Generate a 2-D fractional Brownian motion walk using the the Davies-Harte algorithm. (see FGP, model 346 for details).
Model number: 0374
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The Davies-Harte algorithm is used to generate fractional Gaussian noise (fGn) and fractional Brownian Motion (fBm). The algorithm is described in model #346 and is named FGP (Fractional Gaussian Process). Two copies of the algorithm are employed here, one for the x increments and one for the y increments. Two copies were employed so that the X and Y increments would be independent of each other. The relationship between fBm and fGn is given by m ----- \ fBm(m) = ) fGn(j), for m=0 to N. / ----- j = 0 The JSim project contains the following plots run with the default parameter set DBM2D: The 2-D plot of fractional Brownian Motion, FBMX_and_FBYM: Plots of the x-traces and y-traces for the 1-D fractional Brownian Motions for X and Y and the fractional Gaussian noise (fGn) increments that are summed to produce the fBm. GaussianIncrements: The fGn series are sorted in ascending order and plotted as (m/m.max, sorted fGn(m)) where m runs from 1 to m.max. This demonstrates that the increments are Gaussian. For Hurst coefficients > 0.5, each individual realization can depart from Gaussian. There are two nested plot plots run with the correlationfGnRunLoops parameter set. The user should load that parameter set and run loops. The two plots are: fGn_correl: Shows that for Hurst coefficients <0.5, adjacent points are likely to be negatively correlated, while for Hurst coefficients >0.5, adjacent points are more likely to be positively correlated. fGnx_fGny_Xcorrel: Shows that for an individual realization, the fGnx and the fGny are independent of each other. CAVEAT: Model is slow for Npoints >2,000 as the Fourier transforms used here are not the complex Fast Fourier Transforms. The code utilizes discrete cosine and sine transforms and treats all quantities as real variables because JSim does not support complex variables and the FFT cannot be written as a single simple equation.