Addressing external distorted heterogeneity: Input–output disentangled causal representation for mixed time series

In causal analysis, it is common for industrial systems to have a mixture of continuous and discrete variables, called distribution heterogeneity. In fact, discrete variables typically serve as external inputs to modulate continuous variables in these systems. Existing methods for causal discovery encounter the External Distorted Heterogeneity challenge. The challenge is defined as the difficulty of correcting the statistical relationships distorted by discrete inputs, interfering with the identification of causality within systems. To overcome the challenge, we propose a method called Input–Output Disentangled Causal Representation. The key idea is to reveal the continuous external control effects from discrete inputs and exclude the control effects from observed outputs to decouple the inference of causality. Technically, a reversible external control converter is designed to recover the continuous control effects from discrete input signals through affine processes, bridging the heterogeneity. In addition, we construct an additive causal model to distinguish between causal effects from inputs and outputs, capturing disentangled representations in a unified space through feature distribution alignment and discrimination. Dual predictions are designed to exclude the regulatory influences from observed outputs using gradient truncation, thereby decoupling the inference of causality. The proposed method demonstrates robust causal identification accuracy across diverse datasets and scenarios, outperforming existing approaches in high-dimensional input–output systems. These results highlight its potential for industrial applications in the causal discovery of input–output systems.