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
12 EOFs
• Data compression
...to “carry less luggage”
Original pictures
6 EOFs
24 EOFs
Motivation
QG model (231 var.)
• Data
compression...
to simplify
with the hope
of better
understanding
and
forecasting
Selten (1995)
20-EOF model
Mean Z300 (CI=100m)
Mean Z300 (CI=100m)
r.m.s Z300 (CI=10m)
r.m.s Z300 (CI=10m)
Motivation
• Identify “modes” empirically from data
“Annular modes” in
pressure data
Thompson and Wallace (2000)
Some examples of calculations
Pictures
EOF1
Mean “picture”
EOF2
EOF3
North Atlantic sea surface temperature variability
(Deser and Blackmon 1993)
EOF1
45%
EOF2
12%
PC1
PC2
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!