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Abstract

This thesis presents the simulation of approximated Gaussian process autoregressive models.
Gaussian process models are a Bayesian nonparametric regression method, the main
advantage of which is the quantification of uncertainty in closed-form. However, the closedform
solution for the marginal likelihood results in a cubic computational complexity with
respect to the data set size. An additional downside is the analytically intractable propagation
of the uncertain inputs through the nonlinear covariance function, which implies that
the training of the dynamical Gaussian process models cannot be obtained in closed-form.
Autoregressive models reduce the training to that of the static case. In Gaussian
processes, this allows for an analytical expression of the marginal likelihood. Consequently,
they allow for an effortless extension of the existing approximations of Gaussian processes,
reducing the computational complexity and permitting non-Gaussian likelihoods. However,
the poor computational complexity in the numerical estimation of the simulation persists.
In this thesis, we present an approximation to the simulation of Gaussian process models.
We propose a unified view of the simulation for the pseudo-input-based Gaussian
processes, invariant to the specific approximation up to taking a static sample from the
pseudo-input posterior and the choice of the covariance function. We propose an algorithm
where a single parameter controls the trade-off between the computational complexity and
the accuracy. Our vectorized implementation allows for the acceleration of the unified simulation
algorithm on general-purpose graphics processing units, wrapping the contemporary
software frameworks specific to Gaussian processes.
Practical justification of the autoregressive Gaussian process regression is demonstrated
on two case studies where the quantification of uncertainty is important. The first case
study considers modeling the atmospheric variables near a nuclear power plant where our
forecasts are used to predict the condition of the atmosphere during the passage of the
radioactive pollution cloud. In the second case study, modeling of the electrical load and
photovoltaic generation in the greater area of Sydney is presented, where the forecasts of
the respective outputs are used to estimate the dynamic export limits of households to
preserve the stability and robustness of the electrical grid.