Modeling Multivariate Time Series With Spatio-Temporal Simplicial Complexes

Abstract

Multivariate time series modeling requires capturing complex dependencies both within individual time series and across different variables. Existing graph-based approaches are limited to pairwise interactions, while recent product cell complex methods assume homogeneous higher-order relationships. This thesis proposes the Simplicial Product Complex (SPC), a topological framework that constructs simplicial complexes in the product space to capture heterogeneous higher-order interactions across space and time. The key innovation is the ability to distinguish between different relationship types and learn their relative importance from data. We develop the Simplicial Product Complex Convolutional Neural Network (SPCCNN) to perform data-adaptive learning over these structures. Experimental evaluation shows SPCCNN achieves competitive performance with state-of-the-art methods while offering enhanced flexibility through parameterized structures. The model adapts to dataset-specific patterns and maintains computational efficiency through sparsity regularization. Our findings demonstrate the effectiveness of higher-order simplicial modeling for capturing complex temporal dynamics in multivariate time series data.