OK take the following piece of Matlab code and put it in the

directory where you intend to work. Name it "eoftest.m".

% eoftest.m

% This generates a data set of a sine wave with random amplitude

% lx=number of spatial points; lt=number of sample points

% kw=wavenumber of sine wave in spatial direction; xn=noise level

% The amplitude of the basic sine wave is one, so the noise should

% be of the same order, say zero to ten or so.

function x=mydata(lx, lt, kw, xn)

% get x shape

for i=1:lx

xx(i)=sin((i-1)*kw*2*3.141593/lx);

end

% get time shape of Gaussian white noise

rn=randn(lt);

% combine spatial and temporal structures

for i=1:lx

for j=1:lt

x(i,j)=rn(j)*xx(i) +xn*randn(1);

end

end

Now start Matlab in the same directory where you have put eoftest.m

>> x=eoftest(10, 100, 1, 0.) ;

>> c=x*x';

>> [u,s,v]=svd(c);

>> plot(s)

>> plot(u(1:10,1:3))

>> plot(v(1:10,1:3))

>>

The eigenvalue spectrum, s, should have only one nonzero value, the first.

The first vectors of u and v should be the first eof. The row space and

column space of a symmetric matrix are identical.

Next do the same analysis, but put in some significant noise.

>> clear all

>> x=eoftest(10, 100, 1, 2.) ;

>> c=x*x';

>> [u,s,v]=svd(c);

>> plot(s)

>> plot(u(1:10,1:3))

>> plot(v(1:10,1:3))

What changes? Does eof1 still look the same. How does the eigenvalue

spectrum look?

You can do the same analysis by direct svd of the data matrix

>> clear all

>> x=eoftest(10, 100, 1, 2.) ;

>> [u,s,v]=svd(x);

>> plot(s)

>> plot(u(1:10,1:3))

>> plot(v(1:100,1:3))

Are the functions the same as before?

How do the singular values change?

What are the v vectors now?