190
7 Integration of Functionals, PCM and Stochastic IntegralEquations
We use the Chebychev Inequality [63] to quantify the probability of H 0 . This
inequality states that for a random variable, Z, and a real number, , we have
P [|Z| ≥ ] ≤
VAR(Z)
2
.
(7.57)
Z is the impedance measured due to the random surface, and the term on the right
side of (7.57) can be called the ‘maximum uncertainty’ associated with Z for a given
. This is the metric to be associated with H 0 .
Because H 0 and H 1 are complementary processes (their sample spaces are
disjoint), it follows that their probabilities must sum to unity. Hence, we can assign
a ‘minimum certainty’ metric to H 1 by
Minimum Certainty(() = 1 −
VAR(Z)
2
.
(7.58)
By referring to this as the ‘minimum certainty,’ we are claiming that we are at least
this certain of a correct decision.
Figure 7.20 shows ‘probability of detection’ (POD) curves for the detectability
of a flaw located at two different points within the random surface. These curves
are plots of (7.58), in which plays the role of a ‘threshold variable’ or ‘decision
boundary,’ that will allow us to determine whether to choose H 0 or H 1 (see
Fig. 7.21).
As we suspected, a flaw located at the center of the random surface is much more
likely to be detected than one located away from the center, because its threshold
of detectability is much smaller. This does not address the question of how the size
of the flaw enters the picture. That can only be determined by solving a series of
forward problems for a given probe, frequency and size and shape of the flaw.
7.11 Consistency of Calculations
We know that the mean of the ANOVA expansion is obtained by substituting the
anchor point, a, for the generic point in ξ -space in the general ANOVA expansion.
We get the same result by analytically integrating the expressions for the firstorder and second-order functions given by the quadratic polynomial interpolation
expansions. This yields
Z(ξ ) = (Z −0.5 + 4Z 0 + Z 0.5 )/6 − Z(a)
Z(ξ 1 , ξ 2 ) = (z 00 + 4z 01 + z 02 + 4z 10 + 16z 11 + 4z 12
+ z 20 + 4z 21 + z 22 )/36 − Z(ξ 1 ) − Z(ξ 2 ) − Z(a) .
(7.59)
The mean values vanish when numerical values for Z −0.5 , Z 0 , and Z 0.5 , as well as
z 00 , · · · , z 22 , as presented in (7.56), are used.
7 Integration of Functionals, PCM and Stochastic IntegralEquations
We use the Chebychev Inequality [63] to quantify the probability of H 0 . This
inequality states that for a random variable, Z, and a real number, , we have
P [|Z| ≥ ] ≤
VAR(Z)
2
.
(7.57)
Z is the impedance measured due to the random surface, and the term on the right
side of (7.57) can be called the ‘maximum uncertainty’ associated with Z for a given
. This is the metric to be associated with H 0 .
Because H 0 and H 1 are complementary processes (their sample spaces are
disjoint), it follows that their probabilities must sum to unity. Hence, we can assign
a ‘minimum certainty’ metric to H 1 by
Minimum Certainty(() = 1 −
VAR(Z)
2
.
(7.58)
By referring to this as the ‘minimum certainty,’ we are claiming that we are at least
this certain of a correct decision.
Figure 7.20 shows ‘probability of detection’ (POD) curves for the detectability
of a flaw located at two different points within the random surface. These curves
are plots of (7.58), in which plays the role of a ‘threshold variable’ or ‘decision
boundary,’ that will allow us to determine whether to choose H 0 or H 1 (see
Fig. 7.21).
As we suspected, a flaw located at the center of the random surface is much more
likely to be detected than one located away from the center, because its threshold
of detectability is much smaller. This does not address the question of how the size
of the flaw enters the picture. That can only be determined by solving a series of
forward problems for a given probe, frequency and size and shape of the flaw.
7.11 Consistency of Calculations
We know that the mean of the ANOVA expansion is obtained by substituting the
anchor point, a, for the generic point in ξ -space in the general ANOVA expansion.
We get the same result by analytically integrating the expressions for the firstorder and second-order functions given by the quadratic polynomial interpolation
expansions. This yields
Z(ξ ) = (Z −0.5 + 4Z 0 + Z 0.5 )/6 − Z(a)
Z(ξ 1 , ξ 2 ) = (z 00 + 4z 01 + z 02 + 4z 10 + 16z 11 + 4z 12
+ z 20 + 4z 21 + z 22 )/36 − Z(ξ 1 ) − Z(ξ 2 ) − Z(a) .
(7.59)
The mean values vanish when numerical values for Z −0.5 , Z 0 , and Z 0.5 , as well as
z 00 , · · · , z 22 , as presented in (7.56), are used.