StatisticalInference,LearningandModelsinBigData

Franke,Beate;Plante,Jean–François;Roscher,Ribana;Lee,Annie;Smyth,Cathal;Hatefi,Armin;Chen,Fuqi;Gil,Einat;Schwing,Alexander;Selvitella,Alessandro;Hoffman,MichaelM.;Grosse,Roger;Hendricks,DietrichandReid,Nancy.

AbstractTheneedfornewmethodstodealwithbigdataisacommonthemeinmostscientificfields,althoughitsdefinitiontendstovarywiththecontext.Statisticalideasareanessentialpartofthis,andasapartialresponse,athematicprogramonstatisticalinference,learning,andmodelsinbigdatawasheldin2015inCanada,underthegeneraldirectionoftheCanadianStatisticalSciencesInstitute,withmajorfundingfrom,andmostactivitieslocatedat,theFieldsInstituteforResearchinMathematicalSciences.Thispapergivesanoverviewofthetopicscovered,describingchallengesandstrategiesthatseemcommontomanydifferentareasofapplication,andincludingsomeexamplesofapplicationstomakethesechallengesandstrategiesmoreconcrete.

KeywordsAggregation,computationalcomplexity,dimensionreduction,high–dimensionaldata,streamingdata,networks

1.IntroductionBigdataprovidesbigopportunitiesforstatisticalinference,butperhapsevenbiggerchallenges,especiallywhencomparedtotheanalysisofcarefullycollected,usuallysmaller,setsofdata.FromJanuarytoJune2015,theCanadianStatisticalSciencesInstituteorganizedathematicprogramonStatisticalInference,LearningandModelsinBigData.Itbecameapparentwithinthefirsttwoweeksoftheprogramthatanumberofcommonissuesaroseinquitedifferentpracticalsettings.Thispaperarosefromanattempttodistillthesecommonthemesfromthepresentationsanddiscussionsthattookplaceduringthethematicprogram.Scientifically,theprogramemphasizedtherolesofstatistics,computerscience,andmathematicsinobtainingscientificinsightfrombigdata.Twocomplementarystrandswereintroduced:cross–cutting,orfoundational,researchthatunderpinsanalysis,anddomain–specificresearchfocusedonparticularapplicationareas.Theformercategoryincludedmachinelearning,statisticalinference,optimization,networkanalysis,andvisualization.Topic–specificworkshopsaddressedproblemsinhealthpolicy,socialpolicy,environmentalscience,cyber–securityandsocialnetworks.Thesedivisionsarenotrigid,ofcourse,asfoundationalandapplicationareasarepartofafeedbackcycleinwhicheachinspiresdevelopmentsintheother.Someveryimportantapplicationareaswherebigdataisfundamentalwerenotabletobethesubjectoffocussedworkshops,butmanyoftheseapplicationsdidfeatureinindividualpresentations.Theprogram

startedwithanopeningconference1thatgaveanoverviewofthetopicsofthe6–monthprogram2.AllthetalkspresentedattheFieldsInstituteareavailableonlinethroughFieldsLive3.

Technologicaladvancesenableustocollectmoreandmoredata–alreadyin2012itwasestimatedthatdatacollectionwasgrowingat50%peryear(Lohr,2012).Whilethereismuchimportantresearchunderwayinimprovingtheprocessing,storingandrapidaccessingofrecords(Chenetal.,2014;Schadtetal.,2010),thisprogramemphasizedthechallengesformodelling,inferenceandanalysis.

Twostrikingfeaturesofthepresentationsduringtheprogramwerethebreadthoftopics,andtheremarkablecommonalitiesthatemergedacrossthisbroadrange.Weattemptheretoillustratemanyofthecommonissuesandsolutionsthatarose,andprovideapictureofthestatusofbigdataresearchinthestatisticalsciences.Whilethereportislargelyinspiredbytheseriesoftalksattheopeningconferenceandrelatedreferences,italsotouchesonissuesthatarosethroughouttheprogram.Inthenextsectionweidentifyanumberofchallengesthatarecommonlyidentifiedintheworldofbigdata,andinSection3wedescribesomeexamplesofgeneralinferencestrategiesthatarebeingdevelopedtoaddressthesechallenges.Section4highlightsseveralapplications,tomakesomeoftheseideasmoreconcrete.

2.“BigData,it’snotthedata”Thereisnouniquedefinitionofbigdata.ThetitleofthissectionistakenfromtheopeningpresentationbyBell(2015),whoparaphrasedSupremeCourtJusticePorterStewartbysaying“Ishallnottodayfurtherattempttodefinebigdata,butIknowitwhenIseeit”.Bellalsopointedoutthatthedefinitionof‘big’isevolvingwithcomputationalresources.Gaffield(2015)emphasizedthatpeopleandcompaniesareoftencaptivatedbytheconceptofbigdatabecauseitisaboutus,humans.Bigdataprovidesuswithdetailedinformation–ofteninrealtime–makingitpossibletomeasurepropertiesandpatternsofhumanbehaviourthatformerlycouldnotbemeasured.AnanalogysuggestedbyKeller(2015)istotheHubbletelescope:wecannowseeconstellationswithgreatcomplexitythatpreviouslyappearedonlyasbrightdots.Beyondnaturalhumancuriosity,bigdatadrawsinterestbecauseitisvaluable.Alargepartofthehypearoundbigdatacomesfromrapidlyincreasinginvestmentfrombusinessesseekingacompetitiveadvantage(e.g.McAffeeandBrynjolfsonn,2012;Bell,2013;KironandShockley,2011).NationalStatisticsAgenciesareexploringhowtointegratebigdatatobenchmark,complement,orcreatedataproducts(CappsandWright,2013;Struijsetal.,2014;LetouzéandJütting,2014;TamandClarke,2015).Inmanyfieldsofresearchcostsofdataacquisitionarenowlower,sometimesmuchlower,thatthecostsofdataanalysis.Thedifficultyoftransformingbig

1 The program for the opening conference is at http://www.fields.utoronto.ca/programs/scientific/14–15/bigdata/boot/ 2 A description of the complete thematic program is at https://www.fields.utoronto.ca/programs/scientific/14–15/bigdata/ 3 The Fields Institute video archive link is http://www.fields.utoronto.ca/video–archive

dataintoknowledgeisrelatedtoitscomplexity,theessenceofwhichisbroadlycapturedbythe“FourVs”:volume,velocity,varietyandveracity.

