Spatio-Temporal Delay Aware Causality: A Self-Interpretable Framework for Soft Sensing

With the increasing complexity of industrial systems and process data, deep learning has achieved superior performance in soft sensing but remains constrained by limited interpretability. Most existing interpretability techniques are correlation based, capturing statistical dependencies but providing little insight into underlying mechanisms. Causal modeling, by contrast, offers stronger interpretability by revealing directional and temporal influences, thereby improving both reliability and understanding. Although some recent methods consider time delays, their treatment of lags remains coarse and limited, and cannot adequately capture heterogeneous cross variable delay patterns in industrial time series. To address these limitations, we propose spatio-temporal causal learning with delay annotation (STCLD), which introduces a spatio-temporal delay attention (STDA) module to explicitly learn delay annotated spatio temporal causal graphs for soft sensing. STDA minimizes a maximum mean discrepancy objective to discover causal relations with edge specific delays, while attention path strength and multidimensional dynamic complexity are used to infer causal directions in a model-based way. The learned causal graph and delay information then guide a delay aware prediction module to build a self-interpretable soft sensor. Experiments on two real-world industrial datasets show that STCLD consistently outperforms strong baselines in both predictive accuracy and causal interpretability, providing a robust and general framework for interpretable soft sensor modeling in complex process industries.