In this paper, we propose a precise definition of multivariate Gaussian processes based on Gaussian measures on vector-valued function spaces, and provide an existence proof.

In this paper, we apply entropy and predictability measures to analyse and bound the predictive information of an individual’s mobility pattern and the flow of that information from their top social ties and from their non-social colocators.

we propose that the predictability in human mobility is a state and not a static trait of individuals. First, we show that time (of the week) explains people’s whereabouts more than the sequences of locations they visit. Then, we show that not only does predictability depend on time but also the type of activity an individual is engaged in, thus establishing the importance of contexts in human mobility..

In this paper, we focus on mitigating the harm incurred by a biased machine learning system that offers better outputs (e.g., loans, job interviews) for certain groups than for others. We show that bias in the output can naturally be controlled in probabilistic models by introducing a latent target output.

In this paper, we propose a unified framework which is used not only to introduce a novel multivariate Student-t process regression model (MV-TPR) for multi-output prediction, but also to reformulate the multivariate Gaussian process regression (MV-GPR) that overcomes some limitations of the existing methods.

Gaussian process regression (GPR) is a kernel-based nonparametric method that has been proved to be effective and powerful in many areas, including time series prediction. In this thesis, we focus on GPR and its extensions and then apply them to financial time series prediction. We first review GPR, followed by a detailed discussion about model structure, mean functions, kernels and hyper-parameter estimations. After that, we study the sensitivity of hyper-parameter and performance of GPR to the prior distribution for the initial values, and find that the initial hyper-parameters’ estimates depend on the choice of the specific kernels, with the priors having little influence on the performance of GPR in terms of predictability. Furthermore, GPR with Student−t process (GPRT) and Student−t process regression (TPR), are introduced. All the above models as well as autoregressive moving average (ARMA) model are applied to predict equity indices. We find that GPR and TPR shows relatively considerable capability of predicting equity indices so that both of them are extended to state-space GPR (SSGPR) and state-space TPR (SSTPR) models, respectively. The overall results are that SSTPR outperforms SSGPR for the equity index prediction. Based on the detailed results, a brief market efficiency analysis confirms that the developed markets are unpredictable on the whole. Finally, we propose and test the multivariate GPR (MV-GPR) and multivariate TPR (MV-TPR) for multi-output prediction, where the model settings, derivations and computations are all directly performed in matrix form, rather than vectorising the matrices involved in the existing method of GPR for multi-output prediction. The effectiveness of the proposed methods is illustrated through a simulated example. The proposed methods are then applied to stock market modelling in which the Buy&Sell strategies generated by our proposed methods are shown to be profitable in the equity investment.

In this paper, we provide the first empirical study on the problem of how prior affect Gaussian process regression using simulated and real data experiments.