PLS weights are not orthagonal

When I compute PLS weights using matrix operations, the weights are orthagonal to each other.

When I compute them using plsregress they are not.

See example code below:

load spectra
X = NIR;
Y = octane;

% Compute first two weights using matrix operations
X1 = X - mean(X, 1)
Y1 = Y - mean(Y, 1)
w1 = X1'*Y1/(norm(X1'*Y1));
t1 = X1*w1;
c1 = t1'*Y1/(t1'*t1);
p1 = X1'*t1/(t1'*t1);

X2 = X1 - t1*p1';
Y2 = Y1 - t1*c1;

w2 = X2'*Y2/(norm(X2'*Y2));
t2 = X2*w2;
c2 = t2'*Y2/(t2'*t2);
p2 = X2'*t2/(t2'*t2);

W = [w1, w2];
W = W./vecnorm(W); % normalize

% Compute first two weights using plsregress
ncomp = 2;
[XL, YL, XS, YS, BETA, PercentVar, ~, stats] = plsregress(X, Y, ncomp);
W_mat = stats.W./vecnorm(stats.W)

% compare weights (first weights are identical, second are not)
[W, W_mat]

The dot product of two orthagonal vectors is 0. Checking the weights for orthagonality:

dot(W(:,1), W(:,2))

    ans = 
     -5.6812e-17

dot(W_mat(:,1), W_mat(:,2))

    ans = 
       0.4488

Shouldn't the weights be orthagonal, or am I missing something? Can anyone explain what is going on here?

[Arthur Ryzak]

You are using the NIPALS algorithm and Matlab plsregress uses the SIMPLS algorithm. These two algorithms compute the weights in different ways. I do not believe the weights in SIMPLS are supposed to be orthogonal.

Quoted here is a very relevant excerpt from a webpage that explains some of the differences: "So the somewhat odd thing about the weight vectors w derived from NIPALS is that each one applies to a different X, i.e. tn+1 = Xnwn+1. This is in contrast to Sijmen de Jong’s SIMPLS algorithm introduced in 1993 [3]. In SIMPLS a set of weights, sometimes referred to as R, is calculated, which operate on the original X data to calculate the scores. Thus, all the scores T can be calculated directly from X without deflation, T = XR. de Jong showed that it is easy to calculate the SIMPLS R from the NIPALS W and P, R = W(P’W)-1. (Unfortunately, I have, as yet, been unable to come up with a simple expression for calculating the NIPALS W from the SIMPLS model parameters.) "

S. de Jong, “SIMPLS: an alternative approach to partial least squares regression,” Chemo. and Intell. Lab. Sys., Vol. 18, 251-263, 1993.

https://eigenvector.com/different-kinds-of-pls-weights-loadings-and-what-to-look-at/

[Atticus Beachy]

Thanks! That clarifies things.