Methods of Nonlinear Time Series Analysis and Applications: A Review

Abstract

Many of the real world systems are highly complex with little or no apriori information about the underlying dynamics. We have to depend mostly on measurements or observations of their average responses to study them. These measured or observational data come as a sequence of values at intervals, called time series. A few typical examples are sunspot data, variable star data, x-ray variability of black holes, climate or rainfall data, earth quake data, combustion data, thermoacoustic data, physiological data like EEG, ECG and fMRI, financial market data, output of agricultural crops, gene expression data etc. We present an overview of the techniques used for nonlinear time series analysis, to detect nontrivial structures in such time series data that will indicate the nature of underlying dynamics that produce the data. We start with the method of time delay embedding that can be used to re-construct the dynamics in higher dimension. The geometry and intricate structure of the re-constructed dynamics can then be characterized using two powerful techniques. The first one aims at computing the measures of the fractal geometry of the structure and its scaling properties. The resultant multi-fractal spectrum is uniquely characterized by four parameters that can be computed for each time series or data. The second method involves generating a complex network from the recreated phase space using its recurrence properties. The measures of the recurrence network then helps to identify the dynamical states of the system and their possible transitions. The applications of these techniques to several different types of real data are also included as illustrations.