## AbstractThis thesis explores latent-variable probabilistic models for the analysis and classification of electroenchephalographic (EEG) signals used in Brain Computer Interface (BCI) systems. The first part of the thesis focuses on the use of probabilistic methods for classification. We begin with comparing performance between `black-box' generative and discriminative approaches. In order to take potential advantage of the temporal nature of the EEG, we use two temporal models: the standard generative hidden Markov model, and the discriminative input-output hidden Markov model. For this latter model, we introduce a novel `apposite' training algorithm which is of particular benefit for the type of training sequences that we use. We also asses the advantage of using these temporal probabilistic models compared with their static alternatives. We then investigate the incorporation of more specific prior information about the physical nature of EEG signals into the model structure. In particular, a common successful assumption in EEG research is that signals are generated by a linearmixing of independent sources in the brain and other external components. Such domain knowledge is conveniently introduced by using a generative model, and leads to a generative form of Independent Components Analysis (gICA). We analyze whether or not this approach is advantageous in terms of performance compared to a more standard discriminative approach, which uses domain knowledge by extracting relevant features which are subsequently fed into classifiers. The user of a BCI system may have more than one way to perform a particular mental task. Furthermore, the physiological and psychological conditions may change from one recording session and/or day to another. As a consequence, the corresponding EEG signals may change significantly. As a first attempt to deal with this effect, we use a mixture of gICA in which the EEG signal is split into different regimes, each regime corresponding to a potentially different realization of the same mental task. An arguable limitation of the gICA model is the fact that the temporal nature of the EEG signal is not taken into account. Therefore, we analyze an extension in which each hidden component is modeled with an autoregressive process. The second part of the thesis focuses on analyzing the EEG signal and, in particular, on extracting independent dynamical processes from multiple channels. In BCI research, such a decomposition technique can be applied, for example, to denoise EEG signals from artifacts and to analyze the source generators in the brain, thereby aiding the visualization and interpretation of the mental state. In order to do this, we introduce a specially constrained form of the linear Gaussian state-space model which satisfies several properties, such as flexibility in the specification of the number of recovered independent processes and the possibility to obtain processes in particular frequency ranges. We then discuss an extension of this model to the case in which we don't know a priori the correct number of hidden processes which have generated the observed time-series and the prior knowledge about their frequency content is not precise. This is achieved using an approximate variational Bayesian analysis. The resulting model can automatically determine the number and appropriate complexity of the underlying dynamics, with a preference for the simplest solution, and estimates processes with preferential spectral properties. An important contribution from our work is a novel `sequential' algorithm for performing smoothed inference, which is numerically stable and simpler than others previously published.
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