August 2025 — Volume 83, Issue 8 page 62

The variance and covariance estimates for each param-
eter are obtained when fitting the model (see Equation 25).
However, Equation 27 is written with sums of variances and
covariances. Statistics theory [19] provides the equations (see
Appendix A: Sums of Variances and Covariances). All the
sums needed for Equation 27 are provided in Equation 28.
Also note that, since is a scalar, ar(mβi) = m 2 Var(βi) and
Cov(βi,mβj) =mCov(βi,βj).
(28) Var[ˆ 0 ] =Var[ˆ 0 +mˆ 2 ] Var[ˆ0] + m2Var[ˆ2] +2mCov[ˆ 0 ,ˆ 2 ]
Var[ˆ1] =Var[ˆ1 +mˆ3] Var[ˆ 1 ] +mVar[ˆ3] +2mCov[ˆ 1 ,ˆ3]
Var[ˆε] =Var[ˆε]
Cov[ˆ0,ˆ1] =Cov[ˆ0 +mˆ2,ˆ1 +mˆ3] Cov[ˆ 0 ,ˆ1] +mCov[ˆ 0 ,ˆ3]
+mCov[ˆ2,ˆ 1 ] + m2Cov[ˆ 2 ,ˆ3]
Cov[ˆ0,ˆε] =Cov[ˆ 0 +mˆ 2 ,ˆε] =Cov[ˆ 0 ,ˆε] +mCov[ˆ 2 ,ˆε]
Cov[ˆ1,ˆε] =Cov[ˆ 1 +mˆ 3 ,ˆε] Cov[ˆ 1 ,ˆε] +mCov[ˆ 3 ,ˆε]
Then, to calculate 90/95 one can follow the steps in
Equations 16–19, using the pod values from Equation 27.
3.2. Polynomial Alternatives to the Simple Linear
Model Setup
The conversion from a linear model to a POD curve is nontriv-
ial for any linear model that extends beyond the simple linear
model. This section describes how to perform POD for a linear
model defined by an invertible function of discontinuity size ,
using a change-of-variable methodology.
First, fit a linear model relating to (x) where is a dif-
ferentiable and invertible function. For example, consider
a linear model where is a second-order polynomial.
Then =f(x) = α0 + α1x + α2x 2 and the estimated model
would be y = ˆ 0 + ˆ 1 x + α2x 2 The next step in POD estima-
tion requires writing yl = ˆ 0 + ˆ 1 xi + ˆ 2 xi2 in terms of prob-
ability (as in Equation 6). However, as shown in Equation 29,
the quadratic form does not allow for to be separated into the
form of ([x − ˆ μpod]/ˆ) σpod so that ˆ μpod and ˆ2 σpod are able to
be estimated separately.
(29) POD(a) =Φ( ˆ − ydec _
ˆε
)=Φ( ˆ1x +ˆ2x2 − (ydec − ˆ0) _____________
ˆε
)
Define a new variable, =f(x) to facilitate a change of
variables so that a separable version of Equation 29 is possible.
However, to use the variable we need to know its mean and
variance. The next section will show how to estimate them.
3.2.1. MOMENTS OF Y: EXPECTED VALUE AND VARIANCE
Consider the fitted model, f( ˆ ) which describes the mean
behavior of (x) with respect to Then, a good estimate of the
expected value of comes from estimating f( ˆ ) In practice,
one can define the function = β0z + β1zz =f(x) and fit a linear
model. As shown in Equation 30, the form of the expected
value is identical to the simple linear case, except is replaced
by so ˆ μpod = (ydec − β0z)/ˆ ˆ β1z Since
[ z
] = ˆ f( ) then
ˆ β0z ≈ 0 and ˆ β1z ≈ 1 (which are approximate only because of
potential rounding errors during estimation), so μpod ˆ ≈ ydec.
(30) E[y] =E[β0z + β1zz] = β0z + β1zE[z]
Next, the variance of is needed. A naive approach would
use the residual variance σz2 provided during model fitting.
However, since the variable depends on the predicted values
(i.e., the mean prediction) of f( ˆ ) this may underestimate the
variance of Equation 31 provides the variance of in terms
of (x):
(31) Var[y] =Var[β0z + β1zf(x)]=β1z2Var[f(x)]
Next, the variance of (x) is needed. According to the Delta
method, for a differentiable function with derivative (T)
where [x] =µ then ar[f(X)] = ∑ i=0 fi(T) (μ)2 Var[xi] +
2∑i j fi(T)fj(T)Cov[xi,xj] This requires taking derivatives of (x)
with respect to each parameter ( i in (x) and similarly, the
variance and covariance of each parameter.
Besides differentiability of the regression function (which
is assured for the models we propose), the Delta method also
assumes asymptotic normality for the statistic of interest. Since
the statistic of interest is the predicted response, when normal-
ity is not met, transformations found through methods such
as the Box-Cox transformation can help justify the use of the
Delta method for variance estimation.
For the example where =f(x) = ˆ 0 + ˆ 1 x + ˆ 2 x 2 ,
the necessary first derivatives are given in Equation 32. All
higher-order derivatives are zero in this case, so they do not
contribute to the sum. Note, however, that in some functions,
these higher-order derivatives may need to be included. We
assume that normality is either directly met in the model or
through the use of a transform such as Box-Cox.
(32) (
∂ f(x) _
∂ ˆ0
) =1, (
∂ f(x) _
∂ ˆ1
) =x, (
∂ f(x) _
∂ ˆ2
) = x2
Thus, using the Delta method, the variance of (x) is given
in Equation 33:
(33) Var[f(x)]=∑fi(T)(μ)2Var[xi]
i=0
2
+2∑fi
i j
(T) Cov[xi,xj]
=(_ ∂ f(x)
∂ ˆ0
)
2 Var[α0] + (
∂ f(x) _
∂ ˆ1
)
2 Var[α1]+(_ ∂ f(x)
∂ ˆ2
)
2 Var[α2]
+2(_ ∂ f(x)
∂ ˆ0
∂ f(x) _
∂ ˆ1
Cov[α1,α0]) 2(_ ∂ f(x)
∂ ˆ0
∂ f(x) _
∂ ˆ2
Cov[α2,α0])
+2 (
∂ f(x) _
∂ ˆ1
∂ f(x) _
∂ ˆ2
Cov[α2,α1] ) =Var[α0] + x2Var[α1] + x4Var[α2]
+2xCov[α1,α0] +2 x2Cov[α2,α0] +2 x3Cov[α2,α1]
ME
|
PODMODELING
62
M AT E R I A L S E V A L U AT I O N • A U G U S T 2 0 2 5

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

Previous Page Next Page

ASNT grants non-exclusive, non-transferable license of this material to  .
All rights reserved. © ASNT 2025. To report unauthorized use, contact: customersupport@asnt.org

August 2025 — Volume 83, Issue 8 Page 62 (64 of 112)

August 2025 — Volume 83, Issue 8 resources

Help