Journal of Global Positioning Systems (2003)
Vol. 2, No.2:117-124
On the approximationof the integer least-squares success rate:
which lower or upper bound touse?
Sandra Verhagen
Delft InstituteofEarthObservation and SpaceSystems,Delft University of Technology, Kluyverweg 1, 2629 HSDelft
Received:16 November 2003/ Accepted:28 November2003
Abstract. The probability of correctinteger estimation,
the success rate,is an important measure when the goal
isfast and highprecision positioning witha GlobalNav-
igation SatelliteSystem.Integer ambiguity estimation is
the process of mappingthe least-squares ambiguity esti-
mates, referredto as thefloat ambiguities, toaninteger
value.Itisnamely known that the carrier phaseambigu-
ities are integer-valued, and it isonly after resolution of
these parameters that the carrierphase observationsstart
tobehave asvery precisepseudorange measurements.
The success rate equals the integral oftheprobabilityden-
sity functionof the float ambiguitiesover thepull-in region
centered at the true integer, which is the regionin which all
real values are mappedto this integer.The success ratecan
thus be computed without actualdata and isvery valuable
as ana prioridecision parameter whether successful ambi-
guity resolution is feasible or not.
The pull-in region is determined by the integer estimator
that is used and therefore the success rate also depends on
the choice of the integer estimator.It is known that the in-
teger least-squares estimatorresults in the maximum suc-
cessrate.Unfortunately, itisverycomplex toevaluate the
integral whenintegerleast-squares is applied.Therefore,
approximations have to beused.In practice, forexample,
the successrate of integer bootstrapping is oftenusedas
a lower bound.But more approximations have been pro-
posed which areknownto be either a lower orupper bound
of the actual integer least-squares success rate.
In this contributionan overview of the most important
lower and upper bounds will be given.These bounds are
comparedtheoretically aswellasbasedon theirperfor-
mance.Theperformanceis evaluated using simulations,
since it isthen possible to compute the ’actual’ success
rate.Simulations are carriedout for thetwo-dimensional
case, since its simplicity makesevaluationeasy, but also
for thehigher-dimensional geometry-based case,since this
gives aninsight tothe performance thatcan beexpected in
practice.
Keywords:ambiguity resolution,success rate,integer
least-squares.
1 Introduction
Fast and high precision positioning with a GlobalNavi-
gation Satellite System is only feasible when the verypre-
cise carrier phase observationscan be used.Unfortunately,
these observations are ambiguous byanunknown, inte-
ger numberofcycles.These integerambiguityparameters
need to be resolved, before thecarrier phase observations
startto behave asvery precisepseudorange measurements.
The procedure to solve the GNSSmodel is to firstapply
a standard least-squares adjustment so that a real-valued
float solutionis obtained. Thenextstepis thentomapthe
real-valued ambiguity estimates tointeger values.Several
integer estimators can be usedforthatpurpose:integer
rounding, bootstrapping (conditionalrounding), orinteger
least-squares (ILS).The optimalchoice is thelatter,since
this estimator maximizes the probability of correct inte-
ger estimation as wasproven in Teunissen (1999).The
laststep istocorrect theremaining real-valued parame-
ters, such asthe baseline parameters, by virtue of their cor-
relation with the ambiguities, and then the so-called fixed
solution is obtained.For that purpose, itisassumed that
the integer-valued ambiguityestimatesaredeterministic.
However,this is actuallynot thecase and thisassumption
can onlybemade foravery high probability ofcorrect
integer estimation, i.e.success rate.
The success rateis thus a very important measure in order
to decidewhetheror notan attemptshouldbe madeto fix

