Heteroclinic orbits between static classes of time periodic Tonelli Lagrangian systems

1.   Introduction

Bearings are one of the important parts in machinery such as motors, gearboxes or wind turbines, and their functional state directly determines the performance and efficiency of the mechanical system ^[1]. Often working for long periods of time in environments with variable loads, speeds and vibrations, they are prone to failure, which can lead to significant economic losses and catastrophic accidents. Therefore, in order to detect potential faults in a timely manner and reduce their impact, advanced fault diagnosis technology must be developed ^[2].

In order to accurately detect bearing faults and with advances in computational intelligence technology, data-driven machine learning (ML) methods have been widely used in fault diagnosis because of their ability to characterize the intrinsic correlation between measurements and fault conditions ^[3]. Once a ML model is established, fault diagnosis can be effectively implemented ^[4]. ML techniques can be divided into supervised, unsupervised, semi-supervised and reinforcement learning. A variety of ML techniques are used for different types of problems. In general, supervised learning solves regression and classification processes, and common algorithms are k-nearest neighbor (KNN), support vector machines (SVM), decision tree (DT), random forest (BF) and neural network (NN). Liang et al. ^[5] integrated wavelet transform into the improved residual neural network and used the global singular value decomposition adaptive strategy for feature extraction for fault diagnosis of rolling bearings in noisy environments. The improved model has good anti-noise robustness. Xu et al. ^[6] developed a hybrid deep learning model based on a convolutional neural network and deep forest model. Compared with other existing methods, although the computational efficiency is significantly reduced, it has better detection accuracy. The typical unsupervised method is the clustering algorithm, including k-means clustering (KCM), fuzzy clustering algorithm and density-based clustering method (DBSCAN). Yuan et al. ^[7] proposed a fault diagnosis algorithm based on adaptive hierarchical clustering and subset to extract features and apply them to gearbox fault diagnosis. Hou et al. ^[8] proposed a clustering fault diagnosis method for rolling bearings based on ensemble empirical mode decomposition, permutation entropy, linear discriminant analysis and the Gath-Geva clustering algorithm, which has the advantage of better class clustering compactness.

Faced with the lack of labeled data for the training of fault diagnosis models, researchers often use clustering algorithms. It is a data analysis problem based on the given sample being grouped into several "classes" or "clusters" according to the similarity or distance of their features ^[9]. The purpose of cluster analysis is to discover the characteristics of the data or process the data through the obtained "classes" or "clusters, " and it has been widely used in data analysis problems in many fields such as image segmentation ^[10], wind speed forecasting ^[11], fault diagnosis ^[12] and so on.

Cluster analysis algorithms are divided into two categories: partition clustering algorithms and hierarchical clustering algorithms ^[13]. The partition based method is a simple and commonly used clustering method that clusters objects by dividing them into mutually exclusive clusters, with each object belonging to only one cluster. The partitioning results aim to make the similarity between clusters low and the similarity within clusters high. Common partitioning methods include k-means, k-medoids, K-prototype and other methods. The application of hierarchical clustering is second only to the partition-based clustering. The core idea is to divide the data into clusters of different layers according to the hierarchy of the data set, to form a tree-shaped cluster structure. According to the process of hierarchical clustering, it can be divided into bottom-up clustering and top-down splitting clustering. Aggregation clustering is represented by balanced iterative reducing and clustering using hierarchies (BIRCH), there is a robust clustering algorithm for categorical attributes (ROCK), and split clustering is represented by a divisive analysis (DIANA) algorithm.

As an excellent clustering algorithm, the KCM ^[14] algorithm has the characteristics of fast computing speed, strong scalability and being simple, fast and easy to understand. It has been widely used in research of rotating machinery fault data analysis. Xue et al. ^[15] combined KCM with AdaBoost to optimize artificial hydrocarbon networks. They proposed an intelligent diagnosis based on double-optimized artificial hydrocarbon networks to identify mechanical faults of an in-wheel motor. Mariela et al. ^[16], inspired by KCM and the one nearest neighbor algorithm, put forward a framework to detect a new mode of abnormal conditions in gearboxes. Wan et al. ^[17] proposed a Spark-based parallel ant colony optimization KCM algorithm for rolling bearing fault diagnosis based on massive running state monitoring data of rolling bearings, to realize efficient and accurate fault diagnosis of rolling bearings.

