VAE-Based Interpretable Latent Variable Model for Process Monitoring

Latent variable-based process monitoring (PM) models have been generously developed by shallow learning approaches, such as multivariate statistical analysis and kernel techniques. Owing to their explicit projection objectives, the extracted latent variables are usually meaningful and easily interpretable in mathematical terms. Recently, deep learning (DL) has been introduced to PM and has exhibited excellent performance because of its powerful presentation capability. However, its complex nonlinearity prevents it from being interpreted as human-friendly. It is a mystery how to design a proper network structure to achieve satisfactory PM performance for DL-based latent variable models (LVMs). In this article, a variational autoencoder-based interpretable LVM (VAE-ILVM) is developed for PM. Based on Taylor expansions, two propositions are proposed to guide the design of appropriate activation functions for VAE-ILVM, allowing nondisappearing fault impact terms contained in the generated monitoring metrics (MMs). During threshold learning, the sequence of counting that test statistics exceed the threshold is considered a martingale, a representative of weakly dependent stochastic processes. A de la Peña inequality is then adopted to learn a suitable threshold. Finally, two chemical examples verify the effectiveness of the proposed method. The use of de la Peña inequality significantly reduces the minimum required sample size for modeling.