Causal feature selection framework for stable soft sensor modeling based on time-delayed cross mapping

Soft sensor modeling plays a crucial role in process monitoring. Causal feature selection can enhance the performance of soft sensor models in industrial applications. However, existing methods ignore two critical characteristics of industrial processes. Firstly, causal relationships between variables always involve time delays, whereas most causal feature selection methods investigate causal relationships in the same time dimension. Secondly, variables in industrial processes are often interdependent, which contradicts the decorrelation assumption of traditional causal inference methods. Consequently, soft sensor models based on existing causal feature selection approaches often lack sufficient accuracy and stability. To overcome these challenges, this paper proposes a causal feature selection framework based on time-delayed cross mapping. Time-delayed cross mapping employs state space reconstruction to effectively handle interdependent variables in causality analysis, and considers varying causal strength across time delay. Time-delayed convergent cross mapping (TDCCM) is introduced for total causal inference, and time-delayed partial cross mapping (TDPCM) is developed for direct causal inference. Then, in order to achieve automatic feature selection, an objective feature selection strategy is presented. The causal threshold is automatically determined based on the model performance on the validation set, and the causal features are then selected. Two real-world case studies show that TDCCM achieves the highest average performance, while TDPCM improves soft sensor stability and performance in the worst scenario. The code is publicly available at https://github.com/dirge1/TDPCM.