A forecasting method for time series with fractal geometry and its application

Abstract

This paper deals with the prediction of time series that have fractal geometry. We also describe the prediction error, the estimation of fractal dimension of the times series, and several applications. First, we assume that the time series is represented by a convolution of the input signal and the impulse response function, expanded by using a set of scaling functions. Then, a prediction method is derived by using the fact that the impulse response retains self-similarity even if the time scale is expanded, due to the fractal geometry of the time series. We demonstrate the ability of the prediction method for various fractal dimensions by showing that the one-step-ahead prediction error is very small, and also that the n-step-ahead prediction error can be made small by using adaptive correction of the data in which the prediction is successively used as an observation. We also describe the estimation of the fractal dimension by using the covariance matrix and the shape of the spectrum of the time series. As an application, the option price of the stock index estimated by the method is compared to the price obtained by conventional estimation, and our strategy is shown to give a greater profit. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 82(3): 31–39, 1999