STUDIES IN ASTRONOMICAL TIME-SERIES ANALYSIS .4. MODELING CHAOTIC AND RANDOM-PROCESSES WITH LINEAR FILTERS

While chaos arises only in nonlinear systems, standard linear time series models are nevertheless useful for analyzing data from chaotic processes. This paper introduces such a model, the chaotic moving average. This time-domain model is based on the theorem that any chaotic process can be represented as the convolution of a linear filter with an uncorrelated process called the chaotic innovation. We also present a technique, minimum phase-volume deconvolution, to estimate the filter and innovation. The algorithm measures the quality of a model using the volume covered by the phase portrait of the innovation process. Experiments on synthetic data demonstrate the following properties of the algorithm: It accurately recovers the parameters of simple chaotic processes. Though tailored for chaos, the algorithm can detect both chaos and randomness, distinguish them from each other, and separate them if both are present. It can also recover non-minimum-delay pulse shapes in non-Gaussian processes, both random and chaotic.