Functional data classification and covariance estimation

Focusing on the analysis of functional data, the first part of this dissertation proposes three statistical models for functional data classification and applies them to a real problem of cervical pre-cancer diagnosis; the second part of the dissertation discusses covariance estimation of functional data. The functional data classification problem is motivated by the analysis of fluorescence spectroscopy, a type of clinical data used to quantitatively detect early-stage cervical cancer. Three statistical models are proposed for different purposes of the data analysis. The first one is a Bayesian probit model with variable selection, which extracts features from the fluorescence spectroscopy and selects a subset from these features for more accurate classification. The second model, designed for the practical purpose of building a more cost-effective device, is a functional generalized linear model with selection of functional predictors. This model selects a subset from the multiple functional predictors through a logistic regression with a grouped Lasso penalty. The first two models are appropriate for functional data that are not contaminated by random effects. However, in our real data, random effects caused by devices artifacts are too significant to be ignored. We therefore introduce the third model, the Bayesian hierarchical model with functional predictor selection, which extends the first two models for this more complex data. Besides retaining high classification accuracy, this model is able to select effective functional predictors while adjusting for the random effects. The second problem focused on by this dissertation is the covariance estimation of functional data. We discuss the properties of the covariance operator associated with Gaussian measure defined on a separable Hilbert Space and propose a suitable prior for Bayesian estimation. The limit of Inverse Wishart distribution as the dimension approaches infinity is also discussed. This research provides a new perspective for covariance estimation in functional data analysis

KurSL: a model of coupled oscillators based on Kuramoto's coupling and Sturm-Liouville theory

Methods commonly used to analyse oscillatory systems, such as short-time Fourier or wavelet transforms, require predefined oscillatory structures or ne-tuning of method's parameters. These limitations may be detrimental for an adequate component description and can introduce bias to the interpretation. This thesis addresses the challenge of identifying interacting components in a signal by introducing a model of coupled oscillators. The proposed model consists of two parts: Sturm-Liouville self-adjoint ordinary differential equation (ODE) and Kuramoto's coupling model. The resulting model, KurSL, is described by a set of coupled ODEs producing general amplitude- and frequency-modulated mutually interacting oscillations. The complexity of these equations depends on the definition of the coupling function, the number of oscillators and the initial state of each oscillator. Thus, the performance of the KurSL decomposition can be characterised in terms of the model parameters optimisation. After introducing the model, the thesis provides analysis and discussion of the KurSL with examples of its usage. The method is firstly tested on various synthetic data that were generated from simulated stationary and dynamical processes. Such testing allows capturing various characteristics that are desirable in coupled oscillatory components such as phase and amplitude dynamics. Subsequently, experiments were performed on empirical EEG signals recorded from patients with epilepsy. Validation of these experiments is through comparisons to different orders of the KurSL and to other time-frequency methods. Overall results indicate that the KurSL method provides a more detailed description of oscillatory processes than the Huang-Hilbert transform and it provides insights comparable to manually tuned short-time Fourier transform and Morlet-based wavelet time-frequency representations. However, the advantage of the KurSL is that the similar results can be achieved with a finite number of components. Moreover, in contrast to the mentioned representations which, due to finite resolution, are unable to localise time-frequency events precisely, the KurSL provides an instantaneous description. This exactness allows to identify any modulations in both time and frequency domains and thus better describe the behaviour of the analysed system