# Fisher Discriminative Sparse Representation Based on DBN for Fault Diagnosis of Complex System
## 3 Fisher Discriminative Sparse Representation (FDSR) of the Features Extracted from DBN
### Sparse Dictionary Constructed Based on Fisher Discriminative Dictionary Learning
As the optimal dictionary to represent \(Y_{i}\):  
\[Y_{i}=DX_{i}=D_{1}X_{i}^{1}+D_{2}X_{i}^{2}+\cdots+D_{c}X_{i}^{c} \tag{9}\]  
where \(X_{i}\) is the representation of \(Y_{i}\) in the dictionary \(D\), and \(X_{i}^{j}\) is the representation of \(Y_{i}\) in the dictionary \(D_{j}\). The [PERSON] discriminative fidelity term \(r(Y,D,X)\) includes three parts: (1) The structured dictionary \(D\) should be able to well represent \(Y_{i}\), meaning \(\|Y_{i}-DX_{i}\|_{F}^{2}\) is small; (2) Dictionary \(D\) in Figure 4a satisfies this claim. Then, the subdictionary \(D_{i}\) that corresponds to \(Y_{i}\) is optimal in \(D_{j}\), \(1\leq j\leq c\). This limitation can be represented as a mathematic function: \(\|Y_{i}-D_{i}X_{i}\|_{F}^{2}\), which is small enough. Figure 4b shows this constraint; (3) \(X_{i}^{j}\) should have nearly zero coefficients sothat representation \(\left\|D_{i}X_{i}^{j}\right\|_{F}^{2}\), \(1\leq j\leq c\) and \(j\
eq i\) are small enough, as shown in Figure 4c. Above all, the [PERSON] discriminative fidelity term [46] is as follows:  
\[r(Y,D,X)=\left\|Y_{i}-DX_{i}\right\|_{F}^{2}+\left\|Y_{i}-D_{i}X_{i}\right\|_{F }^{2}+\sum_{ \begin{array}{c}1\leq j\leq c\\ j\
eq i\end{array} }\left\|D_{i}X_{i}^{j}\right\|_{F}^{2} \tag{10}\]  
The idea above limits the discrimination of the dictionary. We defined \(f(X)\) to constrain the discriminative of the sparse coefficient \(X\). Learning from the criterion, such as the Fisher discriminative criterion, we introduced between-class scatter and within-class scatter to illustrate the discrimination of \(X\). Within-class scatter \(T_{w}(X)\) is recognized as the sum of distance between column vectors \(x_{i}\) in the coefficient matrix that correspond to class \(i\) and the mean vector \(m_{i}\) of \(x_{i},x_{i}\in X_{i}\). Between-class scatter \(T_{b}(X)\) is defined as the sum of the distance between the mean vector of \(x_{i},x_{i}\in X_{i}\) and the mean vector of \(X_{i}\), \(1\leq i\leq c\) multiplied by the number of vectors in \(X_{i}\). \(T_{w}(X)\) and \(T_{b}(X)\) are defined as:  
\[T_{w}(X)=\sum_{i=1}^{c}\sum_{x_{i}\in X_{i}}(m_{i}-x_{i})(m_{i}-x_{i})^{T} \tag{11}\]  
\[T_{b}(X)=\sum_{i=1}^{c}n_{i}(m-m_{i})(m-m_{i})^{T} \tag{12}\]  
Denoted by \(T_{w}(X)\) and \(T_{b}(X)\), to ensure the convex and stability of \(f(X)\), we introduced an elastic term \(\left\|X\right\|_{F}^{2}\) and a scalar parameter into \(f(X)\). Then, \(f(X)\) is defined as:  
\[f(X)=tr(T_{w}(X))-tr(T_{b}(X))+\eta\left\|X\right\|_{F}^{2} \tag{13}\]  
The definition of the discriminative of dictionary \(D\) and sparse coefficient \(X\) ensure both dictionary and coefficient can be applied to fault diagnosis. Then, we have the FDSR model:  
\[J_{(D,X)}=\underset{D,X}{\text{argmin}}\left( \begin{array}{c}\sum\limits_{ i=1}^{c}r(A_{i},D,X_{i})+\lambda_{1}\left\|X\right\|_{1}+\\ \lambda_{2}\left(tr(T_{w}(X))-tr(T_{b}(X))+\eta\left\|X\right\|_{F}^{2}\right) \end{array} \right) \tag{14}\]