# 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  
In the quest for the proper dictionary to use in fault classification, preconstructed dictionaries, adapted dictionaries, and redundant dictionaries are recognized as three options of dictionary learning. Preconstructed dictionaries, such as steerable contourlets, wavelets, and curvelets, are tailored specifically to imaging, and in particular, to stylized cartoonlike image content. Alternatively, adapted dictionaries such as wavelet packets and bandelets provide a tunable selection of a dictionary, in which the basis or frame is generated under the control of a particular parameter. The wavelet packets introduced by [PERSON] et al. assumed control over the time-frequency subdivision, tuned to optimize performance over a specified instance of a signal. The bandelets introduced by [PERSON] et al. were spatially adapted to better treat images by orienting the regular wavelet atoms along the principal direction of the local area processed [43]. The Method of Optimal Directions (MOD) and K-singular value decomposition (K-SVD) [44; 45] are popular dictionary learning algorithms. Fisher discriminative dictionary, using three dictionary training mechanisms, has attracted considerable attention. To improve the performance of previous deep learning methods, we propose a novel Fisher discrimination dictionary in this study. Instead of learning a sharing dictionary for all classes, we learned a structured dictionary \(D=[D_{1},D_{2},\ldots D_{c}]\), where \(c\) is the total number of classes, and \(D_{i}\) is the subdictionary that is class-specific and associated with class \(i\). Apart from \(D_{i}\) requiring powerful reconstruction capability of \(Y_{i}\), \(D\) also should have powerful discriminative capability of \(Y_{i}\). To obtain the sparse representation of the signal, an overcomplete dictionary is often learned from training samples based on the sparse model. The dictionary and sparse coefficients are key for the sparse representation of the signal. K-SVD is not suitable for classification tasks because it only requires that the learned dictionary to accurately represent the training data, but not necessarily have classification characteristics. To improve the performance of the classification ability, we propose a novel Fisher discriminate dictionary \(D=[D_{1},D_{2},\ldots D_{c}]\). We get the features \(Y=[Y_{1},Y_{2},\ldots Y_{c}]\) from DBNs, in which \(Y_{i}\) is the subset of features from class \(i\), which is then used as the training samples to construct 22 subdictionaries. These subdictionaries are the representation of 21 fault types and one normal condition, and they are sequentially concatenated columnwise to form an overcomplete dictionary to fully represent all states of TE process. Denoted by \(X=[X_{1},X_{2},\ldots X_{c}]\), where the coding coefficients are \(Y\) over \(D\) and \(X_{i}\) is the submatrix containing the coding coefficients of \(Y_{i}\). \(D\) should be able to reconstruct \(Y\) effectively. Additionally, \(D_{i}\) can represent samples of class \(i\) sparsely with minimal error, whereas \(D_{j}(j\
eq i)\) cannot. That is, \(D\) has powerful discriminative ability for signals in \(Y\). As such, we propose the discriminative dictionary learning model:  
\[J_{(D,X)}=\operatorname*{argmin}_{(D,X)}\{r(Y,D,X)+\lambda_{1}\|X\|_{1}+ \lambda_{2}f(X)\} \tag{8}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) are scalar parameters, \(r(Y,D,X)\) is the Fisher discriminative fidelity term, which limits the learned dictionary \(D\), which optimally represents \(Y\), and \(\|X\|_{1}\) and \(f(X)\) are the sparse constraint and discrimination constraint applied to coefficient matrix \(X\), respectively.