Solar energy in progress and future research trends
Unrestricted model
An alternative unrestricted method (UM) is proposed by §en [128] for preserving the means and variances of the global irradiation and the sunshine duration data.
In the restrictive regression approach (Angstrom equation), the crosscorrelation coefficient represents only linear relationships. By not considering this coefficient in the UM, some nonlinearity features in the solar irradiationsunshine 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 straightline or a curve, then the use of UM is recommended for parameter estimations.
All the aforementioned equations are based on the classical least squares estimation, which has hidden restrictive assumptions of the linearity, normality, homo  scadascity, correlation coefficient and error free data measurements [15]. Nonlinear models (Eqs. (41)(44)) are based on nonlinear regression technique, which impose further assumptions on the parameter estimations. Hence, it is expected that all these models are affected by a set of assumptions, the validity of which are almost not checked in any practical study. It is therefore, necessary to provide a simple model, which will not require the restrictive assumptions.
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 probability distribution function. 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 is more skewed compared to the monthly or annual distributions. On the other hand, the application of the regression technique to Eq. (36) imposes linearity by the crosscorrelation coefficient between the solar irradiation and sunshine duration data. However, the averages and variances of the solar irradiation and sunshine duration data play predominant role in many calculations and the conservation of these parameters becomes more important than the crosscorrelation coefficient in any prediction model. In Gordon and Reddy [48], 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.
The application of the regression technique to Eq. (36) for estimating the model parameters from the available data



























on a Cartesian coordinate system (Fig. 18). 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 unrestricted model overestimates compared to the Angstrom estimations for sunshine duration data greater than the average value and underestimates the solar irradiation for sunshine duration data smaller than the average. On the other hand, Eq. (55) shows that a' < a. Furthermore, the summation of model parameters is
a + b 1 H
rhs rhs H0
These last expressions indicate that the two approaches are completely equivalent to each other for rhs = 1. The new model is essentially described by Eqs. (50), (54) and (55). 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 coefficients estimation with restrictions is almost equivalent to a' and b'. Otherwise, the new model results should be considered for application as in Eq. (50).
Through the unrestricted model, it has been observed that in the classical regression technique, requirements of normality in the frequency distribution function and of linearity and the use of the crosscorrelation coefficient are imbedded unnecessarily in the parameter estimations. Assumptions in the restrictive (Angstrom) model cause overestimations in the solar irradiance amounts as suggested by Angstrom for small (smaller than the arithmetic average) sunshine duration and underestimations for large sunshine duration values. Around the average values solar irradiation and sunshine duration values are close to each other for both models, however, the unrestricted approach alleviates these biasedestimation situations. Additionally, the suggested unrestricted model includes some features of nonlinearity in the solar energy data scatter diagram by ignoring consideration of crosscorrelation coefficient.