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Title: Multiplicative latent force models
Author: Tait, Daniel J.
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2020
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Latent force models (LFM) are a class of flexible models of dynamic systems, combining a simple mechanistic model with the flexibility of an additive inhomogeneous Gaussian process (GP) forcing term. These hybrid models achieve the dual goal of being flexible enough to be broadly applied, even for complex dynamic systems where a full mechanistic model may be hard to motivate, but by also encoding relevant properties of dynamic systems they are better able to model the underlying dynamics and so demonstrate superior generalisation. In this thesis, we consider an extension of this framework which keeps the same general form, a linear ordinary di↵erential equation with time-varying behaviour arising from a set of smooth GPs, but now we allow for multiplicative interactions between the state variables and the GP terms. The result is a semi-parametric modelling framework that allows for the embedding of rich topological structure. Following a brief review of the latent force model, which we note is a particular case of the GP regression model, we introduce our extension with multiplicative interactions which we refer to as the multiplicative latent force model (MLFM). We demonstrate that this class of models allows for the possibility of strong geometric constraints on the pathwise trajectories. This will enable the modelling of systems for which the GP trajectories of the LFM are unsatisfactory. Unfortunately, and as a direct consequence of the strong geometric constraints we have introduced, it is no longer straightforward to carry out inference in these models; therefore the remainder of this thesis is primarily devoted to constructing two methods for carrying out approximate inference for this class of models. The first is referred to as the Bayesian adaptive gradient matching method, and the second is a novel construction based on the method of successive approximations; a theoretical construct used in the standard classical existence and uniqueness theorems for ODEs. After introducing these methods, we demonstrate their accuracy on simulated data, which also allows for an investigation into the regimes in which each of the respective methods can be expected to perform well. Finally, we demonstrate the utility of the MLFM on motion capture data and show that, by using the framework developed in this thesis to allow for the sharing of a smaller number of latent forces between distinct trajectories with specific geometric constraints, we can achieve superior predictive performance than by the modelling of a single trajectory.
Supervisor: Worton, Bruce ; Aitken, Colin Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
Keywords: latent force models ; hybrid model ; Gaussian process ; statistical inference ; modelling ; multiplicative latent force model ; human motion capture data