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Latent force models (LFM) are principled approaches to incorporatingsolutions to differential equations within non-parametric inference methods.Unfortunately, the development and application of LFMs can be inhibited bytheir computational cost, especially when closed-form solutions for the LFM areunavailable, as is the case in many real world problems where these latentforces exhibit periodic behaviour. Given this, we develop a new sparserepresentation of LFMs which considerably improves their computationalefficiency, as well as broadening their applicability, in a principled way, todomains with periodic or near periodic latent forces. Our approach uses alinear basis model to approximate one generative model for each periodic force.We assume that the latent forces are generated from Gaussian process priors anddevelop a linear basis model which fully expresses these priors. We apply ourapproach to model the thermal dynamics of domestic buildings and show that itis effective at predicting day-ahead temperatures within the homes. We alsoapply our approach within queueing theory in which quasi-periodic arrival ratesare modelled as latent forces. In both cases, we demonstrate that our approachcan be implemented efficiently using state-space methods which encode thelinear dynamic systems via LFMs. Further, we show that state estimates obtainedusing periodic latent force models can reduce the root mean squared error to17% of that from non-periodic models and 27% of the nearest rival approachwhich is the resonator model.

Efficient State-Space Inference of Periodic Latent Force Models