PCA Eigenvector Averaging

loesung · August 26, 2022, 18:15

I'm reading a paper where PCA (basically just doing a SVD) is done on a set of 100 velocity fluctuations of turbulent data (basically they do PCA on each slice, ergo 100 times), then the found principle eigenvector, for each of the 100 slices, are averaged.

Does this even make sense to do? Since the eigenvectors are only unique up to a $\pm$ sign -- so there are billions of potential ways to choose the sign, and therefore billions of averages, depending on how one chooses the sign for each eigenvector!

The authors do this averaging of the 1st eigenvector (meaning the one corresponding to the largest eigenvalue) in two different papers. So it seems there must be a way to consistently choose the sign for each of the eigenvectors such that averaging makes sense? It is not clear 100% of the time how to choose the sign if I am just looking at its graph for each of the 100 eigenvectors..

One ad hoc method I read on stackexchange was: "sum the components of a given eigenvector, and if the sum is negative, multiply it by -1, in order to insure sign consistency among all the eigenvectors." However my (fourier transformed) velocity data is complex, so this trick doesn't apply (which assumed real numbers). A analogous trick could be to make sure the sum of the eigenvector's elements is in the upper half complex plane, but I'm not so certain about this. I also could not find additional sources which discuss this 'trick', so i'm skeptical about it!

LuckyTran · August 26, 2022, 18:32

The sign is not important, only the "size" is and ordering them from largest based on this measure of size. You can always force all the eigenvectors to be positive by shifting the negative sign away from the eigenvector and gifting the negative sign to the eigenvalue. Regardless of + or -, the largest eigenvector is still the largest eigenvector. What you are looking for is the axis that explains the measure, and + or - numbers both lie on the same axis.

August 26, 2022, 18:15	PCA Eigenvector Averaging	#1
loesung New Member Join Date: Mar 2022 Posts: 23 Rep Power: 4	I'm reading a paper where PCA (basically just doing a SVD) is done on a set of 100 velocity fluctuations of turbulent data (basically they do PCA on each slice, ergo 100 times), then the found principle eigenvector, for each of the 100 slices, are averaged. Does this even make sense to do? Since the eigenvectors are only unique up to a $\pm$ sign -- so there are billions of potential ways to choose the sign, and therefore billions of averages, depending on how one chooses the sign for each eigenvector! The authors do this averaging of the 1st eigenvector (meaning the one corresponding to the largest eigenvalue) in two different papers. So it seems there must be a way to consistently choose the sign for each of the eigenvectors such that averaging makes sense? It is not clear 100% of the time how to choose the sign if I am just looking at its graph for each of the 100 eigenvectors.. One ad hoc method I read on stackexchange was: "sum the components of a given eigenvector, and if the sum is negative, multiply it by -1, in order to insure sign consistency among all the eigenvectors." However my (fourier transformed) velocity data is complex, so this trick doesn't apply (which assumed real numbers). A analogous trick could be to make sure the sum of the eigenvector's elements is in the upper half complex plane, but I'm not so certain about this. I also could not find additional sources which discuss this 'trick', so i'm skeptical about it!

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
PLS and PCA	Pargaonkar Vijay	Main CFD Forum	2	February 8, 2002 06:19

August 26, 2022, 18:32		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,747 Rep Power: 66	The sign is not important, only the "size" is and ordering them from largest based on this measure of size. You can always force all the eigenvectors to be positive by shifting the negative sign away from the eigenvector and gifting the negative sign to the eigenvalue. Regardless of + or -, the largest eigenvector is still the largest eigenvector. What you are looking for is the axis that explains the measure, and + or - numbers both lie on the same axis.