170
7 Integration of Functionals, PCM and Stochastic IntegralEquations
7.6 HDMR and ANOVA
High-dimensional model representation (HDMR) and analysis of variance
(ANOVA) 2 seek to reduce the complexity of problems with a large number
of dimensions, and are the subject of considerable contemporary research in
computational methods [15, 37, 46, 65].
Instead of the gPC expansion of (7.22), we consider an ANOVA decomposition
as (we suppress the scan variable, l, to simplify notation):
Z(ξ 1 , · · · , ξ M ) = Z 0 +
M
j 1
Z j 1 (ξ j 1 ) +
M
j 1
+
j 1
(7.27)
where Z 0 is a constant function, the {Z j 1 } are one-dimensional functions, {Z j 1 ,j 2 }
are two-dimensional functions, and so on, yielding 2 M different terms [46]. For
example, the expansion in five-dimensional space is given by
Z(ξ 1 , · · · , ξ 5 ) = Z 0 + Z 1 (ξ 1 ) + Z 2 (ξ 2 ) + Z 3 (ξ 3 ) + Z 4 (ξ 4 ) + Z 5 (ξ 5 )
+Z 12 (ξ 1 , ξ 2 ) + Z 13 (ξ 1 , ξ 3 ) + Z 14 (ξ 1 , ξ 4 ) + Z 15 (ξ 1 , ξ 5 )
+Z 23 (ξ 2 , ξ 3 ) + Z 24 (ξ 2 , ξ 4 ) + Z 25 (ξ 2 , ξ 5 ) + Z 34 (ξ 3 , ξ 4 )
+Z 35 (ξ 3 , ξ 5 )+Z 45 (ξ 4 , ξ 5 )+Z 123 (ξ 1 , ξ 2 , ξ 3 ) + Z 124 (ξ 1 , ξ 2 , ξ 4 ) + Z 125 (ξ 1 , ξ 2 , ξ 5 )
+Z 134 (ξ 1 , ξ 3 , ξ 4 ) + Z 135 (ξ 1 , ξ 3 , ξ 5 ) + Z 145 (ξ 1 , ξ 4 , ξ 5 ) + Z 234 (ξ 2 , ξ 3 , ξ 4 )
+Z 235 (ξ 2 , ξ 3 , ξ 5 ) + Z 245 (ξ 2 , ξ 4 , ξ 5 ) + Z 345 (ξ 3 , ξ 4 , ξ 5 ) + Z 1234 (ξ 1 , ξ 2 , ξ 3 , ξ 4 )
+Z 1235 (ξ 1 , ξ 2 , ξ 3 , ξ 5 ) + Z 1245 (ξ 1 , ξ 2 , ξ 4 , ξ 5 ) + Z 1345 (ξ 1 , ξ 3 , ξ 4 , ξ 5 )
+Z 2345 (ξ 2 , ξ 3 , ξ 4 , ξ 5 ) + Z 12345 (ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 ) .
(7.28)
The higher-order terms in (7.27) and (7.28) express the effects of correlations
between the random variables, and if only a few of them are non-negligible, then we
have a good shot at breaking the ‘curse of dimensionality’ that haunts approximation
theory in high-dimensional spaces, but first we will give an algorithm for generating
the terms in the ANOVA expansion [46], [42].
Define a one-dimensional projection operator
Pf (x) =
f (x)dμ(x) ,
(7.29)
2 ANOVA is often referred to in the mathematics literature as Kolmogorov’s superposition theorem.
7 Integration of Functionals, PCM and Stochastic IntegralEquations
7.6 HDMR and ANOVA
High-dimensional model representation (HDMR) and analysis of variance
(ANOVA) 2 seek to reduce the complexity of problems with a large number
of dimensions, and are the subject of considerable contemporary research in
computational methods [15, 37, 46, 65].
Instead of the gPC expansion of (7.22), we consider an ANOVA decomposition
as (we suppress the scan variable, l, to simplify notation):
Z(ξ 1 , · · · , ξ M ) = Z 0 +
M
j 1
Z j 1 (ξ j 1 ) +
M
j 1
+
j 1
(7.27)
where Z 0 is a constant function, the {Z j 1 } are one-dimensional functions, {Z j 1 ,j 2 }
are two-dimensional functions, and so on, yielding 2 M different terms [46]. For
example, the expansion in five-dimensional space is given by
Z(ξ 1 , · · · , ξ 5 ) = Z 0 + Z 1 (ξ 1 ) + Z 2 (ξ 2 ) + Z 3 (ξ 3 ) + Z 4 (ξ 4 ) + Z 5 (ξ 5 )
+Z 12 (ξ 1 , ξ 2 ) + Z 13 (ξ 1 , ξ 3 ) + Z 14 (ξ 1 , ξ 4 ) + Z 15 (ξ 1 , ξ 5 )
+Z 23 (ξ 2 , ξ 3 ) + Z 24 (ξ 2 , ξ 4 ) + Z 25 (ξ 2 , ξ 5 ) + Z 34 (ξ 3 , ξ 4 )
+Z 35 (ξ 3 , ξ 5 )+Z 45 (ξ 4 , ξ 5 )+Z 123 (ξ 1 , ξ 2 , ξ 3 ) + Z 124 (ξ 1 , ξ 2 , ξ 4 ) + Z 125 (ξ 1 , ξ 2 , ξ 5 )
+Z 134 (ξ 1 , ξ 3 , ξ 4 ) + Z 135 (ξ 1 , ξ 3 , ξ 5 ) + Z 145 (ξ 1 , ξ 4 , ξ 5 ) + Z 234 (ξ 2 , ξ 3 , ξ 4 )
+Z 235 (ξ 2 , ξ 3 , ξ 5 ) + Z 245 (ξ 2 , ξ 4 , ξ 5 ) + Z 345 (ξ 3 , ξ 4 , ξ 5 ) + Z 1234 (ξ 1 , ξ 2 , ξ 3 , ξ 4 )
+Z 1235 (ξ 1 , ξ 2 , ξ 3 , ξ 5 ) + Z 1245 (ξ 1 , ξ 2 , ξ 4 , ξ 5 ) + Z 1345 (ξ 1 , ξ 3 , ξ 4 , ξ 5 )
+Z 2345 (ξ 2 , ξ 3 , ξ 4 , ξ 5 ) + Z 12345 (ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 ) .
(7.28)
The higher-order terms in (7.27) and (7.28) express the effects of correlations
between the random variables, and if only a few of them are non-negligible, then we
have a good shot at breaking the ‘curse of dimensionality’ that haunts approximation
theory in high-dimensional spaces, but first we will give an algorithm for generating
the terms in the ANOVA expansion [46], [42].
Define a one-dimensional projection operator
Pf (x) =
f (x)dμ(x) ,
(7.29)
2 ANOVA is often referred to in the mathematics literature as Kolmogorov’s superposition theorem.