The non-linear least squares technique depends on the minimization of the prediction error square summation from a non-linear equation. The non-linearity exists in the power term of the solar radiation model as presented in the text by Eq. 5.12. In order to predict the solar radiation amount (H/H0) at any time instant, say i, from the fractal exponent model there is an error, є;, involved as follows:
(H)-ap+41)’+й (B1)
or the error term is calculated as
S p
H0] ; S0, ;
and the sum of error squares for n predictions becomes notationally as
2
The partial derivatives of this expression with respect to model parameters a, b, and c leads to
In order to find the optimum solution of parameter estimates these three differentials must be set equal to zero:
and
Hence, there are three unknowns and three equations. However, the analytical and simultaneous solution of these three equations is not possible, and therefore, the numerical solution is sought. For this purpose, first of all it is possible to obtain from Eqs. B.6 and B.7 by elimination the following parameter estimations:
These are the two basic equations that reduce to the linear regression line coefficient estimations for p = 1. This situation is equivalent with the AM parameter estimation. The third equation of the non-linear least squares technique can be ob-
[1] Dust, snow, dew, water droplets, bird droppings, etc.
[2] Complete or partial shade-ring misalignment
[3] Incorrect sensor leveling
[4] Shading caused by building structures
[5] All the SIPMs have high H/H0 values for high S/S0 values, and therefore, solar radiation increases with an increase in sunshine duration.
