Empirical values of the parameters α and β in eq 1 can be evaluated from the
sample mean of x, x , and the sample standard deviation, s:
(
)
x 1- x
α = x
- 1 ,
^
(2a)
s2
(
)
x 1- x
(
)
^ = 1- x
- 1 .
β
(2b)
s2
^
Table 4 shows average values of the α and β parameters for winter calculated
^
from eq 2 using the monthly 3-hour series of total cloud amount described above.
In other words, in preparing Table 4, we had about 70 months of data to use in
^
computing each month's average α and β values. We did, however, exclude
^
approximately 10 monthly values from each set of calculations, either because the
correlation between T and n was weak or because the cloud distribution was not
obviously U-shaped.
^
Table 4. Values of α and β for the beta distributions describing
^
total cloud amount in the winter.
November
December
January
February
March
Mean* Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
α
^
0.24
0.18
0.18
0.16
0.19
0.18
0.23
0.18
0.31
0.21
^
β
0.17
0.11
0.17
0.13
0.19
0.15
0.22
0.15
0.25
0.16
* "Mean" is the value averaged from roughly 70 months of fitted beta distribu-
tions; "Std" is the standard deviation of the values used to create the means.
Our method for statistically modeling cloud amount compatible with a beta
~
distribution goes as follows. Let T be the normalized surface-layer temperature.
~
As such, T is a random variable that is approximately normally distributed with
~
mean 0 and standard deviation 1. On assuming that the total cloud amount n( T )
~
is a monotonic function of T , we know from mathematical statistics that the fol-
~
lowing expression relates n to T (Ventcel 1964, p. 263 ff.):
[ ( )] ,
()
-
~
~
n T = Φ 21 Φ1 T
(3)
~
~
Φ 21(n) is the inverse function of Φ2(n).
-
Since Φ2(n) describes a beta distribution, eq 3 can be approximated as (Aivazyan
et al. 1983):
()
α
^
~
nT ≅
[ ( )] ,
^
(4)
~
α + β exp 2 w T
^
^
where α and β are the parameters of the beta distribution given in Table 4, and
^
~
w( T ) is a function of the normalized air temperature. Appendix A fills in the math-
~
ematics on which eq 3 and 4 are based and gives the functional form for w( T ). In
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