#### Abstract

As an entropy representing the complexity of sequence, slope entropy (SloE) is applied to feature extraction of bearing signal for the first time. With the advantage of slope entropy in feature extraction, the effectiveness of bearing fault signal diagnosis can be verified. Five different kinds of entropy are selected to be comparative methods for experiments, and they are permutation entropy (PE), dispersion entropy (DE), a version of entropy adapted by PE, which is weighted permutation entropy (WPE), and two versions of entropy adapted by DE, which are fluctuating dispersion entropy (FDE) and reverse dispersion entropy (RDE). A method of extracting a single feature of bearing fault signals based on SloE is carried out. Firstly, the features of the bearing signals are extracted by the six kinds of entropy. Then, some relevant data are computed, and the identification ratios are calculated by the K-nearest neighbor (KNN) algorithm. The experimental result indicated that the identification ratio of SloE is the highest at 97.71% by comparing with the identification ratios of the other five kinds of entropy, which is higher by at least 13.54% than the others and 27.5% higher than the lowest one.

#### 1. Introduction

Bearing is an indispensable part of modern mechanical equipment, and its roles in modern mechanical equipment are the support of the mechanical rotating body, the sliding part in motion, and the guarantor of rotation accuracy [1–3]. Self-aligning ball bearing is the research object of this paper, which is a kind of rolling bearing equipped with spherical balls installed between the inner race with two raceways and the outer race with a spherical raceway. Its curvature center of the outer race is consistent with the bearing center, so it has the same centering function as the automatic centering ball bearing. It can be adjusted automatically when the shaft and housing deflect, and this process will not increase the bearing burden. Self-aligning ball bearing ball bearings can bear the loads in two directions, which are radial and axial. Compared with the axial bearing capacity, the self-aligning ball bearing has a greater radial bearing capacity, so it can well deal with heavy load and impact load. The self-aligning ball bearing belongs to the bearing that has tapered holes in the inner diameter of the inner race. It can be installed directly or installed on the cylindrical shaft with a remove pipe or fastening sleeve. The cage is stamped with a steel plate and formed with polyamide. Due to its strong ability to bear heavy load and impact load, self-aligning ball bearings are used in sugar pressing, papermaking, precision instrument, petroleum, cement, metallurgy, mines, low noise motor, motorcycle, rolling mill, automobile, and such industries. Therefore, as such a widely used, fine, and important component, bearing fault diagnosis is a subject that needs to be studied carefully [4–6].

Due to the nonlinear stiffness and bearing clearance of rolling bearing [7, 8], the vibration signal generated by its operation often shows nonstationary and nonlinear. Therefore, extracting useful fault feature information from nonstationary and nonlinear signals is the focus and difficulty of rolling bearing fault diagnosis. Aiming at the problem of bearing fault, many scholars have done lots of research on the early diagnosis of bearing fault [9, 10].

Many scholars apply the methods commonly used in signal processing to bearing fault diagnosis. Many methods of nonlinear dynamic are proposed [11], such as fuzzy entropy (FE) [12], permutation entropy (PE) [13], Rényi entropy (RE) [14], dispersion entropy (DE) [15], sample entropy (SE) [16], Wiener entropy (WE) [17], and instantaneous spectral entropy (ISE) [18], which can extract the nonlinear feature of the signals and characterize the health state of the equipment. For SE, its calculation time is long, real-time performance is poor, and similarity measurement is prone to mutation [19]. PE has the advantage of simple calculation, but it ignores the amplitude information of dynamic number sequences [20–25]. Although FE is obtained by improving SE, it still has the disadvantages of slow calculation speed and so on [26–28]. DE has the advantages of small influence by burst signals and better stability, which make up for the problem of PE [29–32]. WE is very sensitive to small alterations. ISE has the relatively low computational burden and fast execution time. Reverse dispersion entropy (RDE) [33] as an improved version of DE has the advantages of both PE and DE and has stronger stability and noise robustness [34, 35]. Similarly, FDE [36] is also an improved version of DE [37], and WPE [38] is an improved version of PE [39, 40]. There are also many other kinds of entropy that are not listed here [41–44].

Slope entropy (SloE) [45] is a new entropy estimator proposed in recent years, which is based on only the vibration amplitude of dynamic number sequences and five symbol patterns. In the three years since it came out, it has been used in many fields, such as medicine and underwater acoustic signal, and achieved excellent results. SloE is applied to the field of medicine by David Cuesta-Frau in 2020 [46, 47], and it is applied to the underwater acoustic signal processing field by Li et al. in 2021 [48]. SloE has a good feature extraction effect in various fields, so it is introduced into the field of fault diagnosis for the first time in this paper.

