An introduction to chaotic and random time series analysis

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., { Y n , 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 ( X n , X n +1 , X n +2 , …, X n+M −1 ). The resulting dimension D either (1) increases continuously with M or (2) levels off and remains constant (= D max ) beyond a certain point. In case (1) or if D max is quite large X is random; if case (2) holds and D max 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 .