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1 3 T im e S e rie s, D e p e n d e n t O b se rv a tio n s
13.1
M o d e ls
aClassicalconceptsfortimeseries
X1,X2,...,Xt,...observationsFulldescription:needjointdistributionofallXtExpectationEhXti,variancevarhXti,Autocovariance:covhXt,Xt+hi−→autocorrelation.
Stationarytimeseries:jointdistributionsdonotdependont:
FhXt,Xt+1,...,Xt+hi=FhX1,X1+1,...,X1+hi
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eaklystationarytimeseries:Momentsareindependentoft:
Xti=µ,varhXti=σ 2,covhXt,Xi+hi=...autocorrelationfunctionρhhi=covhXt,Xt+hi/σ 2. ussiantimeseriesintdistributionismultivariatenormal.First&secondmomentsdescribedistributioncompletely.onlyneedµ(oftenassumed=0);σ 2;ρhhi.
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cAutoregression
Xt=αXt−1+Et,
Et∼N 0,σ 2e independent,indep.ofX1,...,XtMoregeneral:ARhpi:Xt= Pph=1 αh Xt−h +Et“Prototype”ofstationarytimeseriesmodelStationarityneedsassumptionsaboutα1,....
µ=0,σ 2=σ 2e /(1−α 2).Forp=1,ρhhi=α h
dBasicconceptsbecomedifficultwhenobs.aredependent:mainlycontamination,InfluenceFunction
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.2
C o n ta m in a tio n
bservationOutlierseplaceXtbygrosserrorYt=replacementoutlier,orby eYt=Xt+Yt=additiveoutlierwithYt∼Hodel:BinaryvariableZt,
Xt= eXt+Zt·Yt
:un-contaminatedtimeseries.canbeindependent−→“isolatedoutliers”orstronglydependent−→“patchyoutliers”
22313.2 bInnovationOutliers...forARmodels:Etisreplacedbygrosserror
−→thefollowingobservationswillalsobeaffected.Effectdecaysexponentiallyforisolatedoutliers.
cLevelshiftAratherdifferentdeviationfromassumptions:
EhXtiisonlypiecewiseconstant.
dMessageTheconceptofcontamination/grosserrorisambiguous!
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.3
In fl u e n c e F u n c tio n , B re a k d o w n
?IFandabreakdownpointforeachtypeofcontamination?
→Manyapproachesto“robustification”!SeeMaronnaetal.,h.8.
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13.4
O th e r ty p e s o f d e p e n d e n t o b se rv a tio n s
aSpatiallycorrelateddata
Xiismeasuredatlocationsi.CorrelationofXiandXhdependsonsi−sh,or,moresimply,ondih=ksi−shk
−→spatialautocorrelationρhdi.
−→“spatialdata”,“geostatistics”.
bMixedEffectsModelsseeChapter11.6.
