Statistics and Analysis of Scientific Data

491 Pages·2022·9.207 MB·English

by  Massimiliano Bonamente

Graduate Texts in Physics
Massimiliano Bonamente
Statistics and
Analysis of
Scientific Data
Third Edition
Graduate Texts in Physics
SeriesEditors
KurtH.Becker,NYUPolytechnicSchoolofEngineering,Brooklyn,NY,USA
Jean-MarcDiMeglio,MatièreetSystèmesComplexes,BâtimentCondorcet,
UniversitéParisDiderot,Paris,France
SadriHassani,DepartmentofPhysics,IllinoisStateUniversity,Normal,IL,USA
MortenHjorth-Jensen,DepartmentofPhysics,Blindern,UniversityofOslo,Oslo,
Norway
BillMunro,NTTBasicResearchLaboratories,Atsugi,Japan
RichardNeeds,CavendishLaboratory,UniversityofCambridge,Cambridge,UK
WilliamT.Rhodes,DepartmentofComputerandElectricalEngineeringand
ComputerScience,FloridaAtlanticUniversity,BocaRaton,FL,USA
SusanScott,AustralianNationalUniversity,Acton,Australia
H.EugeneStanley,CenterforPolymerStudies,PhysicsDepartment,Boston
University,Boston,MA,USA
MartinStutzmann,WalterSchottkyInstitute,TechnicalUniversityofMunich,
Garching,Germany
AndreasWipf,InstituteofTheoreticalPhysics,Friedrich-Schiller-UniversityJena,
Jena,Germany
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Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/8431
Massimiliano Bonamente
Statistics and Analysis
of Scientific Data
Third Edition
MassimilianoBonamente
UniversityofAlabamainHuntsville
Huntsville,AL,USA
ISSN 1868-4513 ISSN 1868-4521 (electronic)
GraduateTextsinPhysics
ISBN 978-981-19-0364-9 ISBN 978-981-19-0365-6 (eBook)
https://doi.org/10.1007/978-981-19-0365-6
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Numbersdon’tlie
Foreword
ItisnowtheeraofBigData.Hugetrovesofquantitativeinformationresideevery-
whereandcanbeaccessedfromanywhereintheworldinaheartbeat.Whileourtools
for the collection and analysis of data have rocketed forward over a single human
lifetime, we remain human. As such, we can only hold a few simple numbers and
conceptsinourpoor,analogbrains.Werefertothoseconceptsasourunderstanding.
Statistical analysis of data allows us to make quantitative assessments of our
concepts,hopefullykeepingourpreconceptionsandmisconceptionsatbay.Butusing
statisticalmethodsisneverjust“turningacrank”.Wemustapplyjudgment.Wemust
becareful.Computerscaneasilylulloneintoafalsesenseofsecurity.Inhisthird
edition,ProfessorBonamenteprovidestheclearestexpositionofstatisticalanalysis
and the probability theory upon which it is based. He provides a roadmap to the
properuseofstatistics.
Intheopeningchapterofthisbook,theauthordiscussesBayesianversusFrequen-
tist statistical approaches. The Frequentist approach can be useful if one has no
underlying model for the mechanisms at play. But, in the Physical Sciences and
Engineering,sincethetimeofIsaacNewton,wehavemodelsthatrelyonparame-
ters.Andthoseparameters candistilltheunderstanding weseek.How tomeasure
thoseparametersandplaceerrorestimatesupontheresultsisthecentralthemeof
thisbook.
Today,therearesurprisinglyfewcoursesinstatisticsofferedatuniversities.And
thereareevenfewertextbooksavailabletosupportthoseclasses.Manyofthebooks
onstatisticsarewrittenforthebiologicalandmedicalfieldsandaresuitableforuse
inresearcharenaswheretheparametersandmechanismsarepoorlyunderstood.In
this text, you will find the approaches suitable for the analysis of data in physical
fieldsasopposedtothelifesciences.
Thisthirdeditionisthetextbookwehavebeenawaiting.Everyindividualwho
encountersdatainaprofessionalcapacityshouldreadit.Thematerialrangesfrom
thebasicsofprobabilitytotheseriousanalysisofdata.Anyindividualwhoworks
withdatadailyshouldmasterthematerialsinthisbook,keepingacopyforreference.
Ifrequentlyteachourdepartment’scourseindataanalysisandstatistics.Ihave
usedBonamente’ssecondeditiontogoodeffect,butthethirdeditionisthetextbook
vii
viii Foreword
Ihaveawaited.IonlywishIhadhaditbackwhenIwasastudent.Iamoldenough
thatIhadtouseaslideruleasanundergraduateanddidnotgainaccesstoahand
calculatoruntilgraduateschool.Computerswererunondecksofpunchcards.How
far our hardware tools have come! Yet the statistical tools employed today have
hardlychanged.
Even the data have changed. When I was young, astronomers employed film.
Today we use solid-state detectors that count individual photons at light levels so
lowthatmostofthedetectorbinshavecountednophotonsatall.Theswitchfrom
film(analog)tophoton-counting(digital)hascompletelychangedthenatureofthe
appropriatestatisticalanalysis.
In Chaps. 15 and 16, the author presents an approach to quantitative statistical
