Environmental Decision-Making - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 235

u p =

For c 16 we have:

\

u a \ u b ; c = x p ð c Þ \ c = ð 3 : 20 Þ

where a = 0.5pexp( − 2c) and b = p/2.

When c → ∞ , we have:

x p ðÞ ¼1 þ u p c 1 = 2

þ O 1 ðÞ = 3 : 21 Þ

From Eqs. ( 3.13 ) and ( 3.21 ) we obtain:

h i

Þ c 1 = 2

W c ðÞ ¼ U ð x 1 þ O 1 ðÞ = 3 : 22 Þ

m= E m is asymptotically normal with average value and
dispersion equal to 1 and 1/c, respectively, when c → ∞

Thus, the random value

. The expressions ( 3.13 )
and ( 3.22 ) can receive various analytical approximations of W c . For example, let us
represent

ʦ (y) by the following:

U ðÞ ¼ U y ðÞþu y ðÞ y y 0 ð Þ 1 0 ½ : 5y 0 y y 0 ð Þ ; ð 3 : 23 Þ

1 = 2 1 = 2

where y ¼ x 1 and x 0 is some point where

ð Þ c ðÞ = ; y 0 ¼ x 0 1 ð Þ c ð = x 0 Þ

the value of was estimated.

The following formula can be easily derived using Eqs. ( 3.12 ), ( 3.13 ) and ( 3.23 ):

ʦ

W c ðÞ ¼W c x ðÞþu y ðÞ y y 0 ð Þ 1 0 ½ : 5y 0 y y 0 ð Þ

þumðÞm m 0 ð Þ exp 2 ðÞ 1 0 ½ : 5 m 0 ð m m 0 Þ ; ð 3 : 24 Þ

where

1 = 2 1 = 2 and

m ¼ x þ 1 ð Þ c ðÞ = ; m 0 ¼ x 0 þ 1 ð Þ c ð = x 0 Þ

h i

z z 0

5 z 0 1

2

u ðÞ ¼ u z ðÞ 1 z 0 z z 0 ð Þ 0 : ð Þ :

Let us designate

Z

x

exp t 2 dt

2
p

H ð x Þ ¼ p ð 3 : 25 Þ

0

We have