118 Journalof Global Positioning Systems
the ambiguities.The integer ambiguities can only be con-
sidered deterministic when the successrate isvery close to
one, and thenevaluationbymeans of discernibility tests is
possible.
The success rate equals the integral oftheprobabilityden-
sity function of the float ambiguities overthe pull-in re-
gion,which is theregion in whichallreal values are
mapped tothesame integer.The pull-in region isdeter-
mined by the integer estimator that isused, and therefore
the success rate alsodepends on the choice of the integer
estimator.Unfortunately, it isvery complex to evaluate
the integral when integer least-squares isapplied.There-
fore,a numberof approximations havebeen proposed.For
example, the successrate ofinteger bootstrapping is of-
tenused, andisintroduced as alower bound inTeunissen
(1999).In Teunissen (1998a) lower and upper bounds were
obtained by bounding the region of integration.Another
upper bound was given in Teunissen (2000) based on the
Ambiguity Dilution of Precision.A final interesting lower
bound isderived in Kondo (2003).These approximations
of theILS successrate areconsideredin this paperdueto
their improved performance.Notethat inThomsen (2000)
some lower and upper bounds for theILS successrate
are evaluated.However, the evaluation is based only on
two-dimensional examples, which showed that theboot-
strapped lower bound, and the ADOP based upper bound
performed very well.
This paper isorganized as follows.The problem of integer
estimationand theintegerleast-squares estimatorare de-
scribed in section 2.The lower and upper boundsfor the
success rateare presentedin section3.An evaluationof
these bounds is made based on simulations in section 4.
2Integer least-squares estimation
Thegeneral GNSSobservation model canbe writtenin the
form:
y=Aa +Bb +e, Qy(1)
where yisthe random vector withmdouble difference
code and phase observations,athen-vectorwith unknown
integer carrierphase ambiguities, i.e.a∈Zn,bis a p-
vector withtheunknown real-valued parameters, andeis
the noise vector.The real-valuedparameters are referred
toasthe baselineunknowns, although bmay also contain
for example atmospheric delays.The variance-covariance
(vc-) matrix of the observation vector is given by Qy.
Optimizingonthe integernatureof theambiguityparame-
ters,(cf.Teunissen,1999),involves solving a non-standard
least-squares problem, referred to asinteger least-squares
(Teunissen, 1993).Thesolution of model (1) is then ob-
tained by the followingminimization problem:
min
a,b y−Aa −Bb2
Qy,a ∈Zn, b∈Rn(2)
where ·2
Q= (·)TQ−1(·).
The followingorthogonal decomposition can be used:
y−Aa −Bb2
Qy=
ˆe2
Qy+ˆa−a2 Qˆa+ˆ
b(a)−b2
Qˆ
b|ˆa(3)
with the residual estimator ˆe=y−Aˆa−Bˆ b, thecondi-
tionalbaselineestimator ˆ b(a) = ˆ b−Qˆ
bˆaQ−1
ˆa(ˆa−a), and
corresponding vc-matrix Qˆ
b|ˆa=Qˆ b−Qˆ bˆaQ−1 ˆaQˆaˆ b.
It followsfrom eq.(3)that the solution ofthe minimiza-
tion problem ineq.(2) isobtained usingathree stepproce-
dure.Theunconstrained least-squares solutionisreferred
to as the floatsolution, with estimators ˆaand ˆ b, and resid-
ual vector ˆe.Taking into account theinteger nature of the
ambiguities means that thesecond term on the right-hand
side of eq.(3) needs tobe minimized and thelastterm is
set to zero.This isthe integer estimation step, providing
the fixed ambiguities ˇa:
ˇa= arg min
z∈Znˆa−z2
Qˆa(4)
Finally, solvingfor thelasttermineq.(3) corresponds to
fixingthebaseline, ˇ b=ˆ b−Qˆ
bˆaQ−1
ˆa(ˆa−ˇa).
The integerestimation step involvesa mapping fromthe
n-dimensional spaceof reals to the n-dimensional space
of integers.In the integer least-squaresapproach a subset
Sz⊂Rnisassigned toeachinteger vector z∈Zn. This
subset is called the pull-inregion andis definedas the col-
lection of all x∈Rnthat are closer to zthan toanyother
integergrid point inRn,where thedistanceis measured
in the metric of Qˆa.Thepull-in region that belongs to the
integer afollows thus as:
Sa=ˆa∈Rn| ˆa−a2
Qˆa≤ ˆa−z2 Qˆa,∀z∈Zn(5)
Since
ˆa−a2
Qˆa≤ ˆa−z2 Qˆa
⇐⇒ (z−a)TQ−1
ˆa(ˆa−a)≤1
2z−a2
Qˆa,∀z∈Zn
it followsthat
Sa={ˆa∈Rn||w| ≤1
2cQˆa,∀c∈Zn}(6)
with
w=cTQ−1
ˆa(ˆa−a)
cQˆa
(7)

