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What is EOF analysis?

What is EOF analysis?. EOF = Empirical Orthogonal Function Method of finding structures (or patterns) that explain maximum variance in (e.g.) 2D (space-time) dataset Mathematically EOFs are eigenvectors of the covariance matrix of a dataset. Math.

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What is EOF analysis?

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  1. What is EOF analysis? • EOF = Empirical Orthogonal Function • Method of finding structures (or patterns) that explain maximum variance in (e.g.) 2D (space-time) dataset • Mathematically EOFs are eigenvectors of the covariance matrix of a dataset

  2. Math • Any (space-time) dataset can be represented as a matrix: X = M = Xij N

  3. Math • Define XT XT = N = Xji M

  4. Math • And covariance matrix C=XXT C = M N = M N M M

  5. Math • EOFs (ei) are the eigenvectors of C C ei =  ei

  6. Math • Principal components: Fourier coefficients of the corresponding EOFs in the time expansion of the dataset PCi (t) = (XT, ei) Too easy, huh?

  7. Math • Why does 1st EOF maximize explained variance? Answer: by construction. (eTX,Xte) = ||eTX|| = max (eT,e) = 1 Or: (eT,Ce) =  , ( C = XXT ), Ce = e This maximizes on the eigenvector corresponding to the greatest eigenvalue. Amazing!

  8. What do EOFs and PCs mean? • EOF – a coherent orthogonal spatial pattern. • First EOF explains most variance in a physical field • PC – time behavior of the corresponding EOF (=spatial pattern) Stunning! Let’s EOF everything!

  9. EOF interpretation Direction of maximum variance

  10. Example 1. El-Nino.

  11. Example 2. Arctic Oscillation.

  12. Tropospheric Winter Trends Cohen et al, 2012, ERL

  13. Northern Hemisphere Land Temperatures 1987-2010 Data: CRU temperature Alexeev et al, 2012, Clim Change; Cohen et al, 2012, ERL

  14. Major modes in the Northern Hemisphere

  15. Major modes in the Northern Hemisphere

  16. Major modes in the Northern Hemisphere

  17. Why EOFs are not physical modes? Your equations: dx/dt + Ax = f Physical modes: eigenvectors of A. (Solve Ay =  y) Physical modes are not orthogonal (generally speaking)

  18. Why EOFs are not physical modes? Your equations: dx/dt + Ax = f EOFs – eigenvectors of a matrix derived from A AT EOFs: orthogonal by construction

  19. Other methods • SVD = Singular Value Decomposition, aka MCA = Maximum Correlation Analysis • Method is looking for correlated spatial patterns in two different fields

  20. Math • Correlation matrix CXY=XYT CXY = M N = M N L L

  21. Math • SVD vectors of C: in U (X-field) and V (Y-field) matrices CXY = UVT

  22. Other methods • CCA = Canonical Correlation Analysis: SVD over space of Fourier coefficients of EOFs

  23. Other methods • POP = Principal Oscillation Pattern Analysis FDT over space of Fourier coefficients of EOFs (FDT = Fluctuation-Dissipation Theorem)

  24. POP = Principal Oscillating Patterns xn+1 = C xn +  (C = ‘step forward’ operator) Assume < x, > = 0 < xn+1, xn > = C < xn, xn > + < x, > We can approximate C from: C = C 0 C-11 Where C0 = < xn, xn > , C1 = < xn+1, xn >

  25. Other methods • Varimax, Quartimax, rotated EOF analysis EOF modifications

  26. Other methods • MTM = Multi-Taper Method Combination of EOF and Wavelet analyses

  27. Other methods • SSA = Singular Spectrum Analysis • MSSA = Multi-channel SSA • MTM-SVD • EEOF = Extended EOF • FDEOF = Frequency Domain EOF • CEOF = Complex EOF

  28. When is EOF analysis useful? • Analysis of repeating pronounced patterns over long time series • Image/data compression • Filtering Not so fast….

  29. When EOF use is inappropriate? • Short time series, lots of missing and/or inconsistent data • Absence of a prominent signal • Presence of a dominant trend in the data (e.g. seasonal cycle is dangerous!)

  30. Why do people get so excited about EOFs? • EOFs can be applied to any dataset • Simplicity of the analysis is very appealing. Everyone does EOFs. • Patterns are often tempting to analyze (because of method’s simplicity)

  31. Do not overdo it with EOFs! • “New” patterns sometimes turn out to be not so new. • Artificial (mechanistic) data de-trending can lead to surprises (example: removal of seasonal cycle does not remove changing seasonal variability in most of the fields)

  32. Do scientists have problems interpreting EOF results? • Saying “I performed EOF analysis on my data” does not mean you explained any physics • EOFs usually do not coincide with eigen-modes of the physical process you are trying to interpret/explain. POP analysis does not give you orthonormal modes, but it might approximate your physical modes

  33. Are results of EOF analysis accurate? • Statistical significance is always an issue. • If something correlates (even very well) with something else (or appears to be systematically preceding/following), this does not mean one causes the other. They both can be caused by something else.

  34. What are EOF maniacs? • People who eof (svd, cca …) everything with everything just for the sake of it

  35. Are there many EOF/SVD/CCA maniacs out there? • Yes, there are! (I am one of them)

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