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1、PREDICTABILITY:RECENTINSIGHTSFROMINFORMATIONTHEORY1,23TimothyDelSoleandMichaelK.TippettReceived25April2006;revised27December2006;accepted17April2007;published21November2007.[1]Thispapersummarizesaframeworkforinvestigatinganalysisbetweenforecastandobservations.T
2、heregressionpredictabilitybasedoninformationtheory.Thisframeworkoperatorthatmapsobservationstoforecastsplaysanconnectsandunifiesawidevarietyofstatisticalmethodsimportantroleinthisframework,withtheleftsingulartraditionallyusedinpredictabilityanalysis,includingli
3、nearvectorsofthisoperatorbeingthepredictablecomponentsregression,canonicalcorrelationanalysis,singularvalueandthesingularvaluesbeingthecanonicalcorrelations.decomposition,discriminantanalysis,anddataassimilation.Thiscorrespondencebetweenpredictablecomponentsand
4、Centraltothisframeworkisaprocedurecalledpredictablesingularvectorsoccursonlyifthesingularvectorsarecomponentanalysis(PrCA).PrCAoptimallydecomposescomputedusingMahalanobisnorms,aresultthatshedsvariablesbypredictability,justasprincipalcomponentlightontheroleofnor
5、msinpredictability.Inlinearanalysisoptimallydecomposesvariablesbyvariance.Forstochasticmodelstheforcingthatminimizespredictabilitynormaldistributionsthesamepredictablecomponentsareistheonethatrendersthe‘‘whitened’’dynamicaloperatorobtainedwhetheroneoptimizespre
6、dictiveinformation,thenormal.Thisconditionforminimumpredictabilityisdispersionpartofrelativeentropy,mutualinformation,invarianttolineartransformationandisequivalenttoMahalanobiserror,averagesignaltonoiseratio,normalizeddetailedbalance.Theframeworkalsoinspiresso
7、menewmeansquareerror,oranomalycorrelation.Forjointnormalapproachestoaccountingfordeficienciesofforecastmodelsdistributions,PrCAisequivalenttocanonicalcorrelationandestimatingdistributionsfromfinitesamples.Citation:DelSole,T.,andM.K.Tippett(2007),Predictability:
8、Recentinsightsfrominformationtheory,Rev.Geophys.,45,RG4002,doi:10.1029/2006RG000202.1.INTRODUCTION2002].Despitebeingsmall,predictablevariancebeyond3weeksmaystillbeofinterest
