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I have a dataset composed of 123 rows (time bins) and 20 columns (variables)

The question I have is the following. each row,column pair has a radian value and a radius value. If I convert these pairs to a complex number I have effectively take a 123x20x2 real dataset and converted it to a 123x20 dataset composed of complex numbers. Can I then feed this complex dataset into a pca? If so my loadings are now complex and when I try to determine the R^2 of each variable for the component the sum of all R^2 should add to 1 but no longer does due to the imaginary number part.

Is this allowed? if so why do the loadings of a component not add to 1 ( sum(loadings(:,1).^2) ~= 1 but should equal 1)

user448573
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    Standard PCA doesn't handle complex values. In principle you could include the angle and radius as separate dimensions (or convert to cartesian coordinates first). In either case, you'd get a 123 x 40 data matrix. But, keep in mind that PCA only handles linear structure. Since you're using polar coordinates, I wonder whether your data might be strongly nonlinear, or even lie on a circular manifold (e.g. sphere or torus). If that's the case, then PCA won't work well. There may be other suitable techniques, but we'd need to know more about your data and goals. – user20160 Apr 06 '18 at 03:37
  • As noted above, the standard PCA should not be meaningful if the data lie on a sphere (ie. vector lengths are all the same, or maybe roughly). Just as a pointer, there exists a literature of PCA on spheres, called principal nested hyperspheres. See, for example, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3635703/. – Kees Mulder Apr 06 '18 at 08:17
  • I've been looking into this and it seems as though a complex EOF analysis is what I'm looking for. Does this make sense? – user448573 Apr 06 '18 at 19:01
  • [link] (https://books.google.com/books?id=_VHxE26QvXgC&pg=PA358&lpg=PA358&dq=complex+eof+polar+coordinates&source=bl&ots=Ekp_q4zDdu&sig=Wp6h8EO3MBRGZX_5UzhMV83U5rU&hl=en&sa=X&ved=0ahUKEwix0d-_rabaAhXrhlQKHbCPAc0Q6AEIKTAA#v=onepage&q=complex%20eof%20polar%20coordinates&f=false) @KeesMulder – user448573 Apr 06 '18 at 19:15

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