Var[f(x)] is a function of values, and since there are
values of there are potentially values of ar[f(x)] To be
conservative, use the largest variance value, which occurs
at the largest discontinuity size: max =max(xi) Thus,
σε = ˆ 1
2 Var[f(xmax)] By selecting the max value, the resulting
POD curve will be shifted toward larger discontinuity sizes,
so the resulting 90/95 is an upper bound, and an overesti-
mation of the true 90/95 Therefore, in general terms, using
Equation 33, Equation 34 provides the new estimate of σpod: ˆ
(34) ˆd po = ˆε
ˆ 1z
= 1
ˆz1
√
_ _______________
ˆ β1z2Var[f(xmax)]=√
_____________
Var[f(xmax)]
The POD equation, with ˆ μpod ≈ ydec and ˆ σpod from
Equation 34, is given in Equation 35:
(35) POD(z) =Φ (
ˆ − ydec _
ˆεz
) =Φ (
z + (ˆz 0 − ydec) ˆz 1 __________
ˆεz ˆz 1
) =Φ (
z − ˆdop _
ˆd σpo )
3.2.2. TRANSITION MATRIX
Of note is that ˆ σpod/∂ˆ β1z =0 so that specific term of the
U matrix has changed from the value in Equation 6. The last
row of the matrix must match the parameters in so given
the last row now consists of ˆ μpod/∂ˆ2 ε =0 and ˆ μpod/∂ˆ2 ε =∂√
___________Var[f(xmax)]/∂ˆ β1z 2 Var[f(xmax)] let random variable
v =β1z2Var[f(xmax)] ˆ then √Var[f(xmax)]
___________
= √
_
v /β1z2 ˆ Using this
change of variables, the derivative is given in Equation 36:
(36) ∂ ˆd po
∂ ˆε2
= ∂ √
_____________
Var[f(xmax)] ___________
∂ ˆz 1
2 Var[f(xmax)]
= ∂ √ _
v /ˆz 1
2
_
∂ v
= 1 _
2ˆz√v1 _
=________________ 1 2ˆz√
1
________________
ˆ 1z
2 Var[f(xmax)]
= 1
2ˆzˆε1
The new matrix is shown in Equation 37, and the new
matrix is in Equation 38. The variance in ε can be estimated
from the data to form the last entry in the matrix, and the
covariances involving ε are assumed to be zero because ε
is independent of 0z and 1z Recall that the Delta method
establishes the variance of the POD curve as pod = UTVU.
Matrices and are defined the same way as in the
standard methodology (see Equations 15 and 11) except that
they are calculated in -space rather than -space. Then, to cal-
culate 90/95z one can follow the steps in Equations 16–19.
3.2.3. CONVERTING CRITICAL VALUES FROM Z-SPACE TO X-SPACE
Following the standard equations, the final POD curve will be
expressed in terms of which is a function of For each value
of there is an associated POD. To convert back into -space,
a solver may be used to find the solutions to Equation 39 for
in terms of each value on the POD curve and its confidence
interval.
(39) f(x) =z
β0z + β1zz=β0z + β1zf(x)
In the example, because = α0 + α1x + α2x 2 the quadratic
formula provides two solutions for in terms of as given in
Equation 40. One of these solutions will provide useful results
for matching a discontinuity size to a probability value, while
the other solution will provide values that are unrealistic for
the problem.
(40) β0z + β1zz = β0z + β1z(α0 + α1x + α2x2)
0 = β0z + β0z + β1z(α0 + α1x + α2x2) − z
= (β0z − β1zα0 − z) + (β1zα1)x + (β1zα2)x2
x = − (β1z_____________________________)z−0αz1β−z0β(2αz1β4−21αz21β√±)1α_
_____________________________
2 β1zα2
One thing to note is that when converting from back to
x the resulting curve may contain − POD (that is, one minus
the probability of detection) instead. This occurs because the
POD is a cumulative distribution function ( DF which is
defined for both and by Equation 41:
(41) = FZ(Y) =P(Y ≤ y) CDFZ(Y)
=P(α0 + α1x + α2x2 ≤ β0z + β1zz)
=P((α0 − β0z − β1zz) + α1x + α2x2 ≤ 0)
=P(x1 ≤ X ≤ x2)or P(x2 ≤ X ≤ x1)
Without loss of generality, assume (x1 ≤ X ≤ x2) Then,
the solutions to the DF are (X ≤ x2) or (x_1 ≤ X) =
1 − P(X ≤ x _1 ) (Note that the equality is maintained when
the inequality is switched, since is continuous.) This means
that estimating p/q may require − p instead of and − q
instead of .
