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A Truncated Logistic Regression Model in Probability o

Detection EvaluationYan Guo

a, Kai Yang

b& Adel Alaeddini

c

aMaterials and Technology Department , Siemens Energy Inc. , Orlando, Florida

bDepartment of Industrial and System Engineering , Wayne State University , Detroit,

MichigancDepartment of Industrial and Operations Engineering , University of Michigan , Ann Arbo

Michigan

Published online: 29 Aug 2011.

To cite this article:Yan Guo , Kai Yang & Adel Alaeddini (2011) A Truncated Logistic Regression Model in Probability ofDetection Evaluation, Quality Engineering, 23:4, 365-377, DOI: 10.1080/08982112.2011.603664

To link to this article: http://dx.doi.org/10.1080/08982112.2011.603664

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A Truncated Logistic Regression Model in

Probability of Detection EvaluationYan Guo1,

Kai Yang2,

Adel Alaeddini3

1Materials and Technology

Department, Siemens Energy

Inc., Orlando, Florida2Department of Industrial and

System Engineering, Wayne

State University, Detroit,Michigan3Department of Industrial and

Operations Engineering,

University of Michigan, Ann

Arbor, Michigan

ABSTRACT In nondestructive evaluation (NDE) studies, the probability of

detection curve (POD) is an important performance metric. The traditional

POD estimation is to conduct NDE inspections for artificially fabricated spe-

cimens with known flaws. This approach is often challenges because not

only do fabricated flaws not adequately represent the flaws found in the

field, but the cost and time of fabricating artificial specimens can also be

very high. In practice, field samples and components in service withnaturally occurring defects are readily available and much less expensive

to test. However, the disadvantage of this field approach is that the exact

number and sizes of the flaws in a sample (especially for flaws with small

sizes) are unknown. As a result, serious bias in estimating the POD can

occur. In this article, a truncated logistic regression method is developed

that can estimate POD accurately and consistently with field samples based

on multiple inspections. A case study illustrates the successful application of

the proposed approach in a leading manufacturing company. The simula-

tion studies also show that the proposed methods estimate the POD on field

samples with quality comparable to that of the traditional approach on

artificial specimens.

KEYWORDS field inspection, logistic regression, nondestructive evaluation,

POD, simulation

INTRODUCTION

Nondestructive evaluation (NDE) techniques are widely used to evaluate

the states or properties of many kinds of components and structures without

damaging them. The common NDE methods include liquid penetrant, radio-graphic, eddy current, and ultrasonic testing. Effective NDE systems are very

essential for industries such as semiconductor manufacturing, aircraft, and

power generation in which the costs and effects of failures are much higher

than that of inspections and replacements. For example, in semiconductor

manufacturing, ultra-high-frequency ultrasound is extensively employed to

make visible images of the internal voids and defects on the silicon die,

die paddle, and lead frame of the integrated circuits (ICs). Figure 1a illus-

trates the structure of the particle-filled plastic mold compound of an IC,

as well as the circular mold marks at the top surface of the component.

Address correspondence to Kai Yang,Professor, Department of Industrial

and System Engineering, Wayne StateUniversity, 4815 Fourth St., Detroit,

MI 48201. E-mail: [email protected]

Quality Engineering, 23:365377, 2011

Copyright# Taylor & Francis Group, LLC

ISSN: 0898-2112 print=1532-4222 online

DOI: 10.1080/08982112.2011.603664

365


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The small white features are voids (trapped bubbles)

in the mold compound. Figures 1b and 1c also show

a large crack that could lead to a potential failure of a

connecting rod and several tiny cracks on the surface

of a gear using liquid-penetrant method.

A key advantage of NDE systems is their ability to

detect flaws or abnormal conditions well before the

occurrence of catastrophic failures of key compo-

nents or structures. Specifically, the capability of an

NDE system is measured by its ability to detect aparticular size of flaw, a, which is proportional to

the potential hazard. The larger the flaw size, the

higher chance of failure.

