Multiplicative Latent Force Models Tutorial

This module provides methods to carry out simulation and fitting of latent force models, which are broadly taken to be time dependent linear ODEs driven by a set of smooth Gaussian processes which are allowed to interact multiplicatively with the state variable, and so the name Multiplicative Latent Force Models (MLFM) to differ them from the case with an additive forcing term which are discussed in lfm-tutorials-index.

Model Description

Multiplicative latent force models are time dependent linear ODEs of the form

\[\dot{X}(t) = A(t)X(t), \qquad A(t) = A_0 + \sum_{r=1}^R g_r(t) A_r,\]

where \(\{ g_r(t) \}_{r=1}^R\) are a set of independent smooth scalar Gaussian processes, and \(\{A_r \}_{r=0}^{R}\) are a set of square \(K\times K\) matrices. Furthermore it may also be the case that for each of the structure matrices \(A_r\) we have

\[A_r = \sum_{d} \beta_{rd} L_d,\]

for some common set of shared basis matrices \(\{ L_d \}_{d=1}^{D}\) – typically these will be chosen to form a basis of some Lie algebra.

The following tutorials demonstrate the process of constructing these models as well as demonstrating the possible structure preserving properties of this model as well as how to carry out inference.

Model Fitting

In the near future we consider two methods of fitting these models, more will be added as and when they are dreamt of and coded up

Adaptive Gradient Matching