Solved – How to reverse factor analysis (FA) and reconstruct original variables

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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).

enter image description here

@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.

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Best Answer

@amoeba and @ttnphns have solved my problem in the comments. I posted the solution if someone is interested.

@amoeba:

Turns out, factoran implicitly standardizes all input variables and hence conducts FA on the correlation matrix (it's written in Help: "factoran standardizes the observed data X to zero mean and unit variance"). I could not find any input option that would turn off this behaviour. Hence, to do the "reconstruction", you need to compute stds = std(X); in the beginning and then to do Xfa = bsxfun(@times, Xfa, stds); after you multiplied scores by loadings and before adding the mean."

So the FA method corrected is:

X = input_data; [LoadingsPM,specVarPM,rotationPM,stats, scores] = ...                 factoran(X,2,'rotate','promax'); Xfa = scores*LoadingsPM';  Xfa = bsxfun(@times, Xfa, std(X)); Xfa = bsxfun(@plus, Xfa, mean(X)); `

enter image description here

To complete this post, I recommend you this nice explanation made by @ttnphns: What are the differences between Factor Analysis and Principal Component Analysis?

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