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Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs) Arnaud Czaja (SPAT Data analysis lecture Nov. 2011)

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Outline Motivation Mathematical formulation (on the board) Illustration: analysis of ~100yr of sea surface temperature fluctuations in the North Atlantic How to compute EOFs Some issues regarding EOF analysis

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Motivation Data compression...to “carry less luggage” Original pictures 6 EOFs 12 EOFs 24 EOFs

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Motivation Data compression... to simplify with the hope of better understanding and forecasting Selten (1995) Mean Z300 (CI=100m) r.m.s Z300 (CI=10m) 20-EOF modelQG model (231 var.)

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Motivation Identify “modes” empirically from data “Annular modes” in pressure data Thompson and Wallace (2000)

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Some examples of calculations

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Pictures Mean “picture” EOF1 EOF2EOF3

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North Atlantic sea surface temperature variability (Deser and Blackmon 1993) PC2PC1 EOF2 12% EOF1 45%

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How to compute EOFs Compute the covariance matrix Σ of the observation matrix X Compute its eigenvalues (variance explained) and eigenvectors (=eof) The principal component is then obtained by “projection”: pc(t) = X * eof Another (more efficient) method: singular value decomposition of X (come and see me if you are interested)

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Main issues with EOF analysis Sensitivity to size of dataset (“sampling” issues) See North et al. (1982)

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Main issues with EOF analysis Sensitivity to size of dataset (“sampling” issues)

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Main issues with EOF analysis Sensitivity to size of dataset (“sampling” issues)

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Main issues with EOF analysis Orthogonality constraint is not physical. Methods have been developed to deal with this (“rotated EOFs”) The link between EOFs and physical modes of a system is not clear

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Main issues with EOF analysis Orthogonality constraint is not physical. Methods have been developed to deal with this (“rotated EOFs”) The link between EOFs and physical modes of a system is not clear Good luck if you try EOFs... Do not hesitate to come and see me!

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