Notations:
V is the original (or raw) yearly time series to decompose;
Vcc is the part of the signal associated with global warming;
V10 is the part of the signal associated with decadal variability;
Via is the part of the signal associated with interannual variability;
CC is the global warming signal;
D is the operator for the low-pass filter;
rA/B designates the correlation between A and B;
σA designates the standard deviation of A;
σA2 designates the varaince of A;
stdA designates the standardized A;
μA designates the mean of A;
cov(A,B) designates the covariance between A and B.
Under those notations, the definition of the decomposition is as follows: V = Vcc + V10 + Via where: Vcc is a regression of V on CC so that: Vcc = rV/CC σV stdCC + μV
V10 is the result of the application of the low-pass filter so that: V10 = D(V - Vcc)
and by linearity of D and the fact that it leaves unchanged low frequencies: V10 = DV - rV/CC σV stdCC - μV
Via are the residuals of the above so that: Via = V - Vcc - V10 = V - DV (after development).
Independance and covariance
Under those notations, the definition of the decomposition is as follows: V = Vcc + V10 + Via where:
and by linearity of D and the fact that it leaves unchanged low frequencies: V10 = DV - rV/CC σV stdCC - μV
Independance and covariance
An ideal method would provide a decomposition with three independant components so that:
σV2 = σVcc2 + σV102 + σVia2
The method is very close to achieving that but not perfectly, so we have covariance terms remaining in the variance decompostion:
σV2 = σVcc2 + σV102 + σVia2 + 2 covar(Vcc,V10) + 2 covar(Vcc,Via) + 2 covar(V10,Via)
The covariance terms represent a very small portion of the total variance, but since they are not null, let's examine them more in details to better understand what we're doing here.
The covariance between Vcc and V10 and between Vcc and Via, given the definitions above, happened to be quite close to 0 (but not 0) and opposed, so that:
covar(Vcc,V10) = - covar(Vcc,Via) = rV/CC σV (rDV/CC σDV - rV/CC σV)
That's what the maths say. Now let's look at maps of those covariance terms for precipitation for Jul-Jun season. Figure 1 shows that both covariance terms are almost everywhere very close to 0, but have sginificant values in some areas. Their sum is almost 0. I suppose it is not exactly 0 because of the approximation I made stating that DVcc = Vcc.
σV2 = σVcc2 + σV102 + σVia2
The method is very close to achieving that but not perfectly, so we have covariance terms remaining in the variance decompostion:
σV2 = σVcc2 + σV102 + σVia2 + 2 covar(Vcc,V10) + 2 covar(Vcc,Via) + 2 covar(V10,Via)
The covariance terms represent a very small portion of the total variance, but since they are not null, let's examine them more in details to better understand what we're doing here.
The covariance between Vcc and V10 and between Vcc and Via, given the definitions above, happened to be quite close to 0 (but not 0) and opposed, so that:
covar(Vcc,V10) = - covar(Vcc,Via) = rV/CC σV (rDV/CC σDV - rV/CC σV)
That's what the maths say. Now let's look at maps of those covariance terms for precipitation for Jul-Jun season. Figure 1 shows that both covariance terms are almost everywhere very close to 0, but have sginificant values in some areas. Their sum is almost 0. I suppose it is not exactly 0 because of the approximation I made stating that DVcc = Vcc.
Figure 1 - From left to right: covariance between Vcc and V10; covariance between Vcc and Via; and their sum
It is interesting to see that the independance of Vcc from V10 and Via relies on the following same condition:
rV/CC σV = rDV/CC σDV
multilying the equation by σCC, we obtain a condition in terms of covariance:
covar(V,CC) = covar(DV,CC)
Regardless, the fact that their sum adds up to 0 means that the variance decomposition is simpler:
σV2 = σVcc2 + σV102 + σVia2 + 2 covar(V10,Via)
The remaining covariance term develops as follows:
covar(V10,Via) = rDV/V σDV σV - σDV2 - rV/CC2 σV2 + rV/CC rDV/CC σV σDV
My primary concern is that that term could become negative, mostly because I have a hard time interpreting what that means physically speaking. Figure 2 shows us that it does happen, and so in more than one location.
rV/CC σV = rDV/CC σDV
multilying the equation by σCC, we obtain a condition in terms of covariance:
covar(V,CC) = covar(DV,CC)
Regardless, the fact that their sum adds up to 0 means that the variance decomposition is simpler:
σV2 = σVcc2 + σV102 + σVia2 + 2 covar(V10,Via)
The remaining covariance term develops as follows:
covar(V10,Via) = rDV/V σDV σV - σDV2 - rV/CC2 σV2 + rV/CC rDV/CC σV σDV
My primary concern is that that term could become negative, mostly because I have a hard time interpreting what that means physically speaking. Figure 2 shows us that it does happen, and so in more than one location.
Figure 2 - Covariance between V10 and Via masking where it is greater than or equal to 0
Now this is rather complex. Let's make our life simpler for a start by supposing that the equality that conditions the independance of Vcc from V10 and Via is always verified. Given Figure 1, it seems to be a reasonable approximation and the last two terms of the covariance term drop:
covar(V10,Via) = rDV/V σDV σV - σDV2
With some development, one can find out that this covariance term is negative if:
covar(DV,V) < covar(DV,DV) ( = σDV2 )
Now let us map this condition in Figure 3: it's very similar to Figure 2, which simply helps believing that all of the above is not crap.
covar(V10,Via) = rDV/V σDV σV - σDV2
With some development, one can find out that this covariance term is negative if:
covar(DV,V) < covar(DV,DV) ( = σDV2 )
Now let us map this condition in Figure 3: it's very similar to Figure 2, which simply helps believing that all of the above is not crap.
Figure 3 - Conditions based on V and DV for covariance between V10 and Viato be lesser than 0