forming the co-variance matrix
DESCRIPTION
Forming the co-variance matrix. of the data. Multiplying both sides times f ik , summing over all k and using the orthogonality condition:. Canonical form of eigenvalue problem. eigenvectors. eigenvalues. I is the unit matrix and are the EOFs. - PowerPoint PPT PresentationTRANSCRIPT
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Forming the co-variance matrix of the data
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Multiplying both sides times fik, summing over all k and using the orthogonality condition:
Canonical form of eigenvalue problemeigenvectors 2taii eigenvalues
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I is the unit matrix and are the EOFs
Eigenvalue problem corresponding to a linear system:
Matrix = [6637,18]
rows > columns
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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
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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 185.3%
Mode 213.2%
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Mode 185.3%
Mode 213.2%
>> ts=ul*v;
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ts=[6637,18]
Mode 185.3%
Mode 213.2%
>> for k=1:nzvt(k,:,:)=ts(:,k)*v(:,k)';end
vt=[18, 6637,18]
mode #evolution in time
time series #
>> v1=squeeze(vt(1,:,:))’;>> v2=squeeze(vt(2,:,:))’;
Dep
th (m
)
Dep
th (m
)
Dep
th (m
)
Complex Empirical Orthogonal Functions – James River Data
Linear combination of spatial predictors or modes that are normal or orthogonal to each other
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Streamwise
Cross-stream
Rotated 49 degrees
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>> ul=complex(u,v);>> uc=cov(ul);>> [v,d]=eig(uc);
Mode 196.5%
Mode 1
>> lambda=diag(d)/sum(diag(d));>> v=fliplr(v);
Mode 22.5%
Mode 2
Mode 196.5%
Streamwise
cross-stream
>> ts=ul*v;
Mode 196.5%
Principal-axis
cross-axis
Mod
e sc
alin
g
>> ts=ul*v;
Mode 22.5%
Streamwise
Cross-stream
Mod
e sc
alin
g
>> ts=ul*v;
Mod
e sc
alin
g
Mode 22.5%
Streamwise
cross-stream
>> ts=ul*v;
Low-pass filtered data in James River
m/s
m/s
Streamwise
cross-stream
Mode 175%
m/s
m/s
m/s
Streamwise
cross-stream
Mode 222%
m/s
m/s
m/s
Streamwise
cross-stream
Dep
th (m
)
radians
Phase of EOFS
Mode 1
Mode 2
Mode 1 Mode 2
streamwise
cross-stream
>> ts=ul*v;
streamwise
cross-stream
>> ts=ul*v;
Modes 1 + 2 (75% + 22%)
Original
>> for k=1:nzvt(k,:,:)=ts(:,k)*v(:,k)';end
>> v1=squeeze(vt(1,:,:))’;>> v2=squeeze(vt(2,:,:))’;
Modes 1 + 2 + 3 + 4 (77%+22% +2% + 0.6%)
Original