2.1VolumeHandlingmassivedatasetsmayrequireaspecialinfrastructuresuchasadistributedHadoopcluster(e.g.Shvachkoetal.,2010)whereaverylargenumberofrecordsisstoredinseparatechunksscatteredovermanynodes.Asthecostsofprocessors,memoryanddiskspacecontinuestodecrease,thelimitingpiecenowisoftencommunicationbetweenmachines(Scottetal.2013).Asaconsequence,itisimpossibleorimpracticaltoaccessthewholedatasetonagivenprocessorandmanymethodswillfailbecausetheywerenotbuilttobeimplementedfromseparatepiecesofdata.Developmentsarerequired,andbeingmade,tomakewidely–usedstatisticalmethodsavailableforsuchdistributeddata.TwoexamplesarethedevelopmentofestimatingequationtheoryinLinandXi(2011),andtheconsensusMonteCarloalgorithmofScottetal.(2013).

Exploratorydataanalysisisaveryimportantstepofdatapreparationandcleaningthatbecomessignificantlymorechallengingwithincreasingsizeofthedata.Beyondthetechnicalcomplexityofmanagingthedata,graphicalandnumericalsummariesareessentiallylowdimensionalrepresentations–forexample,graphicaldisplaysarephysicallylimitedtoafewmillionpixels.Further,thehumanbrainrequiresaverylowdimensionalinputtoproceedtointerpretation(Scott,2015).Oneconsequenceofthislimitationisthatweoftenmustcompromisebetweentheintuitivenessofavisualizationandthepowerofavisualization(Carpendaleetal.,2004;Nacentaetal.,2012).Anotherconsequenceisthatwetypicallyvisualizeonlyasmallsubsetofthedataatonetime,andthismaygiveabiasedview(Kolaczyk,2009;p.91).

2.2VarietyFormanybigdataproblems,thevarietyofdatatypesiscloselyrelatedtoanincreaseincomplexity.Muchofclassicalstatisticalmodellingassumesa“tallandskinny”datastructurewithmanymorerows,indexingobservationalorexperimentalunitsn, thancolumns,indexingvariablesp.Images,bothtwo–andthree–dimensional,blogposts,twitterfeeds,socialnetworks,audiorecordings,sensornetworks,videosarejustsomeexamplesofnewtypesofdatathatdon’tfitthis“rectangular”model.Oftenrectangulardataiscreatedsomewhatartificially:forexamplefortwo–dimensionalimages,byorderingthepixelsrowbyrowandrecordingoneormorevaluesassociatedwitheachpixel,hencelosinginformationaboutthestructureofthedata.

Formanytypesofbigdatathenumberofvariablesislarger,oftenmuchlarger,thanthenumberofobservations.Thischangeintherelationshipbetweennandpcanleadtocounter–intuitivebehaviour,oracompletebreakdown,ofconventionalmethodsofanalysis.Problemsinwhichpscaleswithnareverycommon:forexample,inmodelsfornetworksthatassignoneparametertoeachnode,therearep=nparametersforagraphwithn×npossibleedges(ChungandLu,2002).Similarly,Bioucas–Diasetal.(2013)discusshyperspectralremotesensingimagedata,whichmayhavehundredsorthousandsofspectralbands,p,butaverysmallnumberoftrainingsamples,n.Inarecentgenomicsstudyreportedbythe1000GenomesProjectConsortium(2012),geneticistsacquireddataon𝑝 = 38 millionsinglenucleotidepolymorphismsinonly𝑛 = 1092individuals.

Theasymptoticpropertiesofclassicalmethodsinaregimewhere𝑝increaseswith𝑛arequitedifferentthanintheclassicalregimeoffixed𝑝.Forexample,if𝑋! , 𝑖 = 1,… ,𝑛areindependentandidenticallydistributedas𝑁(0, 𝐼!),and𝑝/𝑛staysconstantas𝑝and𝑛increase,theeigenvaluesof

theempiricalcovariancematrix𝑆 = 𝑋𝑋′/𝑛donotconvergetothetruevalueof1,butratherhavealimitingdistributiondescribedbytheMarcenko–Pasturlaw.DonohoandGavish(2014)showhowthislawisrelevantforunderstandingtheasymptoticminimaxriskinestimatingalowrankmatrixfromnoisymeasurementsbythresholdingthesingularvalues.Donohoetal.(2013)obtainsimilarresultsfortheproblemofmatrixrecovery.

2.3VeracitySomebigdataisobtainedbypiecingtogetherseveraldatasetsthathavenotbeencollectedwiththecurrentresearchquestioninmind:thisissometimesreferredtoas“founddata”or“conveniencesamples”.Thistypeofdatawilltypicallyhaveextremelyheterogeneousstructure(HodasandLerman2014),andmaywellnotbeatallrepresentativeofthepopulationofinterest.Thevastmajorityofbigdataisobservational,butmoreimportantlymaybecollectedveryhaphazardly.Classicalideasofexperimentaldesign,surveysampling,anddesignofobservationalstudiesoftengetlostintheexcitementoftheavailabilityoflargeamountsofdata.Issuesofsamplingbias,ifignored,canleadtoincorrectinferencejustaseasilywithbigdataaswithsmall.Harford(2014)presentsanumberofhigh–profilefailures,fromtheLiteraryDigestpollof1936,whichhadtwomillionparticipants,toGoogleFluTrendsin2010,whichinbasingitspredictionsoninternetsearchtermsseverelyover–estimatedtheprevalenceofinfluenza.Despitethoselimitations,NationalStatisticalAgenciesaredevelopingmethodsforthecarefuluseofsuchdata,forexampleasanaidtoimputationonsmallareas(Marchettietal.,2015).

Cleaninglargevolumesofdataischallengingandveracityisaffectedbyincompleteorincorrectcleaning,andbythelackoftoolstoevaluatedataquality.Forexample,toconfirmthatameasureofsentimentcalculatedfromthesocialmediawassound,Daasetal.(2015)evaluateditsconsistencywithawell–validatedindicatorofconsumerconfidence.Whileitisclearthatsocialmediadataarenoisyandvolatile,datacollectedfromsensorsmayalsocomewithseveralproblems,includingalargeproportionofmissingvalues.Daasetal.(2015)consideredasanexampletheissuesinvolvedincleaningtrafficloopdetectordata.

