Improved accuracy in the singularity spectrum of multifractal chaotic time series

Existing algorithms for accurately estimating the f(α) singularity spectrum from the samples of generalized dimensions D_q of a multifractal chaotic time series use either linear interpolation of the known D_q values or finely sample the D _q curve. Also, the derivative in the expression for the Legendre transform necessary to go from D_q to f(α) is approximated using first order centered difference equation. Finely sampling the D_q is computationally intensive and the crude linear approximations to interpolation and differentiation give erroneous end points in the f(α) curve. We propose using standard min-max filter design methods to more accurately interpolate between known samples of the D_q values and evaluate the Legendre transform. We use optimum (min-max) interpolators and differentiators designed with the Parks-McClellan (1972) algorithm. The new min-max approach exhibits a computational reduction and an improved accuracy. Examples are provided that show improved accuracy for attractors that contain multifractal behaviour