The above research shows that KCM and rotary fault diagnosis technology have a good combination and can effectively identify the fault state. However, most of these require some domain knowledge to select the appropriate cluster number for the dataset containing data objects with different densities and sizes. The requirement of predefined cluster number makes KCM inefficient for automatic cluster analysis and prone to problems such as slow convergence and low accuracy, thus affecting the efficiency and accuracy of fault diagnosis ^[18]. In recent years, KCM has been combined with naturally inspired metaheuristic optimization algorithms to overcome the challenges of traditional clustering algorithms in processing automatic data clustering. Metaheuristic optimization algorithms have global search capability for optimizing the number of clusters and the clustering centroid ^[19]. The metaheuristic algorithm is mainly used to help the K-means algorithm to get rid of locally optimal convergence by searching the globally optimal initial cluster centroid. Its effectiveness has been widely verified in the literature. Using global-local optimization as an idea, Zhang et al. ^[20] constructed a hybrid genetic algorithm-quasi-Newton method optimization parameter model, which can effectively identify a faulty combustor can based on gas temperature profile. In Yang and Sutrisno ^[21], the symbiotic organisms search (SOS) algorithm was applied to the initial solution of automatic K-means to create subpopulations, thereby improving the quality and efficiency of the search. The subeconomies created by automatic K-means enables the clustering-based symbiotic organisms search (CSOS) algorithm to combine local and global searches on datasets.

The marine predators algorithm (MPA) is a novel and efficient metaheuristic algorithm proposed by Faramarzi et al. ^[22] in 2020. It is primarily inspired by the movement between predator and prey in the ocean and the optimal encounter rate strategy in biological interaction. MPA has many advantages, including fewer parameters, higher flexibility and ease of implementation ^[23], and it has been applied to solve many problems in different fields ^[24,25].

Therefore, this study aims to improve the MPA algorithm for better automatic clustering results. The initialization of MPA enters the high-speed ratio phase with random factors, greatly reducing optimization efficiency and robustness and slowing the initial convergence. In the intermediate stage of optimization, the population wandering strategy needs to be more flexible to lead to an imbalance between exploration and exploitation. On the other hand, the MPA will converge prematurely if prey forages successfully. Therefore, an improved MPA algorithm is proposed and applied to the automatic clustering of bearing faults. The main contributions of this paper can be summarized as follows:

•An improved marine predator algorithm (IMPA), combining refracted opposition-based learning (OBL), sine-cosine algorithm (SCA) and Gaussian-Cauchy mutation (GCM), is developed to improve the convergence accuracy of standard MPA and the ability to handle high-dimensional problems. The efficiency of the IMPA is tested by solving 10 classical optimization functions.

•The proposed IMPA is applied to the automatic clustering of 7 common datasets and Xi'an Jiaotong University bearing data set, and the results are compared with the whale optimization algorithm (WOA), sparrow search algorithm (SSA) and k-means and fuzzy C-means (FCM) algorithms of automatic clustering. The results show that IMPA is more competitive than other optimization methods.

The rest of this article is summarized as follows: Section 2 introduces the concept of cluster correlation, and Section 3 introduces MPA, IMPA and an improved hybrid algorithm for automatic clustering based on IMPA. Section 4 validates the hybrid algorithm. The conclusions and future research directions are presented in Section 5.

2.   Cluster analysis

2.1.   Description of clusters

For a given dataset $X = \left\{{x}_{1}, {x}_{2}, \dots, {x}_{N}\right\}$, the clustering algorithm divides it into K classes, that is, $C = \left\{{C}_{1}, {C}_{2}, \dots, {C}_{K}\right\}$. Get the partitioning matrix $U\left(X\right)$, and the partition matrix $U = {\left[{\mu }_{ij}\right]}_{K\times N}, i = {\mathrm{1, 2}}, \dots, K, j = {\mathrm{1, 2}}, \dots, N$, where ${\mu }_{ij}$ is the attachment of the sample ${x}_{j}$ to the ${C}_{i}$ class.

