www.pudn.com > FitFunc.zip > fit_ML_normal.m

function result = fit_ML_normal( x,hAx ) 
% fit_ML_normal - Maximum Likelihood fit of the normal distribution of i.i.d. samples!. 
%                  Given the samples of a normal distribution, the PDF parameter is found 
% 
%    fits data to the probability of the form:  
%        p(r) = sqrt(1/2/pi/sig^2)*exp(-((r-u)^2)/(2*sig^2)) 
%    with parameters: u,sig^2 
% 
% format:   result = fit_ML_normal( x,hAx ) 
% 
% input:    x   - vector, samples with normal distribution to be parameterized 
%           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 
%                      sig^2,u          - fitted parameters 
%                      CRB_sig2,CRB_u   - Cram?r-Rao Bound for the estimator value 
%                      RMS              - RMS error of the estimation  
%                      type             - 'ML' 
% 
% example:  fit_ML_normal( randn(1,10000)*3 - 1 ) 
%               or 
%           figure;  
%           fit_ML_normal( randn(1,10000)*3 - 1,gca ); 
% 
 
% 
% Algorithm 
% =========== 
% 
% We use the ML algorithm to estimate the PDF from the samples. 
% The normal destribution is given by: 
% 
%    p(x;u,sig^2) = sqrt(1/2/pi/sig^2)*exp(-((x-u).^2)/(2*sig^2)) 
% 
%    where x are the samples which distribute by the function p(x;u,sig^2) 
%            and are assumed to be i.i.d !!! 
% 
% The ML estimator is given by: 
% 
%    a         = parameters vector = [u,sig^2] 
%    f(Xn,a)   = sqrt(1/2/pi/sig^2)*exp(-((Xn-u).^2)/(2*sig^2)) 
%    L(a)      = f(X,a) = product_by_n( f(Xn,a) ) 
%              = (2*pi*sig^2)^(-N/2)*exp(-sum((Xn-u)^2)/(2*sig^2)) 
%    log(L(a)) = -N/2*log(2*pi) - N/2*log(sig^2) - sum((Xn-u)^2)/(2*sig^2) 
% 
%    The maximum likelihood point is found by the derivative of log(L(a)) with respect to "a": 
% 
%    diff(log(L(a)),u) = -2*sum(Xn-u)/(2*sig^2) = N/sig^2 * ( u - sum(Xn)/N ) 
%                    = J(u) * (u_estimation - u)  
%    diff(log(L(a)),sig^2) = -N/(2*sig^2) + sum((Xn-u)^2)/(2*sig^4) 
%                          = N/(2*sig^4) * ( sum((Xn-u)^2)/N - sig^2 )
%                          = J(sig^2) * (sig^2_estimator - sig^2)
%
%    Therefore, the (efficient) estimators are given by: 
% 
%               u     = sum(Xn)/N 
%               sig^2 = sum((Xn-u)^2)/N
% 
%    The Cram?r-Rao Bounds for these estimator are: 
% 
%               VAR( u )     = 1/J(u)     = (sig^2) / N 
%               VAR( sig^2 ) = 1/J(sig^2) = (2*sig^4) / N 
% 
%    NOTE: the ML estimator does not detect a deviation from the model. 
%          therefore, check the RMS value ! 
% 
 
if (nargin<1) 
    error( 'fit_ML_normal - insufficient input arguments' ); 
end 
 
% Estimation 
% ============= 
x       = x(:);                 % should be column vectors ! 
N       = length(x); 
u       = sum(x)/N; 
sig2    = (x-u)'*(x-u)/N; 
CRB_u   = sig2 / N; 
CRB_sig2= (2*sig2^2) / N; 
[n,x_c] = hist( x,100 ); 
n       = n / sum(n*abs(x_c(2)-x_c(1))); 
y       = sqrt(1/2/pi/sig2)*exp(-((x_c-u).^2)/(2*sig2)); 
RMS     = sqrt( (y-n)*((y-n)')/ (x_c(2)-x_c(1))^2 / (length(x_c)-1) ); 
 
% finish summarizing results 
% ============================ 
result = struct( 'u',u,'sig2',sig2,'CRB_u',CRB_u,'CRB_sig2',CRB_sig2,'RMS',RMS,'type','ML' ); 
 
% plot distribution if asked for 
% =============================== 
if (nargin>1) 
    xspan = linspace(min(x),max(x),100); 
    if ishandle( hAx ) 
        plot_normal( xspan,result,hAx,1 ); 
    else 
        figure; 
        plot_normal( xspan,result,gca,1 ); 
    end 
end