**To**:**cccma-student-003@xxxxxxxx****Subject**:**Re: [ferret_users] Spatial correlations[EOF PC, field]****From**:**Ryo Furue <furue@xxxxxxxxxx>**- Date: Mon, 09 Nov 2009 23:53:00 -1000 (HST)
- Cc: ferret_users@xxxxxxxx
- In-reply-to: <4AF87757.5090502@xxxxxxxx>
- References: <4AF87757.5090502@xxxxxxxx>
- Sender: owner-ferret_users@xxxxxxxx

Hi Fabian, | When I plot the spatial correlations between the leading principal | component of an SST dataset | (inferred from Ferret's EOF functions based on correlation matrix) | and the SST dataset itself, the correlations in the domain of the | EOF analysis are very low (see attached Figure). In the North | Pacific they are supposed to explain 20% of the total | variance. Shouldn't R-squared averaged over the domain be about 0.2? | Therefore the correlation should be 0.45 on average in this domain. I quick calculation seems to show (I hope I didn't make mistakes in the derivation) that <T(t,i;n) T(t,i)> / intgr_i[ <T(t,i)T(t,i)> ] = [A(n) / sum_i A(i)] g(i;n) g(i;n) where t is time, i is an index for spatial points, n is the EOF mode number we are focusing on, <> is time average, A(n) is the variance of mode n, g(i;n) is the spatial structure of mode n, "intgr_i" is a symbolical representation of spatial integration. If there is no spatial weights, "intgr_i" is "sum_i". I've assumed that g's are normalized in such a way that intgr_i g(i;n) g(i;n) = 1 for each n. I've also assumed that average is already subtracted from T. Notice that the factor [A(n) / sum_i A(i)] is the contribution of mode n to the total variance, which is "20%" in your case. Since we have a factor "g(i;n) g(i;n)", the correlation can be much smaller than 20% because of the way g is normalized. Basically this is what your are seeing, I think (It's not exactly that, though. See below.) If you integrate the expression in space, we recover the contribution of mode n: \intgr_i <T(t,i;n) T(t,i)> / intgr_i[ <T(t,i)T(t,i)> ] = A(n) / sum_i A(i) So, clearly, the correlation between the mode-n part of T and the original T field contains the desired, correct information. Another issue here is that my correlation coefficient is defined as <T(t,i;n) T(t,i)> divided by the total variance, whereas the ordinary correlation coefficient between T(t,i;n) and T(t,i) is defined as <T(t,i;n) T(t,i)> divided by the standard deviation of T(t,i;n) and that of T(t,i). The spatial integral of such a correlation yields sqrt[A(n) / sum_i A(i)], I think. I can send you (maybe personally) my derivation if you like. (But, I may not be able to respond quickly; I'll be offline for a while.) Regards, Ryo

**References**:**[ferret_users] Spatial correlations[EOF PC, field]***From:*Fabian Lienert

- Previous by thread:
**Re: [ferret_users] Spatial correlations[EOF PC, field]** - Next by thread:
**[ferret_users] Extracting average values over irregular gid points**

Contact Us

Dept of Commerce / NOAA / OAR / PMEL / TMAP

Privacy Policy | Disclaimer | Accessibility Statement