An interpretable temporal convolutional framework for Granger causality analysis

Most existing parametric approaches for detecting linear or nonlinear Granger causality (GC) face challenges in estimating appropriate time delays, a critical factor for accurate GC detection. This issue becomes particularly pronounced in nonlinear complex systems, which are often opaque and consist of numerous components or variables. In this paper, we propose a novel temporal convolutional network (TCN)-based end-to-end GC detection approach called the Interpretable Temporal Convolutional Framework (ITCF). Unlike conventional deep learning models, which act like a “black box” and are difficult to analyse the interactions between variables, the proposed ITCF is able to detect both linear and nonlinear GC and automatically estimate time delay during the multivariant time series prediction. Specifically, GC is obtained by employing the Least Absolute Shrinkage and Selection Operator (Lasso) regression during the prediction of multivariate time series using TCN. Then, time delays can be estimated by interpreting the TCN kernels. We propose a convolutional Hierarchical Group Lasso (cHGL), a hierarchical regularisation approach to effectively utilise temporal information within each TCN channel for enhanced GC detection. Additionally, as far as we are concerned, this paper is the first to integrate the Iterative Soft-Thresholding Algorithm into the backpropagation of TCN to optimise the proposed cHGL, which enabling causal channel selection and inducing sparsity within each TCN channel to remove redundant temporal information, ultimately creating an end-to-end GC detection framework. The testing results of four experiments, involving two simulations and two real data, demonstrate that the proposed ITCF, in comparison with state-of-the-art, offers a more reliable estimation of GC relationships in complex systems featuring intricate dynamics, limited data lengths, or numerous variables.