calc_qt_limits.m

function [tlim,qlim] = calc_qt_limits(E,comp,nobj)

% calc Q and T2 limits of confidence for PCA and PLSDA model
% requires statistics_toolbox
% 
% This is an internal routine of the toolbox.
% The main routine to open the graphical interface is class_gui
%
% HELP:
% note that a detailed HTML help is provided with the toolbox,
% see the HTML HELP files (help.htm) for futher details and examples
%
% LICENCE:
% This toolbox is distributed with an Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence: https://creativecommons.org/licenses/by-nc-nd/4.0/
% You are free to share - copy and redistribute the material in any medium or format. The licensor cannot revoke these freedoms as long as you follow the following license terms:
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% REFERENCE:
% The toolbox is freeware and may be used if proper reference is given to the authors, preferably refer to the following paper:
% Ballabio D, Consonni V, (2013) Classification tools in chemistry. Part 1: Linear models. PLS-DA. Analytical Methods, 5, 3790-3798
% 
% Classification toolbox for MATLAB
% version 5.3 - May 2019
% Davide Ballabio
% Milano Chemometrics and QSAR Research Group
% http://www.michem.unimib.it/

% T2 limit
lev_conf = 0.95;
if license('test','statistics_toolbox')
    F = finv(lev_conf,comp,nobj-comp);
    tlim = comp*(nobj - 1)/(nobj - comp)*F;
else
    tlim = NaN;
end

% Q limit
t1 = sum(E(comp+1:end).^1);
t2 = sum(E(comp+1:end).^2);
t3 = sum(E(comp+1:end).^3);
ho = 1 - (2*t1*t3)/(3*t2^2);
ca = norminv(0.95, 0, 1);
term1 = (ho*ca*(2*t2)^0.5)/t1;
term2 = (t2*ho*(ho - 1))/(t1^2);
qlim = t1*(term1 + 1 + term2)^(1/ho);