Qualityofdatainlargeadministrativedatabasesisapressingproblem:additionalheterogeneitymaybeintroducedwhenthedataarerecordedbynumerousstakeholderswithvaryingprioritiesandincentives.Lixetal.(2012)usemodellingtoevaluateandimprovethequalityofthedatainlargehealthservicesdatabases.

2.4VelocityInmanyareasofapplication,datais“big”becauseitisarrivingatahighvelocity,fromcontinuouslyoperatinginstrumentation,e.g.,whenconsideringautonomousdrivingscenarios(Geigeretal.2012),orthroughanon–linestreamingservice.Sequentialoron–linemethodsofanalysismaybeneeded,whichrequiresmethodsofinferencethatcanbecomputedquickly,thatrepeatedlyadapttosequentiallyaddeddata,andthatarenotmisleadbythesuddenappearanceofirrelevantdata,asdiscussed,e.g.,byJainetal.(2006)andHoensetal.(2012).Theareasofapplicationsusingsequentiallearningarediverse.Forinstance,LiandFei–Fei(2010)useobjectrecognitiontechniquesinanefficientsequentialwaytolearnobjectcategorymodelsfromanyarbitrarilylargenumberofimagesfromtheweb.Monteleonietal.(2011)usesequentiallearningforclimatechangemodellingtoestimatetransitionprobabilitiesbetweendifferentclimatemodels.Forsomeapplications,suchashighenergyphysics,burstsofdataarrivesorapidlythatit

isinfeasibletoevenrecordallofit.Inthiscaseagreatdealofscientificexpertiseisneededtoensurethatthedataretainedisrelevanttotheproblemofinterest.

2.5BeyondtheVsBigdataisabouthumans,itoffersnewpossibilities,andhence,itraisesnewethicalquestions.Duhigg(2012),forinstance,reportedhowtheTargetretailstoresperformedanalyticstosendtargetedmarketingmaterialtotheircustomers,includingpromotionsrelatedtopregnancytoateenagerwhosefatherwasnotyetawareofthecomingbaby.AtFacebook,Krameretal.(2014)conductedalargescaleexperimentonsentimentswithoutobtainingaspecificuseragreement(seealsoMeyer,2014).Privacyisalsoabigconcern,especiallywhenconsideringthatthelinkingofdatabasescandiscloseinformationthatwasmeanttoremainanonymous.ThelawsuitfollowingtheNetflixChallengeisastrikingexampleofthatwherelinkingtheprovideddatatotheIMDBmoviereviewsallowedtoidentifysomeusers(Singel,2009;2010).Ingeneral,ifreleaseddatacanbelinkedtopubliclyavailabledata,suchasmediaarticles,thenprivacycanbecompletelycompromised.Anotherhigh–profileexampleofthiswasthepublicationofthehealthrecordsoftheGovernorofMassachusetts,whichinturnledtostrictlawsonthecollectionandreleaseofhealthdataintheUnitedStatesknownasthe“HIPAArules”(Tarran,2014).Inadditiontoprivacy,legalquestionsarealsounanswered:copyrightandownershipofthedata,forinstance,isnotalwaysclear(seee.g.Struijsetal.,2014).

Sub–samplingfromalargedatasetisoftenpresentedasasolutiontosomeoftheproblemsraisedintheanalysisofbigdata;oneexampleisthe“bagoflittlebootstraps”ofKleiner,etal.(2014).Whilethisstrategycanbeusefultoanalyseadatasetthatisverymassive,butotherwiseregular,itcannotaddressmanyofthechallengesthatwerepresentedinthissection.

3.StrategiesforbigdataanalysisInspiteofthewidevarietyoftypesofbigdataproblems,thereisacommonsetofstrategiesthatseemessential.Asidefrompre–processingandmulti–disciplinarity,manyofthesefollowthegeneralapproachofidentifyingandexploitingahiddenstructurewithinthedatathatoftenisoflowerdimension.Wedescribesomeofthesecommonalitiesinthissection.

3.1DataWranglingAverylargehurdleintheanalysisofdata,whichisespeciallydifficultformassivedatasets,ismanipulatingthedataforuseinanalysis.Thisisvariouslyreferredtoasdatawrangling,datacarpentry,ordatacleaning4.Bigdataisoftenvery“raw”:considerablepre–processingsuchasextracting,parsing,andstoring,isrequiredbeforeitcanbeconsideredinananalysis.

Volumeposesachallengetodatacleaning:whilemanualorinteractiveeditingofadatasetareusuallypreferredforreasonablysizeddata,itquicklybecomesimpracticalwhenthedatagrows.Largeadministrativedatabaseswillrequirespecialtools(e.g.thetableplotsofPutsetal.,2015)

4 The R packages dplyr and tidyr were expressly created to simplify this step, and are highly recommended. The new “pipe” operator %>%, operating similarly to the old Unix pipe command, is especially helpful. (Wickham, 2014).

toassistinthecleaning.Asthesizeorthevelocityincrease,automatizedsolutionsmaybecomenecessary.Putsetal.(2015),forinstance,useaMarkovmodeltoautomatizethecleaningoftrafficloopdata.Varietymakesoftenmakesdatawranglingdifficult,forinstancewhenselectingvariablesorfeaturesfromimageandsounddata(e.g.GuyonandElisseeff,2003),integratingdatafromheterogeneoussources(Izadietal.,2013),orparsingfree–formtextfortextminingorsentimentanalysis(e.g.,GuptaandLehal,2009;PangandLee,2008).Amoreadvancedapproachintextminingusesthesemanticsoftextratherthanthesimplecountofwordstodescribetexts(e.g.,Brienetal.,2013).

Oneexampleofadatapreparationproblemarisesinthemergingofseveralsetsofspatio–temporaldata,whichmayhavedifferentspatialunits–forexampleonedatabasemaybebasedoncensustracts,anotheronpostalcodes,andstillanotheronelectoraldistricts.Brownetal.(2013)havedevelopedanapproachtoheterogeneouscomplexdatabasedonalocalEMalgorithm.Thisstartsbyoverlayingallpartitionsthatareusedforaggregation,andthencreatingafinerpartitionsuchthatallelementsofthatpartitionsarefullyincludedinthelargerones.ThelocalEMalgorithmbasedonthisisparallelizable,andhenceprovidesaworkablesolutiontothiscomplexproblem.ArelatedissuearoseinthepresentationofGaffield(2015),wherehistoricalcensusdataisbeingusedaspartofastudyoftheimpactofindividualsonthehistoryofCanada.Thecensustractshavechangedovertime,andthecurrentdatabaseresolvesthisbymappingbackwardsfromthecurrentcensustracts.

