1. EMPIRICAL ORTHOGONAL FUNCTIONS
2 different modes
Sabrina
Krista
Gisselle
Lauren

2. Principal Component Analysis or Empirical Orthogonal Functions
Subtidal Flow at Chesapeake Bay Entrance
cm/s
Linear combination of spatial predictors or modes that are normal or orthogonal to each other
EOF is equivalent to “factor analysis” a data reduction method in social sciences
Gives a compact representation of the temporal and spatial variability of several (or many) time series in terms of orthogonal functions (statistical modes)

3. Drum Head (circular membrane) vibrating modes

4. Write data series Um(t) = U(zm, t) as:
fim are orthogonal spatial functions, also known as eigenvectors or EOFs
m are each of the time series (function of depth or horizontal distance)
ai(t) are the amplitudes or weights of the spatial functions as they change in time
are the eigenvalues of the problem (represent the variance explained
by each mode i)

5. Subtidal Flow at Chesapeake Bay Entrance (cm/s)

6. 85% of variability
Eigenvectors (spatial functions) or EOFs 1%
f3m
f2m
f1m a1 a2
13% of variability

7. Measured
Mode 1+2
Mode 1+2+3

9. ai is the amplitude of ith orthogonal mode at any time t
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
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
Orthogonality condition means that the time-averaged covariance of the amplitudes satisfies:
(overbar denotes time average)
variance of each orthogonal mode

10. If we form the co-variance matrix
of the data
use
to get
to get
use
Multiplying both sides times fik, summing over all k and using the orthogonality condition:
eigenvectors
Canonical form of eigenvalue problem
eigenvalues
eigenvalues of mean product
Covariance matrix if means of Um(t) are removed

11. Cmk is the covariance matrix; I is the unit matrix and  are the EOFs
Eigenvalue problem corresponding to a linear system of equations:

12. For a non-trivial solution (  0):
time-dependent amplitudes of ith mode
Sum of variances in data = sum of variance in eigenvalues

13. Matrix = [6637,18]
rows > columns

14. Matrix ul = [6637,18]
>> uc=cov(ul);
>> u1=ul(:,1);
>> sum((u1-mean(u1)).^2)/(length(u1)-1)
ans =
9.6143
>> u2=ul(:,2);
>> sum((u1-mean(u1)).*(u2-mean(u2)))/(length(u1)-1)
ans =

15. Covariance Matrix
Maximum covariance at surface

16. >> uc=cov(ul);
>> [v,d]=eig(uc);
eigenvalues (or lambda)
>> lambda=diag(d)/sum(diag(d));

17. >> uc=cov(ul);
>> [v,d]=eig(uc);

18. >> uc=cov(ul);
>> [v,d]=eig(uc);
>> v=fliplr(v);
%flips matrix left to right

19. Mode 1
85.3%
Mode 2
13.2%

20. >> ts=ul*v; ts=[6637,18] Mode 1 85.3% Mode 2 13.2% Mode 2 13.2%

21. vt(k,:,:)=ts(:,k)*v(:,k)'; end vt=[18, 6637,18]
>> for k=1:nz
vt(k,:,:)=ts(:,k)*v(:,k)';
end
vt=[18, 6637,18]
mode #
evolution in time
time series #
>> v1=squeeze(vt(1,:,:))’;
>> v2=squeeze(vt(2,:,:))’;
Depth (m)

22. Depth (m)

23. Depth (m)

25. Suggestions for Final Project:
Calculate Complex EOFs of separate records (raw and filtered)
Calculate Complex EOFs of all records at the same time (raw and filtered)
Describe and understand spatial variability of EOF modes
Describe and understand temporal variability of EOF coefficients (amplitudes)
Perform wavelet analysis (with coherence & cross-wavelet) of the EOF coefficients (vary in time) and possible parameters (e.g wind) linked to EOF coefficient temporal variability
You could also calculate coherence squared between EOF coefficients and possible parameters causing the variability
Write up your story