## 3 Incipient Fault Detection Method Based on LOPV and KL Divergence
### Optimum PV for Incipient Fault Detection
The KL divergence between \(PDF_{Xi}\) and \(f\) can be calculated by Equation (13). \(h_{Xi}(w)=KLD(f||PDF_{Xi})\), \(i\in[1,r]\). The process used to obtain vector \(h_{X}(w)\) is shown in Figure 1.  
\[h_{X}(w)=[h_{X1}(w),\cdots,h_{Xi}(w),\cdots,h_{Xr}(w)] \tag{14}\]  
\[J(w)=\frac{h(w)-E(h_{X}(w))}{\sigma(h_{X}(w))} \tag{15}\]  
The relative deviation in KL divergence of the on-line faulty data matrix \(Y\) and normal data matrix \(X\) on the PV \(w\) is defined as shown in Equation (15). \(E(h_{X}(w))\) and \(\sigma(h_{X}(w))\) in Equation (15) are the mean and standard deviation of the vector \(h_{X}(w)\). \(E(h_{X}(w))\) and \(\sigma(h_{X}(w))\) reflect the center and range of the interval of variation range of the KL divergence for normal data after projection on \(w\), respectively. Once the on-line faulty data matrix \(Y\) and normal data matrix \(X\) are given, the KL relative deviation \(J(w)\) still remains as the function about \(w\). Figure 1: The process of obtaining vector \(h_{X}(w)\). With the given significance level \(\alpha\), the matrix \(Y\) and the matrix \(X\), The larger the value of the \(J(w)\), the easier it is for that fault to be detected. On the contrary, if the value of the \(J(w)\) is small, it is likely to be drowned out by the noise and this leads to poor performance in detection of the fault. Thus, the problem of finding the optimum PV for fault detection can be modeled as an optimization problem, as shown in Equation (16). \[\left\{ \begin{array}{c}\min_{w}-J(w)\\ s.t.w^{T}w=1;\forall w_{i},-1\leq w_{i}\leq 1,i\in[1,m]\end{array} \right. \tag{16}\]  
If the PV \(w\) is directly optimized in Equation (16) so that \(h(w)\) is maximized, and the optimum solution may continue to converge on the PV corresponding to the smallest eigenvalue, thus making the optimization process meaningless. The above mentioned problem arises because the variation ranges of the eigenvalues of the different PVs are different. A small eigenvalue leads to small \(w^{T}\Sigma_{x}w\) and \(w^{T}\Sigma_{y}w\) in Equation (13), which are exactly in the denominator position, and slight noise can cause large fluctuations in the KL divergence on the corresponding PVs of the small eigenvalue. Therefore, it is necessary to standardize \(h(w)\), and the optimization goal is to maximize the relative deviation of KL divergence, \(J(w)\), rather than the absolute deviation of KL divergence, \(h(w)\). The goal of PCA is to maximize the variance of the normal data \(X\) after it is projected on the PV \(w\) and can be modeled as an optimization problem, as shown in Equation (17) [26]. \[\left\{ \begin{array}{c}\min_{w}-w^{T}\Sigma_{x}w\\ s.t.w^{T}w=1\end{array} \right. \tag{17}\]  
We supposed that \(w_{p}\) is the optimum solution of the optimization problem shown in formula (16), \(w_{q}\) is the optimum solution of the optimization problem shown in formula (17). Since the objective functions of the two optimization problems are different, \(w^{T}\Sigma_{x}w\
eq J(w)\), generally, \(w_{p}\) is not equal to \(w_{q}\). However, solving formula (16) solves the optimization problem, and \(w_{p}\) is the optimum PV for using KL divergence to detect the fault. Because \(w_{p}\
eq w_{q}\), \(w_{q}\) is not the optimum PV for using KL divergence to detect a fault in general. Therefore, the PVs obtained by PCA are not necessarily the optimum PV for using KL divergence to detect the fault. Equation (16) converts the problem of finding the maximum \(J(w)\) into a standard constrained optimization problem by adding a negative sign and constrained conditions. The search for the optimum PV \(w\) that maximizes \(J(w)\) is on the surface of a hypersphere with a radius of 1. Equation (16) is a typical constrained optimization problem, which can be solved using readily available tools for solving optimization problem, such as the fmincon function built into MATLAB 2019b.