AFFINE FRACTAL LEAST SQUARES REGRESSION MODEL

Abstract

This paper develops a method to find fractal curves to fit real data. With a formulation for fractal functions through a type of affine systems of iterative functional equations, we apply the procedure of minimizing the sum of square residuals that is used in the classical linear regression. We develop formulas for approximation and exact fractal functions for various fractal levels and number of equations. This method gives estimates for the parameters of the equations and corresponding functions, namely the fractal coefficients (vertical scaling factors) and directional coefficients. As a consequence, it is possible to upper estimate the Hausdorff dimension of the fitted curve. We provide worked examples, including estimating a fractal curve for a time series real data set of exchange rates between USD and EUR currencies.