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

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