N

where rhs is the cross-correlation coefficient between global solar radiation and sunshine duration data, Var (...) is the variance of the argument, and the over-bars indicate arithmetic averages. As a result of the classic regression technique, the variance of the predictand, given the value of predictor is

Var [(h/hOO) / (S/SQ) = S/S0] = (1 - r^Var (h/hQ) . (4.41)

This expression prov(des the mathematical basis for interpreting r(s as the proportion of variability in (H/H0) that can be explained provided that (S/S0) is given. From Eq. 4.41, after rearrangement, one can obtain

2 _ Var (H/Hp) - Var [(h/hQ) / (S/SQ) = S/S0] (4 42)

rs Var(HWo)

If the second term in the numerator is equal to zero, then the regression coefficient will be equal to one. This is tantamount to saying that given S/S0 there is no variability in (H/H0). Similarly, if it is assumed that Var[(H/Hp) / (S^ = (S/So^ — Var(H/H0), then the regression coefficient will be zero. This means that given (S/S0) the variability in (H/H0) does not (hange. In this manner, r^ can be in-(erpreted as the proportion of variability in (H/H0) that is explained by knowing (S/S0). The requirement of normality is not satisfied, especially if the period for taking averages is less than one year. Since, daily or monthly data are used in most practical applications it is over-simplification to expect marginal or joint distributions to abide by the Gaussian (normal) PDF.

The UM parameter estimations require two simultaneous equations since there are two parameters to be determined. The average and the variance of both sides in Eq. 4.30 lead without any procedural restrictive assumptions to the following equations (§en 2001b):

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