## 3 Incipient Fault Detection Method Based on LOPV and KL Divergence
### Optimum PV for Incipient Fault Detection  
We assume that there is an optimum PV that is more sensitive to the incipient fault than the existing PVs \(v_{1},v_{2},\cdots,v_{m}\) and the optimum PV is \(w\in R^{m}\). To find the optimum PV, this paper assumes the \(m\) dimensional feature parameters to be monitored in their normal state, obeying the \(m\) dimensional joint Gaussian distribution \(X\in R^{n_{0}\times m}\sim\text{N}(\mu_{x},\Sigma_{x})\). Let the projection of \(X\) on \(w\) be \(F\in R^{n_{0}}\). By the nature of the \(m\) dimensional joint Gaussian distribution, \(F\) obeys the one-dimensional Gaussian distribution \(F\sim\text{N}\big{(}\mu_{x}^{T}w,w^{T}\Sigma_{x}w\big{)}\)[26]. \[F=Xw=x_{1}w_{1}+x_{2}w_{2}+\cdots+x_{m}w_{m} \tag{8}\]  
We assumed that after the incipient fault occurs, the on-line faulty data \(Y\) still obey the \(m\) dimensional joint Gaussian distribution \(Y\in R^{n_{1}\times m}\sim\text{N}\big{(}\mu_{y},\Sigma_{y}\big{)}\). However, due to the fault, the mean vector \(\mu_{y}\in R^{m}\) and the covariance matrix \(\Sigma_{y}\in R^{m\times m}\) of \(Y\) deviated and that their deviations were \(\Delta\mu\) and \(\Delta\Sigma\), respectively. \[\mu_{y}=\mu_{x}+\Delta\mu \tag{9}\]  
\[\Sigma_{y}=\Sigma_{x}+\Delta\Sigma \tag{10}\]  
The projection of \(Y\) on \(w\) is \(G=Yw\). Then, from the nature of the joint Gaussian distribution, \(G\) obeys the one-dimensional Gaussian distribution \(G\sim\text{N}\Big{(}\mu_{y}^{T}w,w^{T}\Sigma_{y}w\Big{)}\). Assume that there are two PDFs \(f\) and \(g\) for two Gaussian distribution signals \(F\sim\text{N}\big{(}\mu_{1},\sigma_{1}^{2}\big{)}\) and \(G\sim\text{N}\big{(}\mu_{2},\sigma_{2}^{2}\big{)}\), respectively. The KL divergence of \(f\) and \(g\) can be expressed as follows [21]:  
\[KLD(f||g)=\frac{1}{2}\Bigg{[}\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}+\frac{ \sigma_{1}^{2}}{\sigma_{2}^{2}}+(\mu_{1}-\mu_{2})^{2}\Bigg{(}\frac{1}{\sigma_{1 }^{2}}+\frac{1}{\sigma_{2}^{2}}\Bigg{)}-2\Bigg{]} \tag{11}\]  
The projections of the normal data matrix \(X\) and the faulty data matrix \(Y\) on \(w\) obey the one-dimensional Gaussian distribution \(F\sim\text{N}\big{(}\mu_{x}^{T}w,w^{T}\Sigma_{x}w\big{)}\) and \(G\sim\text{N}\Big{(}\mu_{y}^{T}w,w^{T}\Sigma_{y}w\Big{)}\)respectively. Consequently, the mean and variance of \(F\) and \(G\) can be brought into Equation (11), respectively, obtaining:  
\[ \begin{array}{ll}KLD(f||g)&=\frac{1}{2}\bigg{[}\frac{w^{T}\Sigma_{y}w}{w^{T} \Sigma_{x}w}+\frac{w^{T}\Sigma_{x}w}{w^{T}\Sigma_{y}w}+\left(\mu_{x}^{T}w-\mu_{ y}^{T}w\right)^{2}\left(\frac{1}{w^{T}\Sigma_{x}w}+\frac{1}{w^{T}\Sigma_{y}w} \right)-2\bigg{]}\\ &=\frac{1}{2}\bigg{[}\frac{w^{T}\Sigma_{y}w}{w^{T}\Sigma_{x}w}+\frac{w^{T} \Sigma_{x}w}{w^{T}\Sigma_{y}w}+\left(\Delta\mu^{T}w\right)^{2}\left(\frac{1}{w ^{T}\Sigma_{x}w}+\frac{1}{w^{T}\Sigma_{y}w}\right)-2\bigg{]}\end{array}  \tag{12}\]  
In Equation (12), the mean deviation vector \(\Delta\mu\), normal data covariance matrix \(\Sigma_{x}\), and on-line faulty data sample covariance matrix \(\Sigma_{y}\) can be obtained through the on-line faulty data matrix \(Y\) and normal data matrix \(X\). Therefore, when historical normal data \(X\) and the on-line faulty data \(Y\) to be verified are obtained, only the PV \(w\) is unknown in Equation (12). In other words, \(KLD(f||g)\) is a function \(h(w)\) on the PV \(w\) as shown in Equation (13). \[h(w)=KLD(f||g)=\frac{1}{2}\bigg{[}\frac{w^{T}\Sigma_{y}w}{w^{T}\Sigma_{x}w}+ \frac{w^{T}\Sigma_{x}w}{w^{T}\Sigma_{y}w}+\left(\Delta\mu^{T}w\right)^{2}\left( \frac{1}{w^{T}\Sigma_{x}w}+\frac{1}{w^{T}\Sigma_{y}w}\right)-2\bigg{]} \tag{13}\]  
Sliding windows with the length of \(n_{1}\) are used to divide the normal data matrix \(X\) and submatrices of \(X\) can be obtained \(X_{1},\cdots,X_{i},\cdots,X_{r},X_{i}\in R^{n_{1}\times n},i\in[1,r]\). We project all submatrices \(X_{1},\cdots,X_{i},\cdots,X_{r}\) on \(w\) to obtain r PDFs \(PDF_{X1},\cdots,PDF_{Xi},\cdots,PDF_{Xr}\). We assume that each \(PDF_{Xi}\) obeys a one-dimensional Gaussian distribution.