3.2VisualizationVisualizationisveryoftenthefirstformalstepoftheanalysisandisoftenatoolofchoicefordatacleaning(seee.g.Putsetal.,2015).Theaimistofindefficientgraphicalrepresentationsthatsummarizethedataandemphasiseitsmaincharacteristics(Tukey,1977).Visualizationcanalsoserveasaninferentialtoolindifferentstagesoftheanalysis––understandingthepatternsinthedatahelpsincreatinggoodmodels(Bujaetal,2009).Forexample,Albert–Greenet.al(2014),WoolfordandBraun(2007)andWoolfordetal.(2009)presentexploratorydataanalysistoolsforlargeenvironmentaldatasets,whicharemeanttonotonlyplotthedata,butalsotolookforstructuresanddeparturesfromsuchstructures.Theauthorsfocusonconvergentdatasharpeningandthestalagmiteplotinthecontextofforestfirescience.

Evenif“apictureiswortha1000words”,onehastorealizethatgoingfrombigdatato1000wordisheavycompression.Visualizationis“bandwidth–limited”,inthesensethatthereisnotenoughscreenspacetovisualizelarge,orevenmoderatelysized,data.Interactivityprovidesmanywaystohelpwiththis;themostfamiliariszoomingintolargevisualizationstoreveallocalstructure,muchinthemannerofGoogleMaps.Newtechniquesarebeingdevelopedintheinformationvisualizationcommunity:Neumannetal.(2006)describedtakingcuesfromnature,andfromfractals,indevelopingPhylloTreesforillustratingnetworkgraphsatdifferentscales.Carpendaleinherbootcamppresentationdescribedseveralrecentinnovationsininteractivevisualization,manyofwhichareillustratedatthewebsiteoftheInnovisResearchGroup(Carpendale,2015).

3.3ReducingDimensionalityDirectreductionofdimensionalityisaverycommontechnique:inadditiontofacilitatingvisualization(Huronetal.,2014;Wickhametal.,2010),itisoftenneededtoaddressissuesof

datastorage(WittenandCandès,2013).HintonandSalakhutdinov(2006)describetheuseofdirectdimensionreductioninclassification.Dimensionalityreductionalsooftenincorporatesanassumptionofsparsity,whichisaddressedinthenextsubsection.PrincipalComponentAnalysis(PCA)isaclassicalexampleofdimensionalityreduction,introducedbyPearson(1901).Thefirsttwo,orthree,principalcomponentsareoftenplottedinthehopesofidentifyingstructureinthedata.OneoftheissueswithperformingPCAonlargedatasetsistheintractabilityofitscomputation,whichreliesonsingularvaluedecomposition(SVD).WittenandCandès(2013),EldarandKutyniok(2012),andHalkoetal.(2011),amongothers,haveusedrandommatrixprojections.Thisisapowerfultoolbehindcompressedsensingandastrikingexampleofdimensionalityreduction,allowingapproximateQRdecompositionorsingularvaluedecompositionwithgreatperformanceadvantagesoverconventionalmethods.WittenandCandès(2013)approximateaverylargematrixbytheproductoftwo“skinny”matricesthatcanbeprocessedrelativelyeasily.Usingrandomizedmethodsonecaninfertheprincipalcomponentsoftheoriginaldatafromthesingularvaluedecompositionoftheseskinnyfactormatrices,thusovercomingacomputationalbottleneck.

Dependingontheapplication,lineardimensionalityreductiontechniquessuchasPCAmayneedtobeextendedtonon–linearones,e.g.,ifthestructureofthedatacannotbecapturedinalinearsubspace(HintonandSalakhutdinov,2006).However,theyarehardertocomputeandparameterize.Innetworkanalysis,anovelwaytoreducethedimensionalityoflargeandcomplexdatasetsisbyemployinganexplanatorytoolfordataanalysiscalled“networkhistogram”(OlhedeandWolfe,2014)illustratedinFigure1.SofiaOlhedeinjointworkwithPatrickWolfesuggestsanon–parametricapproachtoestimateconsistentlythegeneratingmodelofanetwork.Theideaistoidentifygroupssuchthatnodesfromthesamegroupshowasimilarconnectivitypatternandtoapproximatethegeneratingmodelbyapiecewiseconstantfunctionbyaveragingovergroups.ThepiecewiseconstantfunctionhereiscalledstochasticblockmodelanditapproximatesthegeneratingmodelinthesamewayasahistogramapproximatesapdfortheRiemannsumapproximatesacontinuousfunction.

Figure1.ExampleofanetworkhistogramfromOlhedeandWolfe(2014).Anetworkcanbeexpressedasamatrixwhereeachentryrepresentthestrengthofconnectionbetweentwonodes.Cleverlyorderingthenodesuncoverstheunderlyingstructureofthenetwork.

3.4SparsityandRegularizationIncreasingmodelsparsityenforcesalower–dimensionalmodelstructure.Inturn,itmakesinferencemoretractable,modelseasiertointerpretanditleadstomorerobustnessagainstnoise(Candèsetal.2008,Candèsetal.2006).Regularizationtechniquesthatenforcesparsityhavebeenwidelystudiedintheliteratureformorethanadecade,includingthelasso(Tibshirani,1996),adaptivelasso(Zhou,2006),thesmoothlyclippedabsolutedeviation(SCAD)penalty(FanandLi,2001),andmodifiedCholeskydecomposition(WuandPourahmadi,2003).Thesemethodsandtheirmorerecentextensionsarenowkeytoolsindealingwithbigdata.

Astrongassumptionofsparsitymakesitpossibletouseahighernumberofparametersthanobservations,enablingustofindtheoptimalrepresentationofthedatawhilekeepingtheproblemsolvable.AgoodexampleinthiscontextisthewinningentryfortheNetflixChallengerecommendersystem,whichcombinedalargenumberofmethods,therebyartificiallyincreasingthenumberofparameterstoimprovepredictionperformance(Belletal.2008).

