Error Bars for Distributions of Numbers of Events

0.32 MB·English

by  Ritu Aggarwal

Error bars for distributions of numbers of events
R.Aggarwal1,2,A.Caldwell2
1 PanjabUniversity,Chandigarh,India([e-mail])
2 MaxPlanckInstituteforPhysics,Munich,Germany([e-mail])
Abstract
Thecommonpracticefordisplayingerrorbarsondistributionsofnumbersof
eventsisconfusingandcanleadtoincorrectconclusions. Aproposalismade
for a different style of presentation that more directly indicates the level of
agreementbetweenexpectationsandobservations.
2
1
0 1 Introduction
2
Symmetricerrorbarscenteredontheobservednumberofeventscanbehighlymisleadingandinpractice

n oftengeneratesconfusion. Atypicalexample1 ofsuchadatapresentationisgiveninFig.1.
a
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a E40 E
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p 20
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3
v
3 10
9 1
5
2
.
2 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400
1 Mass(GeV) Mass(GeV)
1
1
:
v
i
X Fig.1: Atypicalpresentationofdata-hereeventcountsasafunctionofmassin25GeVintervals. Thesamedata
√ √
areplottedtwice: left-linearscale; right-logarithmicscale. Theerrorbarscovertherange(o− o,o+ o),
r
a
whereoisthenumberofobservedevents. Thehistogramgivestheexpectations(meansofPoissondistributions).
Therearethreeproblemswiththisstandardstyleofpresentation:
– Firstofall,thereisnouncertaintyonthenumberofobservedevents. Wecertainlydonotmeanthat
thereisahighprobabilitythatwehad2.3ratherthan2eventsinthe7th binintheplot. Actually,
theerrorbarisintendedtorepresenttheuncertaintyonadifferentquantity-theuncertaintyonthe
1Thedataisartificialandinventedforthepurposesofthisnote.
mean of an assumed underlying Poisson distribution. The probability distribution for this mean
givenanobservednumberofeventso,P(θ|o),canbequiteasymmetric,anddifferentchoicescan
bemaderegardingwhattoplotassummary(e.g.,themode,themeanorthemedian)ofP(θ|o).
√
– The second problem arises with the length of the error bar. This is routinely taken as ± o ,
motivatedbythePoissonresultthatthevarianceisequaltothemean,sothattheerrorbarshould
cover±1σ,or68%probabilityforpossiblevaluesofθ. However,theprobabilityrangecovered
by this definition of the error bar varies dramatically as o → 0 and the probability above and
belowthepointishighlyasymmetric. Usuallynoerrorbarisplottedwhen0eventsaremeasured,
although this measurement also yields information on possible values of θ. These problems are
occasionally avoided by using asymmetric error bars, usually covering the central 68 % of the
probability from the cumulative of P(θ|o), but this is still the exception rather than the rule in
experimentalparticlephysics.
– A third problem occurs when data are compared to expectations, as in Fig. 1, and the error bar
is used to determine if the observed number of events represents a significant deviation from the
expectation. The error bar on the plot often gives the completely wrong information in this case,
sincetherelevantprobabilityistheprobabilitythattheexpectationcouldhaveyieldedthenumber
of observed events, not the probability that the observed number of events could have fluctuated
to the expectation. For example, in the next-to-last bin in Fig. 1, the expectation is 0.011 and
two events are observed. It is VERY wrong to conclude that we have slightly more than a 1 σ
discrepancyinthisbin.
A proposal for an alternative presentation is given here. We focus on the case where we are
comparing observations to predictions and the fluctuations can be modeled with a Poisson distribution.
Westartwiththesimplestcase-thatpredictionsareavailablewithnegligibleuncertainty. Wethenshow
how to include the uncertainty due to predictions based on finite Monte Carlo event sets and due to
systematicuncertainties.
2 Negligibleuncertaintyontheprediction
Westartwiththesimplestcase-theexpectations(meansofPoissondistributions)areknownwithvery
small uncertainty, and we want to compare the observed data to these expectations. The plot should
give us an indication of whether the observations are within reasonable statistical fluctuations of the
expectation;i.e.,theplotshouldgivetheuseranindicationofhowrareaparticularobservednumberof
eventsowasexpectedtobegivenaPoissondistributionwithmeannumberofeventsν. Theprobability
distributionforois
e−ννo
P(o|ν) = . (1)
o!
