拟合函数--高斯，拉普拉斯，双高斯拟合

% This folder contains a collection of "fitting" functions. % (Some has demo options - the third section) % The GENERAL input to the functions should be samples of the distribution. % % for example, if we are to fit a normal distribution ('gaussian') with a mean "u" and varaince "sig"^2 % then the samples will distribute like: % samples = randn(1,10000)*sig + u % %fitting with Least-Squares is done on the histogram of the samples. % fitting with Maximum likelihood is done directly on the samples. % % % Contents of this folder % ======================= % 1) Maximum likelihood estimators % 2) Least squares estimators % 3) EM algorithm for estimation of multivariant gaussian distribution (mixed gaussians) % 4) added folders: Create - which create samples for the EM algorithm test % Plot - used to plot each of the distributions (parametric plot) % % % % % % Maximum likelihood estimators % ============================= % fit_ML_maxwell - fit maxwellian distribution % fit_ML_rayleigh - fit rayleigh distribution % (which is for example: sqrt(abs(randn)^2+abs(randn)^2)) % fit_ML_laplace - fit laplace distribution % fit_ML_log_normal- fit log-normal distribution % fit_ML_normal - fit normal (gaussian) distribution % % NOTE: all estimators are efficient estimators. for this reason, the distribution % might be written in a different way, for example, the "Rayleigh" distribution % is given with a parameter "s" and not "s^2". % % % least squares estimators % ========================= % fit_maxwell_pdf - fits a given curve of a maxwellian distribution % fit_rayleigh_pdf - fits a given curve of a rayleigh distribution % % NOTE: these fit function are used on a histogram output which is like a sampled % distribution function. the given curve MUST be normalized, since the estimator % is trying to fit a normalized distribution function. % % % % % Multivariant Gaussian distribution % ================================== % for demo of 1

function result = fit_maxwell_pdf（ xyWhAx ）
% fit_maxwell_pdf - Non Linear Least Squares fit of the maxwellian distribution.
%                   given the samples of the histogram of the samples finds the 
%                   distribution parameter that fits the histogram samples.
%
%    fits data to the probability of the form: 
%        p（r） = sqrt（2/pi）*（a^（-3/2））*（r^2）*exp（-（r^2）/（2*a））
%    with parameter: a
%
% format:   result = fit_maxwell_pdf（ xyWhAx ）
%
% input:    y   - vector samples of the histogram to be fitted
%           x   - vector position of the samples of the histogram （i.e. y = f（xa））
%           W   - matrix or scalar a square weighting matrix of the size NxN where
%                 N = length（y） or 0 to indicate no weighting is needed.
%           hAx - handle of an axis on which the fitted distribution is plotted
%                 if h is given empty a figure is created.
%
% output:   result  - structure with the fields
%                      a   - fitted parameter
%                      VAR - variance of the estimation
%                      type- weighted LS or not weighted LS
%                      iter- number of iteration for the solution
%

%
% Algorithm
% ===========
%
% We use the WLS algorithm to estimate the PDF from the samples.%
% The maxwell distribution is given by:
%
%    p（xa） = sqrt（2/pi）*（a^（-3/2））*（x.^2）.*exp（-（x.^2）/（2*a））
%           = Const * （a^（-3/2）） .* exp（-（x.^2）/（2*a））
%
%    note that X is known and therefore considered a constant vector
%
% The non liner WLS estimator is given by:
%
%    a（n+1） = a（n） + inv（H‘*W*H）*（H‘） * （y-h） = a（n） + G * err
% 
%    where:   h = p（xa）
%             H = diff（ p（xa） ） with respect to “a“
%             W = weighting matrix of size NxN  （N = length（y））
%             a = a single parameter to be estimated
%
% The error estimation is given by:
%
%    VAR（ a ） = G * VAR（ err ） * （G‘）
%
%       or when W=I and the noise is a gaussian noise 
%
%    VAR（ a ） = inv（ H‘ * H ）
%


if （nargin<3）
    error（ ‘fit_maxwell_pdf - insufficient input arguments‘ ）;
end

a       = x（find（y==max（y）））^2;         % initial guess
y       = y（:）;                         % both should be column vectors !
x       = x（:）;
x2      = x.^2;                         % save computation time
C       = sqrt（2/pi）*x2;                % a constant vector
thresh  = 0.995;                        % convergence threshold for the loop
last_cnt= inf;
iter    = 0;

% check weight matrix input
if （size（W1）==length（y）） & （size（W2）==length（y））
    weights_flag    = 1;
    type            = ‘WLS‘;
else
    weights_flag    = 0;
    type            = ‘LS‘;
end


% Estimation
% =============
if （weights_flag）
    % loop for convergence （with weighting matrix）
    % =============================================
    while （1）
        iter    = iter + 1;
        h       = C*（a^（-1.5））.*exp（-x2/（2*a））;
        H       = h.*（ x2/（2*

 属性            大小     日期    时间   名称
----------- ---------  ---------- -----  ----
     目录           0  2009-05-25 05:13  FitFunc\
     目录           0  2004-04-28 13:53  FitFunc\Create\
     文件        1627  2003-11-26 15:15  FitFunc\Create\build_mix_2D_gaussian.m
     文件        1068  2003-11-24 16:53  FitFunc\Create\build_mix_gaussian.m
     文件        4798  2003-11-16 21:49  FitFunc\fit_maxwell_pdf.m
     文件        4276  2003-11-26 15:17  FitFunc\fit_mix_2D_gaussian.m
     文件        2842  2003-11-30 19:10  FitFunc\fit_mix_gaussian.m
     文件        3194  2003-11-17 21:28  FitFunc\fit_ML_laplace.m
     文件        3430  2003-11-17 21:02  FitFunc\fit_ML_log_normal.m
     文件        3017  2003-11-16 21:23  FitFunc\fit_ML_maxwell.m
     文件        3606  2004-04-28 14:16  FitFunc\fit_ML_normal.m
     文件        2867  2003-11-16 21:24  FitFunc\fit_ML_rayleigh.m
     文件        4436  2003-11-16 21:25  FitFunc\fit_rayleigh_pdf.m
     目录           0  2004-04-28 13:53  FitFunc\Plot\
     文件        2088  2003-11-17 21:28  FitFunc\Plot\plot_laplace.m
     文件        2257  2003-11-17 20:04  FitFunc\Plot\plot_log_normal.m
     文件        2105  2003-11-16 23:48  FitFunc\Plot\plot_maxwell.m
     文件        4525  2003-11-26 15:15  FitFunc\Plot\plot_mix_gaussian.m
     文件        2333  2004-04-28 14:11  FitFunc\Plot\plot_normal.m
     文件        2032  2003-11-16 23:48  FitFunc\Plot\plot_rayleigh.m
     文件        2356  2003-12-06 01:42  FitFunc\readme.m
     文件        1329  2009-05-25 05:13  license.txt