Wavelet-based determination of generating matrices for fractal interpolation functions

Fractal interpolation functions have become popular after the works of M.Barnsley and co-authors on iterated function systems (see, e.g., [5]). We consider here the following problem: given a set of values of a fractal interpolation function (FIF), determine the contractive affine mappings generating this function. The suggested solution is based on the observation that the fixed points of some of the affine mappings in question are among the points where the FIF has its strongest singularity. These points may be detected with the aid of wavelet-based techniques, such as modulus maxima lines tracing. After this is done, necessary matrices are computed from a system of linear equations. The method was tested numerically on FIFs with local Holder exponent as low as 0.3, and allowed to recover the generating matrices almost precisely. When applied to segments of financial time series, this approach gave FIFs reproducing some of the apparently chaotic patterns in the series. This suggests the potential usefulness of this techniques for detection of hidden rescaling parameters in the observed data.