Dynamical Decomposition of Multifractal Time Series as Fractal Evolution and Long-Term Cycles: Applications to Foreign Currency Exchange Market

Abstract

The application of the multifractal formalism to the study of some time series with scale invariant evolution has given rise to a rich framework of models and processing tools for the analysis of these signals. The formalism has been successfully exploited in different ways and with different goals: to obtain the effective variables governing the evolution of the series, to predict its future evolution, to estimate in which regime the series are, etc. In this paper, we discuss on the capabilities of a new, powerful processing tool, namely the computation of dynamical sources. With the aid of the source field, we will separate the fast, chaotic dynamics defined by the multifractal structure from a new, so-far unknown slow dynamics which concerns long cycles in the series. We discuss the results on the perspective of detection of sharp dynamic changes and forecasting