I saw this interesting topic: How to reverse PCA and reconstruct original variables from several principal components? and a nice answer with a very useful example of Iris data in Matlab. I would like to do the same using factor analysis instead of PCA. I tried to make it with 'factoran' of Matlab with the help of @ttnphns and @amoeba but I don't obtain a good correlation between my reconstructed data and the original ones.
input_data (*data are EMG measurement from 6 arm muscles in order to identify synergies)
PCA method:
X = input_data;
mu = mean(X);
[eigenvectors, scores] = pca(X);
nComp = 2;
Xpca = scores(:,1:nComp) * eigenvectors(:,1:nComp)';
Xpca = bsxfun(@plus, Xpca, mu);
I obtain good correlation between them.
FA method:
X = input_data;
mu = mean(X);
[LoadingsPM,specVarPM,rotationPM,stats, scores] = ...
factoran(X,2,'rotate','promax');
Xfa = scores*LoadingsPM';
Xfa = bsxfun(@plus, Xfa, mu);
But in this case the correlations are bad. I don't know if I forget something? (I divided per 3 the FA reconstruction in order to see better the 3 curves).
@ttnphns note: word "reverse" in the title should be taken in the technical sense of computing variables as they are returned by the computed factors (their scores), - not in the theoretical sense (in which FA model is nothing but predicting variables by factors, so that there is no a "reverse" direction). In PCA, this prediction/direction indeed could be called "reverse" in a theoretical sense, too.
How to reverse factor analysis (FA) and reconstruct original variables
is incorrect. FA cannot be in reverse. It is PCA which can. FA reconstructs variables by factors - its only theoretical, model "direction". $\endgroup$ – ttnphns Sep 12 '17 at 9:25Whether to call it "reverse" or not...
Yes, of course, true, but it is trivial. It actually is doing with FA scores what we do with PCA scores. It is when we use F. scores in place of F. values in the factor model X_reconstr = F*Loadings. I just didn't think the OP is asking about that triviality. I thought they're asking about that reconstruction which is homologous, not superficially analogous, in FA to what we do in PCA. $\endgroup$ – ttnphns Sep 12 '17 at 10:39