Correlating Time Series with Interpretable Convolutional Kernels

TL;DR

This work introduces a unified framework to learn interpretable convolutional kernels from univariate to multidimensional time series by recasting kernel learning as a non-negative $\tau$-sparse regression problem that leverages circular convolution and circulant matrices. The approach extends naturally to multivariate and multidimensional data through tensor formulations, converting the problem into standard sparse regression via vectorization and tensor unfolding, and solving with non-negative Subspace Pursuit. Empirically, the learned kernels reveal local and periodic temporal patterns (e.g., weekly seasonality) in urban mobility data and improve fluid-flow reconstruction when integrated into tensor factorization. The method provides an interpretable, scalable mechanism for kernel discovery that can benefit downstream predictive and reconstruction tasks across spatiotemporal domains.

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.

Figures (7)

• Figure 1: Linear regression problem $\min_{\boldsymbol{w}}~\|\boldsymbol{x}-\boldsymbol{A}\boldsymbol{w}\|_2^2$ with $\tau$-sparse representation of the coefficient vector $\boldsymbol{w}$. In compressive sensing, the goal is to construct a sparse vector $\boldsymbol{w}$ given measurements $\boldsymbol{x}$ (i.e., the signal) and $\boldsymbol{A}$ (e.g., the dictionary) foucart2013invitation. The vector $\boldsymbol{w}$ is constrained to have no more than $\tau$ non-zero entries in which $\tau\in\mathbb{Z}^{+}$ refers to the sparsity level.
• Figure 2: Illustration of learning $\tau$-sparse vector $\boldsymbol{w}$ from the time series data $\boldsymbol{x}$ with the constructed formula as $\boldsymbol{x}\approx\boldsymbol{A}\boldsymbol{w}$. The $T$-by-$(T-1)$ dictionary matrix $\boldsymbol{A}$ is constructed by the time series $\boldsymbol{x}$, see Eq. \ref{['A_mat']}.
• Figure 3: Illustration of the modal product between a third-order tensor and a matrix. By definition, the entry of resulting tensor $\boldsymbol{\mathcal{X}}$ is the inner product between tensor fiber (i.e., in the form of a vector) of $\boldsymbol{\mathcal{A}}$ and row vector of $\boldsymbol{W}$kolda2009tensorgolub2013matrix.
• Figure 4: Daily average of rideshare pickup and dropoff trips during the first 8 weeks since April 1, 2024 in NYC, USA. There are 37,404,265 trips in total, while the average daily trips are 667,933.
• Figure 5: Rideshare trip time series in the first 3 weeks since April 1, 2024 in NYC, USA.