Dynamic neural orthogonal mapping for fault detection

Abstract

Dynamic principal component analysis (DPCA) and its nonlinear extension, dynamic kernel principal component analysis (DKPCA), are widely used in the monitoring of dynamic multivariate processes. In traditional DPCA and DKPCA, extended vectors through concatenating current process data point and a certain number of previous process data points are utilized for feature extraction. The dynamic relations among different variables are fixed in the extended vectors, i.e. the adoption of the dynamic information is not adaptively learned from raw process data. Although DKPCA utilizes a kernel function to handle dynamic and (or) nonlinear information, the prefixed kernel function and the associated parameters cannot be most effective for characterizing the dynamic relations among different process variables. To address these problems, this paper proposes a novel nonlinear dynamic method, called dynamic neural orthogonal mapping (DNOM), which consists of data dynamic extension, a nonlinear feedforward neural network, and an orthogonal mapping matrix. Through backpropagation and Eigen decomposition (ED) technique, DNOM can be optimized to extract key low-dimensional features from original high-dimensional data. The advantages of DNOM are demonstrated by both theoretical analysis and extensive experimental results on the Tennessee Eastman (TE) benchmark process. The results on the TE benchmark process show the superiority of DNOM in terms of missed detection rate and false alarm rate. The source codes of DNOM can be found in https://github.com/htz-ecust/DNOM.

Appendix

Theorem 3

[27] Reduced Rank Procrustes Rotation. Let M and N be two matrices. Consider the constrained minimization problem

$$\begin{aligned} {\hat{H}}&=\arg \min _{H}\left\| M-NH^{T}\right\| ^{2}\\&subject\ to\ \ \ H^{T}H=I. \end{aligned}$$

Suppose the singular value decomposition (SVD) of $M^{T}N$ is $PDQ^{T}$, then ${\hat{H}}=PQ^{T}$.