Forming the co-variance matrix
description
Transcript of Forming the co-variance matrix
![Page 1: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/1.jpg)
tUtU km
M
i
M
jjkimjikm fftatatUtU
1 1
M
iikimikm fftUtU
1
imi
M
kikkm fftUtU
1
0 C
Forming 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
![Page 2: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/2.jpg)
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:
![Page 3: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/3.jpg)
Matrix = [6637,18]
rows > columns
![Page 4: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/4.jpg)
tUtUtUtUtUtU
tUtUtUtUtUtUtUtUtUtUtUtU
MMMM
M
M
21
22212
12111
Mf
ff
2
1
00
0000
Mf
ff
2
1
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
N
iii uu
N 1
2
11
![Page 5: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/5.jpg)
Covariance Matrix
Maximum covariance at surface
![Page 6: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/6.jpg)
>> uc=cov(ul);>> [v,d]=eig(uc);
eigenvalues (or lambda)
>> lambda=diag(d)/sum(diag(d));
![Page 7: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/7.jpg)
>> uc=cov(ul);>> [v,d]=eig(uc);
![Page 8: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/8.jpg)
>> uc=cov(ul);>> [v,d]=eig(uc);>> v=fliplr(v); %flips matrix left to right
![Page 9: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/9.jpg)
Mode 185.3%
Mode 213.2%
taftUM
iiimm
1
![Page 10: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/10.jpg)
Mode 185.3%
Mode 213.2%
>> ts=ul*v;
taftUM
iiimm
1
ts=[6637,18]
Mode 185.3%
Mode 213.2%
![Page 11: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/11.jpg)
>> 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
)
![Page 12: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/12.jpg)
Dep
th (m
)
![Page 13: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/13.jpg)
Dep
th (m
)
![Page 14: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/14.jpg)
![Page 15: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/15.jpg)
![Page 16: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/16.jpg)
Complex Empirical Orthogonal Functions – James River Data
Linear combination of spatial predictors or modes that are normal or orthogonal to each other
taftUM
iiimm
1
u
v
![Page 17: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/17.jpg)
taftUM
iiimm
1
Streamwise
Cross-stream
Rotated 49 degrees
![Page 18: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/18.jpg)
ttivtutUM
iiimmmm
1
0 C
>> ul=complex(u,v);>> uc=cov(ul);>> [v,d]=eig(uc);
![Page 19: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/19.jpg)
Mode 196.5%
Mode 1
>> lambda=diag(d)/sum(diag(d));>> v=fliplr(v);
![Page 20: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/20.jpg)
Mode 22.5%
Mode 2
![Page 21: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/21.jpg)
Mode 196.5%
Streamwise
cross-stream
>> ts=ul*v;
![Page 22: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/22.jpg)
Mode 196.5%
Principal-axis
cross-axis
Mod
e sc
alin
g
>> ts=ul*v;
![Page 23: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/23.jpg)
Mode 22.5%
Streamwise
Cross-stream
Mod
e sc
alin
g
>> ts=ul*v;
![Page 24: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/24.jpg)
Mod
e sc
alin
g
Mode 22.5%
Streamwise
cross-stream
>> ts=ul*v;
![Page 25: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/25.jpg)
Low-pass filtered data in James River
m/s
m/s
Streamwise
cross-stream
![Page 26: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/26.jpg)
Mode 175%
m/s
m/s
m/s
Streamwise
cross-stream
![Page 27: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/27.jpg)
Mode 222%
m/s
m/s
m/s
Streamwise
cross-stream
![Page 28: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/28.jpg)
Dep
th (m
)
radians
Phase of EOFS
Mode 1
Mode 2
Mode 1 Mode 2
![Page 29: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/29.jpg)
streamwise
cross-stream
>> ts=ul*v;
![Page 30: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/30.jpg)
streamwise
cross-stream
>> ts=ul*v;
![Page 31: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/31.jpg)
Modes 1 + 2 (75% + 22%)
Original
>> for k=1:nzvt(k,:,:)=ts(:,k)*v(:,k)';end
>> v1=squeeze(vt(1,:,:))’;>> v2=squeeze(vt(2,:,:))’;
![Page 32: Forming the co-variance matrix](https://reader036.fdocuments.in/reader036/viewer/2022062310/56816780550346895ddc8a17/html5/thumbnails/32.jpg)
Modes 1 + 2 + 3 + 4 (77%+22% +2% + 0.6%)
Original