I am not sure if I understand what uncorrelation means, but there are lots of statistics that measure association that are not dependent on the relationship being linear or even monotonic. There is some debate about the different measure. Might be worth examining the package matie: https://cran.r-project.org/web/packages/matie/matie.pdf

Sets of scores can be dependent without requiring a specific (e.g., linear) relationship among them.

Hi Daniel:

Thank you very much for your answer. I´ll check the link you sent me. I will try to explain better the situation. We have conducted a research were we have extracted underlying systematic risk factor from the returns on equities of the Mexican Stock Exchange. For this purpose we have carried on:

1) Principal Component Analysis (PCA)

2) Factor Analysis.(FA)

3) Independent Component Analysis. (ICA)

4) Neural Networks Pricipal Component Analysis. (NNPCA)

The latent factors extracted in each technique has the following statistical attributes:

1) PCA: linearly uncorrelated factors.

2) FA: common linearly uncorrelated factors.

3) ICA: statistically independent factors.

4) NNPCA: nonlinearly uncorrelated factors.

We understant that the factors obtained in ICA for example present a better statistical attribute since statistically Independence imply linearly uncorrelation.

However now we are not sure if for example the statistical attribute of the factors obtained by way of NNPCA (nonlinear correlation) are better or superior to that obtained in ICA (statistical Independence).

Best regards,