CDO

hi Sentia!

The point with EOF and the eigen values is:

• The eigen values do not have the unit of the eofs itself.
• both eigen values and eigen vectors are free to be scaled like you wish because of their linear relation: H(v) = k v = k/x (x*v). Thats why you usually normlize vectors and values. the relative difference between the eigen values are in-fact significant. But they do not change with scaling.

IMO the term realistic magnitudes simply does not apply to the results of an EOF analysis.

just my thoughts ;-)

be well
ralf

Dear Ralf,

Thanks for the answer. I think I have a hint here (p14, https://atmos.washington.edu/~dennis/552_Notes_4.pdf):

"One can plot the EOFs directly in their normalized form, but it is often desirable to present them in a way that indicates how much real amplitude they represent. One way to represent their amplitude is to take the time series of principal components for the spatial structure (EOF) of interest, normalize this time series to unit variance, and then regress it against the original data set. This produces a map with the sign and dimensional amplitude of the field of interest that is explained by the EOF in question. The map has the shape of the EOF, but the amplitude actually corresponds to the amplitude in the real data with which this structure is associated. Thus we get structure and amplitude information in a single plot."

For the moment, I did not figure out how to do this, but I am working on it. If somebody has already done it, I will be more than grateful to be helped in that step.

Sentia

For every self-adjoined linear function A between two vector spaces there is set of eigen vectors which form a basis E=(e _i), i=1..n of the domain of the function so that A(e _i) = k _i e _i (k _i being the eigen values).

If you like matrices more than linear functions, you can call A a matrix instead. I just don't because all these coefficient confuse me ...

Since A is linear, the eigenvectors can be normalized without violating the equation: A(r e _i) = r A(e _i)=r k _i e _i. the corresponding eigen values get normalized, too as you can see from that.

Hence the absolute field values of your Temperature-related EOFs have no further meaning. you can make them as large or small as you like. But the ordering of the eigen values will tell you, how large the fraction of corresponding eigen vector (eof) is. This is because each element v of the domain of A (the space of your gridded data) can be written as a linear combination v = a ^j e _j (I use the convention for leaving out the ∑ symbol if the corresponding index appears in sub and super script) because E is a basis. The image of v under A then is

A(v) = A (a ^j e _j) = a ^j A( e _j) = a ^j k _j e _j

so the size of the eigen values determine the fraction of the corresponding eof to the (data) vector v. that's why you sort the eofs according to the eigen values. the one with the biggest eigen value has the largest impact. But the exact values of the components of a single eof do not have a meaning on their own IMO. Instead it is the pattern they describe - the complete vector.

I hope my description is not too confusing

hth
ralf