Generally, the response ′ a vector of Box-Cox transformed
y values, is modeled as a function of β +ε using maximum
likelihood estimation. The method for estimating is described
in [16, 17, 19], and many commercial and open-source software
packages [21] can recommend an optimal by comparing the
sum of squared errors at various values across the range of
power transforms, stepping in increments (e.g., 0.1) between
both positive and negative integers, with the transformation at
a value of 0 representing the natural logarithm.
After fitting the model, the transformation should be
reversed to return to the original data units of the response, .
Once the value is estimated and applied to the response,
then is considered fixed. The residuals are an estimate of the
variability that the model cannot explain. When using a simple
linear model, residuals may be examined, or a lack-of-fit test
may be conducted to determine if additional variables beyond
discontinuity size are necessary [3, 17, 18].
If a transformation is applied to the same transforma-
tion should also be applied to the variable dec (which will
be described in Section 2.2), so that they have the same units
(say, dec ′ For simplicity, the following equations will refer to ,
but if a transformation is necessary, ′ should be substituted.
Lastly, if the data shows evidence of censoring, a modifica-
tion to the likelihood equation is necessary during maximum
likelihood estimation. Censoring often occurs in real-world
NDE measurements. Very large discontinuities may max out
the NDE system’s signals, providing a flat response at the
maximum detection limit, and thus providing a lower bound
on the response, rather than the true value. This is common
if the gain on a system is set high such that small discontinu-
ities are detectable, but large discontinuities may saturate the
detection system. Very small discontinuities that fall below the
detection threshold of the NDE system may return values of
zero, which is also a form of censoring.
2.2. Transforming the Linear Model into Probability
Assume a simple linear model has been fitted using and
(discontinuity sizes): y1 = ˆ 0 + ˆ 1 ai Note that, for convention,
ai is now used in place of i in the simple linear model. Recall
that i is assumed to be normally distributed. A POD curve
consists of a standard normal cumulative distribution function
(CDF), denoted as the response. Therefore, the first step
requires a transformation from the normal distribution (with
estimated mean ˆ and estimated residual error variance σε2)
to the standard normal distribution (with mean 0 and
variance 1), as shown in Equation 5:
(5) i = ˆεZi + ˆ ∼ N(ˆ,ˆε2)⇒Zi = 1
ˆε
ˆ i − ˆ ∼ N(μ =0,σ2 =1)
Before performing a POD evaluation, a predefined proce-
dure incorporating calibration and specific call criteria based
on the system’s noise floor is determined for the NDE system,
and this value is called the decision threshold, dec Therefore,
Equation 6 becomes the formula for the POD of a discontinuity
of size i [1–4]:
(6) POD(ai) =Φ(Zi) =Φ( ˆ i_cedy−
ˆε
)
=Φ (
ai + (ˆ0 − ydec) ˆ1 __________
ˆε ˆ1
) =Φ (
ai_dopˆ−
ˆd σpo )
The final equation provides a new mean, ˆ μpod and a new
variance, ˆ2 σpod as shown in Equation 7:
(7) ˆd po = − (ˆ0 − ydec) _
ˆ1
,and ˆd2 po = (
ˆε
ˆ1)
2
2.3. Estimating a90, the Discontinuity Size at 90% POD
By inverting Equation 6, the discontinuity size associated with
each probability can be calculated. The discontinuity size, ,
with POD %=POD(a) is denoted p (see Equation 8),
where −1(p/100) = zp is calculated using a standard normal
Z-table [1–4].
(8) ap = Φ−1(POD(a))ˆd po + ˆd=Φ−1(p)ˆd po σpo + ˆd po = zpˆd po +ˆdop
Equation 9 shows how to find 90 the discontinuity
size associated with 90% probability, which corresponds to
p =0.90:
(9) a90 = z0.9ˆd po + ˆd po ≈ 1.2816ˆd po +ˆdop
For a simple linear regression model, Equations 8 and 9
simplify to Equation 10:
(10) ap = zp
(
ˆε
ˆ1)
+ (y_, dec − ˆ0)
ˆ1
and 90 = z0.9 (
ˆε ˆ1) + (ydec − ˆ0)
_
ˆ1
≈ 1.2816 (
ˆε
ˆ1) + (y_)0ˆ−ced
ˆ1
2.4. Estimating a 95% Confidence Interval and the a90/95
Discontinuity Size
A confidence interval can be estimated for the linear model, in
terms of the three estimated model parameters: β0 β1 and σε.
Estimates of the variances and the covariances of the param-
eters are also provided in the variance-covariance matrix (see
Equation 11) returned when fitting the linear model.
