Why EOFs ?

Joe Tribbia

NCAR

Random Matrices TOY 5/9/2007

Why EOFs ? outline

• Background history of EOFs in meteorology

• 1 dimensional example-Burger’s eqn

• EOFs as a random matrix

• EOFs for taxonomy

• EOFs for dimension reduction/basis

• Summary

Background in meteorology

• 1956 report by E N Lorenz

• Use EOFs to objectively classify Low –frequency weather patterns

• Application was to “Long range prediction” i.e. monthly weather outlooks

• Through Don Gilman, John Kutzbach and Bob Livezy became the basis for monthly and seasonal outlooks

E N Lorenz : EOFs and dynamical systems

Simplest chaoticsystem : the LorenzAttractor, a metaphorfor the unpredictabilityof weather

E N Lorenz (continued)

Covariance fits anellipsoid to theattractor.

EOFs are the principle axes of the ellipsoid

1 dimensional example:sample over time

1 dimensional example Burger’s equation

llklk

ikxK

k

K

Kkkkk

uluiu

eukxbkxau

xuutu

0

)sin()cos(

0

2K+1 independent degrees of freedom define u on 2K+1 pointson a circle. Defines a random vector u(n) with sample covariance U(n,m)=<u(n)u(m)> Because of sampling U is a random matrix.

Given by orthogonal projectionMultiply eqn by exp(-ikx) and integrate

1 dimensional example (cont.)Diagonalizing U(m,n) determines the eigenvalues andeigenvectors of U(m,n). The sum of the eigenvaluesis the trace of the U and is an invariant correspondingto the total variance. The diagonalization breaks the varianceinto independent pieces and the eigenvalue is the variance in each independent piece. The eigenvectors are the spatialstructures corresponding to each independent variance

kk

kkk

kk

xxxuu

xUx

k

k

a),(

Eigenvectors are orthogonal and can be used as a basis for u

EOF spectrum and wavenumber spectrumLeading EOFs each represent 30% of variance

EOF1

EOF2

EOFs and PCs

EOF1 PC1

EOF2PC2

Looking for variancestructure: taxonomy in climate

ArcticOscillationEOF#1with 19%of Variance

Looking for structure: taxonomy

Looking for dynamicalstructure: bump hunting

Searching for statisticalstructure beyond Gaussian.Is there a reason for EOFdominance beyond lineardynamics?

Comparison ofscatter plots forLorenz attractor and climate data.

Climate data is muchmore homogeneous,i.e. linear dynamics?

Looking for predictable structure

1 dimensional example Burger’s equation

llklk

ikxK

k

K

Kkkkk

uluiu

eukxbkxau

xuutu

0

)sin()cos(

0

2K+1 independent degrees of freedom define u on 2K+1 pointson a circle. Defines a random vector u(n) with sample covariance U(n,m)=<u(n)u(m)>

Given by orthogonal projectionMultiply eqn by exp(-ikx) and integrate

1 dimensional example (cont.)Diagonalizing U(m,n) determines the eigenvalues andeigenvectors of U(m,n). The sum of the eigenvaluesis the trace of the U and is an invariant correspondingto the total variance. The diagonalization breaks the varianceinto independent pieces and the eigenvalue is the variance in each independent piece. The eigenvectors are the spatialstructures corresponding to each independent variance

kk

kkk

kk

xxxuu

xUx

k

k

a),(

Eigenvectors are orthogonal and can be used as a basis for u

Dimension reduction:EOF basis

Sampling strategies for small samples in high dimensional

systems: dimension reduction

From Liouville eqn, importance sampling, entropy consderations

Bred vectors and Singular vectors

Basic state jet

Singular vector (upper)Bred vector (lower)

Singular vectors are the fastest growing structures into the futureBred vectors are the fastest growing structures from the past.

Both are EOFs of linearly predicted error covariance

Concluding remarks

• EOFs can be motivated from a dynamical systems perspective

• EOFs useful for elucidating structure ( taxonomy, predictability, non-gaussianity)

• EOFs useful for dimension reduction (natural basis, importance sampling)

• Limits to utility: intrinsic Gaussianity and linearity, prior information needed