Chiu, YD; (2014) Exploratory studies for Gaussian Process Structural Equation Models. Doctoral thesis , UCL (University College London).

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Abstract

Latent variable models (LVMs) are widely used in many scientific fields due to the ubiquitousness and feasibility of latent variables. Conventional LVMs, however, have limitations because they model relationships between covariates and latent variables or among latent variables with a parametric fashion. A more flexible model framework is therefore needed, especially without prior knowledge of sensible parametric forms. This thesis proposes a new non-parametric LVM for the need. We define a model structure with particular features, including a multi-layered structure constituting of non-parametric Gaussian Processes regression and parametric factor analysis. The connections to existing popular LVMs approaches, such as structural equation models and latent curve models, are also discussed. The model structure is subsequently extended for observed binary responses and longitudinal application. It follows that model identifiability is examined through parameter constraints and algebraic manipulations. The proposed model, despite convenient applicability, has a computational burden for analysing large data sets due to the computation of the inverse of a large covariance matrix. To address the issue, a sparse approximation method using a small number of M selected inputs (inducing inputs) is adopted. The associated computational cost can be reduced to O(M²NQ²) (or O(M²NT²)) where N and Q are the numbers of data points and latent variables (or time points T), respectively. Inference within this framework requires a series of algorithmic developments in a Bayesian paradigm. The algorithms, using Markov Chain Monte Carlo sampling-based methods and Expectation Maximisation optimisation methods with stochastic variant, are presented. A hybrid estimation procedure with two-step implementations is proposed as well, which can further reduce computational cost. Furthermore, a greedy selection scheme for inducing inputs is provided for better model predictive performance. Empirical studies of the modelling framework are conducted for various experiments. Interest lies in inference, including parameter estimation and realization of distribution of latent variables; and assessments and comparisons of predictive performance with two baseline techniques. Discussion and suggestions for improvement are provided based on results.

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