QResidualsLimit (res, confidence)

%Function QResidualsLimit
%
%This function calculates the limit for Q residuals of a PCA decomposition.
%
%Syntax:
%   limit = QResidualsLimit (res, confidence)
%
%Input parameters:
%   res: matrix of residuals
%   confidence: confidence limit for the Q residuals, standard: 0.95 (=95%)
%
%Output parameters:
%   limit: the limit for the Q residuals, higher Q-residuals are considered
%      as outliers

%The Biodata toolbox for MATLAB: a spectral database system for storing and
%processing spectra
%C 2008, Kris De Gussem, Raman Spectroscopy Research Group, Department
%of analytical chemistry, Ghent University
%C 2009 Kris De Gussem
%
%This file is part of Biodata.
%
%Biodata is free software: you can redistribute it and/or modify
%it under the terms of the GNU General Public License as published by
%the Free Software Foundation, either version 3 of the License, or
%(at your option) any later version.
%
%Biodata is distributed in the hope that it will be useful,
%but WITHOUT ANY WARRANTY; without even the implied warranty of
%MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
%GNU General Public License for more details.
%
%You should have received a copy of the GNU General Public License
%along with Biodata.  If not, see <http://www.gnu.org/licenses/>.

%Copyright (c) 2008-2009, Kris De Gussem
%All rights reserved.
%
%Redistribution and use in source and binary forms, with or without 
%modification, are permitted provided that the following conditions are 
%met:
%
%    * Redistributions of source code must retain the above copyright 
%      notice, this list of conditions and the following disclaimer.
%    * Redistributions in binary form must reproduce the above copyright 
%      notice, this list of conditions and the following disclaimer in 
%      the documentation and/or other materials provided with the distribution
%    * Neither the name of Raman Spectroscopy Research Group, Department of
%	  analytical chemistry, Ghent University nor the names 
%      of its contributors may be used to endorse or promote products derived 
%      from this software without specific prior written permission.
%      
%THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 
%AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 
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%ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 
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function limit = QResidualsLimit (res, confidence)
switch nargin
    case 1
        confidence=0.95;
    case 2
end

if confidence > 1
    confidence = confidence/100;
end
confidence = 1-(1-confidence)*2;

si=size(res);

if (si(1) > 1) && (si(2) > 1)
    if si(1) > si(2)
        res = svd(res);
    else
        res = svd(res');
    end
    res = res.*res;
    res = res./(si(1)-1);
end

si=size(res);
if si(2)>si(1)
    res = res';
end

sum1 = res(:,1);
sum2 = sum1' * sum1; %speed optimised calculation of sum of squares
sum3 = sum1' * (sum1.*sum1); %speed optimised calculation of sum of third powers
sum1 = sum(sum1);

if sum1==0;
    limit = 0;
else
    tmp0 = 1-2*sum1*sum3/3/(sum2^2);
    if tmp0<1e-3
        tmp0 = 1e-3;
    end
    tmp1 = sqrt(2)*erfinv(confidence)*sqrt(2*sum2*tmp0^2)/sum1;
    tmp2 = sum2*tmp0*(tmp0-1)/(sum1^2);
    limit = sum1*(1+tmp1+tmp2)^(1/tmp0);
end