Correlating Time Series with Interpretable Convolutional Kernels

TL;DR

This paper introduces a novel method for learning interpretable convolutional kernels from various types of time series data by formulating the problem as a sparse regression, enabling the extraction of meaningful temporal patterns.

Contribution

It reformulates convolutional kernel learning as a sparse regression problem using tensor computations, enhancing interpretability and applicability to multivariate and multidimensional time series.

Findings

Kernels reveal interpretable local correlations and seasonal patterns in real-world datasets.

Method improves fluid flow reconstruction by capturing local and nonlocal correlations.

Approach demonstrates effective kernel learning across diverse time series data types.

Abstract

This study addresses the problem of convolutional kernel learning in univariate, multivariate, and multidimensional time series data, which is crucial for interpreting temporal patterns in time series and supporting downstream machine learning tasks. First, we propose formulating convolutional kernel learning for univariate time series as a sparse regression problem with a non-negative constraint, leveraging the properties of circular convolution and circulant matrices. Second, to generalize this approach to multivariate and multidimensional time series data, we use tensor computations, reformulating the convolutional kernel learning problem in the form of tensors. This is further converted into a standard sparse regression problem through vectorization and tensor unfolding operations. In the proposed methodology, the optimization problem is addressed using the existing non-negative subspace pursuit method, enabling the convolutional kernel to capture temporal correlations and patterns. To evaluate the proposed model, we apply it to several real-world time series datasets. On the multidimensional rideshare and taxi trip data from New York City and Chicago, the convolutional kernels reveal interpretable local correlations and cyclical patterns, such as weekly seasonality. In the context of multidimensional fluid flow data, both local and nonlocal correlations captured by the convolutional kernels can reinforce tensor factorization, leading to performance improvements in fluid flow reconstruction tasks. Thus, this study lays an insightful foundation for automatically learning convolutional kernels from time series data, with an emphasis on interpretability through sparsity and non-negativity constraints.