**1. Introduction**

The development of industrial informatization has given rise to a large amount of data in various fields. This has led to data processing becoming a difficult problem in the industry, especially for fault diagnosis. The explosive growth of data provides more information, and therefore, typical data analysis theories often fail in achieving the necessary results. The main reason for this failure can be attributed to the typical data analysis theory that often sets the data distribution type through prior information and performs analyses based on this assumption. Once the distribution type is set, the subsequent analysis can perform the estimation and parametric analysis based on only that distribution type; however, with the growth of data, more information is provided, and thus, the type of data distribution will need to be modified. As a nonparametric estimation method, kernel density estimation (KDE) is the most suitable method for the massive amount of the current data. KDE does not employ a priori assumption for the overall data distribution, and it directly starts from the sample data. When the sample size is sufficient, the KDE can approximate different distributions. Furthermore, Sheather and Jones [1] provides the optimal bandwidth estimation formula for a one-dimensional KDE and proves that the kernel function is asymptotically unbiased and consistent in the density estimation. However, with the growth of the dimension, the multidimensional KDE becomes more complex, and its optimal bandwidth formula is not provided. The distribution of multidimensional data has been described to a certain extent by estimating the kernel density of

**Citation:** Wei, J.; He, Z.; Wang, J.; Wang, D.; Zhou, X. Fault Detection Based on Multi-Dimensional KDE and Jensen–Shannon Divergence. *Entropy* **2021**, *23*, 266. https://doi. org/10.3390/e23030266

Academic Editors: Yongbo Li, Fengshou Gu and Xihui (Larry) Liang

Received: 30 January 2021 Accepted: 22 February 2021 Published: 24 February 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the reduced data in different dimensions Muir [2], Laurent [3]. In fact, the optimal KDE of multidimensional data is a problem that needs to be studied further.

In the field of fault diagnosis, an essential problem is measuring the difference between samples. A frequency histogram has been used to indicate the distribution difference between two samples Sugumaran and Ramachandran [4], Scott [5]; however, there are three shortcomings to this method: (1) the large number of discrete operations require a higher amount of time; (2) the process depends on the selection of the interval, which is more subjective; (3) there is no intuitive index to reflect this difference. In fact, based on KDE, the JS divergence can be used to measure the difference in data distribution, which can overcome the above shortcomings to a certain extent. For example, the failure of a rolling bearing, which is a key component of mechanical equipment, will have a serious effect on the safe and stable operation of the equipment, and the incipient fault detection of rolling bearings can help avoid equipment running with faults and avoid causing serious safety accidents and economic losses, which has important practical and engineering significance.

In Saruhan et al. [6], vibration analysis of rolling element bearings (REBs) defects is studied. The REBs are the most widely used mechanical parts in rotating machinery under high load and high rotational speeds. In addition, characteristics of bearing faults are analyzed in detail in references Razavi-Far et al. [7], Harmouche et al. [8]. Compared with traditional fault diagnosis, the fault diagnosis of rolling bearings is more complex:


To overcome these problems, in-depth research has been conducted on this topic. Fault detection technology based on trend elimination and noise reduction has been proposed previously He et al. [9], Demetriou and Polycarpou [10]. The signal trend ratio is enhanced by eliminating the trend, and the signal–noise ratio is enhanced by noise reduction, and therefore, the fault detection effect is improved. However, this method uses the traditional detection method and cannot effectively solve the problem of data coupling. In reference Zhang et al. [11], Fu et al. [12], a fault detection method based on PCA dimension reduction and modal decomposition feature extraction is proposed. For multidimensional data, PCA dimension reduction is performed to reduce data dimensions and eliminate correlation between different dimensions. Then, the modal decomposition method is used to extract features among dimensions for fault detection. This method can effectively solve the strong coupling between data; however, it will lose some information in the process of PCA dimension reduction, and it leads to a reduction in the fault detection effect. In reference Itani et al. [13], Kong et al. [14], Jones and Sheather [15], Desforges et al. [16], a bearing fault detection method based on KDE is proposed. These studies analyzed the feasibility of KDE method in fault detection, and combined different classification methods for experiments. However, these methods only use one-dimensional KDE, and cannot directly describe high-dimensional data.

The data distribution is reconstructed by KDE and the cross-entropy function is constructed to measure the distribution difference for improving the fault detection results. However, this method cannot reflect the correlation between different dimensions, and the cross-entropy function is not precise in the description of density distribution, which leads to a reduction in the fault detection effect, especially for unknown fault detection, which is not included in the fault set.

In this study, the KDE method is extended to multidimensional data to avoid information loss caused by the KDE for each dimension, and to better describe the density probability distribution of the data. Meanwhile, this study improves the traditional method using the cross-entropy function as the measurement of density distribution difference, and it uses JS divergence as the measurement of density distribution difference, thereby avoiding the relativity caused by the cross-entropy function. Most fault identification methods are based only on distance measurement; however, only relying on distance measurement cannot effectively detect unknown faults. Based on JS divergence, distribution characteristics of JS divergence between the sample density distribution and population density distribution are derived using the sliding window principle. Thus, the detection threshold of fault identification is assigned to realize the identification of unknown faults.

This paper is based on the following structure. In Section 2, the trend elimination method and detection method are introduced, and the intrinsic and extrinsic signals in the observation data are separated. Then, the fault detection threshold is constructed via statistics. In Section 3, the KDE method is extended to multidimensional data, and the optimal bandwidth is derived. Then, JS divergence is employed to measure the difference between probability distributions of different densities. In Section 4, the sliding window principle is used to sample the training data to obtain the distribution characteristics of JS divergence between the sample density distribution and the overall density distribution, and the detection threshold of fault identification is obtained using the KDE method. In Section 5, the normal data, two known faults, and one unknown fault are identified using the bearing data of the Case Western Reserve University Bearing Data Center as the fault diagnosis data. The experimental results show that the method can identify all types of faults well.