Themostprobablevalueforo(modeoftheprobabilitydistribution)isgivenby
o∗ = (cid:98)ν(cid:99) ; (2)
i.e.,thelargestintegernotgreaterthanν.
Twochoicescanbemadefortheprobabilityintervalstodisplay:
1. The central interval, defined for a probability density P(x) with possible range for the parameter
{x ,x }
min max
(cid:90) x1 (cid:90) xmax
α/2 = P(x)dx = P(x)dx .
xmin x2
The central interval is {x ,x } and contains probability 1 − α. In our case, we have a discrete
1 2
probabilitydistributionP(o|ν)andtheequalitygenerallycannotbesatisfied. Wetakeinstead
o
(cid:88)
o = sup{ P(i|ν) ≤ α/2}+1 (3)
1
o∈Z
i=0
2
∞
(cid:88)
o = inf{ P(i|ν) ≤ α/2}−1 . (4)
2
o∈Z
i=o
IfP(o = 0|ν) > α/2,thenwetakeo = 0.
1
Wedefinethesetofobservationswhichfallintothecentral1−αprobabilityinterval2 as
OC = {o ,o +1,...,o }
1−α 1 1 2
and we display these values of o. Different colors can be used to represent different 1−α prob-
ability ranges. For example, if we take ν = 3.¯3, then we find the values given in Table 1. If we
choose1−α = 0.68,thenourdefinitionsgive
OC = {2,3,4,5}
0.68
whereas
OC = {0,1,2,3,4,5,6,7}
0.95
and
OC = {0,1,2,3,4,5,6,7,8,9,10,11} .
0.999
Table1:Valuesofo,theprobabilitytoobservesuchavaluegivenν =3.¯3andthecumulativeprobability,rounded
tofourdecimalplaces. Thefourthcolumngivestherankintermsofprobability-i.e.,theorderinwhichthisvalue
of o is used in calculating the smallest set OS , and the last column gives the cumulative probability summed
1−α
accordingtotherank.
o P(o|ν) F(o|ν) R F (o|ν)
R
0 0.0357 0.0357 7 0.9468
1 0.1189 0.1546 5 0.8431
2 0.1982 0.3528 2 0.4184
3 0.2202 0.5730 1 0.2202
4 0.1835 0.7565 3 0.6019
5 0.1223 0.8788 4 0.7242
6 0.0680 0.9468 6 0.9111
7 0.0324 0.9792 8 0.9792
8 0.0135 0.9927 9 0.9927
9 0.0050 0.9976 10 0.9976
10 0.0017 0.9993 11 0.9993
11 0.0005 0.9998 12 0.9998
12 0.0001 1.0000 13 1.0000
2. The second option for the probability interval is to use the smallest interval containing a given
probability. Inthecaseofaunimodalcontinuousdistribution,wecanwritetheconditionas
(cid:90) x2
1−α = P(x)dx and P(x ) = P(x ) .
1 2
x1
Forourdiscretecase,thesetmakingupthesmallestintervalcontainingprobabilityatleast1−α,
OS ,isdefinedbythefollowingalgorithm
1−α
(a) Start with OS = {o∗}. If P(o∗|ν) ≥ 1−α, then we are done. An example where this
1−α
requirementisfulfilledfor1−α = 0.68is(ν = 0.001,o∗ = 0,OS = {0}).
0.68
2Notethat1−αistheminimumprobabilitycoveredandthatthesetgenerallycoversalargerprobability.
3
(b) If P(o∗|ν) < 1−α, then we need to add the next most probable number of observations,
whichintheunimodalcaseiseithero∗+1oro∗−1. AssumethatP(o∗+1|ν) > P(o∗−1|ν).
Then,wewouldextendoursettoOS = {o∗,o∗+1}andcheckagainwhetherP(OS ) ≥
1−α 1−α
1−α. WewouldcontinuetoaddmemberstothesetOS untilthisconditionismet,always
1−α
takingthehighestprobabilityoftheremainingpossibleobservations.
Inthespecialcasesthattwoobservationshaveexactlythesameprobability(whenν takeson
anintegervalue,P(o = ν) = P(o = ν −1)),thenbothvaluesshouldbetakenintheset.
IfweconsidertheexamplegiveninTable1,thenusingthecumulativeaccordingtorank,F ,we
R
find
OS = {2,3,4,5}
0.68
whereas
OS = {0,1,2,3,4,5,6,7}
0.95
and
OS = {0,1,2,3,4,5,6,7,8,9,10} .
