EMPIRICAL ORTHOGONAL FUNCTIONS

2 different modes

Sabrina Krista

Gisselle

Lauren

Principal Component Analysis or Empirical Orthogonal Functions

Linear combination of spatial predictors or modes that are normal or orthogonal to each other

Write data series Um(t) = U(z, t) as:

taftUM

iiimm

1

fim are orthogonal spatial functions, also known as eigenvectors or EOFs

are the eigenvalues of the problem (represent the variance explained

by each mode i)

2taii

ai(t) are the amplitudes or weights of the spatial functions as they change in time

m are each of the time series (function of depth or horizontal distance)

Low-passed (subtidal) flow (cm/s)

taftUM

iiimm

1

Eigenvectors (spatial functions) or EOFs

f1m

87% of variability

12% of variability

f2m

f3m taftUM

iiimm

1

a1

a2

MeasuredMode 1Mode 1+2

ij,ij,

ji 01

N

nniii ta

Nta

1

22 1

taftUM

iiimm

1

Goal: Write data series U at any location m as the sum of M orthogonal spatial functions fim:

ai is the amplitude of ith orthogonal mode at any time t

ji

M

mmjmi ff

1

For fim to be orthogonal, we require that:

Two functions are orthogonal when sum (or integral) of their product over a space or time is zero

jiiji tata

Orthogonality condition means that the time-averaged covariance of the amplitudes satisfies:

(overbar denotes time average)

variance of each orthogonal mode

tUtU km

M

i

M

jjkimjikm fftatatUtU

1 1

M

iikimikm fftUtU

1

imi

M

kikkm fftUtU

1

0 C

If we form the co-variance matrix of the data

taftUM

iiimm

1

jiiji tata

Multiplying both sides times fik, summing over all k and using the orthogonality condition:

Canonical form of eigenvalue problemeigenvectors 2taii eigenvalues

tUtUR kmmk eigenvalues of mean product tUtUC kmmk

if means of Um(t) are removed

0 C

tUtUtUtUtUtU

tUtUtUtUtUtUtUtUtUtUtUtU

MMMM

M

M

21

22212

12111

Mf

ff

2

1

00

0000

Mf

ff

2

1

0

00

221

2222112

1221111

MMMMMM

mM

mM

ftUtUftUtUftUtU

ftUtUftUtUftUtUftUtUftUtUftUtU

tUtUC kmmk

I is the unit matrix and are the EOFs

Eigenvalue problem corresponding to a linear system of equations:

0 C

0

00

221

2222112

1221111

MMMMMM

mM

mM

ftUtUftUtUftUtU

ftUtUftUtUftUtUftUtUftUtUftUtU

0

1

2221

11211

MMM

m

CC

CCCCC

det

M

jj

M

m

N

nnm tU

N 11 1

21

M

mimmi ftUta

1

For a non-trivial solution ( 0):

Sum of variances in data = sum of variance in eigenvalues