## 3 Proposed Online Fault Diagnosis Model
### Proposed New Fault Mode Detection
#### 3.3.2 The Proposed MPDFCDF Cluster Evaluation Method  
As mentioned in Section 1, \(k\)-means clustering requires \(k_{kmeans}\) to be specified in advance. However, it is difficult to know the number of clusters (fault modes) of unknown signals in online fault detection and diagnosis system. This problem can be resolved by repeating the clustering process for different values of \(k_{kmeans}\) (e.g., \(k_{kmeans}\) = 2... 7) and then determining \(k_{opt}\) through an efficient cluster evaluation method. To do this, elbow is a well-known unsupervised cluster evaluation technique [39; 40]. The elbow is a visual graph evaluation technique, where the number of clusters is increased one by one and evaluate a cost function of the clusters in every stages. The cost function is basically the calculation of the sum of squared errors of clusters. During the time of increasing the number of clusters, the value of cost function dramatically decreased and then became steady. Based on the curve of the graph, it determines the optimal number of cluster. However, the visual determination is often ambiguous. Also, the evaluation of the cost function of clusters is challenging. The silhouette coefficient [24] and COSES [26] methods work as a cluster evaluation for well-separated data. However, data from faulty bearings have an unusual and complex distribution, which is not handled well by average-distance-based cluster evaluation methods. In Figure 10(a), data are well separated into easily identifiable clusters, but in Figure 10(b), two data groups are in close proximity to each other but far away from the other groups. In such cases, the larger distance dominates over the smaller distance, leading to incorrect identification of the number of clusters. Furthermore, unequal densities of data groups could bias the cluster evaluation. To address these issues, the MPDFCDF is proposed, which considers the multivariate Gaussian distribution and probability density functions (PDFs) of the data in feature space. Figure 10: Initial centroids selection process with an example. The PDF \(\rho(x)\) can be used to express the statistical distribution of data in clusters. The univariate PDF of a normal distribution is given by Equation (7), where \(\mu\) is the mean, and \(\sigma^{2}\) is the variance  
\[\rho(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(\frac{-(x-\mu)^{2}}{2\sigma^{2}} \right)\!. \tag{7}\]  
However, real-world problem sets exhibit behavior that is more suitably represented by multivariate distributions, such as the multivariate normal distribution given by Equation (8)  
\[\rho(X)=\frac{1}{\sqrt{\left|\sum(2\pi)^{d}\right.}}\exp\!\left(-\frac{1}{2}(X -\mu)^{\prime}\sum^{-1}(X-\mu)\right) \tag{8}\]  
where \(d\) is the dimension of data, \(X\) and \(\mu\) are \(1\times d\) vectors, and \(\sum\) is the \(d\times d\) symmetric positive definite covariance matrix. The properties of these distributions and the distance between their means represent key information that can help in identifying \(k_{opt}\). If samples are distributed in a scattered way, the probability that data exist in dense areas is low, and the deviation of samples from the mean is high. On the other hand, if samples are distributed in a compact way, the probability that data exist in dense areas is high and the deviation of samples from the mean is low. Thus, the ratio of the highest PDF value to the variance of the distribution is a good measure of the cluster density, whereas distances between means of clusters measure the cluster separation. \(k_{opt}\) is identified by evaluating clustered samples on these criteria, using different values of \(k\).