A generalized Fellegi-Holt paradigm for automatic error localization
5. Implied edits for general edit operationsA generalized Fellegi-Holt paradigm for automatic error localization
5. Implied edits for general edit operations
In this section, a result will be derived that
establishes whether a given path of edit operations of the form (3.1) can be
used to make a given record consistent with a given system of edit rules (i.e. ,
is a feasible solution to the error localization problem). This result uses the
FM elimination technique discussed in Section 2.
Let
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhaaaa@38CA@
be a given record and let
y
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaaaaa@3A1E@
be any record that can be obtained by
applying, in sequence, the edit operations
g
1
,
…
,
g
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadEgapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab
gAci8kaacYcacaWGNbWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa@3F11@
to
x
:
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhacaGG6aaaaa@3988@
y
t
=
g
t
∘
g
t
−
1
∘
⋯
∘
g
1
(
x
)
.
(
5.1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyypa0Ja
am4za8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqWIyiYBcaWGNb
WdamaaBaaaleaapeGaamiDaiabgkHiTiaaigdaa8aabeaak8qacqWI
yiYBcqWIVlctcqWIyiYBcaWGNbWdamaaBaaaleaapeGaaGymaaWdae
qaaOWdbmaabmaapaqaa8qacaWH4baacaGLOaGaayzkaaGaaiOlaiaa
ywW7caaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaiwdacaGGUaGaaG
ymaiaacMcaaaa@57F4@
Write
g
n
(
x
)
=
T
n
x
+
S
n
α
n
+
c
n
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadEgapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeWaaeWaa8aa
baWdbiaahIhaaiaawIcacaGLPaaacqGH9aqpcaWHubWdamaaBaaale
aapeGaamOBaaWdaeqaaOWdbiaahIhacqGHRaWkcaWHtbWdamaaBaaa
leaapeGaamOBaaWdaeqaaOWdbiaahg7apaWaaSbaaSqaa8qacaWGUb
aapaqabaGcpeGaey4kaSIaaC4ya8aadaWgaaWcbaWdbiaad6gaa8aa
beaakiaacYcaaaa@4AAE@
for
n
∈
{
1
,
…
,
t
}
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaad6gacqGHiiIZdaGadaWdaeaapeGaaGymaiaacYcacqGHMacV
caGGSaGaamiDaaGaay5Eaiaaw2haaiaac6caaaa@41E4@
From (5.1) it follows by induction that
y
1
=
T
1
x
+
S
1
α
1
+
c
1
,
y
2
=
T
2
T
1
x
+
S
2
α
2
+
c
2
+
T
2
(
S
1
α
1
+
c
1
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbuaabaqaciaaaeaacaWH5bWdamaaBaaaleaapeGaaGymaaWdaeqa
aaGcpeqaaiabg2da9iaahsfapaWaaSbaaSqaa8qacaaIXaaapaqaba
GcpeGaaCiEaiabgUcaRiaahofapaWaaSbaaSqaa8qacaaIXaaapaqa
baGcpeGaaCySd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHRa
WkcaWHJbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcaaeaa
caWH5bWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpeqaaiabg2da9i
aahsfapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaaCiva8aadaWg
aaWcbaWdbiaaigdaa8aabeaak8qacaWH4bGaey4kaSIaaC4ua8aada
WgaaWcbaWdbiaaikdaa8aabeaak8qacaWHXoWdamaaBaaaleaapeGa
aGOmaaWdaeqaaOWdbiabgUcaRiaahogapaWaaSbaaSqaa8qacaaIYa
aapaqabaGcpeGaey4kaSIaaCiva8aadaWgaaWcbaWdbiaaikdaa8aa
beaak8qadaqadaWdaeaapeGaaC4ua8aadaWgaaWcbaWdbiaaigdaa8
aabeaak8qacaWHXoWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiab
gUcaRiaahogapaWaaSbaaSqaa8qacaaIXaaapaqabaaak8qacaGLOa
