An introduction to chaotic and random time series analysis

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Abstract
Chaos refers to the paradoxical evolution of a deterministic system in a way that is disordered—to the point that the time dependence of the physical variables appears stochastic. A need for data analysis procedures to detect, model, and separate chaotic and random processes has arisen from this recently understood paradigm. Many special techniques have been designed for chaotic data; the unification of these with conventional time series analysis is a developing field.This tutorial uses examples to explain the origin of chaotic behavior and the relation of chaos to randomness. Two powerful mathematical results are described: (1) a representation theorem guarantees the existence of a specific time‐domain model for chaos and addresses the relation between chaotic, random, and strictly deterministic processes, and (2) a theorem assures that information on the behavior of a physical system in its complete state space can be extracted from time‐series data on a single observable.These theorems form the basis of a practical data analysis scheme, as follows: given N observations of a variable Y, i.e., {Yn, n = 1,2,3, …, N}, define X = A * Y and maximize, with respect to the parameters of A, a function H(X) that measures degree of chaos. This maximization is carried out by minimizing the dimension covered by the data in the M‐dimensional space (Xn, Xn+1, Xn+2, …, Xn+M−1). The resulting dimension D either (1) increases continuously with M or (2) levels off and remains constant (= Dmax) beyond a certain point. In case (1) or if Dmax is quite large X is random; if case (2) holds and Dmax is small, we have chaos. The inverse of A found in this procedure is an estimate of the filter in the moving average model for Y.

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