0.999
The same sets are found as for the central interval for 1−α = 0.68 and 0.95, but is smaller for
0.999.
Ifthepredicteddistributionhasalargemean(ν > 50,say),thenwecanusetheGaussianapprox-
imation. Inthiscase,wetaketheminimalsymmetricrangearoundo∗ suchthat
(cid:90) o∗+(n+0.5) √1 e−(x−2oν∗)2dx ≥ 1−α
o∗−(n+0.5) 2πν
anddefineO = {o∗−n,...,o∗+n}. Thisgivesboththecentralintervalaswellastheminimalinterval.
α
As an example for the procedures defined here, Fig. 2 shows the same distribution of observed
number of events as a function of invariant mass as was shown in Fig. 1. Three different probability
intervals are shown, corresponding to 1−α = 0.68, 1−α = 0.95, and 1−α = 0.999 and follow the
definitionofthesmallestinterval. Notethatthebandsareextendedbeyondtheintegervaluesintheset
by0.5forclarityofpresentation. E.g.,thebandcontainingtheset{2,3,4,5}isdrawnfrom1.5 → 5.5.
The smallest 1−α color is chosen if more than one 1−α set contain the same set members (e.g., this
occurswhentheset{0}contains> 95%probabilityasinthelastbinsintheplot). Thecolorschemeis
meanttobesuggestive. Observedeventcountsoutsidetheshadedbandsshouldindicateunlikelyresults.
3 PredictionsbasedonMonteCarlowithfinitestatistics
We now extend the prescription for cases where the prediction is uncertain. The first case we consider
is that the prediction is based on a Monte Carlo which has non-negligible statistical uncertainties. We
derive the probability distribution for the expected number of events o given that a MC set (consisting
possiblyofdifferentcomponents)givesneventsandtheMCnormalizationfactoriss(definedsuchthat
theMCpredictionisdividedbythefactorswhencalculatingtheexpectedmeanforthedata). Thefactor
sisinitiallytakentobeknownexactly.
TheprocessofassigningMonteCarloeventstobinsindicatestheuseofthemultinomialdistribu-
tionforcalculations(ifwegenerateafixednumberofeventsratherthanafixedluminosity). Weassume
here that the probability for an event to populate any given bin is small so that Poisson statistics can be
usedasavalidapproximation. AssumeourMonteCarlosamplehasresultedinnevents(inabinofin-
terest). Wenowneedtodeterminetheexpectedmeanforthedatasampleandtheprobabilitydistribution
forthismean. WeuseλforthemeanoftheMCdistributioninthebin,andν forthemeanexpectedfor
the observations. Applying Bayes’ Theorem [1] and taking the Jeffreys’ prior [2] on the mean for the
MC,λ,
1
P (λ) ∝ √
0
λ
4
VV8800 VV
ee ee
GG GG
5 5 5 5
22 data 22
s/s/7700 s/s/
ntnt data (0 events) ntnt
ee ee
vv MC vv
EE EE
6600 68 % Prob
95 % Prob
5500 99.9 % Prob 1100
4400
3300
2200
11
1100
00
00 5500 110000 115500 220000 225500 330000 335500 440000 00 5500 110000 115500 220000 225500 330000 335500 440000
MMaassss((GGeeVV)) MMaassss((GGeeVV))
Fig. 2: The proposed style of presentation for the same data as shown in Fig. 1. The shaded bands represent
differentprobabilityintervalsfortheobservednumberofevents: greenisfor1−α=0.68;yellowisfor1−α=
0.95 and red is for 1−α = 0.999. The bands use the smallest interval containing at least this probability, and
extendtothenext1/2integervalue(seetext). Forthelogarithmicscaleplotontheright,thedataareshownasa
solidtriangleifo=0.
thepdfforλis
4nn!
P(λ|n) = √ λn−1/2e−λ
π(2n)!
whichleadsto
E[λ] = n+1/2 and
λ∗ = n−1/2 .
Forscalingthepredictiontothemeanexpectedforthedata(ν = λ/s),wehave:
P(ν|n,s) = P(λ|n)dλ/dν
4nn!
= √ s(sν)n−1/2e−sν
π(2n)!
Forthedistributionofdataevents,weusetheLawofTotalProbability[3]:
(cid:90)
P(o|n,s) = P(o|ν)P(ν|n,s)dν
sn+1/24nn! (cid:90)
P(o|n,s) = √ νo+n−1/2e−(s+1)νdν
πo!(2n)!