GaayzkaaGaaiilaaaaaaa@6487@
and, in general,
y
t
=
T
t
⋯
T
1
x
+
S
t
α
t
+
c
t
+
∑
n
=
2
t
T
t
⋯
T
n
(
S
n
−
1
α
n
−
1
+
c
n
−
1
)
,
(
5.2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyypa0Ja
aCiva8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqWIVlctcaWHub
WdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaahIhacqGHRaWkcaWH
tbWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiaahg7apaWaaSbaaS
qaa8qacaWG0baapaqabaGcpeGaey4kaSIaaC4ya8aadaWgaaWcbaWd
biaadshaa8aabeaak8qacqGHRaWkdaaeWbqaaiaahsfapaWaaSbaaS
qaa8qacaWG0baapaqabaGcpeGaeS47IWKaaCiva8aadaWgaaWcbaWd
biaad6gaa8aabeaaa8qabaGaamOBaiabg2da9iaaikdaaeaacaWG0b
aaniabggHiLdGcdaqadaWdaeaapeGaaC4ua8aadaWgaaWcbaWdbiaa
d6gacqGHsislcaaIXaaapaqabaGcpeGaaCySd8aadaWgaaWcbaWdbi
aad6gacqGHsislcaaIXaaapaqabaGcpeGaey4kaSIaaC4ya8aadaWg
aaWcbaWdbiaad6gacqGHsislcaaIXaaapaqabaaak8qacaGLOaGaay
zkaaGaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaa
iwdacaGGUaGaaGOmaiaacMcaaaa@738D@
where
the sum over
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaad6gaaaa@38BC@
is defined to be zero when
t
=
1.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadshacqGH9aqpcaaIXaGaaiOlaaaa@3B35@
Moreover, all terms involving
S
n
α
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahofapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeGaaCySd8aa
daWgaaWcbaWdbiaad6gaa8aabeaaaaa@3C96@
vanish in these expressions when
g
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadEgapaWaaSbaaSqaa8qacaWGUbaapaqabaaaaa@3A02@
does not contain any free parameters.
The path of edit operations
P
=
[
g
1
,
…
,
g
t
]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadcfacqGH9aqpdaWadaWdaeaapeGaam4za8aadaWgaaWcbaWd
biaaigdaa8aabeaak8qacaGGSaGaeyOjGWRaaiilaiaadEgapaWaaS
baaSqaa8qacaWG0baapaqabaaak8qacaGLBbGaayzxaaaaaa@4317@
can be applied to
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhaaaa@38CA@
to obtain a record that is consistent with the
edits (2.1) if, and only if, there exists a
y
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahMhapaWaaSbaaSqaa8qacaWG0baapaqabaaaaa@3A1E@
of the form (5.2) that satisfies
A
y
t
+
b
⊙
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahgeacaWH5bWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiab
gUcaRiaahkgacqWIzkszcaWHWaaaaa@3F39@
and all relevant additional restrictions of
the form (3.2) on
α
1
,
…
,
α
t
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaahg7apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab
gAci8kaacYcacaWHXoWdamaaBaaaleaapeGaamiDaaWdaeqaaOGaai
Olaaaa@406F@
Using (5.2),
A
y
t
+
b
⊙
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahgeacaWH5bWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiab
gUcaRiaahkgacqWIzkszcaWHWaaaaa@3F39@
can be written as:
(
A
T
t
⋯
T
1
)
x
+
(
A
S
t
)
α
t
+
∑
n
=
2
t
(
A
T
t
⋯
T
n
S
n
−
1
)
α
n
−
1
+
b
t
⊙
0
,
(
5.3
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbmaabmaapaqaa8qacaWHbbGaaCiva8aadaWgaaWcbaWdbiaadsha
a8aabeaak8qacqWIVlctcaWHubWdamaaBaaaleaapeGaaGymaaWdae
qaaaGcpeGaayjkaiaawMcaaiaahIhacqGHRaWkdaqadaWdaeaapeGa
aCyqaiaahofapaWaaSbaaSqaa8qacaWG0baapaqabaaak8qacaGLOa
