Applying non-negative matrix factorization with covariates to multivariate time series data as a vector autoregression model

We propose a novel framework for analyzing multivariate time series (MTS) data by integrating non-negative matrix factorization (NMF) with vector autoregression (VAR). Termed NMF-VAR, this method models the coefficient matrix of NMF as a VAR process, enabling simultaneous extraction of latent components and temporal dependencies. Unlike standard VAR, which struggles with high dimensionality and lacks clarity, our method introduces a low-rank latent structure that reduces the number of parameters while retaining explanatory power. The proposed framework generalizes the standard VAR model to high-dimensional non-negative data, including the standard VAR as a special case. We formulate the estimation as a constrained optimization problem and present multiplicative update rules for NMF based on existing tri-factorization techniques. We evaluate the method on three real-world datasets: quarterly first-differenced macroeconomic indicators of Canada, monthly international airline passenger volumes, and daily COVID-19 infection counts across Japanese prefectures. The results demonstrate that NMF-VAR effectively captures meaningful patterns such as economic cycles, seasonal travel behavior, and regional epidemic trends. Moreover, the method yields a significant reduction in regression parameters, improving both scalability and model transparency. Overall, NMF-VAR provides an efficient and insightful tool for analyzing high-dimensional and large-scale time series data.