Verhagen:On the approximationof the integer least-squares success rate119
−2−10 1 2
−2
−1
0
1
2
Fig. 1 Example of the 2-D integerleast-squarespull-in regions.
An exampleof the two-dimensionalpull-in regions is
shownin Fig. 1.
In order to arriveat the integer least-squares solution, an
integer search isrequired.TheILSprocedure is mecha-
nized in the LAMBDA(Least-SquaresAMBiguity Decor-
relation Adjustment) method (see Teunissen, 1993, 1995;
De Jonge and Tiberius, 1996).The search space is then de-
fined as ann-dimensional ellipsoid centered atˆa, its shape
is governed by the vc-matrix Qˆa.Dueto the high correla-
tion between theindividual ambiguities, the searchspace
in thecase ofGNSS isextremely elongated, sothatthe
search for the integer solution may take very long.There-
fore a very important step is tofirsttransform the search
space toa morespherical shapebymeansofa decorrela-
tion of the original float ambiguities.This decorrelation is
attained by a transformation:
ˆz=ZTˆa, Qˆz=ZTQˆaZ(8)
Thistransformation needs tobeadmissible,whichissaid
to be the case when both Zand itsinverse have integer
entries, so that the integer natureoftheambiguitiesispre-
served.The determinant ofZis thenequal to ±1,so that
the Z-transformation isvolume-preservingwithrespectto
the search space.
3Lower and upper bounds of the ILSsuccess rate
The probability ofcorrectintegerestimationinthecase of
integer least-squares equals:
Ps=P(ˆa=a) = 
Sa
fˆa(x)dx (9)
with Sagiven in (6),and fˆa(x)theprobabilitydensity
function ofthe float ambiguities.In general it is assumed
that thefloat ambiguities arenormally distributed.The
pull-in regionisunfortunately averycomplex region, so
that in practice approximations have to be used for the
integer least-squaressuccess rate.This section gives an
overviewof the most important lower andupper bounds
that are available.
3.1Lower bound based on bootstrapping
Integer bootstrapping meansthatthefloat ambiguitiesare
conditionally rounded tothenearest integers.Onestarts
with the most preciseambiguity,then correctsall other
ambiguities by virtue of theircorrelation with this one, and
continues with rounding the second ambiguity.Thispro-
cess isrepeated until allnambiguities are fixed.Integer
bootstrapping isavery simplemethodof ambiguityreso-
lution, andhasaclose tooptimalperformance after decor-
relation of the ambiguities using the Z-transformation of
the LAMBDA method.In Teunissen (1998b, 1999) it was
shown that the integer least-squares estimator isoptimal
in thesense thatthe successrate ismaximized, andit was
proposed to use thesuccessrate of integer bootstrapping
therefore as a lower boundfor the ILS success rate, since
in Teunissen (1998c) itis shown thatexact and easycom-
putation ofthisbootstrapped successrateispossible:
Ps≥Ps,B=
n

i=1 2Φ 1
2σi|I−1(10)
where σi|Ithe standarddeviation of the ith ambiguity
obtained throughaconditioningon theprevious I=
1,...,(i−1) ambiguities. And
Φ(x) =
x

−∞
1
√2πexp{−1
2v2dv}
3.2Lower and upper bounds based on boundingthe
integration region
In Teunissen (1998a) lower andupper bounds forthe ILS
successratewereobtained bybounding theintegration re-
gion.Obviously, alower bound isobtained ifthe integra-
tion regionis chosen suchthat it is completely contained
by the pull-in region, and an upper bound isobtained if the
integrationregion is chosen such that it completely con-
tainsthe pull-inregion.The integration region canthen be
chosen such that the integral is easy-to-evaluate.In ibid
the integrationregionfor thelowerbound is chosenas an
ellipsoidal regionEa⊂Sa.
The upper bound can thus be obtained by defining aregion
Ua⊃Sa, with Saas definedin (6).Note that the win
this expression can begeometrically interpreted as the or-
thogonal projection of (ˆa−a)onto the direction vector c.

120 Journalof Global Positioning Systems
Hence, Sais theintersectionofbanded subsetscentered
at aand havinga width cQˆa.Any finite intersection of
thesebanded subsetsencloses Sa, and thereforethesubset
Uacould be chosen as
Ua={ˆa∈Rn||wi|≤1
2ciQˆa,i = 1,...,p} ⊃ Sa
(11)
with
wi∼N(0,1)
The choice for pis still open, but a larger pwill result in a
sharper upper bound forthesuccessrate.However, when
p > 1the wiare correlated.This ishandled by defining a
p-vectorvas:
v= (v1,...,vp)Twith vi=wi
ciQˆa
Then Ua={ˆa∈Rn|p ∩
i=1|vi| ≤1
2}.The proba-
bility P(ˆa∈Ua)equals therefore theprobability that
component-wise rounding ofthevector vproduces the
zero vector.This meansthat P(ˆa∈Ua)is bounded from
above by theprobability that conditionalrounding,(cf.Te-
unissen, 1998c), producesthe zero vector,i.e.:
Ps≤P(ˆa∈Ua)≤
p