Sparsity–enforcingalgorithmscombinevariableselectionandmodelfitting:thesemethodstypicallyshrinksmallestimatedcoefficientstozeroandthistypicallyintroducesbiaswhichmustbetakenintoaccountinevaluatingthestatisticalsignificanceofthenon–zeroestimatedcoefficients.Muchoftheresearchinlassoandrelatedmethodshasfocussedonpointestimation,butinferentialaspectssuchashypothesistestingandconstructionofconfidenceintervalsisanactiveareaofresearch.Forinferenceaboutcoefficientsestimatedusingthelasso,vandeGeeretal.(2014),andJavanmardandMontanari(2014a,2014b)introduceanadditionalstepintheestimationprocedurefordebiasing.Lockhartetal.(2014)incontrastfocusontestingwhether

thelassomodelcontainsalltrulyactivevariables.Inasurvivalanalysissetting,Fangetal.(2014)testtheeffectofalowdimensionalcomponentinahighdimensionalsettingbytreatingallremainingparametersasnuisanceparameters.Theyaddresstheproblemofthebiaswiththeaimofimprovingtheperformanceofthetheclassicallikelihood–basedstatistics:theWaldstatistic,thescorestatisticsandthelikelihoodratiostatistic,allofthesebasedontheusualpartiallikelihoodforsurvivaldata.Toofferacompromisetothecommonsparsityassumptionofzeroandlargenon–zerovaluedelements,Ahmed(2014)proposesarelaxedassumptionofsparsitybyaddingweaksignalsbackintoconsideration.Hisnewhigh–dimensionalshrinkagemethodimprovesthepredictionperformanceofaselectedsubmodelsignificantly.

3.5OptimizationOptimizationplaysacentralroleinstatisticalinference,andregularizationmethodsthatcombinemodelfittingwithmodelselectioncanleadtoverydifficultoptimizationproblems.Complexmodelswithmanyhiddenlayers,suchasariseintheconstructionofneuralnetworksanddeepBoltzmannmachines,typicallyleadtonon–convexoptimizationproblems.Sometimesaconvexrelaxationofaninitiallynon–convexproblemispossible:thelassoforexampleisarelaxationoftheproblemthatpenalizesthenumberofnon–zerocoefficientestimates,i.e.itreplacesan𝐿!penaltywithan𝐿!penalty.Whilethisconvexrelaxationmakestheoptimizationproblemmucheasier,itinducesbiasintheestimatesoftheparameters,asmentionedabove.Non–convexpenaltiescanreducesuchbias,butnon–convexoptimizationisusuallyconsideredtobeintractable.However,statisticalproblemsareoftensufficientlywell–behavedthatnon–convexoptimizationmethodsbecometractable.Moreover,duetothestructuresofthestatisticalmodels,LohandWainwright(2014,2015)suggestthatthesemodelsdonottendtobeadversarial,henceitisnotnecessarytolookattheworstcasescenario.Evenwhennon–convexoptimizationalgorithms,suchastheExpectation–Maximizationalgorithm,donotprovidetheglobaloptimum,thelocaloptimumcanbeadequatesincethedistancebetweenthelocalandglobalmaximaisoftensmallerthanthestatisticalerror(Balakrishnanetal.,2015,LohandWainwright,2015).Wainwrightconcludedhispresentationattheopeningconferencewitha"speculativemeta–theorem",thatforanyprobleminwhichaconvexrelaxationperformswell,thereisafirstordersimplemethodonthenon–convexformulationwhichperformsasgood.Applicationsofnon–convextechniquesincludespectralanalysisoftensors,analogoustoSVDofmatrices,whichcanbeemployedtoaidinlatentvariablelearning(Anandkumaretal.,2014).Non–convexmethodscanalsobeusedindictionarylearningaswellasinaversionofrobustPCA,whichislesssusceptibletosparsecorruptions(Netrapallietal.,2014).Insomecases,convexreformulationshavealsobeendevelopedtofindoptimalsolutionsinnon–convexscenarios(GuoandSchuurmans,2007).

Manymodelsusedforbigdataarebasedonprobabilitydistributions,suchasGibbsdistributions,forwhichtheprobabilitydensityfunctionisonlyknownuptoitsnormalizingconstant,sometimescalledthepartitionfunction.MarkovchainMonteCarlomethodsattempttoovercomethisbysamplingfromanunnormalizedprobabilitydensityfunctionbasedonaMarkovchainthathasthetargetdistributionasitsequilibriumdistribution,butthesemethodsdonotnecessarilyscalewell.Insomecases,approximateBayesiancomputation(ABC)methodsprovide

analternativeapproximate,butcomputable,solutiontoamassiveproblem(BlumandTran,2010).

3.6MeasuringdistanceInhigh–dimensionaldataanalysis,distancesbetween“points”,orobservations,areneededformostanalyticprocedures.Forexampleoptimizationmethodstypicallymovethroughapossiblycomplexparameterspace,andsomenotionofstep–sizeforthismovementisneeded.SimulationmethodssuchasMarkovchainMonteCarlosamplingalsoneedto‘move’throughtheparameterspace.InboththesecontextstheuseofFisherinformationasametricintheparameterspacehasemergedasausefultool–itistheRiemannianmetric(Rao,1945;AmariandNagaoka,2000)whenregardingtheparameterspaceasamanifold.GirolamiandCalderhead(2011)exploitedthisinMarkovchainMonteCarloalgorithms,withimpressiveimprovementsinspeedofconvergence.GrosseandSalakhutdinov(2015)showedhowoptimizationofmulti–layerBoltzmannmachinescouldbeimprovedbyusinganapproximationtotheFisherinformation,thefullFisherinformationmatrixbeingtoohigh–dimensionaltouseroutinely.Asimilartechniqueemergedinenvironmentalscience(SampsonandGuttorp,1992),wheretwo–dimensionalgeographicalspaceismappedtoatwo–dimensionaldeformedspaceinwhichthespatialstructurebecomesstationaryandisotropic.Schmidtetal.(2011)describehowthisapproachcanbeexploitedinaBayesianframework,mappingtoadeformedspaceinhigherdimensions,andapplythistoanalysesofsolarradiationandtemperaturedata.

