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Fig. 5.4 Discrete form of air dry bulb temperature at the Brue catchment [ 16 ]
problem. The optimal number of intervals for entropy calculations in a continuous
form is determined base on this objective function:
P
n
j
p ð x i Þ z ð x i Þ
j
i¼1
Min
ð 5 : 10 Þ
n
where p(x i ) is PDF (Probability Density Function), z(xi) i ) is the value of the discrete
function for xi i and n is the number of intervals. PDF is a function which can be
integrated to obtain the probability that the random variable takes a value in a given
interval. By resampling the data, the class interval size in Eq. ( 5.10 ) is assumed to
be the same for all inputs as 5. The marginal entropy refers to the average infor-
mation content of a random variable and it is used as a measure of uncertainty or the
information content. The number of data to compute the values of different
entropies is 1,460 in this case. The first four scenarios in Fig. 5.5 show the values of
different entropy measures. The conditional entropy is a measure of the information
content of input variable which is not contained in the output variable. The join
entropy is the total information content in both input and output variables. The
mutual information, also called transinformation, is the information between each
input and output values. It can be de
ned as the information content of the input
which is contained in the output and can be interpreted as the reduction in uncer-
tainty in the input due to the knowledge of the output variable.
As it can be seen in Fig. 5.5 , based on the transinformation values for different
inputs, horizon extraterrestrial radiation (ETR), air dry bulb temperature (DT) and
wet bulb temperature (WT) are observed as the most signi
cant inputs in solar
radiation modelling (Scenarios 1
3). Atmospheric pressure (p) transfers less
information about solar radiation (Scenario 4), therefore it is not very relevant in the
modelling process. Different combinations of input data were explored to assess
their in
-
uence on the solar radiation modelling (Fig. 5.5 ). The maximum value of
transinformation was observed when we used all the available input data sets, i.e.
the Scenario 15. From the
figure we can deduce that the combination of horizon
 
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