analysis of low-signal-count data. It is often the case today that many events are
recorded,buttheinstrumentstakingthedatahaveevenhigherresolution,i.e.,more
databinsthatevents.Inthesetwochapters,theauthordiscussestheapplicationof
PoissonStatisticstothislow-count-per-bindatausingtheCashStatistic.Henotonly
presentstheuseofthisCashStatisticbutdevelopsittoanewlevelthatshouldprove
usefulacrossarangeofmodernapplications.
Yes,theCashStatisticisnamedafterme,thewriterofthisforeword.Iwaslucky
enoughtoparticipateintheveryearlydaysofx-rayastronomyand,inparticular,be
involvedwiththeveryfirstx-raytelescopesbackinthe1970s.X-raysourceshave
notoriously weak photon fluxes, but every photon packs a punch, so it was x-ray
and gamma-ray astronomers who first had to reckon with the statistics of images
andspectrawhereasignificantfractionofthedatabinswereempty.Thedatawere
fundamentallyofaPoissondistributionnatureandcouldnotbeaddressedwiththe
usualassumptionofaGaussiandistribution.How,forexample,doesonedealwith
animagethathas1024binswithanaverageof.01x-raysperbin,yetthereisone
binwiththreex-raysinit,rightwherethetargetissupposedtobe?Theanswerlays
intheCashStatisticandshowingthatitwasdistributedasaChi-square.Professor
Bonamentehasnow,inthistext,broadenedthescopeoftheapplicationsandprovided
abetterframeworkfortheapplicationofthestatistic.
InPartIIIofthisbook,theauthoraddressesMonteCarlomethods.Theseareof
fundamentalimportancetomuchofmodernmodeling,particularlyinhighlycomplex
modelslikeraytracingofopticalsystems.Again,thisbooktakesthereader(student)
intorigorousanalysistechniquesthataredifficult,ifnotimpossible,tofindexplained
elsewhere.
Statistics can be a hard master. The math can be challenging. But in the age of
easy access to computing, the challenge to the user is to sift from all the numbers
the“understanding”thatissought.Themathbecomessecondary.Judgmentcomes
tothefore.WhatstatisticshouldIuse?Whatdotheresultsactuallymean?Itiseasy
tooverusethestatistics,findingstatisticallysignificantpatternsthatareactuallyof
nosignificance.
Foreword ix
And,astimegoesby,andone’sattentionmoveselsewhere,oneiscertaintoforget
howtoperformthestatisticalanalyses.Butthegeneralknowledgeoftherangeof
statisticaltoolswillstaywithyou.Then,whentheneedarises,pullthisbookoffthe
shelf.ProfessorBonamente’sbookwillbringitallback.
June2022 WebsterCash
ProfessorofAstrophysicalandPlanetarySciences
UniversityofColorado,Boulder
Preface
WiththethirdeditionofStatisticsandAnalysisofScientificData,Ihavecontinued
to pursue the goal of providing a textbook that is both mathematically sound and
easytouseforpracticalapplications.Dataanalysisstraddlestherigorousfieldsof
statisticsandmathematics,andthemessyworldofexperimentsanddatacollection,
inawaythatmakesitoftenunclearastowheretodrawthelinebetweenwhatneeds
tobedonetothedataandwhatshouldbedoneaccordingtotheavailabletheoretical
tools.Thistextbookaimstotreatboththetheoreticalandpracticalaspectsofseveral
keymethodsofdataanalysis,includingthelimitationsofcertaincommonpractices,
suchastheuseofthechi-squaredstatisticfornon-Gaussiandata.
Startingwiththesecondedition,Idecidedtomarkportionsofthetextbookthat
are primarily theoretical with a gray sidebar. These parts are not essential for the
practiceofdataanalysis,andtheycanbeskippedbythereaderwhoisnotinterested
in its theoretical underpinnings, or by the reader who needs a fast reference for a
methodthattheyalreadyknow.Ifindthatthisisaneffectivewayofpresentingcertain
mathematicallyintensivetopics,andIhavecontinuedwiththispracticeinthethird
edition.
Certainkeytheoreticalresultsthathaveagreatinfluenceonstatisticalmethods,
suchasH.Cramér’stheoremonthelimitingdistributionofthechi-squaredstatistic
orS.S.Wilk’stheoremonthedistributionofthelikelihoodratio,arenowpresented
inamoresystematicway.Themathematicaltheorybehindsomeoftheseresultsis
quite complex, and therefore I sometimes found it necessary to describe only the
main results, and then provide a reference to the original work, instead of delving
into too deep a mathematical treatment. The more theoretically inclined reader is
pointedtotherelevantreferencesinstead.Atthesametime,itisimportantthatthe
dataanalysispractitionerisawareofthesemathematicalresultsandthelimitations
they impose. A cautionary tale of what can go wrong when there is a mismatch
betweenthemethodofanalysisandtheunderlyingtheoryisinK.Pearson’suseof
the chi-squared statistic for contingency tables, which are now treated in detail in
Chap.10.Pearsonwaserroneouslyconvincedthat2×2contingencytableswithfixed
xi