GaayzkaaGaaCySd8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGH
RaWkdaGfWbqabSWdaeaapeGaamOBaiabg2da9iaaikdaa8aabaWdbi
aadshaa0WdaeaapeGaeyyeIuoaaOWaaeWaa8aabaWdbiaahgeacaWH
ubWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabl+Uimjaahsfapa
WaaSbaaSqaa8qacaWGUbaapaqabaGcpeGaaC4ua8aadaWgaaWcbaWd
biaad6gacqGHsislcaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaaC
ySd8aadaWgaaWcbaWdbiaad6gacqGHsislcaaIXaaapaqabaGcpeGa
ey4kaSIaaCOya8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqWIzk
szcaWHWaGaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiik
aiaaiwdacaGGUaGaaG4maiaacMcaaaa@73CD@
with
b
t
=
b
+
A
c
t
+
∑
n
=
2
t
A
T
t
⋯
T
n
c
n
−
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahkgapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyypa0Ja
aCOyaiabgUcaRiaahgeacaWHJbWdamaaBaaaleaapeGaamiDaaWdae
qaaOWdbiabgUcaRmaaqadabaGaaCyqaiaahsfapaWaaSbaaSqaa8qa
caWG0baapaqabaGcpeGaeS47IWKaaCiva8aadaWgaaWcbaWdbiaad6
gaa8aabeaak8qacaWHJbWdamaaBaaaleaapeGaamOBaiabgkHiTiaa
igdaa8aabeaaa8qabaGaamOBaiabg2da9iaaikdaaeaacaWG0baani
abggHiLdaaaa@51D5@
a vector of constants.
Interestingly, (5.3) and the possible additional
restrictions of the form (3.2) constitute a linear system of the form (2.1) on
the extended record
(
x
′
,
α
′
1
,
…
,
α
′
t
)
′
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbmaabmaapaqaa8qaceWH4bWdayaafaWdbiaacYcaceWHXoGbauaa
paWaaSbaaSqaaiaaigdaaeqaaOWdbiaacYcacqGHMacVcaGGSaGabC
ySdyaafaWdamaaBaaaleaacaWG0baabeaaaOWdbiaawIcacaGLPaaa
daahaaWcbeqaaOGamai2gkdiIcaaiiaacqWFUaGlaaa@472D@
Therefore, FM elimination may be used to
remove all free parameters from this system. This yields a system of implied
restrictions for
x
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhacaGGUaaaaa@397C@
Moreover, a repeated application of the
fundamental property of FM elimination establishes that
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhaaaa@38CA@
satisfies this system of implied edits if, and
only if, there exist parameter values for
α
1
,
…
,
α
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaahg7apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab
gAci8kaacYcacaWHXoWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa@3FB3@
that, together with
x
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhacaGGSaaaaa@397A@
satisfy (5.3) and (3.2). Hence, it follows
that the path of edit operations
P
=
[
g
1
,
…
,
g
t
]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadcfacqGH9aqpdaWadaWdaeaapeGaam4za8aadaWgaaWcbaWd
biaaigdaa8aabeaak8qacaGGSaGaeyOjGWRaaiilaiaadEgapaWaaS
baaSqaa8qacaWG0baapaqabaaak8qacaGLBbGaayzxaaaaaa@4317@
can lead to a consistent record for
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhaaaa@38CA@
if, and only if,
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaahIhaaaa@38CA@
satisfies the system of implied edits obtained
by eliminating
α
1
,
…
,
α
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaahg7apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiab
gAci8kaacYcacaWHXoWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa@3FB3@
from (5.3) and (if relevant) additional
restrictions of the form (3.2).
