  Nonnegative matrix factorization (NMF) has been actively investigated and
used in a wide range of problems in the past decade. A significant amount of
attention has been given to develop NMF algorithms that are suitable to model
time series with strong temporal dependencies. In this paper, we propose a
novel state-space approach to perform dynamic NMF (D-NMF). In the proposed
probabilistic framework, the NMF coefficients act as the state variables and
their dynamics are modeled using a multi-lag nonnegative vector autoregressive
(N-VAR) model within the process equation. We use expectation maximization and
propose a maximum-likelihood estimation framework to estimate the basis matrix
and the N-VAR model parameters. Interestingly, the N-VAR model parameters are
obtained by simply applying NMF. Moreover, we derive a maximum a posteriori
estimate of the state variables (i.e., the NMF coefficients) that is based on a
prediction step and an update step, similarly to the Kalman filter. We
illustrate the benefits of the proposed approach using different numerical
simulations where D-NMF significantly outperforms its static counterpart.
Experimental results for three different applications show that the proposed
approach outperforms two state-of-the-art NMF approaches that exploit temporal
dependencies, namely a nonnegative hidden Markov model and a frame stacking
approach, while it requires less memory and computational power.
