## Efficient State-Space Inference of Periodic Latent Force Models

*Steven Reece, Siddhartha Ghosh, Alex Rogers, Stephen Roberts, Nicholas R. Jennings*; 15(Jul):2337−2397, 2014.

### 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).

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