SpatialmodellingetcDavidLMiller&MarkVBravington

InternationalWhalingCommissionScientificCommittee2017

Whyarewehere/whydidwedothis?StratifiedHorvitz-Thompsonisworkhorseofmanyabundanceestimates

HowisH-Tgoingtofail?

Whendoweneedtousespatialmodels?

Whatare“Badsurveys”?

Reporthttp://converged.yt/papers/iwc-2017-ht.pdf

OverviewToday:

1. what'swrongwithH-T?

2. Spatialmodelsoverview

3. Whatcangowrongwithspatialmodels?

4. TestingdesignsinR

5. Modelcheckingforspatialmodels

Tomorrow:

1. Whatwemissed,whatishard

2. Tryouttesteronyourdata

3. Othermethods/software,futurework

4. Guidelines

PracticalitiesTrynottotalkformorethananhourwithoutabreak

Webothhavefunnyaccents,yellifyoudon'tunderstand!

Thereismaths–don'tworry

Thisisnotadistancesamplingcourse!Thismaterialusuallytakes4days+toteach

Thiswillnotprepareyoutoanalysespatialdata

BUTyoucandothisinStAndrewsthissummer!

creem2.st-andrews.ac.uk

Whyareweinterestedinspatially-explicitestimation?

Inferentialaims

PartI

Horvitz-Thompsonestimation:thegood,thebadandtheugly

Horvitz-Thompson-likeestimatorsRescalethe(flat)densityandextrapolate

aregroup/clustersizes

isthedetectionprobability(fromdistancesampling)

=N̂ study areacovered area ∑

i=1

n si

p̂ i

si

p̂ i

VarianceofH-TMultiplesourcesofrandomnessinH-Tequation:

-detectability

-dealtwithas ,encounterrate

-groupsize

p̂ i

n n/Ls

Hiddeninthisformulaisasimpleassumption

Probabilityofsamplingeverypointinthestudyareaisequal

Isthistrue?Sometimes.

If(andonlyif)thedesignisrandomised

Manyfacesofrandomisation

Whatdoesthisrandomisationgiveus?Coverageprobability

H-Testimatorassumesevencoverage

(oryoucanestimate)

Otherwisenotreallyvalid

EstimatingcoverageWecanestimatecoverageofanon-uniformdesign!

InDistance!

ExamplefromBC,Canadainthispaper:

Estimatingcoverage

AcomplexsurveyplanThomas,WilliamsandSandilands(2007)

Differentareasrequiredifferentstrategies

Zig-zags,parallellines,census

AnalysisinDistance

Sideline:alternativeterminology“Adesignisanalgorithmforlayingdownsamplersinthesurveyarea”

“Arealization(fromthatalgorithm)iscalledasurveyplan”

LenThomas(Talk@CREEM2004)

H-TestimationagainCan'testimatew/H-Tw/ocoverage

“Fixed”“designs”violateassumptions

Someanimalshave

“Deteriorate”poolingrobustnessproperty

Whatcanwedo?

ℙ(included) = 0

MoreonvarianceEncounterratevariance

Within-transectvariationcanbebad

e.g.,N-Stransect,N-Sdensitygradient

≈ / − n/Lnj lj

StratificationIfwesuspectdensitychangecanstratify!

Preorposthoc(spatialandnon-spatial)

Iamgoingtostoptalkingverysoon

SummaryH-Tisaspatialmodel(sortof)

Violatedanassumptionifnorandomness

Hardtoassesshowbadthisis

Fewsteretal(2009)andFewster(2011)givevarianceapproaches

PartII

Spatialmodels

SpatialmodelsofdistancesamplingdataCollectspatiallyreferenceddata

Whynotmakespatially-explicitmodels?

Gobeyondstratifiedestimates

Relateenvironmentalcovariatestocounts

Thisistherosypicturetalk

We'lltalkaboutthegrimrealitylater

Exampledatainthistalk

SpermwhalesofftheUSeastcoastHangoutnearcanyons,eatsquid

Surveysin2004,USeastcoast

Combinationofdatafrom2NOAAcruises

ThankstoDebiPalka,LanceGarrisonfordata.JasonRobertsfordataprep.

Exampledata

ModelformulationPurespatial,pureenvironmental,mixed?

Mayhavesomepriorknowledge

Biology/ecology

Whataredriversofdistribution?

Inferentialaim

Abundance

Ecology

DensitysurfacemodelsHedleyandBuckland(2004)

Milleretal.(2013)

Ignoringgroupsize(moreonthattomorrow)

PhysetercatodonbyNoahSchlottman

Howdowemodelthat?

SPOILERALERT:yourmodelisprobablyjustaveryfancyGLM

Generalisedadditivemodels(in1slide)Takingthepreviousexample…

somecountdistribution

( ) = exp [ + ( )]nj Ajp̂ j β0 ∑k

sk zkj

∼nj

area of segmentprobability of detection in segment(inverse) link functionmodel terms

Whataboutthosesthingys?

