Time Series Forecasting: A Nonlinear Dynamics Approach

Abstract

The problem of prediction of a given time series is examined on the basis of recent nonlinear dynamics theories. Particular attention is devoted to forecast the amplitude and phase of one of the most common solar indicator activity, the international monthly smoothed sunspot number. It is well known that the solar cycle is very difficult to predict due to the intrinsic complexity of the related time behaviour and to the lack of a succesful quantitative theoretical model of the Sun magnetic cycle. Starting from a previous recent work, we checked the reliability and accuracy of a forecasting model based on concepts of nonlinear dynamical systems applied to experimental time series, such as embedding phase space, Lyapunov spectrum, chaotic behaviour. The model is based on a local hypothesis of the behaviour on the embedding space, utilizing an optimal number k of neighbour vectors to predict the future evolution of the current point with the set of characteristic parameters determined by several previous parametric computations. The performances of this method suggest its valuable insertion in the set of the so-called statistical-numerical prediction techniques, like Fourier analyses, curve fitting, neural networks, climatological, etc. The main task is to set up and to compare a promising numerical nonlinear prediction technique, essentially based on an inverse problem, with the most accurate predictive methods like the so-called "precursor methods" which appear now reasonably accurate in predicting "long term" Sun activity, with particular reference to the "solar" precursor methods based on a solar dynamo theory.