EMPIRICAL ORTHOGONAL FUNCTIONS 2 different modes Sabrina Krista Gisselle Lauren

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)

Drum Head (circular membrane) vibrating modes https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane

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)

Subtidal Flow at Chesapeake Bay Entrance (cm/s)

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

Measured Mode 1+2 Mode 1+2+3

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

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

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

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

Matrix = [6637,18] rows > columns

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 = 10.1154

Covariance Matrix Maximum covariance at surface

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

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

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

Mode 1 85.3% Mode 2 13.2%

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

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)

Depth (m)

Depth (m)

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