Faq how are Q residuals and Hotellings T2 calculated in PLS models

Issue:

How are the Q-residuals and Hotelling's T² values calculated for PLS models?

Possible Solutions:

Q and T² are done in PLS exactly the as they are for PCA (see the PLS_Toolbox manual for a description) except that the basis that is being used for T² is the PLS loads rather than the PCA loads. For Q, we actually calculate the eigenvalues of the residual subspace and it is exactly the same as in PCA.

For T², there is an approximation made of the eigenvalues. If Tcal is the column vectors of scores from your calibration model (extract from the model.loads{1,1} field) :

f = diag(Tcal'*Tcal)/size(Tcal,1);
f = sqrt(1./f);

OR you can use:

f = sum(Tcal.^2)./size(Tcal,1);
f = sqrt(1./f);

Either method gives you f, a normalized vector of lengths of each of your original scores. Next, you take each of those lengths and divide your new scores (Tnew) by the corresponding length: (again, here are two methods of doing it)

Method 1:

T2 = sum( (Tnew * diag(f)).^2 ,2);

Method 2:

for j=1:length(f);
 Tnew(:,j) = Tnew(:,j)*f(j);
end
 T2 = sum( Tnew.^2 ,2);

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