Gaussian Process Kernels for Popular State-Space Time Series Models

Abstract--In this paper we investigate a link between state-space models and Gaussian Processes (GP) for time series modeling and forecasting. In particular, several widely used state-space models are transformed into continuous time form and corresponding Gaussian Process kernels are derived. Experimental results demonstrate that the derived GP kernels are correct and appropriate for Gaussian Process Regression. An experiment with a real world dataset shows that the modeling is identical with state-space models and with the proposed GP kernels. The considered connection allows the researchers to look at their models from a different angle and facilitate sharing ideas between these two different modeling approaches. I NTRODUCTION ANDM OTIVATION Time series modeling and prediction is one of oldest topics in statistics. The very first statisticians already dealt with time dependent data. For example, Beveridge wheat price (years 1500 to 1869) or Wolfer's sunspot number (years 1610-1960) [1] are examples of very early time series. Nowadays time series analysis and forecasting is ubiquitous in many fields of science and engineering. Econometricians, physicists, statisticians, biologists, climatologists etc. encounter time dependent data in their daily work. Since this problem is very old and very widespread, different fields of science developed their own sets of methods for analysis and forecasting of time series. For instance, in statistics and econometrics domains the most common models are state-space (SS) models [2], [3]. In the physics domain the dominating class of models constitute nonlinear dynamical models [4].