7.7 Determining the ANOVA Anchor Point
173
Equation (7.38) allows us to identify the terms beyond Z 0 as contributing partial
variances to the overall variance. With this we can define ‘global sensitivity indices’
as the ratio of each of these partial variances to the overall variance. These indices
describe the contribution of the corresponding inputs, {ξ j 1 , · · · , ξ j k }, to the variance
of the output [46].
It is pointed out in [46] and [65], and the references therein, that there is a
close relationship between the multi-dimensional Taylor expansion and the ANOVA
expansion, (7.27). The infinite number of terms in the Taylor expansion
Z(ξ ) = Z(a) +
∞
j =1
1
j !
M
i=1
∂ j Z
∂ξ
j
i
(a)(ξ i − a i )
j
+
∞
j i ,j 2 >0
1
j 1 !j 2 !
i 1 ∂ j 1 +j 2 Z
∂ξ
j 1
i 1
∂ξ
j 2
i 2
(a)(ξ i 1 − a i 1 )
j 1 (ξ i 2 − a i 2 )
j 2
+ · · ·
(7.39)
are partitioned into a finite number of groups, with each group corresponding to one
of the component functions of (7.27). For example, the first-order function, Z j 1 (ξ j 1 ),
is the sum of all of the Taylor series terms that contain only the variable, ξ j 1 , and so
on. This suggests that a truncated ANOVA expansion should be more accurate than
a truncated Taylor series of the same order [65].
7.7 Determining the ANOVA Anchor Point
There are a number of options for choosing the anchor point [37, 42, 46, 65]. We
will use the point in ξ -space that corresponds to the mean of a ten-sample Monte
Carlo run. This will require a straight-forward inverse problem of the type that
we have done before in other random-characterization problems. The idea is to
replace the random surface with an equivalent homogeneous (non-random) surface
that produces the mean output of the ten VIC-3D® runs. This procedure is called
‘homogenization,’ and is an active area of mathematics research [13].
We use the ten sample functions for L/δ = 1, shown in Fig. 7.7, as the input to
VIC-3D®, which then produces the ten-sample set of impedances responses, shown
in Fig. 7.8. The mean of these ten impedances is shown in Fig. 7.9, and this will be
the input to NLSE to finish the homogenization process.
The best fit to the data of Fig. 7.9 is given when the homogeneous conductivity of
the surface is σ H = 2.8 × 10 5 S/m, and the resulting impedance response when this
conductivity is used is compared with the original data in Fig. 7.10. Clearly, there
is a good fit. Furthermore, we note that the equivalent homogeneous conductivity
173
Equation (7.38) allows us to identify the terms beyond Z 0 as contributing partial
variances to the overall variance. With this we can define ‘global sensitivity indices’
as the ratio of each of these partial variances to the overall variance. These indices
describe the contribution of the corresponding inputs, {ξ j 1 , · · · , ξ j k }, to the variance
of the output [46].
It is pointed out in [46] and [65], and the references therein, that there is a
close relationship between the multi-dimensional Taylor expansion and the ANOVA
expansion, (7.27). The infinite number of terms in the Taylor expansion
Z(ξ ) = Z(a) +
∞
j =1
1
j !
M
i=1
∂ j Z
∂ξ
j
i
(a)(ξ i − a i )
j
+
∞
j i ,j 2 >0
1
j 1 !j 2 !
i 1 ∂ j 1 +j 2 Z
∂ξ
j 1
i 1
∂ξ
j 2
i 2
(a)(ξ i 1 − a i 1 )
j 1 (ξ i 2 − a i 2 )
j 2
+ · · ·
(7.39)
are partitioned into a finite number of groups, with each group corresponding to one
of the component functions of (7.27). For example, the first-order function, Z j 1 (ξ j 1 ),
is the sum of all of the Taylor series terms that contain only the variable, ξ j 1 , and so
on. This suggests that a truncated ANOVA expansion should be more accurate than
a truncated Taylor series of the same order [65].
7.7 Determining the ANOVA Anchor Point
There are a number of options for choosing the anchor point [37, 42, 46, 65]. We
will use the point in ξ -space that corresponds to the mean of a ten-sample Monte
Carlo run. This will require a straight-forward inverse problem of the type that
we have done before in other random-characterization problems. The idea is to
replace the random surface with an equivalent homogeneous (non-random) surface
that produces the mean output of the ten VIC-3D® runs. This procedure is called
‘homogenization,’ and is an active area of mathematics research [13].
We use the ten sample functions for L/δ = 1, shown in Fig. 7.7, as the input to
VIC-3D®, which then produces the ten-sample set of impedances responses, shown
in Fig. 7.8. The mean of these ten impedances is shown in Fig. 7.9, and this will be
the input to NLSE to finish the homogenization process.
The best fit to the data of Fig. 7.9 is given when the homogeneous conductivity of
the surface is σ H = 2.8 × 10 5 S/m, and the resulting impedance response when this
conductivity is used is compared with the original data in Fig. 7.10. Clearly, there
is a good fit. Furthermore, we note that the equivalent homogeneous conductivity