Example. Consider the following
edits in
x
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@39DB@
and
x
2
:
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGG6aaaaa@3AA4@
x
1
≥
0
,
(
5.4
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyzImRa
aGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcaca
aI1aGaaiOlaiaaisdacaGGPaaaaa@4872@
x
2
≥
0
,
(
5.5
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyyzImRa
aGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcaca
aI1aGaaiOlaiaaiwdacaGGPaaaaa@4874@
x
1
+
x
2
≤
5.
(
5.6
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa
amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHKjYOcaaI1a
GaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda
caGGUaGaaGOnaiaacMcaaaa@4B79@
Let
g
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadEgaaaa@38B5@
be the edit operation that transfers an amount
of at most four units between
x
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@39DB@
and
x
2
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGGSaaaaa@3A96@
in either direction:
g
(
(
x
1
,
x
2
)
′
)
=
(
x
1
+
α
,
x
2
−
α
)
′
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadEgadaqadaWdaeaapeWaaeWaa8aabaWdbiaadIhapaWaaSba
aSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8
qacaaIYaaapaqabaaak8qacaGLOaGaayzkaaaccaGae8NmGikacaGL
OaGaayzkaaGaeyypa0ZaaeWaa8aabaWdbiaadIhapaWaaSbaaSqaa8
qacaaIXaaapaqabaGcpeGaey4kaSIaeqySdeMaaiilaiaadIhapaWa
aSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyOeI0IaeqySdegacaGLOa
GaayzkaaGae8NmGikaaa@50CE@
with
−
4
≤
α
≤
4.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiabgkHiTiaaisdacqGHKjYOcqaHXoqycqGHKjYOcaaI0aGaaiOl
aaaa@3FED@
For this single edit operation, the system of
transformed edits (5.3) is:
x
1
+
α
≥
0
,
(
5.7
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa
eqySdeMaeyyzImRaaGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8
UaaGzbVlaacIcacaaI1aGaaiOlaiaaiEdacaGGPaaaaa@4AF6@
x
2
−
α
≥
0
,
(
5.8
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyOeI0Ia
eqySdeMaeyyzImRaaGimaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8
UaaGzbVlaacIcacaaI1aGaaiOlaiaaiIdacaGGPaaaaa@4B03@
x
1
+
x
2
≤
5.
(
5.9
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa
amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHKjYOcaaI1a
GaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda
caGGUaGaaGyoaiaacMcaaaa@4B7C@
I also add the following restrictions of the form (3.2) on
α
:
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabeg7aHjaacQdaaaa@3A26@
α
≥
−
4
,
(
5.10
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabeg7aHjabgwMiZkabgkHiTiaaisdacaGGSaGaaGzbVlaaywW7
caaMf8UaaGzbVlaaywW7caGGOaGaaGynaiaac6cacaaIXaGaaGimai
aacMcaaaa@498D@
α
≤
4.
(
5.11
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabeg7aHjabgsMiJkaaisdacaGGUaGaaGzbVlaaywW7caaMf8Ua
aGzbVlaaywW7caGGOaGaaGynaiaac6cacaaIXaGaaGymaiaacMcaaa
a@4892@
This
yields five linear constraints (5.7)
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa
a@3896@
(5.11) on
x
1
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGccaGGSaaaaa@3A95@
x
2
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGGSaaaaa@3A96@
and
α
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabeg7aHjaacYcaaaa@3A18@
from which
α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabeg7aHbaa@3968@
may be removed by FM elimination to obtain:
x
1
≥
−
4
,
(
5.12
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyzImRa
eyOeI0IaaGinaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl
aacIcacaaI1aGaaiOlaiaaigdacaaIYaGaaiykaaaa@4A1C@
x
2
≥
−
4
,
(
5.13
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyyzImRa
eyOeI0IaaGinaiaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl
aacIcacaaI1aGaaiOlaiaaigdacaaIZaGaaiykaaaa@4A1E@
x
1
+
x
2
≥
0
,
(
5.14
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa
amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHLjYScaaIWa
GaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda
caGGUaGaaGymaiaaisdacaGGPaaaaa@4C3C@
x
1
+
x
2
≤
5.
