Graph Low-Rank Non-Negative Matrix Factorization with Auto-Encoders for Fault Detection

Fault detection is the process of detecting and diagnosing faults or abnormalities in a system by analyzing its operational data. However, with the complexity of modern industrial processes, some faults are difficult to be detected in a timely manner due to various factors such as noise and data nonlinearity. Therefore, data-driven Fault Detection (FD) has become a widely used method to detect abnormal events in functional modules. Non-negative Matrix Factorization (NMF), as an efficient dimensionality reduction technique, has not had potential applications in fault detection (FD) thoroughly explored. In order to improve the FD methods based on NMF, we have developed a new approach, named graph low-rank non-negative matrix Factorization with auto-encoders (GLNMFA). GLN-MFA integrates the Laplacian operator effectively identifies t he l ocal s tructure a mong d ata p oints, enhancing the performance of dimensionality reduction algorithms. It also introduces the nuclear norm to find a low-rank approximation to the original matrix, thereby constraining sparsity. Additionally, auto-encoders are incorporated to learn a low-dimensional representation of the data and extract key features, which are subsequently applied for fault detection purposes. We employ an optimization algorithm based on Alternating Direction Method of Multipliers (ADMM) to optimize this model. Two test statistics <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T^{2}$</tex> (Hotelling's T-squared), SPE (Squared Prediction Error) are used to evaluate detection efficiency. K ernel Density Estimation (KDE) are used to estimate control limits for fault detection. The effectiveness of GLNMFA is validated on two benchmark datasets.