A Bayesian regression approach to the poisson distribution

A Bayesian regression approach to the poisson distribution

著者 平館 道子

journal or

publication title

金沢大学経済学部論集 = Economic Review of Kanazawa University

volume 7

number 2

page range 31‑40

year 1987‑03‑30

URL http://hdl.handle.net/2297/24013

ABayesianregressionapproachtothePoissondistribution（Hiradate）

ABayesianregressionapproachto thepoissondistribution

MichikoHiradate

Countd2t2

Modelamdamethodofestimation Numericalexamples

DiSCuggion RefCreyDces

ＩⅡⅢⅣ

ICountdata

Regressionanalysisofthepoissondistributeddatahasbeendiscussedby

severalauthors（[3],[4],[7])．ButNelderandWedderbumfbrmulated

thegeneralizedlmearmodelinl972([5]）andsmcethenwecanviewthis prob1emmawiderscopeincludingtheusualnormalregression，Thepoisson

is，theoreticaUyandpracticany，themostimportantdistribution，whenwe analysethecountdata・Butweoftenobservecovariables,orfactors,togQther withresponsevariablemeconomlcorsociologicalstudies,ormotherfields、

Inthesecases,multipleregressionanalysiSprovidesanmtereStmgteChniqUe tomterpretedataandobtaininfbnnationaboutanundedyingmechanism6

However,ｗｅｓｏｍｔｉｍｅｓｏｂｔａｉｎｔｈｅｐＯｏrfitinre図essionanalysis,especiaUyin

analysingthecountdata・Thismaybebecausethecountobservationsare aggregatedoversomefactors,orimportantexplanatolyvariableｓａｒｅｎｏｔｏｂ‐

servedorunavaUable、Asaresult,overdiversificationissuspected，

Ｈｅｒｅｗｅｅｘａｍｍｅａｍｅｔｈｏｄｔｏａｃｃｏｍｏdatesuchunexplainedvariations

-31-

金沢大学経済学部論集第７巻第２号１９８７．３

duetomisspecificationbymtroducingnormaUyrandompartmtothelinear

regressionmodelfbrthepoissondistribution．

ⅡModelandamethodofestim2tiOn

NelderandWedderbumproposethegeneralizedlmearmodelwhichre‐

latesthenaturalparameterorsomefimctionofitofexponentialhlmilydis‐

tributionstoexplanatolyvariables（[sDLikelihoodfimctionandlinear

regressionpartaregivenby

f(ｙ１８，`)＝exp（｡[8y-a(8)]}b(｡,ｙ）………(1)

ａｎｄ

７(8)＝h'β………(2) where8isthenaturalparameter,disthescaleparameter,hisapxlvector ofexplanatolyvariables,ａｎｄβisapxlvectorofregressioncoefTicientsm cludmgconstantterm・WhenyfbUowsthestandardpoissondistribution,。

＝１，ａ（８）＝ｅ０．Contmuousdistributionsofexponentialfamnyareas flexibleastoaccountfbrthevariationmthedatabygivingavarietyofvalues tothescaleparameter,However,discretedistributionshavefixedvalueofd andarenotsoflexible・West（[7]）discussesthisproblemindetaUfrom theBayesianpomtofview,andproposesscaledexponentialfamnylikelihood asanapproximatesamplingmodelwhichkeepsthescaleparameterfreeand usesthedeviancefimction、ThescaledexponentialfamUylikelihoodhasthe samemean/variancerelationshipthatoforigmalfbrmulation,andreducesto theoriginalN-Wmodeｌｗｈｅｎｄ＝LIfwekeepdfreeandtlytolearnits behavior,someapproximationisnecessary,becausefactorb（ｄ，ｙ）is usuallydifficulttobemcorporatedintheanalysisofdiscretedata・Sweeting ([6]）discussesthisprobleminmoregeneralcontextwithscaleparameter havmgimproperprior・Ourfbnnulationintroducesnormanyrandompart mtotheregressionrelation(2),fbrthepurposeoffindmgthesourcesofsUch variationsandinterpretmgdatamamoreappropriatemodeL

