172
7 Integration of Functionals, PCM and Stochastic IntegralEquations
−
{i 1 ,··· ,i k−1 }⊂{j 1 ,··· ,j k }
Z i 1 ,··· ,i k−1 (ξ i 1 , · · · , ξ i k−1 )
−
{i 1 ,··· ,i k−2 }⊂{j 1 ,··· ,j k }
Z i 1 ,··· ,i k−2 (ξ i 1 , · · · , ξ i k−2 )
. . .
−
j 1
Z j 1 (ξ j 1 ) − Z 0
· · · = · · ·
(7.34)
where we have introduced the notation
Z(ξ )| ξ =a\{ξ j 1 ,··· ,ξ j k } = Z(a 1 , · · · , a j 1 −1 , ξ j 1 , · · · , ξ j k , a j k +1 , · · · , a M ) .
(7.35)
As an example, we’ll expand Z(ξ 1 , ξ 2 ) = Z 0 + Z 1 (ξ 1 ) + Z 2 (ξ 2 ) + Z 12 (ξ 1 , ξ 2 ),
where:
Z 0
= P 1 P 2 Z(ξ 1 , ξ 2 )
= Z(a 1 , a 2 )
Z 1 (ξ 1 )
= (I − P 1 )P 2 Z(ξ 1 , ξ 2 )
= Z(ξ 1 , a 2 ) − Z(a 1 , a 2 )
Z 2 (ξ 2 )
= P 1 (I − P 2 )Z(ξ 1 , ξ 2 )
= Z(a 1 , ξ 2 ) − Z(a 1 , a 2 )
Z 12 (ξ 1 , ξ 2 ) = (I − P 1 )(I − P 2 )Z(ξ 1 , ξ 2 ) = Z(ξ 1 , ξ 2 ) − Z(ξ 1 , a 2 ) − Z(a 1 , ξ 2 ) + Z(a 1 , a 2 )
= Z(ξ 1 , ξ 2 ) − Z 1 (ξ 1 ) − Z 2 (ξ 2 ) − Z 0 .
(7.36)
Thus, we see that (7.36), and by extension, (7.27), is a multi-scale expansion of an
M−dimensional function along points, lines, faces, hyperplanes, etc., which pass
through the anchor point, a.
We can draw some general conclusions from (7.36). First, we note that the mean
value of the higher-order terms (beyond Z 0 ) is zero, which means that these terms
are orthogonal to Z 0 , a constant. It also means that Z 0 is the mean value of Z.
Recall that to compute the mean, we simply replace the ‘free variable,’ ξ i , by the
fixed anchor point, a i . Secondly, the correlation between each term vanishes; i.e.,
(7.27) is an expansion in orthogonal functions. Finally, it is easy to show that
Z 2
1 (ξ 1 ) = Z 2 (ξ 1 , a 2 ) − Z
2
0
Z 2
2 (ξ 2 ) = Z 2 (a 1 , ξ 2 ) − Z
2
0
Z 2
12 (ξ 1 , ξ 2 ) = Z 2 (ξ 1 , ξ 2 ) − Z 2
1 (ξ 1 ) − Z 2
2 (ξ 2 ) − Z
2
0 ,
(7.37)
from which we draw the important conclusion that
Z 2
1 + Z 2
2 + Z 2
12 = Z 2 − Z
2
0 = VAR[Z] .
(7.38)
7 Integration of Functionals, PCM and Stochastic IntegralEquations
−
{i 1 ,··· ,i k−1 }⊂{j 1 ,··· ,j k }
Z i 1 ,··· ,i k−1 (ξ i 1 , · · · , ξ i k−1 )
−
{i 1 ,··· ,i k−2 }⊂{j 1 ,··· ,j k }
Z i 1 ,··· ,i k−2 (ξ i 1 , · · · , ξ i k−2 )
. . .
−
j 1
Z j 1 (ξ j 1 ) − Z 0
· · · = · · ·
(7.34)
where we have introduced the notation
Z(ξ )| ξ =a\{ξ j 1 ,··· ,ξ j k } = Z(a 1 , · · · , a j 1 −1 , ξ j 1 , · · · , ξ j k , a j k +1 , · · · , a M ) .
(7.35)
As an example, we’ll expand Z(ξ 1 , ξ 2 ) = Z 0 + Z 1 (ξ 1 ) + Z 2 (ξ 2 ) + Z 12 (ξ 1 , ξ 2 ),
where:
Z 0
= P 1 P 2 Z(ξ 1 , ξ 2 )
= Z(a 1 , a 2 )
Z 1 (ξ 1 )
= (I − P 1 )P 2 Z(ξ 1 , ξ 2 )
= Z(ξ 1 , a 2 ) − Z(a 1 , a 2 )
Z 2 (ξ 2 )
= P 1 (I − P 2 )Z(ξ 1 , ξ 2 )
= Z(a 1 , ξ 2 ) − Z(a 1 , a 2 )
Z 12 (ξ 1 , ξ 2 ) = (I − P 1 )(I − P 2 )Z(ξ 1 , ξ 2 ) = Z(ξ 1 , ξ 2 ) − Z(ξ 1 , a 2 ) − Z(a 1 , ξ 2 ) + Z(a 1 , a 2 )
= Z(ξ 1 , ξ 2 ) − Z 1 (ξ 1 ) − Z 2 (ξ 2 ) − Z 0 .
(7.36)
Thus, we see that (7.36), and by extension, (7.27), is a multi-scale expansion of an
M−dimensional function along points, lines, faces, hyperplanes, etc., which pass
through the anchor point, a.
We can draw some general conclusions from (7.36). First, we note that the mean
value of the higher-order terms (beyond Z 0 ) is zero, which means that these terms
are orthogonal to Z 0 , a constant. It also means that Z 0 is the mean value of Z.
Recall that to compute the mean, we simply replace the ‘free variable,’ ξ i , by the
fixed anchor point, a i . Secondly, the correlation between each term vanishes; i.e.,
(7.27) is an expansion in orthogonal functions. Finally, it is easy to show that
Z 2
1 (ξ 1 ) = Z 2 (ξ 1 , a 2 ) − Z
2
0
Z 2
2 (ξ 2 ) = Z 2 (a 1 , ξ 2 ) − Z
2
0
Z 2
12 (ξ 1 , ξ 2 ) = Z 2 (ξ 1 , ξ 2 ) − Z 2
1 (ξ 1 ) − Z 2
2 (ξ 2 ) − Z
2
0 ,
(7.37)
from which we draw the important conclusion that
Z 2
1 + Z 2
2 + Z 2
12 = Z 2 − Z
2
0 = VAR[Z] .
(7.38)