If ${\mu }_{ij}$ satisfies

and the result of cluster division is satisfied, that is, ${C}_{i}\ne {\mathrm{\varnothing }}; {C}_{i}\cap {C}_{j} = {\mathrm{\varnothing }}, {\mathrm{ }}(i, = {\mathrm{1, 2}}, \dots, K, j = {\mathrm{1, 2}}, .., N;\bigcup _{i = 1}^{K}{C}_{i} = X$, then this is called cluster analysis. Each data object is assigned to a certain category according to the principle of similarity, which is generally measured by Euclidean distance ^[26], that is,

2.2.   KCM

Since the proposed new automatic clustering method combines KCM in part, the basic principle of the K-means algorithm is briefly introduced. The goal of K-means is to divide the dataset into K separate parts to minimize the Euclidean distance between each object and the cluster to which it belongs. The main steps of a standard K-means algorithm are usually represented as follows ^[27]:

1) Select K objects from N data as the initial cluster center {Ci}.

2) Calculate the Euclidean distance from each data point to K initial clustering centers, and reclassify the data according to the principle of minimum distance.

3) Calculate the average of each dimension of the cluster to get the new center. Repeat steps 2) and 3) until the cluster center no longer changes or the distance at which the cluster center is less than the set threshold, and output the cluster center of the data set.

As is known from the clustering process, improper selection of initial clustering center can cause falling into local optimization. The calculation formula of cluster center ${C}_{j}$ is as follows:

Therefore, the initialization of KCM has randomness, which may take noise or an abnormal value as the initial center, resulting in a large difference in the results of each clustering and poor algorithm robustness. At the same time, the simple updating process will cause KCM to fall into local optimality.

2.3.   Automate clustering and objective functions

In order to implement automatic clustering in this paper, we need cluster center information and activation threshold or switch vector. By setting a constant value, if the activation threshold is greater than this value, it indicates that the centroid of the cluster is an effective centroid. If we have K centers, the activation threshold part has K elements, and the coordinates of the center are $K\times N$ elements, which is actually $K\times (N+1)$. Overall, we have a matrix with K rows and $N+1$ columns. The following matrix shows the relationship between the clustering centroid and the activation threshold:

where K represents the row, m is the centroid vector, N represents the column, and $a\in \left[{\mathrm{0, 1}}\right]$ is the activation threshold. In this article, the threshold is set to 0.4; if $a < 0.4$, the center is invalid. Figure 1 shows how to activate or deactivate a cluster center case.

In automatic clustering, the commonly used objective function is the optimization cluster validity index (CVI), used to evaluate clustering ^[28]. The validity index is a criterion for the degree of completion of cluster analysis. The objective function (CVI) needs to be minimized, and the lower the CVI value is, the better the clustering result, with better compactness and further cluster separation. In automatic clustering, if the number of clusters is specified, the previous objective function (Eq (2)) works fine; but if the number of clusters is unknown, another objective function must be used. To date, researchers have provided effectiveness indices for a number of clusters, such as the Davis-Bouldin index (DB) ^[29], Calinski-Harabasz index (CH) ^[30], Pakhira Bandyopadhyay Maulik index (PBM) ^[31], Silhouette ^[32] and compact separation index (CS) ^[33].

In this article, we use the CS index as the objective function to measure the effectiveness of the cluster. CS measures the quality of the clustering results using the ratio of scatter points within a cluster to the sum of separations between clusters. The lower the value of CS is, the better the separation, and more compact clustering is reflected. It has been reported to be a better CVI in terms of efficiently handling clusters with different densities, dimensions and sizes ^[34]. CS is expressed in Eq (5) below:

where the number of data points in cluster C is expressed as $Q$, and the distance between intra-cluster scattering ${X}_{i}$ and inter-cluster separation ${X}_{j}$ is expressed as the function $V\left({X}_{i}, {X}_{j}\right)$. The distance between the data point and its center of mass is expressed as $V\left({x}_{i}, {x}_{j}\right)$.

3.   MPA

3.1.   Standard MPA

MPA is a new metaheuristic algorithm that draws inspiration from nature and is based on the various foraging techniques used by marine predators and the best encounter rate policy in biological interaction. Like most metaheuristics, the initial solution is evenly distributed in the search space. The initialization formula is as follows:

where ${u}_{j}$ and ${l}_{j}$ are the lower and upper bound of the variable, and ${R}_{1}$ is a uniform random vector in the range of 0 to 1. The initialization creates the initial Prey matrix, as shown below:

where n is the population number, and d is the problem dimension.

