2.2 The Powers of the Transition Matrix.doc

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1、2.2ThePowersoftheTransitionMatrix2.2.1AFormulaforthePowersoftheTransitionMatrixInsection2.1.1wesawthatprobabilitiesoffutureeventsinaMarkovchaincanbecomputedfromthepowersPnofthetransitionmatrixP.OnewaytocomputePnistomultiplyPbyitselfntimes.Thereisanotherwaytocomput

2、ePnusingtheeigenvaluesandeigenvectorsofPthatshedslightonthebehaviorofPnforlargen.InfactthismethodleadstoaformulaforPnthatshowsthatPnapproachesalimitingmatrix,whichweshalldenotebyP¥,asntendsto¥.Weillustratetheseconceptsusingthefollowingexamplewithtwostates.Example1

3、.InsunnySouthernCalifornia,ifitissunnyonedaythereisa90%chancethatitwillbesunnythenextdayanda10%chancethatitwillbecloudy.Ontheotherhandifitcloudyonaparticularday,thereisa60%chancethatitwillbesunnythenextdayanda40%chancethatitwillbecloudy.WemodelthisbyaMarkovchainwi

4、thtwostates.State1correspondstobeingsunnyonaparticulardayandState2tobeingcloudy.Weletncountthedayswithn=0beingtoday.AlsoletXndenotethestateofdayn,eithersunnyorcloudy.AssumethattheassumptionsforaMarkovchainaresatisfied.Thetransitionmatrixis(1)P=ThenPn=i.e.Pncontain

5、stheprobabilitiesofbeingsunnyorcloudyndaysfromnowgiventhatitissunnyorcloudytoday.TheformulaforPnthatweareinterestedininvolvestheeigenvaluesandeigenvectorsofP.RecallthataneigenvectorvofPhasthefollowingtwopropertiesv¹0Pv=lvforsomenumberlliscalledaneigenvalueofP.Itis

6、nothardtoshowthataneigenvaluemustsatisfytheequationdet(P-lI)=0whichiscalledthecharacteristicequation.Forthematrix(1)inExample1onehas2.2.1-7P-lI=IngeneraltoformP-lIfromPonesimplysubtractslfromthediagonalentries.Thecharacteristicequationis0=det(P-lI)==(0.9-l)(0.4-l)

7、–(0.1)(0.6)=l2-1.3l+0.3=(l-1)(l-0.3)Sotheeigenvaluesarel1=1andl2=0.3Ingeneral,ifPisann´nmatrix,thendet(P-lI)isannthdegreepolynomial,thecharacteristicequationisannthdegreepolynomialequationandthereareneigenvalues,althoughsomeofthemmayberepeated.Itturnsoutthatoneisa

8、lwaysaneigenvalueforthetransitionmatrixofaMarkovchainandtherestoftheeigenvalueshaveabsolutevaluelessthanorequaltoone.Infact,usuallytherestoftheeigenvalu