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

• How to compute EOFs

• Some issues regarding EOF analysis

Motivation

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

Original pictures

6 EOFs

12 EOFs

24 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.)

Motivation

• Identify “modes” empirically from data

“Annular modes” inpressure data

Thompson and Wallace (2000)

Some examples of calculations

Pictures

Mean “picture”

EOF1 EOF2 EOF3

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

PC2PC1

EOF212%

EOF145%

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)

Main issues with EOF analysis• Sensitivity to size of

dataset (“sampling” issues)

See North et al. (1982)

Main issues with EOF analysis• Sensitivity to size of

dataset (“sampling” issues)

Main issues with EOF analysis• Sensitivity to size of

dataset (“sampling” issues)

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

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!