Modeling high‐dimensional dynamic processes is a challenging task. In this regard, probabilistic slow feature analysis (PSFA) is revealed to be advantageous for dynamic soft sensor modeling, which can extract slowly varying intrinsic features from high‐dimensional data. However, nonlinearities prevalent in industrial processes are not considered, which could lead to unsatisfactory prediction performance. In this paper, a weighted PSFA (WPSFA)‐based soft sensor model is proposed for nonlinear dynamic chemical process. In WPSFA, a weighted log‐likelihood function of complete data is constructed to linearize the nonlinear state emission equation. Then, the expectation maximization algorithm is applied to estimate the model parameters and a locally weighted regression model is established for quality prediction. Finally, the feasibility and effectiveness of the proposed approach are well illustrated through a numerical example and a real industrial process.

In practical process industries, the measurements coming from different sources are collected at different sampling rates, thereby soft sensors developed using uniformly sampled measurements may result in poor prediction performance. Besides, industrial processes are inherently stochastic and most of them present dynamic characteristic. To cope with these issues, a multi-rate probabilistic slow feature regression (MR-PSFR) model is proposed in this paper for dynamic feature learning and soft sensor development in industrial processes. In the MR-PSFR, both input and output observation datasets with different sampling rates are used to extract the slow features, which can separate slowly and fast changing features and have a better interpretation of the outputs. Then, the expectation-maximization algorithm is modified to derive the model parameters of MR-PSFR and the quality prediction strategy for multi-rate processes is constructed. Finally, the proposed method is investigated through a numerical example and a real industrial process. The simulation results show that the extracted slow features better represent the intrinsic characteristics of the processes and the proposed model has better prediction performance for multi-rate dynamic processes than other methods.INDEX TERMS Soft sensor, feature learning, multi-rate probabilistic slow feature regression, expectationmaximization algorithm; multi-rate dynamic process.