For an ideal NDE system, it is desirable to catch all

of the potential harmful flaws in an inspection run as

illustrated in Figure 2a. In Figure 2a, ath denotes a

threshold value of flaw size. If the flaw size a is

greater thanath, the corresponding flaw is a potential

concern and should be detected. On the other hand,

ifa is smaller thanath, the corresponding flaw is not

potentially critical to the applications. POD(a) inFigure 2 represents the probability of detection as a

function of flaw size a. Therefore, for an ideal NDE

system, all flaws with sizes a greater than ath would

be detected with probability one; that is, POD(a) 1

for all a ath. In contrast, flaws with sizes a smallerthan ath would not be reported; that is, POD(a) 0for all a


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assess the POD curve, artificial specimens are usually

designed and fabricated carefully with artificial flaws

whose sizes and locations are known on specimens

verified by destructive analysis or advanced labora-

tory methods after the inspection (MIL-HDBK-1823

1999). The obvious advantage of using artificial

specimens is that we will know all of the true infor-

mation about flaws and their sizes, so we can know

exactly what flaws are detected and what flaws aremissed by an NDE system.

The aforementioned POD curve estimation pro-

cedure based on the evaluation of artificial speci-

mens is often challenged because the inspection

conditions in service are quite different from the

simulated in-service inspection environment. In such

trials, artificial flaws (electrical discharge machining

notches for eddy current inspections) or fabricated

flaws (fatigue cracks in coupons for penetrant

inspections) do not adequately represent the flaws

found in the field (Thompson et al. 2006). Further-more, the cost and time of fabricating these artificial

specimens required for a POD study is very high,

which is a significant obstacle to the routine assess-

ments of POD. These problems may even delay or

prevent a new inspection process or new technology

from being implemented.

In practice, field samples, components in service

with naturally occurring defects, are readily available

and much less expensive. Field samples are real com-

ponents or structures that are picked for routine non-

destructive evaluations. The flaws in field samples arealso real flaws that we are trying to catch. However,

one major disadvantage of field samples in POD esti-

mation is that we may not know the number and sizes

of the flaws in a sample exactly, in particular for flaws

with small sizes, because currently there is no NDE

system that can guarantee detection of all of the flaws.

These undetected flaws in field samples will create

complications in POD curve estimation. Specifically,

the traditional standard approach in estimating the

POD curve, such as the ones described in MIL-HDBK-

1823 (1999), will not give the best POD curve esti-mation results when we are using field samples

instead of artificial specimens. Brewer (1994) pro-

posed that a POD curve is attainable if crack growth

data are available, which use the basic principle that

the size of a previously missed flaw can be estimated

from currently observed flaw size. One approach pro-

posed by Meeker et al. (2001) utilized the analytical

model of the physical system to estimate the signal

of system response and the probability of detection.

Smith et al. (2007) proposed combining the empirical

data and the analytical model to reduce the

complexity of determining POD.

This article will describe an approach for applying

a truncated logistic regression (TLR) model to adjust

the bias introduced by the undetected flaws when

the NDE inspection techniques are applied in the fieldsamples. In particular, the proposed method uses TLR

to combine different NDE measurements into one

optimal POD curve that is substantially superior to

the POD curve from standard logistic regression. This

approach will overcome the uncertainties introduced

by naturally occurring flaws and will cancel out the

influence of lacking knowledge of undetectable flaws

to increase the accuracy and reliability. Such method-

ology is particularly appropriate for situations where

there are multiple NDE methods available and the

methods detect different defects.In the next section, the basic probability of detec-

tion models are reviewed and the effect of unde-

tected flaws in field samples on models are

discussed. Next, the truncated logistic regression

model for probability of detection is presented. We

further show the effects of bias adjustment of the

truncated logistic regression model on POD esti-

mation and compare these results with the POD

estimation model without considering truncation.

PROBABILITY OF DETECTION

Data from NDE inspections can be classified into

two categories according to the specific responses

from an NDE technique to a defect. The first category

is calledadata, whereais a continuous variable that

represents the intensity measurement data from NDE

system signals when it measures a flaw with size a.

The flaw size,a, could be measured by an advanced

measurement system or destructive analysis. In this

case, the expected magnitude of a will be pro-

portional to the flaw size a. The second categoryinvolves binary data (also called hit=miss data)

representing the presence or absence of a flaw. In

this case, when a flaw is detected by an NDE system,

it is a hit. When the NDE system fails to detect a flaw,

it is a miss. The hit=miss category includes NDE data

produced by systems that generate visual images of

targets and their backgrounds, because these images

367 A Truncated Logistic Regression Model for POD


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will be analyzed by pattern recognition techniques to

be translated into binary results.

The statistical estimation methods of POD models

are different for these two categories of NDE data.

