Tensor slow feature analysis and its applications for batch process monitoring

•A high-order tensor slow feature model for the 3D batch data is proposed.•The 3D batch data are analyzed directly without unfolding operation.•The slowly varying dynamics within batch processes is efficiently extracted.•Two sub-optimal problems are iteratively used to solve the propsoed method.•Within-batch detection can be conducted based on the applied wavelet functions. How to establish efficient monitoring models for complex batch processes is a challenging problem because of the three-dimensional (3-D) data array and serious dynamics. The unfolding operation in most methods would destroy the batch data structure and increase the dimensions of modeling data obviously. Meanwhile, it is more practical to perform within-batch detection rather than end-of-batch detection. In this work, a novel high-order tensor slow feature analysis model is proposed to handle the 3-D and dynamical issues simultaneously. The proposed model can be solved by iteratively tackling two sub-optimal problems. Within-batch detection is performed based on the properly defined monitoring statistics to timely detect the abnormal situation. The multi-phase property is further considered in this work to improve detection accuracy. A simulated case and an industrial case are taken to show the merits of the proposed method.