# Fisher Discriminative Sparse Representation Based on DBN for Fault Diagnosis of Complex System
## 1 Introduction
[PERSON] combined a particle swarm optimization algorithm with DBNs to particle DBNs (PDBNs), and determined the PDBN model with optimized network parameters performed better and had higher identification precision [23]. To take advantage of DBNs and quantum inspired neural networks (QINN), [PERSON] et al. proposed the linear superposition of multiple DBNs with quantum intervals in the last hidden layer to construct a deep quantum inspired neural network and applied it to the fault diagnosis of aircraft fuel systems [24]. To decrease the DBN pretraining time, [PERSON] proposed two complementary approaches. An adaptive step size technique was first introduced to enhance the convergence of contrastive divergence algorithms, and then a highly scalable graphics process unit parallel implementation of the CD-k algorithm was presented, which notably increased the training speed [25]. [PERSON] et al. proposed a novel supervised local multilayer perceptron classifier architecture integrated with ICA models for fault diagnosis and detection of industrial systems. The Expectation-Maximizing clustering technique was applied to improve single neural network (SNN) classifiers. The effectiveness of the proposed method was illustrated with the Tennessee Eastman (TE) process [26]. Since the deep belief network extracts features layer by layer, it has unique advantages in the feature extraction of high-dimension and nonlinear data processing. Additionally, the TE process is a complex industrial process with many variables and a large amount of data. To reduce the dimension of variables and determine the relationship with interpretability, we introduced deep belief networks to extract features of raw data from industrial processes. The other drawback of the above dimensionality reduction methods is that they are based on a statistical model, which assumes the probability distribution of the process variables first, and then sets the corresponding threshold to complete fault diagnosis, whereas the accuracy of the methodsmostly depends on the threshold and statistical model. To create more interpretable results and promote better generalization, we introduce Fisher discrimination sparse representation (FDSR) to diagnose faults that can be seen as a supervised classification task, the target of which is to assign new observations to one of the existing classes. Various supervised learning models have been introduced, such as error propagation, deep neural networks [27], support vector machine (SVM), and logistic regression. Deep neural networks perform well when sufficient normal and fault data are available, whereas the features extracted by DBNs are dimension reduced, meaning there is insufficient input data for deep neural networks. SVM, another efficient classifier, performs well with limited training data, whereas the lack of a rejection mechanism limits the application in fault diagnosis, especially for some industries such as nuclear areas and chemical processes with high danger probability [28]. Sparse representation is widely used in pattern recognition, image processing, and fault diagnosis [29; 30; 31; 32; 33; 34; 35], since it can include danger rejection as a restriction in the representation model and iterate training data to learn the discriminative dictionaries. Then, the corresponding sparse coefficients can be used to perform fault diagnosis in a complex system. [PERSON] et al. proposed a sparse PCA algorithm for nonlinear fault diagnosis and robust feature discovery in industrial processes, with the robustness of this method being proved with the TE process [29]. To study the relationship between monitoring parameters and faults, [PERSON] et al. proposed a new sparse dimensionality reduction method called sparse global-local preserving projections, which greatly improved the interpretability of fault diagnosis [30]. [PERSON] et al.