The model for a data is called the a vs. a model,

and the model for the second category is a binary

response regression model for hit=miss data.

The a vs. a ModelIn NDE testing process, the NDE device will create

a signal when a sample is tested. The signalstands

for the output response that can be used to charac-

terize flaws. In detection theory, the probability of

detection with the given probability density function

fsignalassuming the signal amplitudes as random vari-

ables can be expressed as (Rummel and Matzkanin

1997; Yang and Donath 1983)

POD a P aa aadec Z1

aadecfsignal aas;a daas 1

whereadecis the critical NDE output response magni-

tude above which an indication is considered relevant

to the flaw and is reported. The signal measure of the

NDE system is given by a and a is the real size of

flaw. The signal amplitudes are often approximately

normally distributed (Berens 1987) with respective

characteristics N ls a ;r2s a

. As a result, the prob-

ability of detection in Eq. [1] takes the form

PODa Z1a

1ffiffiffiffiffiffi2p

p rs a

exp aasls a 2

2r2s a !

daas

1 U a ls a rs a

2

From empirical studies (Berens 1987), it has been

shown that a logarithmic transformation on a vs. a

data often allows the use of normal theory regression

model. LetY log(a) andx log(a), then the simpleversion of theavs.amodel is

Y b1 b2x e 3

Equations [2] and [3] are the basis for the statistical esti-

mation of POD model for avs.a data when artificial

specimens with known flaws and flaw sizes are used.

However, when service samples are used, where the

number of flaws and flaw sizes are unknown, the

POD model estimated by using Eqs. [2] and [3] may

have serious bias resulting in incorrect estimation of

the regression parameters (Meeker and Thompson

2007). Figure 3a illustrates an example of such bias,

which is due to the existence of unknown missed

flaws, concentrated at the low end ofavs.a and cor-

responding to small undetectable flaws. Meeker and

Thompson (2007) developed a left-truncated

linear regression model to estimate POD curves for

a vs. a data with field samples, and this truncated

linear regression model is reported to achieve good

results.

Binary Regression Model

for Hit/Miss Data

Binary responses are most common for NDE

methods such as liquid penetrant inspection, radio-graphic methods, and the newly developed sonic

infrared methods (Han and He 2006). For hit=miss

testing, the whole set of binary response data is

binomial in nature with detection probability given

by POD(a). Maximum likelihood is used to estimate

FIGURE 3 Effect of undetectable flaws on the bias of the regression: (a)a vs.a model and (b) binary regression model for hit/miss data(the dashed line is the regression fit without undetectable flaws and the solid line is the true regression model).

Y. Guo et al. 368


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the parameters of a POD curve (Chambers and

Hastie 1992). The likelihood of detection probability

using a single observation I is

LiPi; ai;xi Pxii1 Pi1xi 4

where xi is a Bernoulli distributed random vari-

able, PiP(ai) is the probability of detection atflaw size ai, and Li is the maximum likelihood ofPi. The log odds function is often suggested to

model binary data (Chambers and Hastie 1992)

and is given as

Pi expb1 b2log ai1 expb1 b2log ai

5

The parameter to be estimated is b b1 b2Y.Similar to the avs. a data, Eqs. [4] and [5] are the

bases for the statistical estimation of the POD model

for hit=miss data when artificial specimens with

known flaws and flaw sizes are used. Again, whenservice samples are used in which the number of

flaws and flaw sizes are unknown, the POD model

estimated using Eqs. [4] and [5] may also have serious

bias due to unknown missed flaws, an example of

which is shown in Figure 3b. In this article, we

develop a left-truncated logistic regression model

to estimate POD curves for hit=miss data. This pro-

posed truncated logistic regression model will be

thoroughly presented, discussed and evaluated in

subsequent sections.

TRUNCATED LOGISTIC REGRESSION

MODEL

There are two approaches to POD evaluation

according to how the samples are collected. The

first approach is conducted in laboratories, whereas

the other approach is conducted in the field. In the

laboratory approach, the samples are fabricated

with artificial flaws whose sizes and locations

are designed to simulate the real flaw and are

under control while fabricating. For these artificialflaws, the sizes are known before the inspections.

The binary inspection results (hit=miss) are

collected by comparing the results to the actual

flaws.