(
5.15
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIa
amiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHKjYOcaaI1a
GaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiwda
caGGUaGaaGymaiaaiwdacaGGPaaaaa@4C33@
According to the theory, any record
(
x
1
,
x
2
)
′
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbmaabmaapaqaa8qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqa
aOWdbiaacYcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpe
GaayjkaiaawMcaaGGaaiab=jdiIcaa@3FFD@
that satisfies (5.12)
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa
a@3896@
(5.15) can be made
consistent with the original edits (5.4)
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa
a@3896@
(5.6) by transferring a
certain amount
−
4
≤
α
≤
4
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiabgkHiTiaaisdacqGHKjYOcqaHXoqycqGHKjYOcaaI0aaaaa@3F3B@
between
x
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@39DB@
and
x
2
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGccaGGUaaaaa@3A98@
The example record
(
x
1
,
x
2
)
′
=
(
−
2
,
3
)
′
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbmaabmaapaqaa8qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqa
aOWdbiaacYcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpe
GaayjkaiaawMcaaGGaaiab=jdiIkabg2da9maabmaapaqaa8qacqGH
sislcaaIYaGaaiilaiaaiodaaiaawIcacaGLPaaacqWFYaIOaaa@473A@
is inconsistent with the original edit rules (5.4)
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa
a@3896@
(5.6) but satisfies (5.12)
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislaa
a@3896@
(5.15). This implies
that the record can be made consistent with the original edits by applying
g
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadEgacaGGUaaaaa@3967@
It is easy to see that this is true; any
choice
2
≤
α
≤
3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaaikdacqGHKjYOcqaHXoqycqGHKjYOcaaIZaaaaa@3E4B@
will do.
It is interesting to note that, for the special case that
P
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiaadcfaaaa@389E@
consists of the single FH operation that
imputes
x
j
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaWGQbaapaqabaGccaGGSaaaaa@3AC9@
the transformed system of edits (5.3) is
obtained by replacing every occurrence of
x
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaWGQbaapaqabaaaaa@3A0F@
in the original edits by an unrestricted
parameter
α
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabeg7aHjaac6caaaa@3A1A@
Eliminating
α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabeg7aHbaa@3968@
from (5.3) is equivalent in this case to
eliminating
x
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9
Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadIhapaWaaSbaaSqaa8qacaWGQbaapaqabaaaaa@3A0F@
directly from the original edits. In this
sense, the above result generalizes the fundamental property of FM elimination
for FH operations to all edit operations of the form (3.1).
In general, the set of records defined by expression (5.2)
depends on the way the edit operations are ordered. Thus, two paths consisting
of the same set of edit operations in a different order need not yield the same
solution to the error localization problem. In this respect, general edit
operations differ from FH operations (Scholtus 2014).
ISSN : 1492-0921
Editorial policy
Survey Methodology publishes articles dealing with various aspects of statistical development relevant to a statistical agency, such as design issues in the context of practical constraints, use of different data sources and collection techniques, total survey error, survey evaluation, research in survey methodology, time series analysis, seasonal adjustment, demographic studies, data integration, estimation and data analysis methods, and general survey systems development. The emphasis is placed on the development and evaluation of specific methodologies as applied to data collection or the data themselves. All papers will be refereed. However, the authors retain full responsibility for the contents of their papers and opinions expressed are not necessarily those of the Editorial Board or of Statistics Canada.
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Copyright
Published by authority of the Minister responsible for Statistics Canada.
© Minister of Industry, 2016
Use of this publication is governed by the Statistics Canada Open Licence Agreement .
Catalogue No. 12-001-X
Frequency: semi-annual
Ottawa
Date modified:
2016-06-22