Supposeweobservensamplesofy,andlety＝（ｙ１…….,y､).Wesuppose thecomponentsofyfbUowindependentlytheexponentialfhmUｙｄｉｓｔｒｉｐ

－３２－

ABayesianregresslonapproachtothepoissondistributioH1（Hiradate）

butionsofthesametype､Then,hkelihoodfimctionisgivenby

。(y'@ルなIexp[川y,－３(``D]b(Ｍ)|………(3)

where,＝(8,,………,０，）isavectorofthenaturalparameters・Wesuppose thefbUowmglinearrelationmsteadof(2)．

７(,‘)＝ｈ'β＋ｕｊ・………･………･………(4) Theaddedtennuaccomodateserrorsduetothemisspecificationorthefail‐

ureofobservingsomeimportanthlctors．Ｉnthecaseofthepoissonof

pammeterル，relation（４）mayreduceto

８`＝logeルーh/β＋ｕｉ………(5) Letubeavectorofuterm､AsfOrthepriordistributions,wehaVenoconju・

gateanalysisheleexceptthesimplestcase,ａｎｄｗｅａｓｓｕｍｅβanduare

mdependentandnormallydistributed，andalsouhavezeromeans・Then jomtpriordistributionisasfbUows

p(β,ulb,Ｂ,Ｖ)＝p(β|ｂ,Ｂ)ｐ(ulV）

…p{-÷(β-b)B(β-b)}･exp{-÷ｕｗ}……(6)

wherebisavectorofpriormeansofβ，BandVareprecisionmatricesof βandurespectively・ItisnotpracticaltoassumethatplecisionmatrixB

andVareknowncompletely．Ａnunknownscalefactorisassumedtoｅｘｉｓｔ ｉｎｅａｃｈｍａｔｒｉｘａｎｄｉｔｈａｓｓｏｍｅｈyperpriordistlibutionNamely，ｗｅｓｅｔ Ｂ＝γＢｏ，Ｖ＝シＶｏ，whereBo,VoarespeciHed、Unknownfactors7,y

fbUowchi-sqUaredistributionsfbrsomefo,ｄｏ,ｋｏ,ｇｏ，thatis，ｆｏ７～Ｚ２ａｏ，

koy～X２３。、Thenintegrating（６）withrespectp（γ),ｐ（ソ）gives

p(β,ulhfo,｡｡,k･’9b）

。c[fo＋(β-b)．Ｂ･(β-b)]~苧・[ko＋ｕＶｏｕ]~且夢上…………(7) Thekemelofposteliordistributionisgivenbymakingproductof（３）and (7),andincorporatingtherelation（４）．Posteriormeansofparametersare

－３３－

金沢大学経済学部論築第７巻第２号１９８７．３

notobtaindanalytiCallyinthisfbrmulation,sowetaketheposteriormodesaS convenientestimates6Letp'＝(β,,ｕ')，thentheposteriorscorefimction

andinfOrmationfimctionaredefinedrespectivelyby

＆(刺y俳念Iogp〃側………'8）

and

Myルー為g(畷',小………(9)

Inthecaseofrelation（５）,theyare

…作|:雪歴貯｡Ⅲ）

whereHispxnmatrixwithcolumnhf，

zisnxlvecterwithelements（yi-a'(h；β＋u‘))，ｉ＝1..…､’

7(硬)=f･+(蒜B･(β_b)=E(''y川………(u）

7(璽)一意語荒ＴＥ(,Ｍ）………('，

Ｇ(Ｆｌｙ,‘）

-(鰍讐)…B-2M1;隅"Wo-21）

………(lD

whereA（Ｐ）isnxnmatrixwithdiagonalelementsa,'(h;β＋ｕｊ)，

and

D戸~:;筈←B伽Ⅲ-b川

D2＝ＺＵ２ＬｖｏｕｎＷ・

_ｇｏ＋n

TheposteriormodesareobtainediterativelybyusualNewton‐Raphson method・Ｌｅｔ蛭bethevalueofpofi，thiteration,andtheiterationeqUation

isaSfbUowS

酪十,＝巫＋Ｇ(野|ｙ,`)-19(蛭Ｉｙ,｡）………00

－３４－

ABayesianregressionapproachtothepoissondistribution（Hiradate）

mNumericalexamples l・AmodiffedfbrmuIation

Weshowtwonumericalexamplesmthissection・Forconvemence,wemod‐

ifythefbrmulationAssumewehavemgroupsofobservationsonthepoL ssonvariableyandi，thgroupconsistsofrfobservations，andwehavep

explanatoryvariableswithobservationsh,`＝(h】`,………,hpf),whereh,`＝１

fbralli,i=1,.…,ｍ・Thenwehavempoissonparametersル，ｉ＝1,….mand logeルー＆・Ojisexpressedbyregressionmodelas（５）．

