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    %% Multifractal Random Walk Cumulants
% Compute the cumulants for a multifractal random walk. The
% multifractal random walk is a realization of a random process with a
% theoretical first cumulant of 0.75 and a second cumulant –0.05.
% The second cumulant value of –0.05 indicates that the scaling
% exponents deviate from a linear function with slope 0.75.
%%
% Load a random walk signal.
load mrw07505;
%%
% Obtain and display the first and second cumulants.
[~,~,cp,tauq] = dwtleader(mrw07505);
cp([1 2])
%%
% For monofractal processes, the scaling exponents are a linear function of
% the moments. Linearity is indicated by the second and third cumulants 
% being close to zero. In this case, the nonzero second cumulant indicates
% that the process is multifractal.
%%
% Plot the scalaing exponents for the _q_ th moments.
plot(-5:5,tauq,'bo--'); title('Estimated Scaling Exponents');
grid on
xlabel('qth Moments')
ylabel('\tau(q)')
%%
% The scaling exponents are a nonlinear function of of the moments.