## 1 Introduction
proposed an adaptive frequency band selection for an empirical wavelet decomposition model for bearing fault diagnosis. In this paper, the harmonic significance index and particle swarm optimization is used for selecting an optimal sub band. There are many kinds of research where the traditional fault diagnosis models are introduced and have achieved high accuracy for detecting faults based on trained knowledge. The traditional fault diagnosis methods use prior knowledge of failure modes to diagnose bearing condition. However, in complex modern industrial processes, bearing often operates continuously under heavy loads, which causes material fatigue and leads to faults (i.e., cracks or spalls). Thus, it is difficult to have complete failure knowledge in advance, which means traditional diagnostic systems make decisions based on incomplete knowledge. Due to the progressive nature of bearing faults, new fault modes can appear during continuous operation. Discounting information about new faults degrades the effectiveness of fault diagnosis systems. In contrast, an online fault diagnosis system considers new faults continually for enhancing the fault knowledge in reliable fault diagnosis [17]. [PERSON] et al. proposed a progressive fault diagnosis model, where a new hybrid ensemble auto-encoder (HEAE) is used to increase the feature quality for detecting a fault [18]. In this research, several classification algorithms are used for validating the performance. However, this work still considers a background trained model for identifying the new fault. Also, the fault classification model is not updated by incorporating the new fault information. To address this issue, an online fault diagnosis model is proposed in this paper that can detect new faults as they appear using unsupervised clustering technique, update its knowledge in real time, and use the updated knowledge for a reliable diagnosis. The number of faults can be determined from a set of condition monitoring data through clustering. The \(k\)-means clustering algorithm is a simple and popular way to solve the unsupervised clustering problem [19] without the need for training on existing knowledge. It can, therefore, be used to cluster unknown bearing signals and detect new faults. The \(k\)-means algorithm first selects \(k\) initial centroids as the centers of \(k\) clusters. Each sample is then assigned to the nearest cluster through an iterative process. The algorithm is sensitive to the initial centroids, which are generally selected randomly [19; 20]. Several methods have been proposed to select optimal centroids. [PERSON] et al. proposed a cluster center initialization algorithm, which divides the overall dataset into subsamples using \(k\)-means and then selects final centroids based on minimum clustering error [21]. [PERSON] et al. introduced global \(k\)-means, a tree-based center initialization algorithm to select initial centroids independently of initial position and empirical parameters [22]. [PERSON] et al. proposed the \(k\)-means \(+\) +, which yield better clustering performance using random seeds [23]. Those algorithms, however, suffer from randomness, are computationally expensive, and introduce new parameters. In [20], [PERSON] proposed a simple and effective initial centroid selection method that reduces randomness, improves clustering performance, and does not introduce any empirical variables. Thus, this method proposed in [20] is used as a baseline for initial centroid selection. In \(k\)-means clustering, the number of possible clusters \(k\) is an important parameter. Since the number of fault modes in a bearing cannot be known in advance, an efficient cluster evaluation technique is required to determine \(k\). The silhouette coefficient is a popular cluster evaluation metric [24; 25] that quantifies the proximity of points in a cluster to one another, which helps in identifying the number of visible clusters.