Asfbrpriors,wesupposeinsection2βanduhaveindependentnormal distributions，butbetweencomponentsofβａｎｄｕ，theremaybecorre‐

lationsrespectively・Ｈｅｒｅwesimplyassumethatcomponentsofβandu distributeindependentlyeachotherwithdifferentscaleparameters”,ツガ、

Then，thepriordistributionofβａｎｄｕｉｓ

,(β皿川,)｡cJi小xp[－，(β号Ｍ１ルトexp[-竺乳

Ashyperpriorsofyi，ハ，weassessthatfo”～Z｡.;koy`～Z図｡2．Then priordistributionofβａｎｄｕｉｓ

ｐ(β,ulfok…｡)"Jil[(昨b臘川｡}樂堂[u`圏十k．１－鶚

…=舅y腱小P｡‘t･…giv･ｎｂｙ

ｐ(β,ｕｌｙ,h）。C

oxp[魚{s`(h/β+uJ-r，e…}］

xhI(〃M’十fo1樂立[u'十k．1鵯 L蝕加)-f･器M製,k-い…。

耐(ｼ)=鵲戸i-い…､m

zbemxlvectorwithelements(sf-r：eh/β+ｕ')，

Ｐ(Ｆ)bepxpmatrixwithdiagonalelementｓ７ＭＰ)，

－３５－

金沢大学経済学部論巣第７巻第２号１９８７，３

andＮ(ｐ)bemxmmatrixwithdiagonalelementｓｙ,(〃)．

ItfbUowssimUarrelationslike(11)and(12)．

…ＷＷＩ'㈹

･川棚lwwﾄ☆塊削ﾄ☆・）

whereA（Ｐ）ispxpmatrixWithdiagonalelementsrjeh…u`，

Ｄ,＝（ｐ)2(β－b)(β－b)’

Ｄ２＝NGPP)ｚｕｕ,．

２Electronicequl1pment●

Jorgenso、（[4]）discussesthenumberoffailuresofacomplexpieceof electroniceqUipmentusmgregressionanalysisofthepoissondistribution・

Explanatoryvariablesaretimesspentmtwooperatmgreglmesatthecycleof operationTheobservationsareshownmTableLDependentvariableisthe numberoffanuresmthei，thcycle・

Jorgenson,sfbrmulationis

ｙｉ＝β1t,`＋β2t2i，ｉ＝１，………,n Altematively,ｗｅset

logルーβ･＋βltli＋β2t2i＋U`．

PosteriormodesandestimatesofAａｒｅａｌｓｏｓｈｏｗｎｍＴａｂｌｅｌ，wherewe takefo,ｋｏ,。｡,ｇｏａｓ５．

－３６－

ABayesianregressionapproachtothepoissondistribution（Hiradate）

Tablel

〈、八 八ｕ

Ｔ１ Ｔ２ ^ｙ

14.10 ８．７９ １３．６３ ２４．１３ ２７．３５ ２６．８８ ２３.3４ １８．０５ 21.66

６８２１９０９４９８１３１２１２０２ＯＧ●●●●●●■０００００００００

一一 一一

３３．３ ５２．２ ６４．７ １３７．０ １２５．９ １１６．９ １３１．７ ８５．０ ９１．，

３４５５６６６３５●●●、■●●●●５４２０７３６７７２１３２９５５８４ ５９４４７７３８２１１２２２２１２

β,＝0.015,八

β･＝1.135,八 ^{β2＝０．００６八}

３Quine，ssociolOgicaldata

Aitken（[1DanalysesdataofasociologicalstudyaboutAustralianschool childrenAresponsevariableisthenumberofdaysabsentfromschool dulingschoolyear・Childrenweresampledbyfburfactors，Ｌｅ・age，sex，

