What is EOF analysis? EOF = Empirical Orthogonal Function Method of finding structures (or patterns) that explain maximum variance in (e.g.) 2D (space-time) dataset Mathematically EOFs are eigenvectors of the covariance matrix of a dataset

Any (space-time) dataset can be represented as a matrix: X = M = X ij N Math

Define X T X T = N = X ji M Math

And covariance matrix C=XX T C = M N = M N M M

Math EOFs (e i ) are the eigenvectors of C C e i = e i

Math Principal components: Fourier coefficients of the corresponding EOFs in the time expansion of the dataset PC i (t) = (X T, e i ) Too easy, huh?

Math Why does 1st EOF maximize explained variance? Answer: by construction. (e T X,X t e) = ||e T X|| =  max (e T,e) = 1 Or: (e T,Ce) =, ( C = XX T ), Ce = e This maximizes on the eigenvector corresponding to the greatest eigenvalue. Amazing!

What do EOFs and PCs mean? EOF – a coherent orthogonal spatial pattern. First EOF explains most variance in a physical field PC – time behavior of the corresponding EOF (=spatial pattern) Stunning! Let ’ s EOF everything!

EOF interpretation Direction of maximum variance

Example 1. El-Nino.

Example 2. Arctic Oscillation.

Tropospheric Winter Trends Cohen et al, 2012, ERL

Northern Hemisphere Land Temperatures Data: CRU temperature Alexeev et al, 2012, Clim Change; Cohen et al, 2012, ERL

Major modes in the Northern Hemisphere

Influence of Cosmic Rays on Earth's Climate (H.Svensmark)

Why EOFs are not physical modes? Your equations: dx/dt + Ax = f Physical modes: eigenvectors of A. (Solve Ay = y) Physical modes are not orthogonal (generally speaking)

Why EOFs are not physical modes? Your equations: dx/dt + Ax = f EOFs – eigenvectors of a matrix derived from A A T EOFs: orthogonal by construction

Other methods SVD = Singular Value Decomposition, aka MCA = Maximum Correlation Analysis Method is looking for correlated spatial patterns in two different fields

Math Correlation matrix C XY =XY T C XY = M N = M N L L

Math SVD vectors of C: in U (X-field) and V (Y-field) matrices C XY = U  V T

Other methods CCA = Canonical Correlation Analysis: SVD over space of Fourier coefficients of EOFs

Other methods POP = Principal Oscillation Pattern Analysis FDT over space of Fourier coefficients of EOFs (FDT = Fluctuation-Dissipation Theorem)

POP = Principal Oscillating Patterns x n+1 = C x n +  (C = ‘step forward’ operator) Assume = 0 = C + We can approximate C from: C = C 0 C -1 1 Where C 0 =, C 1 =

Other methods Varimax, Quartimax, rotated EOF analysis EOF modifications

Other methods MTM = Multi-Taper Method Combination of EOF and Wavelet analyses

Other methods SSA = Singular Spectrum Analysis MSSA = Multi-channel SSA MTM-SVD EEOF = Extended EOF FDEOF = Frequency Domain EOF CEOF = Complex EOF

When is EOF analysis useful? Analysis of repeating pronounced patterns over long time series Image/data compression Filtering Not so fast….

When EOF use is inappropriate? Short time series, lots of missing and/or inconsistent data Absence of a prominent signal Presence of a dominant trend in the data (e.g. seasonal cycle is dangerous!)

Why do people get so excited about EOFs? EOFs can be applied to any dataset Simplicity of the analysis is very appealing. Everyone does EOFs. Patterns are often tempting to analyze (because of method ’ s simplicity)

Do not overdo it with EOFs! “ New ” patterns sometimes turn out to be not so new. Artificial (mechanistic) data de-trending can lead to surprises (example: removal of seasonal cycle does not remove changing seasonal variability in most of the fields)

Do scientists have problems interpreting EOF results? Saying “ I performed EOF analysis on my data ” does not mean you explained any physics EOFs usually do not coincide with eigen- modes of the physical process you are trying to interpret/explain. POP analysis does not give you orthonormal modes, but it might approximate your physical modes

Are results of EOF analysis accurate? Statistical significance is always an issue. If something correlates (even very well) with something else (or appears to be systematically preceding/following), this does not mean one causes the other. They both can be caused by something else.

What are EOF maniacs? People who eof (svd, cca …) everything with everything just for the sake of it

Are there many EOF/SVD/CCA maniacs out there? Yes, there are! (I am one of them)