Theintegralgives
√
[2(o+n)]! π
4n+o(1+s)o+n+1/2(n+o)!
5
so
sn+1/2 n![2(o+n)]!
P(o|n,s) = . (5)
(1+s)o+n+1/24oo!(2n)!(n+o)!
and
n+1/2
E[o] = .
s
Foro = 0,wehave
(cid:18) s (cid:19)n+1/2
P(o = 0|n,s) = . (6)
1+s
Theprobabilityforthesucceedingvaluesofocanthenbeeasilycalculatedas
(2n+2o+2)(2n+2o+1)
P(o+1|n,s) = P(o|n,s)· .
4(o+1)(1+s)(n+o+1)
We would now use these probabilities of o for the procedure described in the previous section
ratherthanthePoissondistributionforP(o|ν).
Table 2: Values of o, the probability to observe such a value given n = 10 and s = 3 and the cumulative
probability, rounded to four decimal places. The fourth column gives the rank in terms of probability - i.e., the
orderinwhichthisvalueofoisusedincalculatingthesmallestsetOS ,andthelastcolumngivesthecumulative
1−α
probabilitysummedaccordingtotherank.
o P(o|n,s) F(o|n,s) R F (o|n,s)
R
0 0.0488 0.0488 7 0.9072
1 0.1280 0.1768 4 0.6654
2 0.1840 0.3608 2 0.3757
3 0.1917 0.5525 1 0.1917
4 0.1617 0.7142 3 0.5374
5 0.1173 0.8315 5 0.7827
6 0.0757 0.9072 6 0.8584
7 0.0446 0.9519 8 0.9519
8 0.0244 0.9763 9 0.9763
9 0.0125 0.9888 10 0.9888
10 0.0061 0.9949 11 0.9949
11 0.0028 0.9978 12 0.9978
12 0.0013 0.9991 13 0.9991
13 0.0006 0.9996 14 0.9996
14 0.0002 0.9998 15 0.9998
15 0.0001 0.9999 16 0.9999
AnexampleoftheeffectoffiniteMonteCarlostatisticsonP(o)isgiveninFig.3. Asisclear,if
theMonteCarlousedtoderivethepredictionforthenumberofeventshasaneffectiveluminosityonlya
factor3largerthanthedata,significantdifferencescanresultinprobabilitiesforobservedevents. Ifwe
considertheexamplegiveninTable2,wheren = 10ands = 3,themeanvalueofν isν¯ = 3.5andthe
finiteMCstatisticsgivesslightlydifferentresultsforoursets:
OC = {1,2,3,4,5,6} OS = {1,2,3,4,5}
0.68 0.68
6
OC = {0,1,2,...,8} OS = {0,1,2,...,7}
0.95 0.95
OC = {0,1,2,...,13} OS = {0,1,2,...,12} .
0.999 0.999
As expected, the set of possible values of o has increased for a given 1−α probability due to the extra
uncertaintyintroducedbythefinitenumberofMonteCarlocounts.
0.25
0.2
0.15
0.1
0.05
0
0 5 10 15
Fig.3: ComparisonofthedistributionsP(o|ν =10/3)andP(o|n=10,s=3.).
4 Predictionwithsystematicuncertainties
If the predictions have systematic uncertainties, then this can also be taken into account in defining the
probabilitiesfortheobservednumberofevents. Theprobabilitydensityforν willnowdependonextra
quantities,andwewrite
P(ν|n,s,θ(cid:126))
where θ(cid:126)is a set of nuisance parameters used to describe the systematic uncertainties (e.g., energy scale
parameters). ItmaybepossibletodetermineP(ν|n,s,θ(cid:126))directlyfromMonteCarlosimulationswhere
alsothesystematicallyuncertainquantitiesarevariedaccordingtotheirbeliefdistributions. Inthiscase,
wewouldusethisinformationandhave:
(cid:90)
P(o|n,s,θ(cid:126)) = P(o|ν)P(ν|n,s,θ(cid:126))dν .
In general, this integral will need to be solved numerically and the P(o|n,s,θ(cid:126)) then input into the pre-
scriptionsabove.