i=1 2Φ( 1
2σvi|I
)−1(12)
with σvi|Ithe conditional standard deviation of vi. The
conditional standarddeviations areequal to the diagonal
entries of thematrix DfromtheLDLT-decomposition
of the vc-matrix of v.The elements of this vc-matrix are
given as:
σvivj=cT
iQ−1 ˆacj
ci2
Qˆacj2 Qˆa
In order to avoid the conditional standard deviations be-
coming zero, the vc-matrixof vmust be of full rank, and
thus thevectors ci,i= 1,...,p ≤nneedto belinearly
independent.
Theprocedure forcomputation ofthisupperbound isas
follows.LAMBDA is used to find theq>>n closest
integersci∈Zn\{0}for ˆa= 0.These qinteger vec-
tors areorderedby increasingdistance to thezerovector,
measured in the metric Qˆa.Startwith C=c1, so that
rank(C) =1.Then find the first candidate cjfor which
rank([c1cj]) = 2.Continue with C= [c1cj]and find the
nextcandidatethat results inanincrease in rank. Continue
this process until rank(C)= n.
In Kondo(2003)instead of the conditionalvariances,sim-
plythe variancesof theviare used.Then thefollowing is
obtained:
p

i=1 2Φ( 1
2σvi
)−1=
p

i=1
Ps,i (13)
with the Ps,i
Ps,i =2
√2πσvi
1
2
0
exp −1
2
x2
σ2
vidx (14)
We know,(cf. Teunissen, 1998c),that
p

i=1 2Φ( 1
2σvi
)−1≤P(ˆa∈Ua)(15)
This meansthatit is only guaranteedthat Kondo’sapprox-
imation of thesuccess rateis alower bound ifP(ˆa∈Ua)
is equal tothe success rate.This will be the caseif pis
chosen equal to halfthenumber of facetsthatbound the
ILS pull-in region.So,it isrequired to know thisnumber,
but in practice onlythe boundsare known:
n≤p≤2n−1
If pischosen to besmallerthan half thenumber of bound-
ing facets,it is not guaranteedthat the approximation
gives a lower bound.On the other hand,if pischo-
sen to be larger thanrequired in order to guaranteethat
P(ˆa∈Ua) = P(ˆa∈Sa),the lower bound islessstrict
since it is defined as aproduct of probabilities which are
all smaller or equalto one.Notethat pmay becomevery
large when manysatellitesare visible.For instance, with
6 visible satellites and two frequenciesavailable,thenum-
ber ofunknown ambiguities for oneepoch is n= 10, and
p≤2n−1 = 1023.
It is possibleto findall adjacent integers, but it iscompu-
tationally demanding.First, note that it is not always the
case that the 2padjacent integers are also the 2pclosest
integers.Therefore, alarge setof integers cimust be se-
lected, in the same way as for the computation of the upper
bound described above with q>> 2(2n−1).Foreachin-
teger in thissetitmust bechecked if itisadjacent,which
is the case if1
2cilies onthe boundaryof both the pull-in
regions S0andSci.This isthe case if:
1
2ci−02
Qˆa=1
2ci−ci2
Qˆa= min
z∈Zn1
2ci−z2
Qˆa(16)
Note thatif cjis selectedas adjacent integer, −cjmustnot
be included inthe set C= [c1. . . , cp]adjacentintegers
that is used to compute the lower bound.
3.3Upper bound based on ADOP
The AmbiguityDilution of Precision (ADOP) is definedas
a diagnostic that triesto capture the maincharacteristics of

