Local and global analysis of multifractal singularity spectrum through wavelets

This paper presents the implementation aspects pertinent to the computation of the multifractal singularity spectrum through wavelets, and the methods of overcoming them. Multifractals are mixtures of monofractals, and monofractals are self-affine objects that hold power law relationships over several scales. Such multifractals can be detected and measured through a singularity spectrum. Many natural and artificial phenomena such as turbulence, diffusion limited aggregates, and electrical discharges exhibit multifractality. Since these phenomena are highly nonlinear and nonstationary, regular analyses such as Fourier decomposition cannot characterize them effectively. In order to characterize, compare and quantify multifractal objects, appropriate measures such as the Renyi fractal dimension spectrum (RS) and Mandelbrot singularity spectrum (MS) are required. There are two major methods for calculating a singularity spectrum; one is through Legendre transform of the RS, and the other is through the wavelet transform modulus maxima (WTMM) method. This paper provides solutions to the difficulties that arise in the computation of the MS through WTMM, for one-dimensional signals and compares them to the existing multifractal literature and software implementations. Appropriate mother wavelets, continuous wavelet implementation, and thresholding of the wavelet coefficients are also discussed