hurst指数matlab程序

下面是一份计算Hurst指数的MATLAB程序：

function [H, R, S] = hurst(X)
% HURST  Computes the Hurst exponent of time series data.
%
%   H = HURST(X) computes the Hurst exponent of a one-dimensional time
%   series X. The input data should be a vector of length N, where N is a
%   power of 2.
%
%   [H, R, S] = HURST(X) returns the rescaled range R and standard deviation S
%   used in the Hurst exponent calculation.
%
%   References:
%   [1] E. Peters, "Fractal Market Analysis: Applying Chaos Theory to
%       Investment and Economics," John Wiley & Sons, Inc., 1994.
%   [2] J. Feder, "Fractals," Plenum Press, 1988.
%   [3] B. B. Mandelbrot, "The Fractal Geometry of Nature," W. H. Freeman
%       and Company, 1983.
%
%   See also FBM, HFD, and IAAFT.


% Check for valid input
if ~isvector(X)
    error('Input data must be a vector');
end

% Compute range and standard deviation of data
R = max(X) - min(X);
S = std(X);

% Compute the cumulative deviation of the time series from the mean
Y = cumsum(X - mean(X));

% Initialize output variables
M = length(X);
H = zeros(log2(M)-4,1);

% Loop over scales
for k = 5:log2(M)

    % Number of segments and segment length
    n = 2^k;
    l = M/n;

    % Initialize rescaled range
    Rn = zeros(n,1);

    % Compute the range of each segment
    for i = 1:n
        idx = (i-1)*l + (1:l);
        Xn = Y(idx);
        Mn = mean(Xn);
        Rn(i) = max(Xn) - min(Xn);
    end

    % Compute rescaled range
    Rn = Rn./std(Y);

    % Compute Hurst exponent
    H(k-4) = log2(mean(Rn));
end

% Fit line to log2(R/S) versus log2(n) plot
p = polyfit(log2((5:2:length(H)+4)'),log2(H),1);
H = p(1);

该程序实现了计算Hurst指数的算法，使用的是累计离差序列法。

在程序中，输入数据X应该是一个长度为N的向量，其中N为2的幂次。程序首先计算输入数据的范围和标准差，然后计算累计离差序列Y，接着在不同的尺度下计算数据的范围，最后使用最小二乘法拟合得到Hurst指数。

使用该程序计算Hurst指数的示例代码如下：

% Generate fractional Brownian motion with H=0.6
X = fbm(1024, 0.6);

% Compute Hurst exponent
[H, R, S] = hurst(X);

% Display results
fprintf('Hurst exponent: %f\n', H);
plot(log2(5:2:length(H)+4),log2(H),'o',log2(5:2:length(H)+4),polyval(p,log2(5:2:length(H)+4)));
xlabel('log2(n)');
ylabel('log2(R/S)');
legend('Data','Fit');

该示例代码生成了一个具有Hurst指数为0.6的分数布朗运动，并使用hurts函数计算其Hurst指数。程序还绘制了log2(R/S)与log2(n)的图像，其中R和S是在计算Hurst指数时使用的范围和标准差。

注意，该程序使用的是最小二乘法拟合来计算Hurst指数，因此结果可能受到噪声或数据量的影响。因此，建议在使用该程序计算Hurst指数时，应在多个尺度下计算并取平均值，以获得更加准确的结果。