（分形）布朗运动的Matlab生成代码

DoubleX

本人是研究分形理论的，看到论坛上有人求助如何生成布朗运动，特贡献出一段代码。

生成（分形）布朗运动的近似方法主要有随机中点位移法（RMD），快速付立叶换（FFT），后者精度很高且性能好。我所写代码是paxson论文中的S语言代码翻译过来的，希望对大家有所帮助。


***********************刚才传txt文件说附件内容非法，只好贴在下面***************************



function similar_sequence = generator_FFT(n,H)
%--------------------------------------------------------------------------
% GENERATOR_FFT Use fast fourier transform to generate normalized FGN
%     and FBM. Then use Norrs method to generate normalized
%     similar_sequence. Finally, the average of similar_sequence was set to
%     1 through normaliztion.
%  
%     Note:
%     1. The input argument n is the number of point of sequence. It must
%     be even. H is the objective similarity you want.
%     2. The output argument similar_sequence is a similar_sequence with
%     average equal to 1. The FGN and FBM are normalized FGN and FBM
%     respectively.
%     3. This routine is a matlab version of paxson's R routine. For more
%     details, see "Fast, approximate synthesis of fractional Gaussian
%     noise for generating self-similar network traffic"
%--------------------------------------------------------------------------

%--------------------------------------------------------------------------
%
%    generator_FFT
%    Edit by Chu Chen, 07/07/2007
%    Should you have any suggestion for improving the code, please contact:
%    chuch@scut.edu.cn.
%--------------------------------------------------------------------------

DoubleX

if mod(n,2) ~= 0
    error('The input argument "n" must be even');
else
    % Returns a Fourier-generated sample path of a "self similar" process  
    % Consisting of n points(n should be even) and Hurst paramenter H  
    n = n/2;
    lambda = [1:n]*pi/n;

    % Approxiamte ideal power spectrum.
    f = FGNspectrum(lambda,H);

    % Adjust for estimating power spectrum via periodogram
    f = f.*exprnd(1,1,n);

    % Construct corresponding complex numbers with randm phase
    alpha = 2*pi.*unifrnd(0,1,1,n);
    a = sqrt(f).*cos(alpha);
    b = sqrt(f).*sin(alpha);
    z = complex(a,b);

    % Last element should have zero phase
    z(n) = abs(z(n));

    % Expand z to correspond to a Fourier transform of a real-valued signal.
    zprime = [0,z,conj(fliplr(z(1:n-1)))];

    % Inverse FFT gives sample path.
    FGN = real(ifft(zprime));
     
    % Standardize FGN and create FBM.
    FGN = (FGN-mean(FGN))/std(FGN);
    FBM = cumsum(FGN);

    % Use Norrs method to generate normalized similar_sequence
    similar_sequence = FGN;
     
    % M = 30;
    % a = 5;
    % similar_sequence = M + sqrt(a*M)*similar_sequence;
    % similar_sequence = max(0,similar_sequence);
    % similar_sequence = similar_sequence*2*n/sum(similar_sequence);
end;


%----------------------------subfunction1----------------------------------
function f = FGNspectrum(lambda,H)
% Returns an approximation of the power spectrum of FGN at the given
% frequencies lambda and the given Hurst parameter H.
f = 2*sin(pi*H)*gamma(2*H+1).*(1-cos(lambda)).*(lambda.^(-2*H-1) + FGNest(lambda,H));


%----------------------------subfunction2----------------------------------
function est = FGNest(lambda,H)
% Returns the estimate for B(lambda,H).
d = -2*H-1;
dprime = -2*H;
a1 = 2*1*pi+lambda;
b1 = 2*1*pi-lambda;
a2 = 2*2*pi+lambda;
b2 = 2*2*pi-lambda;
a3 = 2*3*pi+lambda;
b3 = 2*3*pi-lambda;
a4 = 2*4*pi+lambda;
b4 = 2*4*pi-lambda;
est = a1.^d + b1.^d + a2.^d + b2.^d + a3.^d + b3.^d + (a3.^dprime+b3.^dprime+a4.^dprime+b4.^dprime)/(8*pi*H)
本帖转自52mc论坛