3.7RepresentationLearningRepresentationlearning,alsocalledfeaturelearningordeeplearning,aimstouncoverhiddenandcomplexstructuresindata(Bengioet.al,2013)andfrequentlyusesalayeredarchitecturetouncoverahierarchicaldatarepresentation.Themodelforthislayeredarchitecturemayhavehigherdimensionalitythantheoriginaldata.Inthatcase,onemayneedtocombinerepresentationlearningwiththedimensionalityreductiontechniquesdescribedaboveinthisreport.Large,well–annotateddatasetsfacilitatesuccessfulrepresentationlearning.Butlabelingdataforsupervisedlearningcanconsumeexcessivetimeandresources,andmayevenproveimpossible.Tosolvethisproblem,researchershavedevelopedtechniquestouncoverdatastructurewithunsupervisedtechniques(Anandkumaretal.,2013;SalakhutdinovandHinton,2012).

3.8SequentialLearningAstrategytodealwithbigdatacharacterizedbyahighvelocityissequentialorincrementallearning.Thistechniquetreatsthedatasuccessively,eitherbecausethetaskwasdesignedinthiswayorthedatadoesnotallowanyotheraccess,asitisthecaseforcontinuousdatastreamssuchaslongimagesequences(Kalaletal.2012).Sequentiallearningisalsoappliedifthedataiscompletelyavailable,buthastobeprocessedsequentially.Thisoccurswhenthedatasetistoolargetofitintothememoryortobeprocessedinapracticallytractableway.Forexample,Scottetal.(2013)introduceconsensusMonteCarlo,whichoperatesbyrunningseparateMonteCarloalgorithmsondistributedmachines,andthenaveragingindividualMonteCarlodrawsacrossmachines;theresultcanbenearlyindistinguishablefromthedrawsthatwouldhavebeenobtainedbyrunningasinglemachinealgorithmforaverylongtime.

3.9Multi–disciplinarityAcommonthemeinagreatmanypresentationsonbigdataisthenecessityofaveryhighlevelofmulti–disciplinarity.Statisticalscientistsarewell–trainedinbothplanningofstudies,andininferenceunderuncertainty.Computerscientistsbringasophisticatedmixofcomputationalstrategiesanddatamanagementexpertise.Boththesegroupsofresearchersneedtoworkcloselywithscientists,socialscientists,andhumanistsintherelevantapplicationareastoensurethatthestatisticalmethodsandcomputationalalgorithmsareeffectiveforthescientificproblemofinterest.Forinstance,Deanetal.(2012)describeanexcitingcollaborationinforestry,wheretheyareworkingonlinkingstatisticalanalysis,computationalinfrastructurefornewinstrumentation,withsocialagencywiththegoalofmaximizingtheirimpactonthemanagementofforestfires.Thesynergyspacebetweenthemiscreatedinanintegrativeapproachinvolvingscientistsfromallfields.Thisintegrationisnotaninventionoftheworldofbigdatabyanymeans––statisticsisaninherentlyappliedfieldandcollaborationshavebeenamajorfocusofthedisciplinefromitsearliestdays.Whathaschangedisthespeedatwhichnewtechnologyprovidesnewchallengestostatisticalanalysis,andthemagnitudeofthecomputationalchallengesthatgoalongwiththis.Muchofthebigdatanowavailableinvolvesinformationaboutpeople––thisisoneoftheaspectsofbigdatathatcontributestoitspopularity.Thesocialsciencescanhelptoelucidatepotentialbiasesinsuchdata,aswellashelptofocusonthemostinterestingandrelevantresearchquestions.Humanistscanprovideframeworksforthinkingaboutprivacy,abouthistoricaldataandchangesthroughtime,andaboutethicalissuesarisinginsharingofdataabout,forexample,individuals’health.Individualsmaywellbewillingtosharetheirdataforresearchpurposes,forexample,butnotformarketingpurposes:thehumanitiesofferwaysofthinkingabouttheseissues.

4.ExamplesofApplicationsThebreadthofapplicationsisnearlylimitless;weprovidehereaselectionoftheapplicationareasdiscussedduringtheopeningconference,inparttomaketheideasoutlinedintheprevioussectionmoreconcrete.

4.1PublicHealthDietisanimportantfactorofpublichealthinthestudyofdisabilities(Murrayetal.,2013),butassessingpeople’snutritionalbehaviourisdifficult.Nielsencollectsinformationaboutallproductssoldbyalargenumberofgroceriesandcornerstoresatthe3–digitpostalcodelevel,andpurchasecardloyaltyprogramshavedataonpurchasesatthehouseholdlevel.UsingthesesourcesofinformationalongwithaUPCdatabaseofnutritionalvalues,Buckeridgeetal.(2012)matchnutritionvariablestoexistingmedicalrecordsintheprovinceofQuébec,incollaborationwiththeInstitutdesantépubliqueduQuébec.Sincethepurchasedataistemporal,Buckeridgeetal.(2014)measurethedecreaseinsalesofsugarycarbonateddrinksfollowingamarketingcampaign,henceassessingtheeffectivenessofthispublichealtheffort.Bigdataisalsobeingusedforsyndromicsurveillance,themonitoringofthesyndromesoftransmittablediseasesatthepopulationlevel.IntheprovinceofQuébec,acentralsystemmonitorsthetriagedatafromemergencyroomsincludingfreeformtextwhichneedstobetreatedappropriately.Whentheautomatedsurveillancesystemissuesanalert,anad–hoc

analysisisperformedtodecidewhetherimmediateactionsareneeded.Buckeridgeetal.(2006)evaluatetheperformanceofsuchsystemsthroughsimulationfordifferentratesoffalsepositivealerts.Socialmediaproducedatawhosevolumeandvarietyareoftenchallenging.Thetraditionalreportingofflufromphysicianreportsgenerallyintroducesa1–2weektimedelay.GoogleFluTrends(Ginsberget.al2009)speedsupthereportingbytabulatingtheuseofsearchkeywords,suchas“runnynose”or“sorethroat”,inordertoestimatetheprevalenceofinfluenza–likeillnesses.Predictionscansometimesgowrong,seeButler(2013),butthegaininreactiontimeissignificant.UsingtheGPSlocationinthemetadataofTwitterpostsRamírez–Ramírezetal.(2013)introduceSimulationofInfectiousDiseases(SIMID)thatimprovegeographicandtemporallocalitybeyondGoogleFluTrends.TheyillustratetheuseofSIMIDtoestimateandpredictfluspreadinthePeelsub–metropolitanregionnearToronto.