Calculate the fitness of each prey, select the most fit prey individual as the top predator, and build it into the Elite matrix:

The MPA optimization process is divided into three phases based on the different velocity ratios of the predator and prey. 1) At a high-velocity ratio, the prey moves faster than the predator. 2) At a unit-velocity ratio phase, both prey and predator speeds are similar. 3) At a low-velocity ratio phase, the prey is slower than the predator. At each stage, the movement of the predator and prey in nature is imitated separately.

Phase 1: The high-velocity ratio phase at the beginning of the iteration. The best strategy for the predator is to remain completely immobile, while the prey obeys Brownian motion. The mathematical model for this phase is shown as follows:

where ${s}_{i}$ is the prey search step. ${R}_{B}$ represents the Brownian motion, which is a random vector with a normal distribution. The notion ⨂ indicates entry-wise multiplications. ${R}_{v}\in \left({\mathrm{0, 1}}\right)$ is a uniformly distributed random vector. $Iter$ denotes the current number of iterations, and $Max\_Iter$ is the maximum number of iterations.

Phase 2: The unit-velocity ratio phase is the middle of the iteration, requiring both exploration and exploitation. Thus, half of the population is used for exploitation to perform Lévy movement and the other half for exploration to practice Brownian motion. The following formulas are proposed in this regard:

where ${R}_{L}$ is a vector of random numbers based on Lévy distribution; ${C}_{F}$ is considered an adaptive parameter for controlling the step size of predator movement.

Phase 3: The exploitation process is mainly carried out. The best strategy for the predator is Lévy movement. Its mathematical model is

In addition, environmental issues such as eddy formation or fish aggregating devices (FADs) also can cause behavioral changes in marine predators, which are considered locally optimal. So, longer jumps are used to avoid stagnation when finding the best. The model of the process is as follows:

where Y is the binary vector with arrays including 0 and 1. ${R}_{2}$ is a uniform random number in [0, 1]. ${Prey}_{r1}$ and ${Prey}_{r2}$ denote two individuals randomly selected from the prey matrix. The pseudo-code for MPA is as follows (Algorithm 1).

3.2.   IMPA

In this part of the study, an IMPA is provided, using refracted OBL, SCA and GCM. Initializing MPA with random factors reduces optimization efficiency and robustness, while refracted OBL is used to generate a more widely distributed initial population, which may increase the speed of convergence in the initial search. In the intermediate stages of optimization, the pace of population movement limits the population exchange, thereby reducing the search area, and the SCA helps to achieve a smoother step transition in the balance phase. If the prey succeeds in foraging, it becomes a predator, leading the population to a local optimum. When the algorithm stagnates, Gaussian-Cauchy mutation is utilized to increase the diversity of the population and make the algorithm be out of the local optimum.

3.2.1.   Refracted OBL

OBL is an optimization strategy proposed by Tizhoosh ^[35]. The basic idea is to consider the current solution and its opposite solution in order to obtain optimal solution acquisition, which is calculated as follows:

where l and u are, respectively, the lower and upper bounds. x* represents the opposite solution of x. OBL is a common means of improving population initialization ^[36]. It is a simple approach, but low flexibility makes it detrimental to handling dynamic changes in the population. To improve the performance of OBL, the refraction law of light is added to obtain refraction opposition-based learning (ROBL) ^[37]. It can help generate high-quality populations and increase population diversity, thus avoiding blind search in the initial iterations and accelerating early convergence. The principle of ROBL is shown in Figure 2, and the mathematical model is described as follows:

Then, the refractive medium rate z can be obtained as

Let the scaling factor be $q = \frac{h}{{h}^{{\mathrm{*}}}}$, and Eq (17) is changed to

Equation (13) can extend to the d-dimensional space of the MPA to get the refracting solution as follows:

where ${l}_{j}$ and ${u}_{j}$ are the minimum and maximum of the j-th dimension in the current population, respectively. ${Prey}_{i, j}$ represents the value of the j-th dimension of the i-th particle in the current population. ${Prey}_{i, j}^{{\mathrm{*}}}$ is the refracting solution of ${Prey}_{i, j}$.

