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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. - PowerPoint PPT Presentation

Why EOFs ?Joe TribbiaNCARRandom Matrices TOY 5/9/2007

Why EOFs ? outlineBackground history of EOFs in meteorology1 dimensional example-Burgers eqnEOFs as a random matrixEOFs for taxonomyEOFs for dimension reduction/basisSummary

Background in meteorology1956 report by E N LorenzUse EOFs to objectively classify Low frequency weather patternsApplication was to Long range prediction i.e. monthly weather outlooksThrough Don Gilman, John Kutzbach and Bob Livezy became the basis for monthly and seasonal outlooks

E N Lorenz : EOFs and dynamical systemsSimplest 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 Burgers equation2K+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)= 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 varianceEigenvectors are orthogonal and can be used as a basis for u

EOF spectrum and wavenumber spectrumLeading EOFs each represent 30% of varianceEOF1EOF2

EOFs and PCsEOF1PC1EOF2PC2

Looking for variancestructure: taxonomy in climateArcticOscillationEOF#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 Burgers equation2K+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)=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 varianceEigenvectors 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 reductionFrom Liouville eqn, importance sampling, entropy consderations

Bred vectors and Singular vectorsBasic state jetSingular 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 remarksEOFs can be motivated from a dynamical systems perspectiveEOFs 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