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I am running PCA on a multivariate time-series dataset using observations across time (i.e. w/out time as an explicit variable) as the design matrix. Given this setup, I've found that it is difficult for me to interpret the covariance matrix derived from such a design matrix. For instance, if two variables (e.g. signal A & signal B) positively covary, I think that means that--on average (across observations/time-points)--a positive amplitude of signal A predicts a positive amplitude of signal B. However, it seems evident that some amount of temporal information is lost when only looking at the covariance matrix in this manner. Given that PCA exploits info in the covariance matrix, I'm having trouble intuitively understanding how my end-result PCA components or "virtual signals" would accurately retain most of the phase relationships between variables in the design matrix. If PCA does impact phase relationships substantially, then I may be in trouble, because I am attempting to run VAR models on datasets that have had PCA applied to them. Is this a reasonable concern, or can someone convince me that phase relationships between variables are retained even after running PCA?

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