EOF and SVD

Synthetic Example

• Generate gridded data from three pressure patterns (think of it like a time series)

• Mean field is 1012mb• Input patterns are orthogonal to each other

EOF Analysis of Synthetic Data

• First 3 EOFs explain all the variance in the data• Data can be completely reconstructed using

only 3 patterns and 3 time series• EOF method correctly finds the right number

of independent patterns that make up the variability of the field

EOF Analysis of Synthetic Data

• The anticyclonic pattern is correctly reproduced

• But zonal and meridional patterns are not– We get eigenvectors that are a combination of

these• So even when the EOF method is applied on

orthogonal input fields, we do not necessarily get EOFs that resemble the data

Rotate with varimax function (see Appendix of EOF and SVD packet)

• Form a “truncated eigenvector basis from first three eigenvectors already obtained”– In this case use all 3, but could use only a few

• Now perform varimax rotation– This time we get the 3 types of input height fields

EOF Examples

• Analysis of 40 years of SST in South Atlantic– Monthly SST anomalies from equator to 50S over

a 2x2 grid from 1983-1992• Three leading EOF modes account (in total) for

47% of the monthly SST variance– EOF1: 30%– EOF2: 11%– EOF3: 6%

EOF shown as homogenous correlation map

Temporal variations of each eigenvector, represented by expansion coefficients

EOF1

• Mono-pole pattern over entire domain

• Square of correlation represents the variance explained LOCALLY

• PC is dominated by mixture of interannual and decadal fluctuations

EOF2

• Out of phase relationship between temperature anomalies north and south of ~25S

• PCs consist of interannual and lower frequency oscillations

EOF3

• Three alternating signs of signal

• PCs consist of mostly interannual timescales

SVD Example

• Monthly anomalies of SST and SLP analyzed together over the same spatial and temporal grid as in the EOF example

• The SVD analysis will identify only the modes of behavior in which the SST and SLP variations are strongly coupled

• Together the 1st 3 modes account for 89% of the total square covariance

• SCF=Squared Covariance Fraction• r: Correlation Coefficient between the

expansion coefficients of both variables• Indicates strength of coupling• Parentheses are 95% significance levels

Spatial Pattern 1 and Time Series for Pattern 1

• Strengthening and weakening of the subtropical anticyclone• Accompanied by

fluctuations in a north-south dipole structure in ocean temperature• Both time series are

dominated by interdecadal oscillations

Spatial Pattern 2 and Time Series for Pattern 2

• East-west displacements of the subtropical anticyclone center• Accompanied by large

SST fluctuations over a broad region off the coast of Africa• Low frequency

interannual oscillations

Spatial Pattern 3 and Time Series for Pattern 3

• North-south displacements of the subtropical anticyclone• Accompanied by large

SST fluctuations over a latitudinal band in the central South Atlantic ocean• High frequency

interannual oscillations

Examples from Model Data (CCSM3)

• From project last year using model data from CCSM3 on ENSO

• First thing to do is read in data:

clear all%read in datalat=[-87.8638, -85.0965, -82.3129, -79.5256,-76.7369, -73.9475, -71.1578,-68.3678,...-65.5776, -62.7874, -59.997, -57.2066,-54.4162, -51.6257, -48.8352,-46.0447,...-43.2542, -40.4636, -37.6731, -34.8825,-32.0919, -29.3014, -26.5108,-23.7202,...-20.9296, -18.139, -15.3484, -12.5578,-9.76715, -6.97653, -4.18592, -1.39531,...1.39531, 4.18592, 6.97653, 9.76715, 12.5578,15.3484, 18.139, 20.9296, 23.7202,26.5108,...29.3014, 32.0919, 34.8825, 37.6731, 40.4636,43.2542, 46.0447, 48.8352, 51.6257,54.4162,...57.2066, 59.997, 62.7874, 65.5776, 68.3678,71.1578, 73.9475, 76.7369, 79.5256,82.3129,...85.0965, 87.8638];lon=[0:2.8125:357.1875];%get slpfid=fopen('slp.dat','rb');slpall=fread(fid,'float');slp=reshape(slpall,[128 64 600]);%change 600 based on your timeslp=permute(slp,[2 1 3]);fclose(fid);clear sstall%get sstfid=fopen('sst.dat','rb');sstall=fread(fid,'float');sst=reshape(sstall,[128 64 600]);%change 600 based on your timesst=permute(sst,[2 1 3]);fclose(fid);clear sstall

Example using seasonal analysis

%to get seasons slpDJF(:,:,1)=(slp(:,:,1)+slp(:,:,2))./2;c=13;for yr=2:50 slpDJF(:,:,yr)=(slp(:,:,c-1)+slp(:,:,c)+slp(:,:,c+1))./3; c=c+12;endc=1;for yr=1:50 slpMAM(:,:,yr)=(slp(:,:,c+2)+slp(:,:,c+3)+slp(:,:,c+4))./3; slpJJA(:,:,yr)=(slp(:,:,c+5)+slp(:,:,c+6)+slp(:,:,c+7))./3; slpSON(:,:,yr)=(slp(:,:,c+8)+slp(:,:,c+9)+slp(:,:,c+10))./3; c=c+12;end

Compute EOFs

%to calculate EOFscoslat3d=ones(64,128,50);for j=1:length(lat) coslat3d(j,:,:)=cos(lat(j)*pi/180);endX=reshape(slpDJF.*sqrt(coslat3d),[64*128 50]);

%remove time meanfor i=1:(64*128) X(i,:)=X(i,:)-mean(X(i,:));endX2=permute(X,[2 1]);

[COEFF,SCORE,latent,tsquare]=princomp(X2);

explained = latent/sum(latent(:)) .*100;explained(1:4)

[COEFF,SCORE] = princomp(X) returns SCORE, the principal component scores; that is, the representation of X in the principal component space. Rows of SCORE correspond to observations, columns to components.

[COEFF,SCORE,latent] = princomp(X) returns latent, a vector containing the eigenvalues of the covariance matrix of X.

[COEFF,SCORE,latent,tsquare] = princomp(X) returns tsquare, which contains Hotelling's T2 statistic for each data point.

Compute EOFs (Continued)

%for EOF 1eof1=reshape(COEFF(:,1),[64 128]);pc1=SCORE(:,1);%for EOF 2eof2=reshape(COEFF(:,2),[64 128]);pc2=SCORE(:,2);%for EOF 3eof3=reshape(COEFF(:,3),[64 128]);pc3=SCORE(:,3);%for EOF 4eof4=reshape(COEFF(:,4),[64 128]);pc4=SCORE(:,4);

Some Examples of EOFs

• http://team-climate-diagnostics.wikispaces.com/Seasonal

• http://team-climate-diagnostics.wikispaces.com/Annual

• http://team-climate-diagnostics.wikispaces.com/Decadal

Some More Examples

• http://mpo581-hw3-eofs.wikispaces.com/Assignment