Verhagen:On the approximationof the integer least-squares success rate121
Table 1 Two-dimensionalexample.Mean and maximum differencebetween successrate based on simulationsand the lower and upperbounds.The
success ratefor whichthemaximum difference obtained isgiveninthe lastrow.
LB bootstr.LB regionUB ADOPUB region
mean difference0.00450.01800.00120.0181
maximum difference0.01040.10520.00290.0648
success rate0.80460.38760.83310.5589
the ambiguity precision.It is given as:
ADOP =|Qˆa|
1
n(17)
and has units of cycles.Itisintroduced inTeunissen
(1997),and describedand analyzedinTeunissen and Odijk
(1997).The ADOP measurehas some desirableprop-
erties.First,itis invariant for the class of admissible
ambiguity transformation, e.g.ADOPis independent of
the chosen reference satellite in the double differenceap-
proach, and ADOP will not change after the decorrelating
Z-transformationof the ambiguities.Whenthe ambigui-
ties are completely decorrelated,the ADOP equals the ge-
ometric meanofthe standarddeviationsof theambiguities,
hence it can be considered as a measure of the ambiguity
precision.
In Teunissen (2000) it isproven that anupper bound for
the ILS success rate based on the ADOP can be given as:
Ps≤Pχ2(n, 0) ≤cn
ADOP 2(18)
with
cn=n
2Γ(n 2)
2
n
π
This upper bound isidentical tothe one presented inHas-
sibi and Boyd (1998).
4Evaluation of the bounds
In order toevaluate the lower and upper bounds aspre-
sented in section3, simulations are used.Insection 3.2
the lowerboundbased onboundingtheintegrationregion
with anellipsoidalregion Ea⊂Sawas briefly outlined.
This bound is not includedin the results presented here,
sinceforallexamples thislowerbound performed badly.
Theprocedureis asfollows.Since itis assumedthatthe
float solution is normallydistributed, theprobabilities are
independentof themean, so onecan use N(0,Q)anddraw
samples from this distribution.
The first stepis to use a randomgeneratorto generaten
independent samples from the univariatestandard normal
distribution N(0,1), and then collect these inavector s.
This vector istransformed by means ofˆa=Gs, withG
equal to the Cholesky factor of Qˆa=GGT.The result is a
sample ˆafrom N(0, Qˆa), and this sampleis used asinput
for integer least-squares estimation.Ifthe output of thises-
timator equals thenull vector, then itiscorrect, otherwise
it is incorrect.This processcan be repeatedNnumber of
times, and one can count how manytimes thenull vector
is obtained asa solution, sayNstimes, andhow often the
outcome equals a nonzero integer vector, say Nftimes.
The approximationsof thesuccessrateand fail ratefollow
then as:
Ps=Ns
N, Pf=Nf N
In orderto getgoodapproximations,the number of sam-
ples Nmust be sufficiently large (see Teunissen, 1998a).
Wewillstartherewiththesimpletwo-dimensional case.
Thedual-frequency geometry-free GPSmodelforashort
baseline and for only one satellite-pair is used:
E{



p1
p2
φ1
φ2



}=



1 00
1 00
1λ10
1 0 λ2





ρ
a1
a2
(19)
where piand φiare the double difference (DD) code and
phase observations on frequency Li.Wavelengthsare de-
noted as λi, the range asρ,and the integer ambiguities
as ai.E{·} istheexpectation operator.The variance-
covariance matrix Qyis chosen asa diagonal matrix, with
undifferencedstandard deviations of σp=15 cm and
σφ= 1.5mm for both frequencies.For the simulation
1,000,000 samples were used.Theresulting lower and up-
per bounds are shown in table 2 (first row).
The same approach was followed by using:
Qˆa=1
fQˆa,ref,0<f≤1
for different values of f, and Qˆa,ref the vc-matrixfromthe
example described above.The results are shown in Fig. 2.
The top panels show thetwo upper bounds and the success
rates from the simulations.Obviously, the ADOP-based
upper boundis very strict and is always much better than
the upperboundbased onboundingthe integrationregion.
The bottom panelsshow the lower bounds.It follows that
for lower success rates (<0.93) thebootstrapped success
rate is the best lower bound.For higher success rates (right

122 Journalof Global Positioning Systems
Table2 Approximated successrates usingsimulation,the lowerbounds basedonbootstrapping (LBbootstr.)andboundingthe integration region (LB
region), and the upperbounds based on ADOP (UB ADOP) and bounding theintegration region (UB region).Number of satellites(no.SV) and the
ionospheric standard deviation (σI)are giveninthe firstcolumns.
no. SVσI[cm]simulationLB bootstr.LB regionUB ADOPUB region
200.99960.99920.9996 0.99970.9998
400.81750.74940.6976 0.84800.9420
410.44200.40970.1177 0.47490.6256
500.99890.99790.9989 1.00000.9990
510.87440.83370.8109 0.94700.9388
610.98860.97590.9881 0.99940.9922
630.47630.44160.1256 0.68080.6608
0.1 0.2 0.3 0.4 0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f
success rate
upper bounds
ADOP
region
simulation
0.6 0.7 0.8 0.91
0.985
0.99
0.995
1
f
success rate
upper bounds
0.1 0.2 0.3 0.4 0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f
success rate
lower bounds
bootstrapped
region
simulation
0.6 0.7 0.8 0.91
0.985
0.99
0.995
1
f
success rate
lower bounds
Fig. 2 Upper and lower bounds for the success ratein the 2-D case as functionof fwith vc-matrix1
fQˆa,ref.Left:whole rangeof success rates; Right:
only forhigh successrates.