4.2HealthPolicyManyexamplesinhealthpolicyrelyonthelinkageoflargeadministrativedatasets.InOntario,theInstituteforClinicalEvaluativeSciences(ICES)isaprovince–wideresearchnetworkthatcollectsandlinksOntario'shealth–relateddataattheindividuallevel.Similardatabasesexistsinotherprovinces.ComparingdataondiabetescaseascertainmentfromManitobawiththoseofNewfoundlandandLabrador,Lixetal.(2012)evaluatetheincompletenessofphysicianbillingclaimsandestimatetowhichextentthenumberofnon–fee–for–serviceactsisunderestimated.Withthequalityofthedatainmind,theauthorsalsoimprovediseaseascertainmentwithmodel–basedpredictivealgorithmsasanalternativetoofrelyingonpossiblyincorrectdiseasecodes.

Healthdatacanbespatiotemporalandcomplicationsarisequicklywhenthedataareheterogeneous.Postalcodesandcensustracts,forinstance,defineareasthatdonotmatch,withshapesthatvarythroughtime(seee.g.Nguyenetal.,2012).InastudyaboutcancerinNovaScotia,Leeetal.(2015)haveaccesstolocationdataforthemostrecentyears,butolderdataonlycontainthepostalcodes.Tohandlethedifferentgranularities,theyuseahierarchicalmodelwherethelocationofcancercasesaretreatedasalatentvariableandhigherlevelsofthehierarchyaretheaggregationunits(Lietal.,2012).

4.3LawandOrderInsurgenciesandriotsarefrequentinSouthAmerica,withhundredsorthousandsofoccurrencesineachcountryeveryyear.Korkmazetal.(2015)haveadatabaseofallTwittermessagessentfromSouthAmericaoveraperiodoffouryearsaswellasabinaryvariablenamedGSR,agoldstandardtodeterminewhetheraneventwasaninsurgencyornot.Theoccurrenceofaninsurgencycanbepredictedbythevolumeoftweets,thepresenceofsomekeywordstherein(fromatotalof634whowereinitiallyidentifiedaspotentiallyuseful),andanincreaseduseofTheOnionRouter(TOR),anonlineservicetoanonymizetweets.ThemassivedatabasecontainingallthetweetsisstoredonaHadoopfilesystemanditstreatmentisdonewithPythonthroughmapreduce.Thosetoolswhicharenotpartofatypicalstatisticscursusarestandardinthetreatmentofmassivedatasets.

4.4EnvironmentalSciencesTheworldisfacingglobalenvironmentalchallengeswhicharehardtomodel,giventheirinherentcomplexity.Toestimatethetemperaturerecordsince1000,Barbozaetal.(2014)use

differentproxiesincludingtreerings,pollen,coral,icecores,hencecombiningveryheterogeneousdata.Fromearlierwork,theauthorsfoundthathistoricalrecordscanbemuchbetterpredictedbyincorporatingclimateforcingsinthemodel,forinstancesolarirradiance,greenhousegasconcentrationandvolcanism.

Inanexamplecombininghealthandenvironmentdata,Ensoretal.(2013)analyzethelocationandtimeofcardiacarrestsfrom2004to2011inthecityofHouston,withrespecttoairpollutionmeasurementstakenbyEPAsensorsscatteredacrossthecity.Insteadofconsideringdailyaveragesofexpositiontoairpollution,thisstudymeasuredtheexposureofanindividualinthethreehoursprecedingacardiacarrestandfindasignificanteffectontheriskofanout–of–hospitalcardiacarrest.Inanunpublishedfollow–upstudydiscussedbySallieKeller,asyntheticmodelisusedtosimulatetheactivitiesof4.4millionpeopleinHoustononagivenday.Theindividualexposurevariesgreatly,evenwithinagivenhousehold,aspeople’sexposuretopollutiondependsontheirmovementsinthecity.ThesyntheticdatahelpedoptimizingCPRtrainingsgeographicallyinthecity.

4.5EducationInaresearchprojecttostudythetheoryofJohnson(2003)aboutgradeinflation,Reeseetal.(2014)analyzethehistoryofBrighamYoungUniversitystudentevaluationsfrom2002to2014,matchingitwithstudentrecords.Atotalof2.9millioncourseevaluationsfrom185,000studentwereused.Thevariablesofinterestarethreequestionsansweredonanordinalscaleandafreetextsectionforwhichasentimentanalysisdeterminesifthecommentispositiveornegative.ThemagnitudeoftheBayesianproportionaloddsmodelfittedtotheordinalresponsevariablesistoobigforMCMCmethods,buttheABCapproachleadstoacomputationallyfeasiblesolution.Thedatashowsempiricalevidenceofgradeinflation.Withanexamplebasedoneducationdata,Cook(2014)illustrateshowvisualizationmaybeusedforinference.Basedonthe2012OECDPisatestsresultsfrom485,900studentsin65countries,Cookshowsafigurepresentingthedifferenceinmeanofmathscoresandreadingscoresbetweenboysandgirlsbycountry.Whiletheboysseemtooutperformthegirlsinmath,amuchgreaterdifferenceisseeninfavourofthegirlsforthereadingscores.Tohaveanideaofthesignificanceofthedifferences,Cookproducessimilarplotswherethegenderlabelsarerandomlyassignedtoscoreswithineachcountry.Thefactthatthetruedatastandsoutincomparisontotherandomassignmentsindicatesthatthegapdidnotoccurbychance.

4.6MobileApplicationSecurityConveyinginformationofinterestfrombigdataisachallengewhichmaybeaddressedbyvisualization.Hosseinkhanietal.(2014)proposePapilio,anovelgraphicalrepresentationofanetworkthatdisplaysthedifferentpermissionsofAndroidapps.Thecreationofatractablerepresentationismadepossiblebytakingadvantageofthespecialstructureofthedata.Thewholerepresentationisstilltoolargeforasinglescreen,sohalos(BaudischandRosenholtz,2003)areusedtoindicatethepositionofoffscreennodes,asillustratedinFigure2.Theradiusofthehalosareproportionaltothedistancetothenode.

Figure2:IllustrationoftheHalosinPapilio,whichindicatethepositionofoffscreennodes.ThisPapilio

figurefromHosseinkhanietal.(2014)displaysthedifferentpermissionsofAndroidapps.

