Nonlinear statistical analysis of biological time series

Abstract

In the present thesis, methods of non-linear time series analysis and dynamical systems analysis have been combined with statistical methods, and statistical measures with high discriminating power directly estimated on time series have been studied and developed. Specifically, existing univariate and multivariate linear and nonlinear measures have been thoroughly reviewed, new nonlinear measures have been defined here and existing methods have been extended or modified to gain statistical significance and be more effective on different applications. All the investigated measures were first tested on simulation studies and were then applied on real data. Electroencephalogram (EEG) recordings from epileptic patients have been considered in order to evaluate the effectiveness of the measures to detect dynamical changes in the brain activity of the patients at different states, e.g. in order to discriminate between EEG records many hours before the seizure onset and EEG records from one hour before the seizure onset. The prediction of the onset of an epileptic seizure and the detection of the dynamical changes of the brain dynamics just before the seizure onset were investigated in the frame of univariate time series analysis. The performance of the univariate measures was quite promising and therefore the study was extended beyond the univariate case. Thus, the existence of interdependencies between different brain areas and the identification of the direction of the causal effects among the brain areas was investigated in the frame of multivariate time series analysis. The study focused on information measures, as they are model-free, computationally efficient and do not require any prior knowledge of the distribution of the data. The evaluation of the information measures was first assessed by Monte Carlo simulations on well-known dynamical systems in order to examine their discriminating power and significance. Dynamical characteristics of the systems varied by changing the parameters of the equation of the systems. The discriminating ability of the measures was also assessed in terms of the time series length. The stochasticity of the systems was also a factor that was examined as it can be controlled by variation of the level of noise added in the observations of the systems. The variation of the complexity or stochasticity of a system is considered to simulate the different states of the brain activity and thus of the EEG signal. Mutual information is an essential tool in nonlinear time series analysis and therefore was thoroughly reviewed and tested on different applications, e.g. in detecting dynamical changes of systems. Different estimators of mutual information have been compared and the selection of their parameters was investigated in order to be optimized. The promising performance of mutual information led to a more comprehensive study which included also entropy measures (e.g. Shannon, Tsallis, Permutation). This investigation has led to the extension and modification of many existing measures. The statistical significance and power of the measures in discrimination tasks were assessed using different statistical tests (t-test, Wilcoxon rank sum test, ROC curves, surrogate data test). It was also examined whether the discriminating power of the measures can be improved if the time series were first transformed to have uniform or normal marginal distribution. The results from the simulation studies on known dynamical systems were required in order to interpret the values of the estimated measures on the EEG recordings in terms of their stochasticity and complexity. Thus, the measures were evaluated in discriminating EEG signal from preictal and interictal stages. The effectiveness of the nonlinear measures was compared to that of some linear ones, as there are contradictory studies on this matter. Based on the frame of univariate time series analysis, EEG recordings from different channels were used for the analysis; channels were chosen based on knowledge of the epileptic focus area or were randomly selected in cases of generalized seizures to cover all parts of the brain. EEG recordings are multivariate signals, and therefore one should also test the interdependencies between the recordings from the different channels, i.e. the existence of interactions among the different parts of the brain should be examined. Here, the direction of the information flow was investigated using bivariate nonlinear measures. The study focused again on information measures detecting the direction of interdependencies between interacting systems, however also other types of causal measures were included for comparative studies, e.g. state-space and synchronization measures. The evaluation of the causality measures was initially assessed on well-known nonlinear systems and results were considered in order to apply the measures on EEG signals. This study led us again in the modification of the existing measures and the extraction of measures that seem to improve their performance. There exist measures that can discriminate among different dynamical systems and detect the dynamical changes of a systems and some of the measures that have been defined here may improved the discriminating power of the original measures when tested on synthetic data, however even these improved methods do not perform dramatically better on EEG. EEG signal is very complex and the finite length of the EEG signal, the measurement artifacts and the interactions between the different brain parts render the problem of the EEG analysis to be even harder. There is still a long way to go for the use of any measure in clinical practice. The main contribution of this work is to assess the statistical significance of the information measures for univariate and bivariate time series, propose modifications to attain better significance and show in an objective clinical setting the shortcomings and possible limited success of many existing measures for the problem of seizure prediction.

Στην παρούσα διατριβή, μελετήθηκαν, αναπτύχθηκαν και τροποποιήθηκαν στατιστικά μέτρα πληροφορίας που υπολογίζονται από μονομεταβλητές και πολυμεταβλητές χρονοσειρές. Ιδιαίτερη έμφαση δόθηκε στην μελέτη της αμοιβαίας πληροφορίας, στην μελέτη των στατιστικών ιδιοτήτων της, στην αξιολόγηση των γνωστότερων εκτιμητών της και στην βελτιστοποίηση των παραμέτρων τους. Ενδελεχής αξιολόγηση των στατιστικών μέτρων πληροφορίας/συσχέτισης, ευτροπίας και πολυπλοκότητας πραγματοποιήθηκε σε προσομοιωτικά δεδομένα και σε ηλεκτροεγκεφαλογραφήματα επιληπτικών ασθενών για πρόβλεψη επιληπτικής κρίσης. Τέλος, μελετήθηκαν μέτρα ροής πληροφορίας και τροποποιήθηκαν ώστε να έχουν καλή στατιστική σημαντικότητα. Η αξιολόγηση των μέτρων πραγματοποιήθηκε σε προσομοιωτικά δεδομένα και σε ηλεκτροεγκεφαλογραφήματα επιληπτικών για διερεύνηση αλληλεπιδράσεων περιοχών του εγκεφάλου.