EMPIRICAL ORTHOGONAL FUNCTIONS 2 different modes SabrinaKrista Gisselle Lauren

Principal Component Analysis or Empirical Orthogonal Functions Linear combination of spatial predictors or modes that are normal or orthogonal to each other cm/s 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) Subtidal Flow at Chesapeake Bay Entrance

Drum Head (circular membrane) vibrating modes

Write data series U m (t) = U(z m, t) as: f im are orthogonal spatial functions, also known as eigenvectors or EOFs are the eigenvalues of the problem (represent the variance explained by each mode i) a i (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)

Subtidal Flow at Chesapeake Bay Entrance (cm/s)

Eigenvectors (spatial functions) or EOFs f 1m 85% of variability 13% of variability f 2m f 3m a1a1 a2a2 1%

Measured Mode 1+2 Mode 1+2+3

Goal: Write data series U at any location m as the sum of M orthogonal spatial functions f im : a i is the amplitude of ith orthogonal mode at any time t For f im 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 matrixof the data Multiplying both sides times f ik, summing over all k and using the orthogonality condition: Canonical form of eigenvalue problem eigenvectors eigenvalues eigenvalues of mean product Covariance matrix if means of U m (t) are removed use to get use to get

C mk 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): Sum of variances in data = sum of variance in eigenvalues time-dependent amplitudes of i th mode

Matrix = [6637,18] rows > columns

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

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 % Mode %

Mode % Mode % >> ts=ul*v; ts=[6637,18] Mode % Mode %

>> 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)

Suggestions for Final Project: 1)Calculate Complex EOFs of separate records (raw and filtered) 2)Calculate Complex EOFs of all records at the same time (raw and filtered) 3)Describe and understand spatial variability of EOF modes 4)Describe and understand temporal variability of EOF coefficients (amplitudes) 5)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 6)You could also calculate coherence squared between EOF coefficients and possible parameters causing the variability 7)Write up your story