SloE is applied to the field of bearing fault diagnosis for the first time in this paper. The remaining of the paper is structured as follows: the specific calculation steps of SloE are introduced, and an example of the algorithm is given in Section 2. Section 3 introduces the detailed steps of the experiment and gives a flow chart of the steps. In Section 4, the graphs of the six types of signals are given, the single feature extraction experiment is carried out, and KNN is used for classification. Section 5 is the summary, where the main innovations and conclusions of this paper are given.

#### 2. Slope Entropy

##### 2.1. Basic Principle

SloE is a new algorithm put forward in 2019, which can indicate dynamic number sequence complexity. It is founded on both the vibration amplitude of the dynamic number sequence and the five symbol patterns that have been set. Each symbol pattern is allocated according to the difference between the vibration amplitude of the input dynamic number sequences. The calculation process of the SloE algorithm is simple and easy to understand, whose main operation has only addition and subtraction calculations. SloE is calculated as follows.(1)For given dynamic number sequences , the number of subsequences of *D* is extracted in the light of the embedding dimension *m*, , , …, , where .(2)Diverse symbol modes (+2, +1, 0, −1, −2) are partitioned by the positive and negative values of the two threshold parameters ( and ). Figure 1 is the symbol allotment of SloE. The specific symbol modes of SloE are assigned in a very unequivocal way. The vertical increments of SloE are decided by and , and the horizontal increment between the samples of the continuous dynamic number sequence is always set up as 1. If , the inclination of the borderlines is 16.5° and −16.5°. And if , the boundaries slope of the region of symbol “0” is 5° and -5°. The specific distribution principles are as follows: if , the symbol mode is +2; if , the symbol mode is +1; if , the symbol mode is 0; if , the symbol mode is -1; if , the symbol mode is -2, where .(3)Mode sequences , which are corresponding to , are gained after symbol allotment, , , …, , where , are the symbol modes calculated by , , …, through step (2).(4)Mode sequences have diverse forms. The quantity of each form is . The frequencies of the mode sequences are the proportions of the number of times they appear: , , …, . The calculation formula of SloE is defined as follows in view of the classical Shannon entropy:

##### 2.2. Example

Here is a dynamic number sequence , sequence length . Set up the delay time , the embedding dimension , and the two threshold parameters and . The specific steps of calculating SloE are as follows:(1)According to *D*, get the subsequences , , , , , , , , , , , , .(2)Then, according to the subsequences, obtain the mode sequences: , , , , , , , , , , , , .(3)There are 10 types of mode sequences. The frequency of is 3, the frequency of is 2, and the frequency of other types is all 1. Therefore, the probabilities are , , , , , , , , , .(4)Finally, the value of SloE obtained by equation (1) is .

#### 3. Proposed Method

A single feature extraction method is put forward for the six types of bearing signals in this experiment. As shown in Figure 2 is the flow chart of the single feature extraction method, the particular procedures of the method are as follows.(1)The six types of bearing signals are imported after being normalized.(2)For the normal bearing signals or each type of bearing fault signals, which are normalized, 120 samples are selected, where each sample contains 1000 sample points. The features of PE, WPE, DE, FDE, RDE, and SloE are extracted.(3)K-nearest neighbor (KNN) is chosen as the classifier to classify the features of six kinds of bearing signals. For each type, 40 groups of sample signals are selected as training samples, and 80 groups of sample signals are selected as test samples. The number of nearest samples is set as *k* = 3.(4)The identification ratios are obtained by operation.

Through these steps, we can conclude that SloE is effective in single-feature classification by comparing the identification ratio calculated by SloE and the others of the five other kinds of entropy.

#### 4. Feature Extraction

##### 4.1. Six Types of Signals

The features of the six types of signals are extracted, which are normal signals, bearing inner race, ball, and outer race fault signals [49]. According to the position relative to the load zone, which is centered, orthogonal, and opposite, there are three types of bearing outer race faults signals. The signals come from the same website, and they are acquired under the same fault diameter, motor load, and motor speed. The names of these signals are replaced by 100, 108, 121, 133, 147, and 160 in the paper. Types and codes of the signals are shown in Table 1.

The lengths of sampling points for them are 485643, 122917, 121556, 122571, 122281, and 122136. The normalized six types of bearing signals are shown in Figure 3.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

##### 4.2. Extraction Method

In the feature extraction experiment, for normal signals or each type of bearing fault signal, 120 samples are chosen, and every sample includes 1000 sampling points. These samples almost contain all sampling points of the five bearing fault signals.