3.2.2.   SCA

Based on the position update equations of MPA described in Eqs (10) and (11), MPA converges in advance during optimization iteration without fully exploring the search space. In addition, giant steps may occur in alternating Brownian and Lévy motions for optimization, traversing the optimal solution during the search process. The SCA is introduced in this paper to improve the performance of the MPA. SCA is a new intelligent algorithm proposed in 2016 by Mirjalili ^[38] which uses the oscillation switching characteristics of sine and cosine functions to find the optimum solution. This algorithm has the advantages of fast optimization and strong robustness. At the core of SCA is the location updating method, and its mathematical model of location updating is shown Eq (20):

where ${r}_{2}\in \left[{\mathrm{0, 2}}\pi \right], {r}_{3}\in \left[{\mathrm{0, 2}}\right], {r}_{4}\in \left[{\mathrm{0, 1}}\right]$, and $B$ is the target position, the global optimal position in SCA. The most critical parameter is ${r}_{1}$, which controls the balance between the exploration and exploitation of the algorithm. ${r}_{1}$ is a linearly decreasing function, and the change of function is unique and needs to be more flexible. For this purpose, the non-linear decreasing function ${r}_{1}$ expression in ^[39] is used as follows:

Since C_F is a non-linear descent curve, the rate of the population with Brownian motion decreases with the number of iterations, which follows the standard iteration law. So, only the population with Lévy motion is required to improve the search strategy. Then, the new predator position update equation is obtained as

3.2.3.   GCM

In order to solve the problems of low accuracy and easily falling into a local optimum, MPA needs to incorporate a variational strategy. To this end, the GCM ^[40] strategy is adopted. The Gauss and Cauchy distributions are shown in Figure 3.

The Cauchy distribution is small in the middle and large at both ends, having a larger search step and expanding the search range. The Gaussian distribution is highest in the middle and lowest at both ends, having a smaller search step and improving the ability to search locally. The GCM strategy is outlined as follows:

We have a Gaussian distribution with mean zero ($\mu = 0$) and unit variance (${\sigma }^{2} = 1$). We have a Cauchy distribution with position parameter (${x}_{0} = 0$) and scale parameter ($\gamma = 1$). Therefore, the GCM strategy is used in MPA to improve accuracy, which is described as follows:

where R₃ is a random number uniformly distributed between 0 and 1. $C\left({\mathrm{1, 0}}\right)$ is a random variable based on the Cauchy distribution, and ${\mathrm{G}}\left({\mathrm{1, 0}}\right)$ is based on the Gaussian one. $1-Iter/Max\_Iter$ decreases with increasing $Iter$. In the early iterations, the value of $Iter$ is small, and the predator performs a Cauchy distribution to expand the search. Later in the iteration, the value of $Iter$ increases, and $1-Iter/Max\_Iter$ becomes smaller for the Gaussian distribution. The two variant strategies are used alternately, which amounts to an adaptive dynamic parameter that increases with the number of iterations. It increases the opportunity that the algorithm will jump out of its stagnation and accelerate its convergence later. The pseudo-code of IMPA is as follows (Algorithm 2):

3.3.   Overall scheme of AC-IMPA

In summary, for the AC-IMPA method, after iterative optimization of the algorithm, the position vector that best fits the fitness function is selected as the cluster center representing a specific cluster number. According to the effective clustering centroid obtained, a better cluster grouping can be obtained. Figure 4 shows the flow chart of AC-IMPA, visually showing the whole process.

4.   Experimental results and discussion

This section is divided into two parts. One is the result of IMPA in test functions, and the other is the application of AC-IMPA in common datasets and rotating fault datasets. The simulation results and discussion of the proposed algorithm are introduced and compared with other results in the literature. The experimental running environment is a 64-bit Windows 11 operating system, the CPU is an Intel(R) Core (TM) i7-12650H, and the main frequency is 2.3 GHz. The algorithm is written based on MATLAB R2021b.

4.1.   Test functions

This section uses test functions to evaluate the performance of the proposed IPA model. The experiments were performed on 10 standard benchmark functions, including unimodal, multimodal, and fixed dimension multimodal. The unimodal test functions (F1–F4) are used to test the exploitation ability of the algorithm, while multimodal functions (F5–F19) are used to evaluate the exploration performance. The fixed dimension test function (F10) shows the exploration capability in low dimensions ^[41]. The detailed descriptions and related information are given in Table 1. To ensure fairness of the comparison between algorithms, the basic parameters of the algorithms are set to the same values, including the population size n = 30, the maximum number of iterations $Max\_Iter = 500$, and the dimensionality of the test functions d = 30.