Inmanycases,wehaveafixednumberofMCeventsandthesameeventsareusedrepeatedlywith
different assumptions for the uncertain quantities, so that the systematic uncertainty is on the scaling
parameters. Inthiscase,wecanwrite
(cid:90)
P(ν|n,θ(cid:126)) = P(ν|n,s)P(s|θ(cid:126))ds
wherethesystematicuncertaintyappearsasapdfforthescalefactor. Theprobabilitydistributionforo
isnow
(cid:90) (cid:20)(cid:90) (cid:21)
P(o|n,θ(cid:126)) = P(o|ν) P(ν|n,s)P(s|θ(cid:126))ds dν .
7
Often,weassumethebeliefinvaluesforthescalefactorscanbemodeledasaGaussian:
1 −(s−s0)2
P(s|s0,σs) = √ e 2σs2 .
2πσ
s
Inthiscase,wewouldhave
(cid:34) (cid:35)
P(o|n,s0,σs) = (cid:90) e−ννo (cid:90) √4nn! s(sν)n−1/2e−sν√ 1 e−(s−2σss20)2ds dν . (7)
o! π(2n)! 2πσ
s
This looks rather forbidding but can be solved numerically. Taking our standard example, we now add
totheMCstatisticaluncertaintya30%systematicuncertaintyinthescalefactorsandfind3 theresults
giveninTable3. AgraphicalpresentationoftheeffectonP(o)ofincludingsystematicuncertaintieson
sisshowninFig.4.
Table 3: Values of o, the probability to observe such a value given n = 10 and s = 3 and 30 % systematic
uncertainty on s, together with the cumulative probability, rounded to four decimal places. The fourth column
givestherankintermsofprobability-i.e.,theorderinwhichthisvalueofoisusedincalculatingthesmallestset
OS ,andthelastcolumngivesthecumulativeprobabilitysummedaccordingtotherank.
1−α
o P(o|n,s) F(o|n,s) R F (o|n,s)
R
0 0.0539 0.0539 7 0.8492
1 0.1272 0.1811 4 0. 6111
2 0.1699 0.3511 2 0.3403
3 0.1703 0.5214 1 0.1703
4 0.1436 0.6650 3 0.4839
5 0.1083 0.7733 5 0.7194
6 0.0759 0.8492 6 0.7953
7 0.0509 0.9001 8 0.9001
8 0.0332 0.9333 9 0.9333
9 0.0214 0.9547 10 0.9547
10 0.0138 0.9685 11 0.9685
11 0.0090 0.9776 12 0.9776
12 0.0060 0.9835 13 0.9835
13 0.0040 0.9876 14 0.9876
14 0.0028 0.9904 15 0.9904
Theelementsofourdifferentsetsarenowgivenby
OC = {1,2,3,4,5,6} OS = {1,2,3,4,5}
0.68 0.68
OC = {0,1,2,...,11} OS = {0,1,2,...,9}
0.95 0.95
OC = {0,1,2,...,44} OS = {0,1,2,...,33} .
0.999 0.999
The 68 % interval is now different between the two definitions, and very large values of o are
allowedatprobability1−α = 0.999,inparticularforthecentralinterval.
3notethatwithsuchalargesystematicvariation,theGaussiandistributioninEq.7istruncatedandisrenormalized,leading
toashiftinE[o].
8
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
0 5 10 15 0 5 10 15
Fig.4: ComparisonofthedistributionP(o|n = 10,s = 3,σ = 0.3s )withP(o|ν = 10/3)(top)andP(o|ν =
0 s 0
10/3,σ =0.3s )toP(o|ν =10/3)(bottom).
s 0
5 Summary
We have described an alternative presentation of data for cases where the aim is to judge whether an
observednumberofeventsisconsistentwiththemodelpredictions. Webelievethisstyleofpresentation
ismoreappropriatethanthecommonone,whereerrorbarsareplacedontheobservednumberofevents.
Examplecodesnippetsfortheexamplesdescribedherecanberequestedfromtheauthors.
6 Acknowledgments
TheauthorswouldliketothankFrederikBeaujeanandKevinKröningerforusefuldiscussions.
References
[1] see, e.g., G. D’Agostini, Bayesian Reasoning in Data Analysis - A Critical Introduction, World
Scientific,2003.
[2] H.Jeffreys,AnInvariantFormforthePriorProbabilityinEstimationProblems,Proceedingsofthe
RoyalSocietyofLondon.SeriesA,MathematicalandPhysicalSciences186(1946)453.
[3] see, e.g., D. Zwillinger and S. Kokoska, S. CRC Standard Probability and Statistics Tables and
Formulae,CRCPress.Press(2000).
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