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A Collection of Fitting Functions

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A Collection of Fitting Functions

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05 Dec 2003 (Updated )

A collection of fitting functions for various distributions.

fit_mix_gaussian( X,M )
function [u,sig,t,iter] = fit_mix_gaussian( X,M )
%
% fit_mix_gaussian - fit parameters for a mixed-gaussian distribution using EM algorithm
%
% format:   [u,sig,t,iter] = fit_mix_gaussian( X,M )
%
% input:    X   - input samples, Nx1 vector
%           M   - number of gaussians which are assumed to compose the distribution
%
% output:   u   - fitted mean for each gaussian
%           sig - fitted standard deviation for each gaussian
%           t   - probability of each gaussian in the complete distribution
%           iter- number of iterations done by the function
%


% run with default values
if ~nargin
    M   = round(rand*5)+1;
    sig = rand(1,M)*3;
    u   = randn(1,M)*8;
    prob= rand(1,M);
    [u,sig,t,iter] = fit_mix_gaussian( build_mix_gaussian( u,sig,prob,2000*M ),M );
    return
end

% initialize and initial guesses
N           = length( X );
Z           = ones(N,M) * 1/M;                  % indicators vector
P           = zeros(N,M);                       % probabilities vector for each sample and each model
t           = ones(1,M) * 1/M;                  % distribution of the gaussian models in the samples
u           = linspace(min(X),max(X),M);        % mean vector
sig2        = ones(1,M) * var(X) / sqrt(M);     % variance vector
C           = 1/sqrt(2*pi);                     % just a constant
Ic          = ones(N,1);                        % - enable a row replication by the * operator
Ir          = ones(1,M);                        % - enable a column replication by the * operator
Q           = zeros(N,M);                       % user variable to determine when we have converged to a steady solution
thresh      = 1e-3;
step        = N;
last_step   = inf;
iter        = 0;
min_iter    = 10;

% main convergence loop, assume gaussians are 1D
while ((( abs((step/last_step)-1) > thresh) & (step>(N*eps)) ) | (iter<min_iter) ) 
    
    % E step
    % ========
    Q   = Z;
    P   = C ./ (Ic*sqrt(sig2)) .* exp( -((X*Ir - Ic*u).^2)./(2*Ic*sig2) );
    for m = 1:M
        Z(:,m)  = (P(:,m)*t(m))./(P*t(:));
    end
        
    % estimate convergence step size and update iteration number
    prog_text   = sprintf(repmat( '\b',1,(iter>0)*12+ceil(log10(iter+1)) ));
    iter        = iter + 1;
    last_step   = step * (1 + eps) + eps;
    step        = sum(sum(abs(Q-Z)));
    fprintf( '%s%d iterations\n',prog_text,iter );

    % M step
    % ========
    Zm              = sum(Z);               % sum each column
    Zm(find(Zm==0)) = eps;                  % avoid devision by zero
    u               = (X')*Z ./ Zm;
    sig2            = sum(((X*Ir - Ic*u).^2).*Z) ./ Zm;
    t               = Zm/N;
end

% plot the fitted distribution
% =============================
sig     = sqrt( sig2 );
plot_mix_gaussian( u,sig,t );

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