## 1 Introduction
A silhouette value is calculated for each point by measuring its average dissimilarity \(a_{i}\) to other points in the same cluster and its minimum average dissimilarity \(b_{i}\) to points in other clusters. The silhouette coefficient, calculated by averaging all silhouette values, gives the number of clusters in the data. In [26], [PERSON] et al. proposed a computationally simpler cluster evaluation algorithm called the compactness and separation measure of clusters (COSES). It calculates cluster separability by measuring the minimum distance between cluster seeds and calculates compactness by averaging the sum of the distances of all samples of a cluster to its seed. However, cluster evaluation techniques based on average distances are unsuitable for evaluating the feature distribution of bearings, because samples at larger distances from the cluster center tend to dominate those lying closer to it, and the uneven distribution of densities across the feature space can misrepresent the actual number of clusters in the data. To address these issues, a new cluster evaluation algorithm, i.e., multivariate probability density function's cluster distribution factor (MPFDCF), is proposed here. The MPDFDF identifies the optimal number of clusters and uses that information to detect new faults. Once a new fault mode is detected, the online diagnostic system knowledge (DSK) is updated through heterogeneous feature pool extraction and optimal fault signature selection based on existing knowledge and newly detected fault signals. Subsequently, the online fault diagnosis module uses \(k\)-NN for classification. Key contributions of this paper are:
* Since it is difficult to know in advance what types of faults a healthy bearing can experience, traditional offline fault diagnosis systems classify faults on the basis of incomplete knowledge. To address this issue, we propose an online bearing fault diagnosis model that first detects unknown fault modes in real-time using \(k\)-means clustering and continually updates the DSK for more reliable fault diagnosis. * It is difficult to determine the number of discernible fault modes through \(k\)-means clustering alone without knowing how to define \(k_{kmean}\). To address this issue, we propose a new cluster evaluation method, the MPDFCDF, to identify the optimal number of clusters (\(k_{opt}\)) in the fault signatures. * To evaluate our proposed model, we recorded bearing signals for different shaft speeds and constructed a heterogeneous fault signatures pool to extract the maximum possible fault information. We used a hybrid fault signature selection to create discriminative fault signatures and build a DSK. Finally, we used the \(k\)-NN classification algorithm to estimate the classification performance. The rest of this paper is organized as follows. Section 2 describes the data acquisition from a test rig. Section 3 provides details of the proposed online fault diagnosis model, including the DSK updating process, unknown fault mode detection using the proposed MPDFCDF cluster evaluation method, and bearing fault diagnosis. Section 4 presents the experimental results and analysis of the proposed model, and Section 5 concludes the paper.