principal component analysis (pca) or empirical orthogonal functions (eofs)

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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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Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs). Arnaud Czaja (SPAT Data analysis lecture Nov. 2011). Outline. Motivation Mathematical formulation (on the board) Illustration: analysis of ~100yr of sea surface temperature fluctuations in the North Atlantic - PowerPoint PPT Presentation

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Page 1: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Arnaud Czaja(SPAT Data analysis lecture Nov. 2011)

Page 2: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

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

Page 3: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Motivation

• Data compression...to “carry less luggage”

Original pictures

6 EOFs

12 EOFs

24 EOFs

Page 4: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Motivation

• Data compression... to simplify with the hope of better understanding and forecasting

Selten (1995)

Mean Z300 (CI=100m)Mean Z300 (CI=100m)

r.m.s Z300 (CI=10m)r.m.s Z300 (CI=10m)

20-EOF modelQG model (231 var.)

Page 5: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Motivation

• Identify “modes” empirically from data

“Annular modes” inpressure data

Thompson and Wallace (2000)

Page 6: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Some examples of calculations

Page 7: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Pictures

Mean “picture”

EOF1 EOF2 EOF3

Page 8: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

North Atlantic sea surface temperature variability (Deser and Blackmon 1993)

PC2PC1

EOF212%

EOF145%

Page 9: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

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)

Page 10: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Main issues with EOF analysis• Sensitivity to size of

dataset (“sampling” issues)

See North et al. (1982)

Page 11: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Main issues with EOF analysis• Sensitivity to size of

dataset (“sampling” issues)

Page 12: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

Main issues with EOF analysis• Sensitivity to size of

dataset (“sampling” issues)

Page 13: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

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

Page 14: Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs)

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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