Learning Linear Dynamical Systems from Multivariate Time Series: A Matrix Factorization Based Framework

Abstract

The linear dynamical system (LDS) model is arguably the most commonly used time series model for real-world engineering and financial applications due to its relative simplicity, mathematically predictable behavior, and the fact that exact inference and predictions for the model can be done efficiently. In this work, we propose a new generalized LDS framework, gLDS, for learning LDS models from a collection of multivariate time series (MTS) data based on matrix factorization, which is different from traditional EM learning and spectral learning algorithms. In gLDS, each MTS sequence is factorized as a product of a shared emission matrix and a sequence-specific (hidden) state dynamics, where an individual hidden state sequence is represented with the help of a shared transition matrix. One advantage of our generalized formulation is that various types of constraints can be easily incorporated into the learning process. Furthermore, we propose a novel temporal smoothing regularization approach for learning the LDS model, which stabilizes the model, its learning algorithm and predictions it makes. Experiments on several real-world MTS data show that (1) regular LDS models learned from gLDS are able to achieve better time series predictive performance than other LDS learning algorithms; (2) constraints can be directly integrated into the learning process to achieve special properties such as stability, low-rankness; and (3) the proposed temporal smoothing regularization encourages more stable and accurate predictions.