culturalbackgmundandleamingability・Aitkenanalysesthedatamtennsof theanalysisofvariance，andseekstofIndthemmimaladequatemodeLHe

obtainssixfinalmmimalmodelsandshowsfittedvaluesoneacｈｍｏｄｅＬＦｉｔ

ｓｅｅｍｓｔｏｂｅｎｏｔｓｏｇｏｏｄ、Wesupposetheresponsevariableispoissonvariate，

andthelogarithmofparameterhasaregressionpartanderrorpart，Unfbrtu‐

nately,wecannotrefertoQuine，sorigmaldatabutonlyAitken,ssummarized data(samplemeans)withsamplesizes、Dataareshownintable2、Under

abovemodiiiedfbrmulation，Aisareestimatedandhttedtosamplemeans、

ExplanatoryvariablesareasfbUows，

Ｃ:１＝Aborigina１，２＝white，

Ｓ:１＝Female,２＝Male，

Ａ:１＝Primary，２＝Firstfbnn，３＝Secondfbnn，４＝Thirdfbrm，

Ｌ:１＝slow,２＝averageOeamingability)．

－３７－

金沢大学経済学部鶴築第７巻第２号１９８７．３

WetryalsothecasewithexplanatoryvariablesincludingSxA,CxAwithout u-term，fbrcomparison・Ａｉｔｋｅｎｏｂｔａｍｓｔｈｅｂｅｓｔｆｉｔｍｔｈｉｓｃａｓeunderhis fbImulation・Ｔｈｅｒｅｓｕｌｔｉｓｓｈｏｗｎｉｎｔａｂｌｅ２・Ｈｅｒｅｗｅｔａｋｅｆｏ,ｋｏ,ｄｏ,gbas

alntable2，ハisestimatefbrourfbrmulationandズロisestimatefbi

model(SA,ＣＡ)Withoutu．

Table２

Ｂへ、几

八

八、八

八ｕ

ＣＳＡＬｙｒ

１１１１１１１１１１１１１１２２２２２２２２２２２２２２ ’１１１１１１２２２２２２２１１１１１１１２２２２２２２ ’’２２３３４１１２２３３４１１２２３３４１１２２３３４ ｌ２１２ｌ２２１２１２１２２１２ｌ２ｌ２２１２１２ｌ２２ ０００００９４０５００８０６０３４０３４８００００２０００００５０２１０２６４３０５０３１５３１２０５００２０５●●●●●●●ＤＣ●●●●●●由●●●９０■●●●□●Ｃ９３９０７７７３ｌ２ｌ６２４０５６３９９７５８６１６ｌ３ｌ１３２２２２１３１３２２２１１１ ３５３２４．７７１４０５８１９３６７２３７７１４１６９１０１１１ １７９１７１９８３３５８５７３０５５９７９４８１８６５５７０５２４２６７３２００４８１２２９４００９０６０１１３９９０５７３０８０５４０２５６４２９０２２９３００３３５●Ｏ●●●■０●●●●ＣＯ●□■０●●●●。●●の●●●８２９０６７７３１２１５３４９５６３９９７３８６１６２３１１３２２２２１３１２２２２１１１ ０８３６８５１０９２２１２４３８５９７１３６３４６４８２３９９６８５４３０４３３２１６８５５２３８８１９９００７７９４３１８４３２９２０１４８４０８０８６００９３９３９０３１００６４８８６０９５２２４５９８３４０７５０７８２●□●●●●０●●ＣＤＵ●●●□■、、ＤＣ●●●●●●■００００１０００００００１０１００００００１００００１０

一一 一一 一一一 一一一一一

４６２６１４０５２６５７９３５３３０７１７３３１９９５１３９００８５５３３６５９７０５５４５０８０７７８７８８９●●■ＤＰ●●●●●●●●●●●●●●●●ｃ●●●●●⑪４０１６０３４１６１６１６７８６２９８３０２９２９２９９１１２１３２３２１２１２１１１１１２１１１

