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This thesis details several applications of Gaussian processes (GPs) for enhanced time series modeling. We first cover different approaches for using Gaussian processes in time series problems. These are extended to the state space approach to time series in two different problems. We also combine Gaussian processes and Bayesian online change point detection (BOCPD) to increase the generality of the Gaussian process time series methods. These methodologies are evaluated on predictive performance on six real world data sets, which include three environmental data sets, one financial, one biological, and one from industrial well drilling. Gaussian processes are capable of generalizing standard linear time series models. We cover two approaches: the Gaussian process time series model (GPTS) and the autoregressive Gaussian process (ARGP).We cover a variety of methods that greatly reduce the computational and memory complexity of Gaussian process approaches, which are generally cubic in computational complexity. Two different improvements to state space based approaches are covered. First, Gaussian process inference and learning (GPIL) generalizes linear dynamical systems (LDS), for which the Kalman filter is based, to general nonlinear systems for nonparametric system identification. Second, we address pathologies in the unscented Kalman filter (UKF).We use Gaussian process optimization (GPO) to learn UKF settings that minimize the potential for sigma point collapse. We show how to embed mentioned Gaussian process approaches to time series into a change point framework. Old data, from an old regime, that hinders predictive performance is automatically and elegantly phased out. The computational improvements for Gaussian process time series approaches are of even greater use in the change point framework. We also present a supervised framework learning a change point model when change point labels are available in training.

Visual speech synthesis deals with synthesising facial animation from an audio representation of speech. In the last decade or so, data-driven approaches have gained prominence with the development of Machine Learning techniques that can learn an audio-visual mapping. Many of these Machine Learning approaches learn a generative model of speech production using the framework of probabilistic graphical models, through which efficient inference algorithms can be developed for synthesis. In this work, the audio and visual parameters are assumed to be generated from an underlying latent space that captures the shared information between the two modalities. These latent points evolve through time according to a dynamical mapping and there are mappings from the latent points to the audio and visual spaces respectively. The mappings are modelled using Gaussian processes, which are non-parametric models that can represent a distribution over non-linear functions. The result is a non-linear state-space model. It turns out that the state-space model is not a very accurate generative model of speech production because it assumes a single dynamical model, whereas it is well known that speech involves multiple dynamics (for e.g. different syllables) that are generally non-linear. In order to cater for this, the state-space model can be augmented with switching states to represent the multiple dynamics, thus giving a switching state-space model. A key problem is how to infer the switching states so as to model the multiple non-linear dynamics of speech, which we address by learning a variable-order Markov model on a discrete representation of audio speech. Various synthesis methods for predicting visual from audio speech are proposed for both the state-space and switching state-space models. Quantitative evaluation, involving the use of error and correlation metrics between ground truth and synthetic features, is used to evaluate our proposed method in comparison to other probabilistic models previously applied to the problem. Furthermore, qualitative evaluation with human participants has been conducted to evaluate the realism, perceptual characteristics and intelligibility of the synthesised animations. The results are encouraging and demonstrate that by having a joint probabilistic model of audio and visual speech that caters for the non-linearities in audio-visual mapping, realistic visual speech can be synthesised from audio speech.