A Comparative Study of Gamma Markov Chains for Temporal Non-Negative Matrix Factorization
This work addresses a theoretical limitation in temporal NMF models for time series analysis, though it is incremental as it builds on and compares existing methods.
The paper identified that existing Gamma Markov chains for temporal non-negative matrix factorization lack a well-defined stationary distribution, and introduced the BGAR(1) model to address this limitation, showing improved performance in prediction tasks under a Poisson likelihood framework.
Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.