**1. Introduction**

The automation of techniques takes place around the world in the manufacturing and processing of industrial sectors [1,2]. In the industrial and rotary machines, the main idea of automation is the monitoring without input parameters. The error is human, and the limitation of inexact input parameters affects the accuracy of monitoring, so it was interesting to make an autonomous method. There is a growing demand for real-time monitoring in the rotary machines to facilitate advanced maintenance programs [3]. Rotary machines are most often made of a significant and critical component: the rolling bearings [4].

The monitoring of rolling bearings gets the scientist's attention; so many methods applied to detect defects such as support vector machine [5], Bayesian network [6] and clustering [7]. Numerous literature reviews are available on monitoring methods [8,9]. From all these used methods, clustering analysis is one of the most remarkable approaches [10–12]. The density-based method is one of them, the clusters of dense regions of data separated from the less dense [11]. The method OPTICS (ordering points to identify clustering structure), subdivided from density-based, has the basic idea to separate clusters by density [13]. In addition, it has the advantage to attain clusters with varied data density. Clustering by OPTICS methods is an unsupervised learning method directly implemented to vibration data. Being thus can be applied directly in the industrial environments without trained by data measured on a machine under a fault condition [7]. Further advantages of the method are its ease of programming and the accomplishment of a good trade-off and achieved the best performances.

In addition, it is fast for small data, used with different density to detect and attain arbitrary and sphere-shaped clusters [14].

Within the framework of bearing monitoring, the OPTICS method integrated dynamic classification processes for real-time monitoring [15]. The algorithm proposes to make a detection of faults from two time features (rms and kurtosis). The monitoring is then carried out using three geometric values, the contour, the distance and the density. However, the process was incomplete and not completely automated, which required an expert.

The extracted features play an essential role in the classification, for that many methods used to eliminate unwanted and unimportant features. The relief method is used to select features for the classification of biomedical data. It eliminates the irrelevant features and to prepare data of rolling bearings for the classification [16]. The Chi-square is another method that has the same aim of the relief to reduce and rank features, this method used ranking features to detect the defect in the rolling bearing [17]. After selecting features and reducing them by eliminating the uncorrelated ones, the importance of dimension reduction comes before starting the classification. In the literature, many methods have applied for dimension reduction, principal component analysis (PCA) [18] and kernel principal component analysis (KPCA) [19], to detect the defect in rolling bearings. A recently developed nonlinear dimensionality reduction technique shows its efficiency in the detection of a fault in rotary based on t-distributed stochastic neighbor embedding (t-SNE) [20].

The parameters specific to OPTICS are also subject to automation. The lack of automation concerns the choice of features according to a library and the internal parameters of OPTICS: ε (cluster radius), *MinPts* (the minimum number of data points needed to cluster) and the distance metric used to calculate instances between arrays [15]. The determination of the parameter values of the OPTICS algorithm can be a challenging task because the parameter values affect the accuracy and precision of the clustering. Many researchers have discussed this topic, and they were looking for ways to solve it. An automated algorithm AE-DBSCAN, proposed by [21], defines ε like the K-nearest neighbor for this *MinPts*. Regarding the choice of distance, [22] show the hardness approximation of data with Euclidean distance in k-means clustering. Manhattan outperforms the Euclidean distance with the k-means method. The aim is to automate calculation of all the parameters and to offer a complete real-time monitoring solution dedicated to the bearings.

This paper proposes an Online One Class Monitoring based on OPTICS Classification for Rolling Bearing, automatic online classification monitoring based on ordering points to identify clustering structure (AOC-OPTICS). The input parameters are limited to the initialization time of the method and the number of signals collected at a monitoring time *t*. It integrates the detection and monitoring of the evolution of a fault. After an initialization phase, the detection is carried out by a multidimensional analysis with extraction, ranking (relief method) selection (t-SNE) and classification (OPTICS) of one class clustering type. The follow-up is carried out when creating a new class. In this phase, geometric parameters from this class are proposed and discussed due to regressions models.

This paper is organized as follows. Section 1 introduces the context of the monitoring of the rotating elements and presents the bibliographical review on the contributions and the limits of the classification methods. Section 2 describes the OPTICS method and highlights the parameters to be automated. Section 3 presents the general methodology for automatic monitoring and follow-up of the healthy state of a bearing. Section 4 assesses the relevance of the methodology on data simulating the initiation and growth of a defect on the outer ring of a bearing. The automated parameters and their influences are discussed. Tracking parameters are defined, and mathematical laws are established. Section 5 corresponds to an experimental validation on a test bench. Section 6 concludes this study.

#### **2. Classification Method OPTICS**

OPTICS (ordering points to identify clustering structure) is a hierarchical clustering algorithm that relies on a density notion [13]. The application of this method is not limited to one field. It used in many fields and areas of biology, astronomy, topology, and recently for the detection of the defect in rolling bearings in rotary machines [15]. This method is capable of regrouping the base of data into an order of points with different parameter settings, and then detecting a meaningful difference of data with varied density by producing a request of data that is spatially closed to each other and can become a neighbor. It can separate considerable objects from noise and identify all possible levels of clusters. The main idea for the OPTICS algorithm is that for each point of a cluster the neighborhood of a given radius (ε) has to contain at least a minimum number of points (*MinPts*), where ε and *MinPts* are input parameters. The concept of OPTICS algorithm starts by adding points to the clustered data in arbitrary shape and then to continue by adding points iteratively for developing the final cluster. The addition of points close to each other respecting the ε-neighbor order continues until getting the entire group.

The two-components of OPTICS are the core distance, *Cd*, and the reachability distance, *Rd*, Equations (1) and (2). If the number of points in the vicinity of an object, *<sup>N</sup>*ε(*p*), is less than *MinPts*, *Cd* is the distance from p to its *Minptsth* neighbour, *MinPtsdistance*(*p*). In this case, *p* is a core-object. The reachability distance of an object *o***,** *Rd*, is the maximum of the Core Distance of *p* and the Euclidean distance between *o* and *p*. Figure 1a is a representation of the reachability distance and the core distance objects.

$$\mathbb{C}\_d(\varepsilon\_\prime \textit{MinPts}(p)) = \begin{cases} \textit{!Indefined} & \textit{if } \mathrm{N}\_c(p) < \textit{MinPts} \\ \textit{!dimPts}\_{\mathrm{distance}}(p) & \textit{else} \end{cases} \tag{1}$$

$$R\_d(\varepsilon, MinPs(p, o)) = \begin{cases} \text{ } & \text{if } N\_\varepsilon(p) < MinPs\\ \text{ } & \text{max}(\mathbb{C}\_d; distance(o, p)) \quad \text{else} \end{cases} \tag{2}$$

**Figure 1.** (**a**) Representation of the core distance and reachability distance for *MinPts* = 4. (**b**) Reachability plot.

The number of classes is determined from the reachability plot, Figure 1b. It corresponds to the number of valleys of the graphic representation *Rd* as a function of the points o ordered.
