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1、4.9APPLICATIONSTOMARKOVCHAINS12111011121100210应用物理系王允磊WhatisMarkovchains?TheMarkovchainsdescribedinthissectionareusedasmathematicalmodelsofawidevarietyofsituationsinbiology,business,chemistry,engneering,physics,andelsewhere.Ineachcase,themodelisusedtodescribeanex
2、perimentormeasurementthatisperformedmanytimesinthesameway,wheretheoutcomeofeachtrialoftheexperimentwillbeoneofseveralspecifiedpossibleoutcomes,andwheretheoutcomeofonetrialdependsonlyontheimmediatelyprecedingtrial.Forexample,ifthepopulationofacityanditssuburbswere
3、measuredeachyear,thenavectorsuchascouldindicatethat60%ofthepopulationlivesinthecityand40%inthesuburbs.Thedecimalsinx0addupto1becausetheyaccountfortheentirepopulationoftheregion.Percentagesaremoreconvenientforourpurposeherethenpopulationtotals.Avectorwithnonnegati
4、veentriesthataddupto1iscalledaprobabilityvector.Astochasticmatrixisasquarematrixwhosecolumnsareprobabilityvectors.AMarkovchainisasequenceofprobabilityvectorsx0,x1,x2,…,togetherwithastochasticmatrixP,suchthatThustheMarkovchainisdescribedbythefirst-orderdifferencee
5、quationWhenaMarkovchainofvectorsinRndescribesasystemorasequenceofexperiments,theentriesinxklist,respectively,theprobabilitiesthatthesystemisineachofnpossiblestates,ortheprobabilitiesthattheoutcomeoftheexperimentthatisoneofnpossibleoutcomes.Forthisreason,xkisoften
6、calledastatevector.SothepopulationdistributioncouldbeSimilarly,thedistributionin2002isdescribedbyavectorx2,whereWhatisMarkovmatrix?Annxnmatrixwhoseformsatisfiestwopropertiesbelow1.Allentries≥0;2.Allcolumnsaddto1;iscalledaMarkovmatrix.SuchasAsyoucansee,thedefiniti
7、onoftheMarkovmatrixiscloselyrelatedtoMarkovchainsandprobabilitytheories.Wecanalsoderivatethisresult:Lemma:thepowersofaMarkovmatrixisstillaMarkovmatrixLemma’sproof:Wecouldusemathematicalinductiontocompletethislemma’sproof.Besides,wecouldproveamorestrongerresult:An
8、ytwoMarkovmatrices’productionisstillaMarkovmatrix.Proof:ThinkabouttwoMarkovmatricesAandB,letLetC=AB,wecangetSothematrixChastheproperty2:allcolumnsaddto1.Obviou