However, to calculate a confidence interval on the POD
curve, a variance-covariance matrix is needed in terms of the
parameters μpod ˆ and ˆ σpod as shown in Equation 12:
(12) Vpod =V(ˆd,ˆd)=opop [
Var(ˆd) po Cov(ˆd,ˆd)opop
Cov(ˆd,ˆd) po po Var(ˆd) po ]
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 59
y values, is modeled as a function of β +ε using maximum
likelihood estimation. The method for estimating is described
in [16, 17, 19], and many commercial and open-source software
packages [21] can recommend an optimal by comparing the
sum of squared errors at various values across the range of
power transforms, stepping in increments (e.g., 0.1) between
both positive and negative integers, with the transformation at
a value of 0 representing the natural logarithm.
After fitting the model, the transformation should be
reversed to return to the original data units of the response, .
Once the value is estimated and applied to the response,
then is considered fixed. The residuals are an estimate of the
variability that the model cannot explain. When using a simple
linear model, residuals may be examined, or a lack-of-fit test
may be conducted to determine if additional variables beyond
discontinuity size are necessary [3, 17, 18].
If a transformation is applied to the same transforma-
tion should also be applied to the variable dec (which will
be described in Section 2.2), so that they have the same units
(say, dec ′ For simplicity, the following equations will refer to ,
but if a transformation is necessary, ′ should be substituted.
Lastly, if the data shows evidence of censoring, a modifica-
tion to the likelihood equation is necessary during maximum
likelihood estimation. Censoring often occurs in real-world
NDE measurements. Very large discontinuities may max out
the NDE system’s signals, providing a flat response at the
maximum detection limit, and thus providing a lower bound
on the response, rather than the true value. This is common
if the gain on a system is set high such that small discontinu-
ities are detectable, but large discontinuities may saturate the
detection system. Very small discontinuities that fall below the
detection threshold of the NDE system may return values of
zero, which is also a form of censoring.
2.2. Transforming the Linear Model into Probability
Assume a simple linear model has been fitted using and
(discontinuity sizes): y1 = ˆ 0 + ˆ 1 ai Note that, for convention,
ai is now used in place of i in the simple linear model. Recall
that i is assumed to be normally distributed. A POD curve
consists of a standard normal cumulative distribution function
(CDF), denoted as the response. Therefore, the first step
requires a transformation from the normal distribution (with
estimated mean ˆ and estimated residual error variance σε2)
to the standard normal distribution (with mean 0 and
variance 1), as shown in Equation 5:
(5) i = ˆεZi + ˆ ∼ N(ˆ,ˆε2)⇒Zi = 1
ˆε
ˆ i − ˆ ∼ N(μ =0,σ2 =1)
Before performing a POD evaluation, a predefined proce-
dure incorporating calibration and specific call criteria based
on the system’s noise floor is determined for the NDE system,
and this value is called the decision threshold, dec Therefore,
Equation 6 becomes the formula for the POD of a discontinuity
of size i [1–4]:
(6) POD(ai) =Φ(Zi) =Φ( ˆ i_cedy−
ˆε
)
=Φ (
ai + (ˆ0 − ydec) ˆ1 __________
ˆε ˆ1
) =Φ (
ai_dopˆ−
ˆd σpo )
The final equation provides a new mean, ˆ μpod and a new
variance, ˆ2 σpod as shown in Equation 7:
(7) ˆd po = − (ˆ0 − ydec) _
ˆ1
,and ˆd2 po = (
ˆε
ˆ1)
2
2.3. Estimating a90, the Discontinuity Size at 90% POD
By inverting Equation 6, the discontinuity size associated with
each probability can be calculated. The discontinuity size, ,
with POD %=POD(a) is denoted p (see Equation 8),
where −1(p/100) = zp is calculated using a standard normal
Z-table [1–4].
(8) ap = Φ−1(POD(a))ˆd po + ˆd=Φ−1(p)ˆd po σpo + ˆd po = zpˆd po +ˆdop
Equation 9 shows how to find 90 the discontinuity
size associated with 90% probability, which corresponds to
p =0.90:
(9) a90 = z0.9ˆd po + ˆd po ≈ 1.2816ˆd po +ˆdop
For a simple linear regression model, Equations 8 and 9
simplify to Equation 10:
(10) ap = zp
(
ˆε
ˆ1)
+ (y_, dec − ˆ0)
ˆ1
and 90 = z0.9 (
ˆε ˆ1) + (ydec − ˆ0)
_
ˆ1
≈ 1.2816 (
ˆε
ˆ1) + (y_)0ˆ−ced
ˆ1
2.4. Estimating a 95% Confidence Interval and the a90/95
Discontinuity Size
A confidence interval can be estimated for the linear model, in
terms of the three estimated model parameters: β0 β1 and σε.
Estimates of the variances and the covariances of the param-
eters are also provided in the variance-covariance matrix (see
Equation 11) returned when fitting the linear model.
However, to calculate a confidence interval on the POD
curve, a variance-covariance matrix is needed in terms of the
parameters μpod ˆ and ˆ σpod as shown in Equation 12:
(12) Vpod =V(ˆd,ˆd)=opop [
Var(ˆd) po Cov(ˆd,ˆd)opop
Cov(ˆd,ˆd) po po Var(ˆd) po ]
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 59
