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DARTTutorialSec'on13:HierarchicalGroupFiltersandLocaliza'on

Waystodealwithregressionsamplingerror

1.Ignoreit:ifnumberofunrelatedobserva'onsissmallandthereissomewayofmaintainingvarianceinpriors.Wedidthisinthe3and9variablemodels.

2.Uselargerensemblestolimitsamplingerror(testinlorenz_96).

Thiscangetexpensiveforbigproblems.Trymodifyingens_sizeinfilter_nml(try40,80,160).

3.Useaddi'onalaprioriinforma'onaboutrela'onbetween

observa'onsandstatevariables.Don’tletanobserva'onimpactstateiftheyareknowntobeunrelated.

4.Trytodeterminetheamountofsamplingerrorandcorrectforit.

Therearemanywaystodothis;somesimple,somecomplex. DARTTutorialSec'on13:Slide2

Waystodealwithregressionsamplingerror

Canuseotherfunc'onstoweightregression.Unclearwhatdistancemeansforsomeobs./statevariablepairs.ReferredtoasLOCALIZATION.

−2000 −1000 0 1000 20000

0.5

1

Distance from Observation

Regr

essio

n W

eigh

t3.Useaddi'onalaprioriinforma'onaboutrela'onbetween

observa'onsandstatevariables.

DARTTutorialSec'on13:Slide3

Localiza'onisfunc'onofexpectedcorrela'onbetweenobsandstate.

O_en,don’tknowmuchaboutthis.Horizontaldistancebetweensametypeofvariablemaybeokay.Whatisexpectedcorrela'onforco-locatedtemperatureandpressure?Whataboutver'callocaliza'on?Looksprebycomplex.Whataboutcomplicatedforwardoperators:

Expectedcorrela'onofsatelliteradianceandwindcomponent?Note:DARTdoesallowver'callocaliza'onformorecomplex

models.

DARTTutorialSec'on13:Slide4

Waystodealwithregressionsamplingerror

4.Trytodeterminetheamountofsamplingerrorandcorrectforit:

A.Couldweightregressionsbasedonsamplecorrela'on.Limitedsuccessintests.Forsmalltruecorrela'ons,cans'llgetlargesamplecorrel.

B.Dobootstrapwithsamplecorrela'ontomeasuresamplingerror.Limitedsuccess.Repeatedlycomputesamplecorrela'onwithasampleremoved.

C.UsehierarchicalMonteCarlo.

Havea‘sample’ofsamples.Computeexpectederrorinregressioncoefficientsandweight.

DARTTutorialSec'on13:Slide5

Waystodealwithregressionsamplingerror

4C.UseHierarchicalMonteCarlo:ensembleofensembles.

MIndependentN-Memberensembles

β1

βM

RegressionConfidenceFactor,α

MgroupsofN-memberensembles.Computeobserva'onincrementsforeachgroup.Forgivenobserva'on/statepair:1.  HaveMsamplesofregressioncoefficient,β.

2.  Uncertaintyinβimpliesstatevariableincrementsshouldbereduced.

3.  Computeregressionconfidencefactor,α.

DARTTutorialSec'on13:Slide6

4C.UsehierarchicalMonteCarlo:ensembleofensembles

SplitensembleintoMindependentgroups.Forinstance,80ensemblemembersbecomes4groupsof20.

WithMgroupsgetMes'matesofregressioncoefficient,.Findregressionconfidencefactora(weight)thatminimizes:MinimizesRMSerrorintheregression(andstateincrements).

βi

αβi−β j( )

i=1,i≠ j

M

∑2

j=1

M

∑

DARTTutorialSec'on13:Slide7

4C.UsehierarchicalMonteCarlo:ensembleofensembles

Weightregressionbyα.Ifonehasrepeatedobserva'ons,cangeneratesamplemeanormediansta's'csforα.Meanαcanbeusedinsubsequentassimila'onsasalocaliza'on.

Aisfunc'onofMand(sampleSD/samplemeanregression) Q=Σβ β

DARTTutorialSec'on13:Slide8

&assim_tools_nml cutoff = 1000000.0 …&filter_nml ens_size = 80 num_groups = 4 inf_flavor = 0, 0 …

4C.UsehierarchicalMonteCarlo:ensembleofensembles

Ifwedon’tknowhowtolocalizetostartwith,canusegroupstohelp.Trysplikng80ensemblemembersinto4groupsof20membersforLorenz96.

DARTTutorialSec'on13:Slide9

models/lorenz_96/work/

4C.UsehierarchicalMonteCarlo:ensembleofensembles

Turnonregressionfactordiagnos'cs.A_errunningthe80by4‘group’filter,lookatplotsofα.Essen'allyanes'mateofa‘good’localiza'onforagivenobserva'on.Useplot_reg_factorinMatlab.Selectdefaultinputfilename.Onlyobserva'ons1,2,3,and4areavailable:Locatedat:0.39,0.17,0.64,0.86Thinkaboutvalueof'memedianvs.'memean.Coulduse'memeanormedianaspriorlocaliza'onfunc'onsPlayaroundwithmodelerroragain.Whathappenstolocaliza'on?

DARTTutorialSec'on13:Slide10

&reg_factor_nml select_regression = 1 input_reg_file = “time_mean_reg” save_reg_diagnostics = .true. reg_diagnostics_file = “reg_diagnostics” /

MoreDetailedLookatHierarchicalFilters

AmoredetailedlookatsomefeaturesofgroupfiltersisavailableinthetutorialdirectoryinthefileOLD_sec'on_13.pdf.Pages10-54complementthematerialsinthissec'on.WARNING:Thematerialonpages1-9ofOLD_sec'on_13.pdfisoutdated.

DARTTutorialSec'on13:Slide11

1.   FilteringForaOneVariableSystem2.   TheDARTDirectoryTree3.   DARTRun>meControlandDocumenta>on4.   Howshouldobserva>onsofastatevariableimpactanunobservedstatevariable?

Mul>variateassimila>on.5.   ComprehensiveFilteringTheory:Non-Iden>tyObserva>onsandtheJointPhaseSpace6.   OtherUpdatesforAnObservedVariable7.   SomeAddi>onalLow-OrderModels8.   DealingwithSamplingError9.   MoreonDealingwithError;Infla>on10.   RegressionandNonlinearEffects11.   Crea>ngDARTExecutables12.   Adap>veInfla>on13.   HierarchicalGroupFiltersandLocaliza>on14.   QualityControl15.   DARTExperiments:ControlandDesign16.   Diagnos>cOutput17.   Crea>ngObserva>onSequences18.   LostinPhaseSpace:TheChallengeofNotKnowingtheTruth19.   DART-CompliantModelsandMakingModelsCompliant20.   ModelParameterEs>ma>on21.   Observa>onTypesandObservingSystemDesign22.   ParallelAlgorithmImplementa>on23.  Loca'onmoduledesign(notavailable)24.  Fixedlagsmoother(notavailable)25.   Asimple1Dadvec>onmodel:TracerDataAssimila>on

DARTTutorialIndextoSec'ons

DARTTutorialSec'on13:Slide12