An alternative unrestricted method is proposed by Sen (2001) for preserving the means and variances of the global irradiation and the sunshine duration data. In the restrictive regression approach (Angstrom equation), the cross-correlation coefficient between H/H0 and S/S0 represents linear relationship only. By not considering this coefficient in the UM, some non-linearity features in the solar irradiation-sunshine duration relationship are taken into account. Especially, when the scatter diagram of solar irradiation versus sunshine duration does not show any distinguishable pattern such as a straight-line or a curve, then the use of UM is recommended for parameter estimations.
In practice, the estimation of model parameters is achieved most often by the least squares method and regression technique using procedural assumptions and restrictions in the parameter estimations. Such restrictions, however, are unnecessary because procedural restrictions might lead to unreliable biases in the parameter estimations. One critical assumption for the success of the regression equation is that the variables considered over certain time intervals are distributed normally, i. e. according to Gaussian PDF. As the time interval becomes smaller, the deviations from the Gaussian (normal) distribution become greater. For example, the relative frequency distribution of daily solar irradiation or sunshine duration has
more skewness compared to the monthly or annual PDFs. The averages and variances of the solar irradiation and sunshine duration data play predominant role in many calculations and the conservation of these parameters is regarded as more important than the cross-correlation coefficient in any prediction model. In Gordon and Reddy (1988), it is stated that a simple functional form for the stationary relative frequency distribution for daily solar irradiation requires knowledge of the mean and variance only. Unfortunately, in almost any estimate of solar irradiation by means of computer software, the parameter estimations are achieved without caring about the theoretical restrictions in the regression approach. This is a very common practice in the use of the Angstrom equation all over the world.
The application of the regression technique to Eq. (6.10) for estimating the model parameters from the available data leads to new statistical approach (Sen 2001)
where rhs is the cross-correlation coefficient between global solar irradiation and sunshine duration data, Var(.) is the variance of the argument; and the overbar (-) indicates arithmetic averages during a basic time interval. Most often in solar engineering, the time interval is taken as a month or a day and in rare cases as a season or a year. As a result of the classical regression technique, the variance of predictand, given the value of predictor is
This expression provides the mathematical basis for interpreting r2, as the proportion of variability in (H/H0) that can be explained by knowing (S/S0) from Eq. (6.24), one can obtain after arrangements
In this expression, if the second term in the numerator is equal to 0, then the regression coefficient will be equal to 1. This is tantamount to saying that by knowing (S/S0) there is no variability in (H/H0). Similarly, if it is assumed that
0. This means that by knowing (S/S0) the variability in (H/H0) does not change. In this manner, r2s can be interpreted as the proportion of variability in (H/H0) that is explained by knowing (S/S0). In all the restrictive interpretations, one should keep in mind that the cross-correlation coefficient is defined for joint Gaussian (normal) PDF of the global solar irradiation and sunshine duration data. The requirement of normality is not valid, 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 with Gaussian (normal) PDF. As mentioned before, there are six restrictive assumptions in the regression equation parameter estimations such as used in the Angstrom equation that should be critically taken into consideration prior to any application. 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. (6.10) lead without any procedural restrictive assumptions to the following equations,
Var(H/H0) = b’2Var(S/S0) (6.27)
where for distinction, the UM parameters are shown as a’ and b’, respectively. These two equations are the basis for the conservation of the arithmetic mean and variances of global solar irradiation and sunshine duration data. The basic Angstrom equation remains unchanged whether the restrictive or unrestrictive model is used. Equation (6.26) implies that in both models the centroid, i. e. averages of the solar irradiation and sunshine duration, data are preserved equally, hence both models yield close estimations around the centroid. The deviations between the two model estimations appear at solar irradiation and sunshine duration data values away from the arithmetic averages. Simultaneous solution of Eqs. (6.26) and (6.27) yields parameter estimates as,
H_ _ lVar(H/H0) ҐS H0J Var(S/S0) UJ
Physically, variations in the solar irradiation data are always smaller than the sunshine duration data, and consequently, Var(S/S0) > Var(H/H0) and for Eq. (6.28) this means that 0 < b’ < 1. Furthermore, Eq. (6.28) is a special case of Eq. (6.22) when rsh = 1 and the same is valid between Eqs. (6.23) and (6.29). In fact, from these explanations, it is clear that all of the bias effects from the restrictive
assumptions are represented globally in rsh, which does not appear in the UM parameter estimations. The second term in Eq. (6.29) is always smaller than the first one, and hence a’ is always positive. The following relationships are valid between the restrictive and UM parameters
These theoretical relationships between the parameters of the two models imply that b and b’ are the slopes of the restrictive models. The slope of the restricted (Angstrom) equation is larger than the UM (b’ > b) according to Eq. (6.30) since always 0 < rhs < 1 for global solar irradiation and sunshine duration data scatter on a Cartesian coordinate system. As mentioned previously, two methods almost coincide practically around the centroid (averages of global solar irradiation and sunshine duration data). This further indicates that under the light of the previous statement, the UM over-estimates for sunshine duration data greater than the average value and under-estimates the solar irradiation for sunshine duration data smaller than the average. On the other hand, Eq. (6.31) shows that a’ < a. Furthermore, the summation of model parameters is,
These last expressions indicate that the two approaches are completely equivalent to each other for rhs = 1. The UM is essentially described by Eqs. (6.26), (6.30) and (6.31). Its application supposes that the restricted model is first used to obtain a’, b’ and r. If r is close to 1, then the classical Angstrom equation coefficient estimations with restrictions are almost equivalent to a’ and b’. Otherwise, the UM results should be considered for application as in Eq. (6.26).