# Efficient State-Space Inference of Periodic Latent Force Models

Steven Reece, Siddhartha Ghosh, Alex Rogers, Stephen Roberts, Nicholas R. Jennings.

Year: 2014, Volume: 15, Issue: 68, Pages: 2337−2397

#### Abstract

Latent force models (LFM) are principled approaches to incorporating solutions to differential equations within non- parametric inference methods. Unfortunately, the development and application of LFMs can be inhibited by their computational cost, especially when closed-form solutions for the LFM are unavailable, as is the case in many real world problems where these latent forces exhibit periodic behaviour. Given this, we develop a new sparse representation of LFMs which considerably improves their computational efficiency, as well as broadening their applicability, in a principled way, to domains with periodic or near periodic latent forces. Our approach uses a linear basis model to approximate one generative model for each periodic force. We assume that the latent forces are generated from Gaussian process priors and develop a linear basis model which fully expresses these priors. We apply our approach to model the thermal dynamics of domestic buildings and show that it is effective at predicting day-ahead temperatures within the homes. We also apply our approach within queueing theory in which quasi-periodic arrival rates are modelled as latent forces. In both cases, we demonstrate that our approach can be implemented efficiently using state-space methods which encode the linear dynamic systems via LFMs. Further, we show that state estimates obtained using periodic latent force models can reduce the root mean squared error to 17% of that from non-periodic models and 27% of the nearest rival approach which is the resonator model (Sarkka et al., 2012; Hartikainen et al. 2012).