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

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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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