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

function result = fit_maxwell_pdf( x,y,W,hAx ) 
% 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( x,y,W,hAx ) 
% 
% input:    y   - vector, samples of the histogram to be fitted 
%           x   - vector, position of the samples of the histogram (i.e. y = f(x,a)) 
%           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(x,a) = 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(x,a) 
%             H = diff( p(x,a) ) 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(W,1)==length(y)) & (size(W,2)==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*a^2) - 3/(2*a) ); 
        HTW     = H'*W; 
        e       = inv( HTW * H ) * HTW * (y-h); 
        a       = a + e; 
        control = e*e; 
        if ( control > (last_cnt * thresh) ) 
            break; 
        else 
            last_cnt = control; 
        end 
    end 
 
    % summarize results 
    h           = C*(a^(-1.5)).*exp(-x2/(2*a)); 
    H           = h.*( x2/(2*a^2) - 3/(2*a) ); 
    HTW         = H'*W; 
    G           = inv( HTW * H ) * HTW; 
    err         = ( y - h ); 
    result.a    = a; 
    result.VAR  = G * var( err ) * (G'); 
    result.RMS  = sqrt( (err')*err/ (x(2)-x(1))^2 / (length(err)-1) ); 
    result.iter = iter; 
    result.type = type; 
else 
 
    % loop for convergence (without a weighting matrix) - assume white noise 
    % ======================================================================== 
    while (1) 
        iter    = iter + 1; 
        h       = C*(a^(-1.5)).*exp(-x2/(2*a)); 
        H       = h.*( x2/(2*a^2) - 3/(2*a) ); 
        HT      = H'; 
        control = inv( HT * H ); 
        a       = a + control * HT * (y-h); 
        if ( control>(last_cnt * thresh) ) 
            break; 
        else 
            last_cnt = control; 
        end 
    end 
     
	% summarize results 
    h           = C*(a^(-1.5)).*exp(-x2/(2*a)); 
	H           = h.*( x2/(2*a^2) - 3/(2*a) ); 
    err         = ( y - h ); 
    result.a    = a; 
    result.VAR  = inv( (H') * H ); 
    result.RMS  = sqrt( (err')*err/ (x(2)-x(1))^2 / (length(err)-1) ); 
    result.iter = iter; 
    result.type = type; 
end 
 
 
% plot distribution if asked for 
% =============================== 
if (nargin>3) 
    if ishandle( hAx ) 
        plot_maxwell( x,result,hAx,2 ); 
    else 
        figure; 
        plot_maxwell( x,result,gca,2 ); 
    end 
end