Yes, that sounds like it would be useful to me. Thank you.

Having now had the chance to try the suggestion of @Sangdon Lee, I have found that SingularValueDecomposition works well subject to the following caveats:

1. To get the best result from SVD, it is necessary first to subtract the mean of each variable from its respective values, i.e. to translate the values so that they have zero mean. Otherwise, SVD does not return the same result as PCA and is not as successful in concentrating the variance into a few components.

2. With SVD, the signs of the transformed variables are in general different from those returned by PCA though the absolute values are the same. The sign is essentially arbitrary and this does not affect any conclusions from the analysis.

It seems that PrincipalComponents is a convenience method that automatically adjusts the variables to have zero mean before applying SVD. I now imagine that the reason there is no option to get a vector with the loadings from PrincipalComponents is because the transformed variables are not just a linear combination of the original variables but are related to them by a shift and then a linear combination. It would nevertheless be good if the PrincipalComponents documentation could include some discussion of this and of how to obtain the relationship between the original variables and the transformed variables if that is what one is interested in.

PCA is identical with SVD (Singular Value Decomposition). Therefore,

X=U*S*V'=T*V'= PC Scores * PC Loadings'.  (' is for Transpose).

The PrincipalComponentAnalysis function displays the PC Scores (T=U.S) only and does not provide the "V". Thus use the SingularValueDecomposition function. The "V" is the loadings you want to find.

For example,

  X = N[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}}]

  {U, S, V} = SingularValueDecomposition[X],

  U.S.Transpose[V] reproduces the original matrix (X).  

By the way, I wish that Wolfram would develop functions for PARAFAC.

I found Mathematica functions for Factor Analysis with Varimax rotation and PLS (partial least square) (don't remember the location. search in the Mathematica webpage). Anton Antonov developed ICA (independent component analysis): https://resources.wolframcloud.com/FunctionRepository/resources/IndependentComponentAnalysis/

Hi Daniel,

I believe you and I are discussing different aspects. Your comment pertains to the PC scores (T=US) (e.g., uu.ww in your comments), while my comment is about the PC loadings (V). The initial question was regarding the PrincipalComponentAnalysis function, which only displays the T and does not provide the V. The V shows how the original variables are transformed (as a linear combination of the original variables) and maybe interpreted afterward. The T values represent the outcomes of the transformation (the reduced vector): T=XV, where X=TV' and XV=TV'V=T, as V'V=I.

X=USV’=Left singular vector*Singular values*right singular vector’ = TV’ = PC scores*loadings.  

SVD is related to EVD as follows to clarify terminologies between SVD and EVD:

A=X’X=(USV’)’(USV’)=VS^2V’=eigenvectors*eigenvalues*eigenvectors’, because U’U=I and V’V=I. 

By the way, many dimension reduction methods apply SVD after deriving various “similarity” matrices. X’X represents a covariance or correlation matrix in PCA, depending on normalization or standardization applied to column variables. Experiences regarding the applications of PCA, MDS, and CA have reported producing similar results.

• PCA: X --> covariance or correlation matrix (X’X) --> apply SVD on the covariance matrix.
• Multidimensional Scaling (MDS) : X--> distance matrix (various distance matrices) --> apply SVD on the distance matrix
• Correspondence analysis (CA): X-->profile matrix -->apply SVD on the profile matrix

By the ways, thanks for the detail information about the DimensionReduction function. I am not well familiar with this function and your comments are useful to me.

Hi Daniel,

I posted another question on PCA after reading your comments carefully because I consider PCA as one of the most important methods.

https://community.wolfram.com/groups/-/m/t/2949003?p_p_auth=6pSCpXdg

Yes, mea culpa. In the original post, I referred to a PrincipalComponentsAnalysis function when I should have said PrincipalComponents function. Sorry for the confusion. It does seem that this is kind of an omission in WL. When you're interested in doing a PCA with Mathematica, you naturally reach for the PrincipalComponents function and yet it won't give you the loadings, which is, very often, what is of most interest, i.e. to identify which of your original variables is responsible for most of the variation in the data. While you can, it seems, extract that information with other methods as described by @Sangdon Lee , you don't really want to be reading through MSE or Wolfram Community threads to try and understand exactly how PCA works so that you can apply other more general purpose functions to your problem.

You're right. Thank you very much.

Thanks a lot for sharing! I frequently use your nice MonadicQuantileRegression workflow.