For reasonable and scientific comparison, because all kinds of entropy have the same settable parameters the embedding dimension and the delay time, set up them as and . The number of categories is the same settable parameters of DE and two changed versions of DE, set up it as *c* = 3. DE and FDE have the same mapping format, which is the normal cumulative distribution function (NCDF). The two threshold parameters of SloE are set up as and . Feature distribution of six types of bearing signals is shown in Figure 4.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

It can be inferred from Figure 4, for PE distribution, the entropy points of 100 and 147 and the entropy points of 108, 121, 133, and 160 are near to each other; for WPE distribution, almost all entropy points of 121, 133, and 160 are mixed together; for DE and FDE distribution, the entropy points of all types of signals intersect in varying degrees; for RDE distribution, only the entropy points of 133 are significantly distinguished from those of the other five types of signals; for SloE distribution, only a few entropy points of 160 are close to those of 121 and 133. It indicates that SloE has better classification ability on the six types of bearing signals.

For proving the validity of SloE, the mean and the absolute minimum mean difference (AMMD) of diverse features are computed. AMMD is set to the absolute value of the minimum mean difference, and it can intuitively express the distance between the entropy points of the two types of signals, whose entropy points are the closest to each other in the six types of signals. The larger the MMD, the more reliable the interclass separability of the entropy. Table 2 shows the mean and AMMD of different features.

As Table 2 shows, with regard to the six kinds of entropy, the average value of each type of signal has diverse degrees of diversity. RDE has the minimum AMMD of 0.001, and the AMMD of SloE is the maximum, which is 0.1901. It is preliminarily judged that RDE has the worst interclass separability and SloE has the better.

##### 4.3. Feature Classification

For proving the better effect of bearing fault signals feature extraction and classification based on SloE, KNN classification is led into this experiment. For the normal bearing signals or each type of bearing fault signals, which are normalized, 120 samples are selected, where each sample contains 1000 sample points. For each type, 40 groups of sample signals are chosen as training samples, and 80 groups of sample signals are classified as test samples. The feature classification and recognition distribution are shown in Figure 5.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

As shown in Figure 5, for these six types of signals, PE and DE have diverse quantities of wrongly classified samples, where the ones of 100 and 160 are separately the largest; WPE, FDE, and RDE only correctly classify the samples of 100, 147, and 133, respectively, but a large number of samples in other signals are classified wrongly; SloE has only a few misclassified samples for 121 and 160, which are classified to 160, 121, and 133, and the other four types of signals are classified correctly; among the six feature extraction methods, 160 has the worst classification effect, while SloE has the best average classification effect. Identification ratios of the feature are shown in Table 3.

As shown in Table 3, for 100, WPE and SloE have the highest classification and identification ratio of 100%, that of DE is 90%, and those of PE, FDE, and RDE are less than 80%; for 108, the classification and identification ratios are more than 90% expect FDE and RDE, where that of SloE is 100%; for 121 and 160, only the classification and *C* of SloE are more than 90%, and the other kinds of entropy have the classification and identification ratios less than or equal to 80%; for 133, SloE and RDE have the highest classification and identification ratio of 100%, and WPE has the lowest one of 58.75%; for 147, SloE and FDE have the highest classification and identification ratio of 100%, and RDE has the lowest one of 57.5%; for the six types of signals, SloE has the maximum average identification ratio of 97.71%. The mean identification ratios of the other five kinds of entropy are lower than 85%.

The results show that the SloE classification of six kinds of signal samples is the most accurate, and the average identification ratio is the highest.

#### 5. Conclusions

SloE is applied to the field of bearing fault diagnosis, and a new method of extracting features is put forward. The practicability of the proposed method is proved by the feature distribution and the recognition distribution of the six types of measured bearing signals. The main innovations and conclusions are as follows.(1)SloE is applied to the field of bearing fault diagnosis for the first time.(2)A new single feature extraction method based on SloE is proposed, and all methods in the paper adopt single feature extraction, which saves a lot of time.(3)The proposed single feature extraction method based on SloE in this paper has larger AMMD than the single feature extraction method of the six signals based on PE, WPE, DE, FDE, and RDE, which proves that the interclass separability of SloE is better. Moreover, it has the highest average identification ratio of 97.71%, which is higher by at least 13.54% than the others of the other five kinds of entropy.

#### Data Availability

The data supporting the findings of this study are available within the reference [49].

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.