4. Results
In this section, the statistical methods are applied to two sim-
ulations. These simulations are useful to describe the direction
of bias (since you can simulate the “truth”), but real-world
data has many complexities that these simulations do not
capture. Section 4.1 describes the real-world POD data that
inspired this research, along with a discussion of how the
methods described in this paper were applied to that data
[16]. Simulation 1 (Section 4.2) illustrates fitting the more
complex models discussed in the Methods section, compar-
ing the quality of these different models and how to interpret
A U G U S T 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 63
values of there are potentially values of ar[f(x)] To be
conservative, use the largest variance value, which occurs
at the largest discontinuity size: max =max(xi) Thus,
σε = ˆ 1
2 Var[f(xmax)] By selecting the max value, the resulting
POD curve will be shifted toward larger discontinuity sizes,
so the resulting 90/95 is an upper bound, and an overesti-
mation of the true 90/95 Therefore, in general terms, using
Equation 33, Equation 34 provides the new estimate of σpod: ˆ
(34) ˆd po = ˆε
ˆ 1z
= 1
ˆz1
√
_ _______________
ˆ β1z2Var[f(xmax)]=√
_____________
Var[f(xmax)]
The POD equation, with ˆ μpod ≈ ydec and ˆ σpod from
Equation 34, is given in Equation 35:
(35) POD(z) =Φ (
ˆ − ydec _
ˆεz
) =Φ (
z + (ˆz 0 − ydec) ˆz 1 __________
ˆεz ˆz 1
) =Φ (
z − ˆdop _
ˆd σpo )
3.2.2. TRANSITION MATRIX
Of note is that ˆ σpod/∂ˆ β1z =0 so that specific term of the
U matrix has changed from the value in Equation 6. The last
row of the matrix must match the parameters in so given
the last row now consists of ˆ μpod/∂ˆ2 ε =0 and ˆ μpod/∂ˆ2 ε =∂√
___________Var[f(xmax)]/∂ˆ β1z 2 Var[f(xmax)] let random variable
v =β1z2Var[f(xmax)] ˆ then √Var[f(xmax)]
___________
= √
_
v /β1z2 ˆ Using this
change of variables, the derivative is given in Equation 36:
(36) ∂ ˆd po
∂ ˆε2
= ∂ √
_____________
Var[f(xmax)] ___________
∂ ˆz 1
2 Var[f(xmax)]
= ∂ √ _
v /ˆz 1
2
_
∂ v
= 1 _
2ˆz√v1 _
=________________ 1 2ˆz√
1
________________
ˆ 1z
2 Var[f(xmax)]
= 1
2ˆzˆε1
The new matrix is shown in Equation 37, and the new
matrix is in Equation 38. The variance in ε can be estimated
from the data to form the last entry in the matrix, and the
covariances involving ε are assumed to be zero because ε
is independent of 0z and 1z Recall that the Delta method
establishes the variance of the POD curve as pod = UTVU.
Matrices and are defined the same way as in the
standard methodology (see Equations 15 and 11) except that
they are calculated in -space rather than -space. Then, to cal-
culate 90/95z one can follow the steps in Equations 16–19.
3.2.3. CONVERTING CRITICAL VALUES FROM Z-SPACE TO X-SPACE
Following the standard equations, the final POD curve will be
expressed in terms of which is a function of For each value
of there is an associated POD. To convert back into -space,
a solver may be used to find the solutions to Equation 39 for
in terms of each value on the POD curve and its confidence
interval.
(39) f(x) =z
β0z + β1zz=β0z + β1zf(x)
In the example, because = α0 + α1x + α2x 2 the quadratic
formula provides two solutions for in terms of as given in
Equation 40. One of these solutions will provide useful results
for matching a discontinuity size to a probability value, while
the other solution will provide values that are unrealistic for
the problem.
(40) β0z + β1zz = β0z + β1z(α0 + α1x + α2x2)
0 = β0z + β0z + β1z(α0 + α1x + α2x2) − z
= (β0z − β1zα0 − z) + (β1zα1)x + (β1zα2)x2
x = − (β1z_____________________________)z−0αz1β−z0β(2αz1β4−21αz21β√±)1α_
_____________________________
2 β1zα2
One thing to note is that when converting from back to
x the resulting curve may contain − POD (that is, one minus
the probability of detection) instead. This occurs because the
POD is a cumulative distribution function ( DF which is
defined for both and by Equation 41:
(41) = FZ(Y) =P(Y ≤ y) CDFZ(Y)
=P(α0 + α1x + α2x2 ≤ β0z + β1zz)
=P((α0 − β0z − β1zz) + α1x + α2x2 ≤ 0)
=P(x1 ≤ X ≤ x2)or P(x2 ≤ X ≤ x1)
Without loss of generality, assume (x1 ≤ X ≤ x2) Then,
the solutions to the DF are (X ≤ x2) or (x_1 ≤ X) =
1 − P(X ≤ x _1 ) (Note that the equality is maintained when
the inequality is switched, since is continuous.) This means
that estimating p/q may require − p instead of and − q
instead of .
4. Results
In this section, the statistical methods are applied to two sim-
ulations. These simulations are useful to describe the direction
of bias (since you can simulate the “truth”), but real-world
data has many complexities that these simulations do not
capture. Section 4.1 describes the real-world POD data that
inspired this research, along with a discussion of how the
methods described in this paper were applied to that data
[16]. Simulation 1 (Section 4.2) illustrates fitting the more
complex models discussed in the Methods section, compar-
ing the quality of these different models and how to interpret
A U G U S T 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 63
