Modelling and Inference using Locally Stationary Processes,
This thesis considers statistical methods for non-stationary signals, specifically stochastic modelling, inference on the model parameters and optimal spectral estimation. The models are based on Silverman's definition of Locally Stationary Processes (LSPs). In all the contributions, an example of a biomedical application of the proposed method is provided. In the first two papers, the methods are applied to electroencephalography (EEG) data, while in the third paper the application involves Heart Rate Variability (HRV) data.
In paper A, we propose a method for estimating the parameters of an LSP model. The proposed method is based on the separation of the two factors defining the LSP covariance function, in order to take advantage of their individual structure and divide the inference problem into two simpler sub-problems. We present a simulation study to show the method's performance in terms of speed of convergence, accuracy and robustness. Finally, we provide an illustrative example of parameter estimation from three sets of EEG signals, measured from one person during several trials of a memory encoding task of three different categories of visual memories.
Paper B investigates the estimation of the Wigner-Ville spectrum of time-varying processes, modelled as LSPs. Previous works have provided the theoretical expression of the mean-square error optimal time-frequency kernel, and now, thanks to the introduced inference method, we are able to compute the optimal kernel in real data cases. The derived optimal spectral estimator is compared with the single Hanning spectrogram and the Welch method in a simulation study. As biomedical application, we compute the optimal spectral estimate according to the estimated model parameters for the three EEG data-sets also treated in paper A.
In paper C, we model HRV signals with a model known as Locally Stationary Chirp Processes (LSCP), which is an extension of the LSP model including the presence of a chirp. The inference method proposed in paper A is adapted to take into account the respiratory signal information, in the form of the covariance matrix of the chirp respiratory signal estimated beforehand. We perform least squares regression analysis with each of the LSCP model parameters as the response to explore the correlation of the parameters with several factors of interest. Our results show a statistically significant correlation between the model parameters with age, BMI, State and Trait Anxiety as well as stress level.