Hi Daniel!

There is a lot of variants of MDS, and the one implemented here is Principal Coordinates Analysis. So, It is exactly identical to PCA (up to the sign of eigenvectors), but since you cannot have an access to the loadings of the variable, results are less understandable than those from PCA. So, PCA is best, excepted if one use a special proximity measure (the Rao distance between distributions, the ManhattanDistance between vectors, etc). In such cases, the original table of data is no more considered, but there is NO guarantee to fit an Euclidean structure to the data. For instance, the Torgerson matrix may have large negative eigenvalues, which shows that data don't have an euclidean image - as an example, the Rao's distance is Riemannian.

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.

Thank you very much. I have seen that post and some similar ones. Unfortunately, that post illustrates the problem in that it talks about manually reproducing the effect of the PrincipalComponentAnalysis function rather than pointing to any built-in Mathematica functionality, and in any case it doesn't answer my specific question about getting the coefficients. I know that I can work out the coefficients from first principles. However, I would have thought that there ought to be a function in Mathematica for doing this. The closest solution I have seen is in this MSE post, which suggests using the KarhunenLoeveDecomposition function because that does return the transformed axes as vectors expressed in the original basis, though it also requires some manual manipulation of the data to get it zero-centred before applying the function.

I became interested in this because a friend was doing a PCA on some data in R and I thought I would like to reproduce what he did in Mathematica. At first I thought great, when I saw that there is a PrincipalComponentAnalysis function, but then I could not find any simple way to get the coefficients, which my friend gets easily and automatically in R. I am surprised that my options are either to use a different, more general purpose function, KarhunenLoeveDecomposition, which is not ideal for this application, or to program it manually from first principles. Nevertheless, it seems that must be the case.