what is eof analysis? eof = empirical orthogonal function method of finding structures (or patterns)...

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 = Xij

N

Math

• Define XT

XT = N = Xji

M

Math

Math

• And covariance matrix C=XXT

C = M N = M

N M

M

Math

• EOFs (ei) are the eigenvectors of C

C ei = ei

Math

• Principal components:

Fourier coefficients of the corresponding EOFs in the time expansion of the dataset

PCi (t) = (XT, ei)Too easy, huh?

Math• Why does 1st EOF maximize explained

variance?Answer: by construction.

(eTX,Xte) = ||eTX|| = max(eT,e) = 1

Or:(eT,Ce) = , ( C = XXT ), 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 1987-2010

Data: CRU temperatureAlexeev et al, 2012, Clim Change; Cohen et al, 2012, ERL

Major modes in the Northern Hemisphere

Major modes in the Northern Hemisphere

Major modes in the Northern Hemisphere

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 AT

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 CXY=XYT

CXY = M N = M

N L

L

Math

• SVD vectors of C: in U (X-field) and V

(Y-field) matrices

CXY = UVT

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

xn+1 = C xn + (C = ‘step forward’ operator)

Assume < x, > = 0< xn+1, xn > = C < xn, xn > + < x, >

We can approximate C from: C = C 0 C-1

1

Where C0 = < xn, xn > , C1 = < xn+1, xn >

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)