To verify the overall performance of IMPA, six algorithms are selected for comparison, including the basic MPA, the nonlinear marine predator algorithm (NMPA) ^[42], the quasi-opposition learning and the Q-learning based marine predators algorithm (QQLMPA) ^[43], WOA ^[44], the pelican optimization algorithm (POA) ^[45] and particle swarm optimization (PSO) ^[46]. The internal parameters of each basic algorithm are set as shown in Table 2. All improved MPA algorithms have the same parameters as MPA.

Table 3 shows the comparison results of 7 algorithms such as IMPA, MPA, QQLMPA on the benchmark functions. Figure 5 shows the convergence curves of these meta-heuristic algorithms. According to the experimental results in Table 3 and Figure 5, it can be seen that IMPA and NMPA can achieve the theoretically optimal value in three different evaluation metrics for two unimodal test functions F1–F2 and one multimodal function F8. As seen from the figure, IMPA can reach the ideal value before 150 iterations, indicating the effectiveness of the ROBL strategy. However, for F3 and F4, it is pretty challenging to find the global optimal solution. F3 is known as the banana-type valley function, and the shape of F4 is parabolic. They both have many local optimizations, which can easily lead the algorithm to falling into local optima and stagnate the optimization search. In F5 and F8, all three metrics of the algorithm are 0 except for WOA and PSO. F6 is characterized by an almost flat outer region with a large hole in the center. This function poses a risk for optimization algorithms to be trapped in one of its many local minima. All three improved MPA algorithms have a mean value of 8.88 × 10⁻¹⁶ and a mean difference of 0. However, IMPA can approach the ideal optimum faster in early iterations. There are two round spikes outside the optimal value of F10, and the result is a large fluctuation between the locally optimal position and the globally optimal position. IMPA can find the optimal point for F12 quickly. Generally, for other different types of test functions, IMPA is at the bottom of its iterative curve most of the time. It shows high convergence efficiency and verifies the effectiveness of the algorithm optimization strategy.

The Wilcoxon signed rank test is used to demonstrate the validity of the improved algorithm. Let the best algorithm be IMPA. Paired comparisons are made between RSGMPA and MPA, POA, WOA, PSO, NMPA and QQLMPA with the significance level of p = 5%. The symbols "+", "−", and "=" indicate that IMPA performs better, worse and equivalent. As shown in Table 4, most of the p values among the 10 benchmark functions are less than 5%. This shows that the superiority of the algorithm is statistically significant. In other words, the convergence accuracy of the IMPA algorithm is higher than other algorithms ^[47].

In addition, in order to make the statistical test results more convincing, we performed Friedman tests on the mean values of seven algorithms calculated on 10 benchmark functions. This method is mainly used to detect whether there are differences in the performances of different algorithms. The significance level was set to 0.05. When the p-value is less than 0.05, several algorithms can be considered to have statistically significant differences. Conversely, there is no statistically significant difference between algorithms ^[48]. According to the calculation results in Table 5, the p-value are less than 0.05, indicating that there is a significant difference between the seven algorithms.

4.2.   Datasets

To verify the overall performance of AC-IMPA, six algorithms were selected for comparison, including K-means, FCM ^[49], basic MPA, PSO, WOA and SSA ^[50]. To ensure the fairness of the comparison between the meta-heuristic algorithms, the basic parameters of the algorithms are set to the same values, population size n = 30, and maximum number of iterations $Ma{x}_{Iter} = 100$.

A total of 7 datasets were used, including 1 Labor dataset and 6 UCI datasets. Table 6 gives the summary of the 7 datasets, showing the data type, data set dimension, number of data points and number of clusters. UCI data sets are real classical data sets from different fields, and the selected data dimensions are different.

Tables 7 and 8 show the results of 50 independent runs of these algorithms based on CS. Table 6 shows the average and standard deviation of CS measurements, and Table 7 shows the cluster numbers and standard deviations determined by CS. Figure 6 shows the clustering results of MPA and IMPA on data 1. The data is shaped like two moons, representing the actual classifications. For the data at the intersection of two moons, IMPA has a better classification effect. In this paper, cluster numbers are compared first, followed by CS values. In the Cancer dataset, the numbers of clusters of K-means and AC-IMPA match the actual and are highly stable. In the Jain dataset, AC-IMPA and AC-MPA have the best cluster numbers. In the remaining 5 datasets, AC-IMPA performed best. For the values of CS table, AC-IMPA has the best average value in four datasets, Cancer, Jain, Thyroid, Haberman, and the other datasets rank high, indicating AC-IMPA's superior performance.