Verhagen:On the approximationof the integer least-squares success rate123
Table 3 Overview of lower and upper bounds for the ILS success rateconsidered inthis contribution.
boundbased onreferences
Ps≥
n

i=1 2Φ 1
2σi|I−1bootstrappingTeunissen (1998b, 1998c,1999)
Ps≥
p

i=1 2Φ( 1
2σvi
)−1bounding integration regionKondo (2003)
Ps≤Pχ2(n, 0)≤cn
ADOP 2ADOPHassibi and Boyd (1998);Teunissen (2000)
Ps≤
p

i=1 2Φ( 1
2σvi|I
)−1bounding integration regionTeunissen (1998a)
panel),the lower bound proposed by Kondo works very
well and isbetter thanthe bootstrapped lower bound.Note
that the range of success ratesin the right panel isvery
small.
Table 1 shows the maximum and mean differences of the
lowerand upper bounds with the success rate fromsim-
ulation.From thesedifferencesitfollows that the boot-
strapped lower bound and the ADOP-based upper bound
are best.
Becauseofitssimplicitythegeometry-free modelisvery
suitable for afirstevaluation, though it isof course more
useful to know how well the approximationswork in
practice.Therefore,simulations were carriedfor several
geometry-based models.The GPSconstellation was based
on the Yuma almanac for GPS week 184 and a cut-off
elevationof 15o.Undifferencedstandard deviations of
σp= 30cmand σφ=3mm were used for bothfrequen-
cies.The GPSmodel was setup fora single epoch for
three different times, for which 4, 5 and6 satellites were
visible respectively.A short to medium baseline length
was chosen byvarying the ionospheric standard deviation
σI.Forthesimulation500,000 sampleswereused.The
resulting lower and upper boundsareshown in table 2.
The resultsshow that Kondo’slower bound works very
well for a highsuccess rate, butin general thebootstrapped
lowerboundismuchbetter.It is difficult to say whichup-
per bound isbest.For the examples with only fourvisible
satellitesthe ADOP-basedupper bound isbetter thanthe
one obtained by bounding the integration region, but inthe
examples with more satellites the latter is somewhat better.
All boundsarebest in the caseofhigh precisions, i.e. high
success rates.
5Concluding remarks
In this contributiontwo lower boundsand two upper
bounds for the integerleast-squaressuccess rate were pre-
sented.An overviewof the boundsis givenin table 3.
Theperformance of thedifferent boundswasevaluated by
comparing their outcomes for several geometry-free and
geometry-based examples with thesuccess ratethat isob-
tained by using simulation.
In general,the bootstrappedlowerbound givesthebest re-
sults.Whenthe success rate ishigh, the lower boundpro-
posed byKondo (2003) basedonbounding theintegration
region maywork better.
It can be concludedthatKondo’s lower boundseemstobe
useful onlyin a few cases.Firstly, to obtain a strictlower
bound the precisionshould be high, so that the success rate
is high.Even then,it depends on the minimum required
success rate whether it is really necessary to use the ap-
proximation:if thebootstrapped success rate issomewhat
lower thanthisminimum requiredsuccess rate,Kondo’s
approximation canbeusedtoseeifitislarger.Themin-
imum requiredsuccess rate could be chosensuch that the
fixed ambiguitiescan be considered deterministic.In this
case, thediscernibility testsas used inpractice, suchas the
ratio test, can beused.
Anadvantage of thebootstrapped successrateisthatitis
very easy to compute,since theconditional variances are
already available whenusing the LAMBDAmethod.The
computation of Kondo’slower bound may beslightly more
complex, since for high-dimensional problems the number
of facets thatbound thepull-in region can bevery large,
and this numberneeds to be knownin order to guarantee
that a (strict) lower boundis obtained.
With respectto theupper bounds,one canhave a little
more confidence in the ADOP-based bound, since its over-
all performance, basedon all examples, is slightly better.
However, in thegeometry-based case,the upper bound
based onbounding the integration region oftenperforms
somewhat better.An advantageof the ADOP-based upper
bound is thatit is easy to compute, whereas for the up-
per bound based on bounding the integration regionone
has theproblem ofdetermining thenclosest independent
integers to the zero vector.

124 Journalof Global Positioning Systems
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