Back to PLS HelpPosted on 06/24/15 15:44:38
Number of posts: 2
Dear All,
Currently I am using command line PLS (correlation PLS) to analyze some EEG and fMRI data. I am new to this method and I need to calculate the percentage of covariance (between task design contrast and brain data) explained by each latent variable. I searched this forrum and found that Jimmy suggested that this could be calculated as S.^2/sum(S.^2). where S is the singular value diagonal matrix (https://www.rotman-baycrest.on.ca/index.php?action=view_thread&id=1239&module=bbmodule&src=%40random45c35fcb17881)
However, after I read Krishnan et al. (2011; Neuroimage), I wonder whether we could just use S(i) /sum(S(i)), without the squaring, for this purpose. I am not sure whether I understand this correctly, but it seems that the singular values are already the covariance of the behavior and brain latent variables. For example, following Krishnan et al. (2011; Neuroimage), if Y is behavioral data matrix, X is brain data, and the covariance matrix of Y and X is R = Y'X (' denote transpose), after singlular value decompose, R = USV'. Then we can have the latent variables Ly = YU, and Lx = XV. The covariance between Ly and Lx is (YU)'(XV) = U'Y'XV = U'RV = U'USV'V = S (since U'U = V'V = I). If this is correct, this seems to show that singlular values are already the covariance between the two sets of latent varialbes. If so, to obtain the percentage of explained covariance for each latent variable, would it be fine to just use each singular value devided by the sum of all singlular values?
Because the square of singular value S is the eigen value of the R'R, S./sum(S.^2) seems to reflect the percentage of variance and covariance of matrix R (which itself is the covariance of the behaivoral/design and brain variables) explained by each latent variable.
I somehow feel that the former one, i.e., directly using singluar value without square to calculate explained covariance, is easier to understand. But very likely I may misundstand this method, and I'd like to thank you very much in advance for your time and help.
Zhongxu
Replies:
Posted on 06/27/15 10:45:35
Number of posts: 394
Thanks for the posting. Technically speaking the quantity that is calculated in PLS (s^2/sum(s^2)) is the percent sums of squared cross-block covariance (SSCB), which we explained in the McIntosh, et al 1996 paper. Over the years its has been shortened and unfortunately mislabelled at times as percent covariance (%cov), which as you point out would be better calculated at s/sum(s)
However, in practice neither metric is absolute in the sense that it is not comparable to percent total variance (e.g., R^2) as the SSCB or %cov depends entirely on the dimensions of the matrix you decompose in PLS. Concretely, if you have only one vector, your SSCB or %cov will be 100. As such these measures should only be used as a relative index for Latent variables within an analysis.
Posted on 06/28/15 10:11:39
Number of posts: 2
Dear Randy,
Thank you very much for your explanantion, which is really helpful.
Zhongxu
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