Detecting and describing pseudo-periodic dynamics from time series

Abstract

Time series measurements are sequential records of physical variables such as velocity, temperature, or pressure that are collected over some period of time from a dynamical process of interest. Many time series in the real world display apparent irregular behavior or fluctuations that carry abundant information regarding the underlying process. Given such a time series, a most fundamental question to ask is: Does the underlying dynamics originate from a low-dimensional deterministic (possibly chaotic) process, or is it governed by stochastic ones? If the system is proved to be chaotic, or at least contains deterministic component, how can we characterize such signals and further extract relevant information reliably in the presence of different kinds of noise or nonstationarity? In this thesis, we intend to answer the above questions. In particular, we focus on those time series that demonstrate strong periodic behavior known as pseudoperiodic time series. Such signals are abundant in natural and physiological systems, e.g., annual sunspot numbers, laser output, human electrocardiogram (EGG) and gait data. The paradigm of chaos introduced decades ago has provided a potential explanation for the seemingly erratic and apparently unpredictable behavior of the time series. However, traditional statistics like Lyapunov exponent and correlation dimension cannot confirm chaos absolutely in that noise tends to mask or mimic deterministic behaviors. In addition, the available techniques seeking determinism originated from dynamical system theory are not always suitable or perform poorly for analyzing pseudoperiodic data, because the presence of strong periodicity may hide the underlying fractal structures. We have developed novel methods to address the problem of faithfully detecting and characterizing temporal and spatial correlations in pseudoperiodic time series that are noisy and nonstationary in this thesis. Utilizing the inherent periodicity, we first segment the time series into consecutive cycles and adopt the correlation coefficient as a measure of their distance, which forms the foundation for subsequent analysis. In detecting temporal correlation, we construct a number of cycle-reordered time series that preserves progressively less determinism than the original one. We check the number of cycle pairs in the new time series that share similar dynamical evolution to those in the original time series. Appropriate statistics are then devised to quantify how this correlation decays within and across each new time series, so that the original dynamics can be inferred reliably. In terms of spatial correlation, we have for the first time proposed a transformation from time domain to the complex network domain, with the cycles from the time series represented by nodes in the corresponding network. The dynamics of the time series thus can be readily explored through the topology of the corresponding network, and various topological indices have greatly enhanced our understanding of the complex dynamics of the time series. The validity of the above methods is further verified by using the pseudoperiodic surrogate (PPS) method. The ability of the statistics to capture the correlation among cycles can be validated by finding significant differences between the original time series and its PPS in terms of the new statistics used. This in turn has formed a framework that is more sensitive to examine the subtle changes in the dynamics. Finally we apply the proposed methods to various experimental data such as human EGG, vowel data, gait signal, and laser data. The intrinsic correlations among cycles in such time series are clearly revealed and effectively quantified by our methods, which provide new insights into the understanding of the complex physiological systems. Meanwhile, the new statistics are shown to be very promising in discriminating between healthy and pathological state from visually indistinguishable biomedical recordings.