The Wilcoxon signed rank test is used to demonstrate the validity of the improved algorithm. Let the best algorithm be AC-IMPA. Paired comparisons are made between AC-IMPA and AC-MPA, AC-WOA, AC-SSA and AC-PSO with the significance level of p = 5%. As shown in Table 9, most of the p values among the 7 datasets are less than 5%. This shows that the superiority of the algorithm is statistically significant.

4.3.   XJTU-SY rolling bearing accelerator life data set of Xi'an Jiaotong University

The bearing experiment data set of Xi'an Jiaotong University was used as the data ^[51]. In the experiment, the sampling frequency was set at 25.6 kHz, the sampling interval was set at 1 min, and the sampling duration was set at 1.28 s each time. The bearing speed is 2100 r/min, and the radial force is 12 kN. Bearing1_1, Bearing1_4, and Bearing1_5 in the data set are used. The detailed data are described in Table 10.

Table 11 shows the adaptive cluster values and CS values of 7 algorithms on the XJTU-SY rolling bearing data set. It can be seen from the table that only the cluster numbers of AC-MPA, AC-IMPA and AC-SSA algorithms are close to the optimal category number of the data set. When K = 3, the change curves of the fitness functions of the three algorithms are shown in Figure 7, which once again verifies that the improved algorithm has a relatively high degree of accuracy in solving. Meanwhile, in order to demonstrate the effectiveness of the proposed algorithm, when K = 3, the three algorithms are run independently 10 times, and the recognition rates of all samples are shown in Table 12. It can be seen that AC-IMPA has the highest recognition rate for cluster analysis of such complex data. Although the overall recognition rate of AC-MPA algorithm is also up to 83.33%, which is close to this, the results of 10 runs show that its recognition results are less stable than the algorithm proposed in this paper.

5.   Conclusions and future research directions

In order to solve the clustering problem of bearing fault datasets more effectively, a new automatic clustering algorithm (AC-IMPA) based on improved MPA and K-means algorithm is proposed. In this algorithm, the improved MPA algorithm aims to provide a better initial center of mass for the K-means algorithm, that is, the predator is closer to the center of the data and is able to use this information to move faster and find a better solution than the original predator's movement. For validation, the proposed AC-IMPA algorithm is tested using some benchmark datasets and compared with some automatic clustering meta-heuristic algorithm algorithms, AC-MPA, AC-SSA, FCM, K-MEANS and AC-WOA. The results show that AC-IMPA can obtain better and more stable results in cluster number. Finally, AC-IMPA is applied to the problem of automatic classification of bearing fault, and it is found that the number of clusters obtained by the proposed new method is closer to the true facts.

Although the proposed automatic clustering method AC-IMPA shows advantages in the experiment, it also has some weaknesses. First, the advantage of cluster quality is not obvious on high cluster number data sets, because the number of clusters is small, and once the data has errors, it is easy to limit the performance of the algorithm. Another is that the clustering stability of AC-IMPA is not strong enough. In addition, in future research, more, other clustering effectiveness indicators can be attempted for automatic clustering to make the clustering effect better. Therefore, AC-IMPA provides a new and effective automatic clustering mode, but it still needs in-depth analysis and further improvement.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by the Key Project of Scientific Research of Anhui Provincial Education Department, China, grant No. 2022AH050995 and No. 2022AH050975. It was supported by Anhui Province Intelligent Mine Technology and Equipment Engineering Laboratory Open Fund, grant No. AIMTEEL202201; Project of Science and Technology Bureau of Wuhu City, grant No. KZ32022014; Open Fund of Anhui Key Laboratory of Electric Transmission and Control, grant No. DQKJ202208; Research Project of Anhui Polytechnic University, grant No. Xjky2022007; Research Start-up Fund Project of Anhui Polytechnic University, grant No. 2022YQQ004; Applied Basic Research Project of Wuhu City, grant No. 2022jc20.

Conflict of interest

The authors declare there is no conflict of interest.

Supplementary