4.7ImageRecognitionandLabellingImagesareunstructureddataandtheiranalysisprogressedsignificantlywiththedevelopmentofdeeplearningmodelssuchasdeepBoltzmannmachinesthatareespeciallysuitableforautomaticallylearningfeaturesinthedata.TheTorontoDeepLearninggroupofferanonlinedemo5ofmulti–modallearningbasedontheworkofSrivastavaandSalakhutdinov(2012)thatcangenerateacaptiontodescribeanyuploadedimage.TheimagesearchengineofGooglealsousesdeeplearningmodelstofindimagesthataresimilartoanuploadedpicture.

4.8DigitalHumanitiesAlargebodyofresearchinthehumanitiesisbuiltfromanin–depthstudyofcarefullyselectedauthorsorrecords.Theabilitytoprocesslargeamountsofdataisopeningnewopportunitieswheresocialevidencecanbegatheredatapopulationlevel.Forinstance,Drummondetal.(2006)usethe1901CanadiancensusdatatorevisedifferenttheoriesaboutbilingualisminCanadaandimprovetheiragreementwiththedata.TheyalsoobservethatpatternsinQuébecdiffersignificantlyfromthoseintherestofthecountry.InhisbookCapitalintheTwenty–FirstCentury,Piketty(2014)discussestheevolutionofreferencestomoneyinnovelsasareflectionofinequalitiesinthesociety.Underwoodetal.(2014)digfurtherintothequestionbylistingallreferencestomoneyamountsinEnglish–languagenovelsfrom1750to1950.Theyfindthattherelativefrequencyofreferencestomoneyamountsincreasesthroughtime,buttheamountmentioneddecreasesinconstantdollaramount.Theydonotattempttodrawstrongconclusionsfromthissocialevidence,mentioningforinstancethatthechangesinthereadingaudienceduetoanincreasedliteracyamongtheworkingclassmayalsoplayaroleintheevolutionofthenovels.Onalargerscale,Micheletal.(2011)analyzeabout4%ofallbookseverprintedtodescribesocialtrendsthroughtimeaswellaschangesinlinguistics.Whilepatternsinthedatamatch 5 http://deeplearning.cs.toronto.edu/i2t

milestoneseventssuchastheGreatWarsandsomeglobalepidemics,theevolutionofwordssuchas“feminism”measureimportantsocialtrends.Localeventssuchasthecensorshipofdifferentartistsforanumberofyearsindifferentcountriesarealsovisibleinthedata.Thosearesomeexamplesof‘culturomics’,thequantitativeanalysisofdigitizedtexttostudyculturaltrendsandhumanbehavior.

4.9MaterialsScienceInanapplicationtoMaterialsSciencebyNelsonetal.(2013),thepropertiesofabinaryalloydependontherelativepositionofatomsfrombothmaterialswhosedescriptionreliesonaverylargenumberofexplanatoryvariables.Surprisingly,asCandèsetal.(2006)showed,arandomgenerationofcandidatematerialsismoreefficientthanacarefullydesignedchoice.Thisisanexamplewherebigdatashowsacounterintuitivebehaviour,sinceonecouldthinkthatcarefuldesignofexperimentshouldbeabletobeatrandomselections.

5.ConclusionWhiletechnicalachievementsmaketheexistenceandavailabilityofverylargeamountsofdatapossible,thechallengesemergingfromsuchdatagoesbeyondprocessing,storingandaccessingrecordsrapidly.Asitstitleindicates,thethematicprogramonStatisticalInference,LearningandModelsforBigDatatriedtofocusthediscussionparticularlyonstatisticalissues.Thereareagreatmanyfieldsofresearchinwhichbigdataplaysakeyrole,andithasnotbeenpossibletosurveyallofit.

Inthisreport,weidentifychallengesandsolutionsthatarecommonacrossdifferentdisciplinesworkingoninferenceandlearningforbigdata.Withnewmagnitudesofdimensionality,complexityandheterogeneityinthedata,researchersarefacedwithnewchallengesofscalability,qualityandvisualization.Inaddition,bigdataisoftenobservationalinnature,havingbeencollectedforapurposeotherthantheoneintended,andhencemakinginferenceevenmorechallenging.Emergingtechnologicalsolutionsprovidenewopportunities,buttheyalsodefinenewparadigmsfordataandmodels.Makingthemodelsbiggerwithlayersoflatentvariablesmayprovideaninsightintothefeaturesofthedata,whilereducingthedatatolowerdimensionalmodelscanbekeyingettinganycalculationsthrough.Lowerdimensionalstructureanalysismanifestsitselfnotonlyassparsityassumptionsanddimensionalityreduction,butalsoinvisualization.Othertechniquessuchasnon–convexalgorithms,samplingstrategiesandmatrixrandomizationaidintheprocessingoflargedatasets.Withmassivedata,settlingforanapproximatesolutionseemstobethesavingcompromise,whetherbyallowingformoreflexibilityinanobjectivefunctionorbyreplacingastandardalgorithmthatisinfeasibleatsuchalargescale.

Manyofthemoststrikingexamplesofbigdataanalysispresentedduringthisthematicprogramcamefromteamsofmanyscientistswithcomplementarybackgrounds.Movingforward,researchteamsneedtoscaleup,invitingtalentsfromdifferentfieldstocometogetherandjoinforces.Thephrase“BigData”hashadagreatdealofmediaattentioninrecentyears,creatingahypethatwillcertainlyfade.Thenewproblemsthathaveemerged,however,areheretostayandtheneedforinnovativesolutionswillcertainlykeepourcommunitybusyformanyyearstocome.

AcknowledgmentsWearegratefulfortheFieldsInstituteforResearchintheMathematicalSciences,whichprovidedthemajorportionofthefundingforthethematicprogramonStatisticalInference,LearningandModelsinBigData.TheFieldsInstituteisfundedbytheNaturalSciencesandEngineeringResearchCouncilofCanadaandtheMinistryofTraining,CollegesandUniversitiesinOntario.FundingwasalsoprovidedforspecificworkshopsbythePacificInstitutefortheMathematicalSciences,theCentredeRecherchesMathématiques,theAtlanticAssociationforAdvancementintheMathematicalSciences,theDepartmentofStatisticsatIowaStateUniversity,theNationalScienceFoundation,andtheCanadianStatisticalSciencesInstitute.

Wethankthereviewersofanearlierversionofthepaperforhelpfulandconstructivecommentsthatimprovedthescopeandpresentation.

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