# Interesting Way To Implement Factor Analysis From PCA

Is it possible to modify the PCA algorithm so that it actually implements factor analysis? We can assume that the uniquenesses are known.

I'm aware that for a $$d$$-dimensional data $$x$$, PCA takes the leading $$k eigenvectors of $$\Sigma_X$$ and uses these to define the principal components.

In factor analysis models this data as $$x=Af + \epsilon$$ where $$f$$ is a matrix of factors, $$A$$ is the loading matrix between the latent factors and the observable data, and $$\epsilon$$ is the vector of uniquenesses.

I have read (see slide 18 of link) that we should construct $$A$$ as the leading $$k$$ eigenvectors of $$\Sigma_X - \psi$$ where $$\psi=\text{diag}(\psi_{11}, \dots , \psi_{dd})$$ is the covariance of the $$\epsilon$$ vector.

Is this true? What is the explanation? The slides are very superficial...

• The term factor analysis is used in two ways, one that includes techniques such as PCA and a more narrow technical definition that excludes methods such as PCA. If you search on this site for difference between PCA and factor analysis you might find the answers illuminating. – ReneBt May 2 at 13:56
• Hi. Thanks for the reply. Yes, I have read several of them but they mostly seem to focus on differences in interpretation and application. I am interested in the specific mathematics of how pca can be adapted to perform factor analysis. I didn't find any threads on that - are you able to offer any insight? Thanks. – user11128 May 2 at 16:28