Generate both fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) series using the Davies-Harte algorithm.

Description

The Davies-Harte algorithm is used to generate fractional Gaussian noise (fGn) and
fractional Brownian Motion (fBm). The sequences can be saved for processing with the
other fractal programs.

The Davies-Harte algorithm is realized here with out resorting to complex
quantities: 

Let the autocorrelation sequence for fGn be given as

s(n) = (  |n+1|^2H - 2|n|^2H +|n-1|^2H  ) * var/2, where

n=0,1, ... N, 2H is twice the Hurst coefficient and var is the variance.

Define 
r(n)=(  |N-n+1|^2H-2*|N-n)^H2 +|N-n-1|^H2  )*(var/2) and
and

S(n2) = if(n2<=N) s(n2) else r(n2-N) for n2 = 0 ... 2N-1.

Create C1 from S using the discrete Fourier cosine transform:

                              2 N - 1
                               -----
                                \               n2 k PI
                      C1(k) =    )    S(n2) cos(-------)
                                /                  N
                               -----
                              n2 = 0

Let Vr and Vi be defined by
when(n2=n2.min)			{Vr= randomg()*sqrt(C1(0)*2*N); 
                		 Vi=0;				}
event( (1<=n2) and (n2<N) )	{Vr=randomg()*sqrt(C1(n2)*N);
                                 Vi=randomg()*sqrt(C1(n2)*N);	}
event( N=n2) 			{Vr=randomg()*sqrt(C1(N)*2*N);
                    		 Vi=0; 				}
event( (N<n2) and (n2<=n2.max) ) {Vr = Vr(n2.max+1-n2);
                                  Vi =-Vi(n2.max+1-n2); 	}
where randomg() is a Gaussian distributed random number with mean 0
and variance 1.

Define fgp as
             /2 N - 1                    \   /2 N - 1                    \
             | -----                     |   | -----                     |
             |  \                n2 k PI |   |  \                n2 k PI |
    fgp(k) = |   )    Vr(n2) cos(-------)| + |   )    Vi(n2) sin(-------)|
             |  /                   N    |   |  /                   N    |
             | -----                     |   | -----                     |
             \n2 = 0                     /   \n2 = 0                     /

for k=0 to N.

    fGn(m) = fgp(m), for m=0 to N.

The relationship between fBm and fGn is given by 

                                         m
                                      -----
                                       \
                             fBm(m) =   )   fGn(j),  for m=0 to N.
                                       /
                                      -----
                                      j = 0


The spectra for fGn is given by

SfGn=A/j^(2*Hurst-1) and for
fBm as
SfBm= B/j^(2*Hurst+1) where j is the frequency.

The periodograms of fGn and fBm are calculated. The fGn series is
sorted and used to calculate the cumulative Gaussian distribution.

CAVEAT: Model is slow for N >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. 

Equations

None.

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.