八

J8゜＝1.804,_人 βA＝0.2321,

βt＝－００１９１，ノas＝0.0672八八

Ａ＝0．０３６９

－３８－

ABayesianregressionapprOachtothepoissondistribution（Hiradate）

ｗＤｉｓｃｕｓｓｉＯｎ

ｌｔｉｓｎｏｔａｅａｓｙｔａｓｋｔｏｉnterpretetheestimatedutermsappropriately,but

wemayexamineinausualanalysisofresidualsmanner・Asmentionedabove，

wecamotrefertotheoriginalOuine，sdata,soweestimateハsandexamine

thefittosamplemeansonly･Ｔｈｅｎｗｅｄｏｎｏｔｈａｖｅｔｈｅｐｒｏｂｌｅｍｏｆｏｖｅｒｂ diversification・However，ｉｆｗｅｔｒｙｔｏｆｉｔｔｏｔｈｅｏｒｉｇｍａｌｉｎｄｉｖｉdualobser‐

vations,extravariationmayarise・Thisproblemmaybeapproachedeitherby thecoumpounddistributionwhichisamixturewithrespecttosomedistri‐

butionfbrtheparameter,orbythemethodthatWestproposes,whichkeeps scaleparameterunristrictedanduseshisapproximatelikelihoodBut,ifwe canconsiderextravariationsasconseqUencesofthemisspecification,u-term mayconveysomeinfbnnationaboutthekindofmisspecification,ｏｒ,missing

eXplanatoryvariables・Thatsomeimportantexplanatoryvariablesareover‐

lookedmaymean,mthisexample,observationsofdependentvariablewhich belongtodifferentdistributionsareaggregatedmtothesamedistribution，

andthengiverisetotheextraVariations・Anadhocmethodtodealwiththis proｂｌｅｍｍａｙｂｅｔｏ厚CupObservationsaccordmgtouvalues，Ｂｕｔｓome

fbrmalprocedurefbrexploitingthemfbrmationthatuconveysisnecessaly・

FmaUy,asfbrthepriordistributionofu-telm,somesmoothpriormight

beused,whichisrelatedtovaluesoftheexplanatryvaIiablesmthesenseof expressmgthejudgementthatthechangｅｓｏｆｖａｌｕｅｓｏｆｕａｒｅｎｏｔｓｏｌａｒｇｅ ａｍｏngyhavmgsimilarvaluesofexplanatolyvariables,asinBhghtandOtt ([2])．

RefCrence8

LAitken，Ｍ､Ａ、1178．T11eanalysisofunbalancedcross-cl2IssifHcations（withdis･

cussion)．』.Ｒ､ＳＳＡ141,195--223.

2．ＢｎｇｈｔｂＢＪ．Ｎ､，Ott，Ｌ、１９７５．ABayesianapproachtomodelinadequacymr po1ynomialregressionBiometrika62,７９－８８．

３．Frome,Ｅ、Ｌ，Kutner,Ｍ、Ｈ､,Beauchamp，Ｊ、１９７３．Regressionanalysisofpoisson‐

－３９－

金沢大学経済学部論巣第７巻第２号1987.3

distrmuteddata．』.Ａ､Ｓ､Ａ、68,,0.334,935-940.

4．Jorgenson,，.Ｗ･l96LMultipleregressionanalysisofapoissonpIocess．』.Ａ､Ｓ､Ａ、

235-245.

5．Nelder,』.Ａ､、WedderbumDR.Ｗ､Ｍ、1972．Genera1izedLinearModels．Ｊ､Ｒ､Ｓ､Ｓ、

Ａ･No.135,370-384.

6．Sweeting，Ｔ・l98LScaleParameters；aBayesianTIeatment．』．Ｒ，Ｓ、Ｓ、Ｂ、４３ No.3,333-338

7．West，Ｍ、１９８５．GeneranzedUnearmodels、Scaleparameters，OutUerAccomo‐

dationandPrioldistributionaBayesianstatistic８２，Ｊ.Ｍ・Bemardo,Ｍ､Ｈ､Ａ・DeGroot，

ｎＶ・Lindley,Ａ､Ｆ､Ｍ､Ｓｍｉｔｈ（Eds.）North-HoUand,５３１‐-558.

－４０－