Linear State-Space Model with Time-Varying Dynamics

TL;DR

This paper presents a linear state-space model with smoothly varying dynamics, modeled as a linear combination of matrices with time-dependent weights, enabling better modeling of processes with continuous parameter changes.

Contribution

The paper introduces a novel linear state-space model with time-varying dynamics using a linear combination of matrices, improving over switching models for continuous process changes.

Findings

Model outperforms previous approaches on weather data

Enables smooth transition modeling of physical processes

Uses variational Bayesian inference for posterior estimation

Abstract

This paper introduces a linear state-space model with time-varying dynamics. The time dependency is obtained by forming the state dynamics matrix as a time-varying linear combination of a set of matrices. The time dependency of the weights in the linear combination is modelled by another linear Gaussian dynamical model allowing the model to learn how the dynamics of the process changes. Previous approaches have used switching models which have a small set of possible state dynamics matrices and the model selects one of those matrices at each time, thus jumping between them. Our model forms the dynamics as a linear combination and the changes can be smooth and more continuous. The model is motivated by physical processes which are described by linear partial differential equations whose parameters vary in time. An example of such a process could be a temperature field whose evolution is driven by a varying wind direction. The posterior inference is performed using variational Bayesian approximation. The experiments on stochastic advection-diffusion processes and real-world weather processes show that the model with time-varying dynamics can outperform previously introduced approaches.