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PCA using the augmented correlation matrix #13
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It would be really cool to be able to do a PCA decomposition on the augmented or weighted correlation matrix generated by pairwise_correlations, so that the PCA actually reflects the augmented correlation directly.
There may be a way to do this via eigen and then generating the scores, keeping in mind that PCA on the correlation is already scaled and centered.
Note that I think we would have to set the diagonal to 1 for this to work properly.
Thoughts @hunter-moseley ??
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rmflight commentedon Jan 27, 2017
This could be tested by generating a correlation matrix for data with non-missing values, and verifying that the centered / scaled PCA results match those from the correlation matrix.
rmflight commentedon Jan 27, 2017
Possibly helpful posts:
hunter-moseley commentedon Jan 28, 2017
I think of this from the stand-point of embedding from a distance matrix. The correlation can be viewed as a normalized distance matrix and this is used to embed the rows/columns into an Euclidean space. Starting to understand the link you sent where the covariance matrix or correlation matrix shows dependency between variables which can be used to collapse the number of variables into principal components by calculating significant eigenvectors with large eigenvalues.
hunter-moseley commentedon Jan 28, 2017
Just realized that the correlation matrix needs to be between the features and not the samples. If the current PCA we are using is not dropping zeros, then this approach is going to dramatically change the PCA results, since the correlation will be limited to features the co-occur and not over-weighted by the zeros.
rmflight commentedon Jan 28, 2017