describing a fuzzy region in the feature space where the consequent probability distribution over predicted classes is valid.

Acknowledgments

The work was sponsored in part by the Prometeo Project of the Secretariat for Higher Education, Science, Technology and Innovation (SENESCYT) of the Republic of Ecuador, the National Key Research & Development Program of China (2016YFE0132200), the Chongqing Technology and Business University (CTBU) open grant number 1456027, by CNPq, Brazil, grant number 309197/2014-7, and by FCT, Portugal, grant number SFRH/BSAB/128153/2016. The experimental work was developed at the GIDTEC research group lab

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