The second approach is conducted in the field and

is the approach discussed in this article. The samples

are collected in the field inspection stage, and the

flaws are unknown before inspection. The unde-

tected flaws are also unknown. In order to obtain

information on undetected flaws, multiple NDE

methods are used to determine missed flaws by

individual NDE methods.

Without loss of generality, we illustrate the field

approach with two combined NDE methods as

shown in Figure 4. The second NDE method

(method B) is assumed to have different capabilitiesfor detecting the flaws that were missed by method

A. In addition, we assume that the total number of

detected flaws by multiple NDE methods is greater

than the number of detected flaws by each method.

This assumption is easily met, and the purpose of

setting this assumption is to avoid an unbounded

likelihood function while curve fitting. In addition

to flaw size, many other factors, such as flaw orien-

tation, flaw type, flaw location, etc., will also affect

the probability of detection. Given the same flaw

size, some NDE methods such as penetrant testingare more capable of detecting flaws open to the sur-

face, whereas others are more effective in detecting

tight-closed flaws, such as sonic infrared (IR). There-

fore, even if the flaws are similar in size, different

NDE methods usually pick up different subsets of

flaws.

Based on the characteristics of NDE inspections,

the logic operation is used for inspecting results of

multiple NDE methods, as illustrated in Figure 4,

where the frame represents the whole set of flaws,

but the details of this set are unknown. The area con-sisting of zones I, II, and III is the set of detected

flaws by either one of the NDE methods in use.

FIGURE 4 Illustration of inspections by multiple NDE methods.

369 A Truncated Logistic Regression Model for POD


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Though multiple methods are used for detection,

there are still undetected flaws, shown as zone

IV in Figure 4. The bias introduced by the unde-

tected flaws will be adjusted by the truncated

logistic regression model. Truncated logistic

regression is an improvement of the standard

regression by using multiple NDE methods in the

field.

In NDE inspection, there are two tasks: (1) detec-tion and (2) sizing. Once an indication is detected,

other advanced methods, such as those involving

microscopes or computed tomography, for

example, will be used for sizing. Some NDE meth-

ods have the ability to measure the size at the time

of detection. After inspection, the detected flaws

(shown in zones I, II, and III) are sized for POD

estimation.

The regression model assumes that the sizes of

detected flaws (zones I, II and III) are measured

but not the sizes of the undetected flaws (zone IV).For zones I, II, and III, because at least one of the

NDE methods detected the flaw, sizing can be con-

ducted after detection. Therefore, the POD evalu-

ation model outlined previously could be applied

to field inspections. As a result, the accuracy of esti-

mation will be improved by applying a truncated

logistic regression model.

We assume that Nflaws are to be inspected and

the number of inspection systems is M. Define a

binary random variable yij represent the ith flaw

inspected by the jth inspection system, where

yij 0; undetected flawmiss fori 0;1; . . . ;N;1; detected flawhit j 1; . . . ;M

Letairepresent the size of flaw i. Furthermore, we

define bj bj1 bj2T as the vector of POD curveparameters of the jth inspection system for

j 1, . . .,M, where b bT1 bT2 . . . bTMT is the vectorof POD curve parameters for all inspection systems.

From Eq. [5], the probability of detection for theindividual flaw by an inspection system is

Pyij1 expbj1 bj2log ai

1 expbj1 bj2log aipbj; ai 1 qbj;ai 6

The likelihood ofP, based on a single observation, is

then

Lib; ai;yij YMj1

pbj; aiyijqbj;ai1yij 7

and the estimator bbj is the optimal solution for

maximizing log(Li(b;ai,yij)). Equations [6] and

[7] are the conventional regression models for PODevaluation.

With the multiple NDE inspection strategy in the

field, only the detected flaws are recorded. The set

of flaws for POD curve fitting that contains the set

of flaws that were detected by at least one NDE

inspection system is given by

< iXM

j1yij1; i1; 2; . . . ;N

( ) 8

The flaws that were missed by all inspections

are truncated. This set of truncated data is expressed

as

I iXM

j1yij0; i1; 2; . . . ;N

( ) 9

Because the truncated data are not known in prac-

tice, the conventional logistic regression continu-

ously applied for POD estimation would introduce

a bias because the effect of truncated flaws would

not be taken into consideration. An adjustment is

then made for the conventional regression model

to be conditioned given that a flaw is detected by

at least one NDE inspection system. The resulting

TLR model would be

Lb; ai;yij Yi2

truncated regression model

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