Covariatesspace,time,environmental(remotelysensed?)data

Modellingsmooths1-dimension:notmuchdifference

2Dmoretricky

edgeeffects

trickyboundaries

moretomorrow

Nowgoingtodosomemaths…

(ignoreatwill)

Howdowebuildthem?Functionsmadeofother,simplerfunctions

Basisfunctions,

Estimate

bk

βk

s(x) = (x)∑ Kk=1 βkbk

Straightlinesvs.interpolationWantalinethatis“close”toallthedata

Don'twantinterpolation–weknowthereis“error”

Balancebetweeninterpolationandgenerality

Howwigglyisafunction?

MakingwigglynessmatterFitneedstobepenalised

Somethinglike:

(Canalwaysre-writethisintheform )

Estimatethe termsbutpenaliseobjective

“closenesstodata”+penalty(REML/ML)

dx∫ℝ ( )s(x)∂2

∂x2

2

SββT

βk

Smoothingparameter

Sideline:GAMsareBayesianmodelsGenerally:

penaltiesareimproperpriorprecisionmatrices

(nullspacegivesimproperpriors)

Usingshrinkagesmoothers:

properpriors

empiricalBayesinterpretation

Beyondunivariatesmooths?Canbuild(anisotropic)tensorproductterms

Take2ormoreunivariateterms

Thinplateregressionsplinesallowmultivariateterms(isotropic)

SpatialsmoothingCanjustsmoothinspace

Validabundanceestimationtechnique

UsefulforEDAforenv.cov.models(hardday2!)

Notgoodforextrapolations

Basischoicecanmatter!

WhyGAMsarecool...Fancysmooths(cyclic,boundaries,…)

Fancyresponses(expfamilyandbeyond!)

Randomeffects(byequivalence)

Markovrandomfields

Correlationstructures

SeeWood(2006/2017)forahandyintro

Let'sfitamodel

dsmisbasedonmgcvbySimonWood

library(dsm)# environmental covariatesdsm_env_tw <- dsm(count~s(Depth) + s(NPP) + s(SST), ddf.obj=df_hr, segment.data=segs, observation.data=obs, family=tw())# spacedsm_xy_tw <- dsm(count~s(x, y), ddf.obj=df_hr, segment.data=segs, observation.data=obs, family=tw())

Simple!Done?

NO

Moreonmodelcheckinglater...

Predictions/abundanceestimatesGridofcovariatesmustbeavailable

Predictwithinsurveyarea

Extrapolateoutside(withcaution)

Workingonagridofcells

Plotiss(x,y) +s(Depth)Adduptogetabundance

EstimatingvarianceUncertaintyfrom:

detectionfunctionparameters

spatialmodel

Needtopropagateuncertainty!

MethodsindsmBravington,Hedley&Miller(inprep)

PlottinguncertaintyMapsofcoefficientofvariation

CVforgivenstratum(better)

Visualisationishard

CommunicatinguncertaintyAreanimationsagoodwaytodothis?

Simulatefromposteriorparameterdistribution

Somefeatures(e.g.shelf,N-Sgradient)stickout

β ∼ N( , )β̂ Σ̂

Iamgoingtostoptalkingverysoon

SummaryBuildmodelsinstages(detectionfunction+GAM)

Countsarefunctionsofcovariates

Purespatialmodels

Environmentalcovariatemodels

Mix?!

Fit/checkusingdsmMostofthetheoryisresolved,applicationsarehard

PartIII

H-Torspatialorgiveup?

SpatialmodelscanhelpSpatialmodellingcangiveubiasedabundanceests

evenwithunevencoverage

limitstoextrapolation

V.evencoverage=>HT?

“Evenness”subtle,detectabilityeffect

e.g.,weatherbadineast

Weatherordistribution?Weatherhasabigeffectondetectability

Needtorecordduringsurvey

Disambiguatebetweendistribution/detectability

PotentialconfoundingcanbeBAD

VisibilityduringPOWER2014

ThankstoHirotoMuraseandcoforthisdata!

Covariatescanmakeabigdifference!

Otherstuff

Spatialmodellingwon'tsolveallyrproblems

Designissues

Ludicrousextrapolation

Surveyplannotrobusttoweatherissues

Non-uniformdistributionwrtsampler

Migration

Spatialmodelsalonecan'tsolvetheseissues

Spatialmodellingwon'tsolveallyrproblems

Violationsofsurveyprocedure

Followinganimals

Responsivemovement

Guardingthetrackline

Groupsizeestimation

Spatialmodelsalonecan'tsolvetheseissues

Spatialmodellingwon'tsolveallyrproblems

Detectionfunctions

Notenoughobservations

UncertainspeciesID

Groupsize

Spatialmodelsalonecan'tsolvetheseissues

@kitabet

Shouldeverythingbespatial?Doyouhaveenoughobservations?

Iftheydolookgood(evencoverage,etc)

Isitworthre-analysingfromH-T?

Pointestimatessimilar?

Variancemaywellbedifferent?

Iamgoingtostoptalkingverysoon

SummarySpatialmodelsdon'tsolveallproblems

Complexmodelscanleadtocomplexissues

Recordingweatherconditionsisimportant

Youcanalwaysgiveup!

PartIV

Testingdesigns

Whatcanwedo?Takeasurveyandsimulate

IsH-Trobust?

Howdodifferentspatialmodelscompare?

Onlythinkingabouttotalabundance&CV

Softwareltdesigntester(basedonDSsimbyLauraMarshall,CREEM)

Setupsimulations,testwhatcanbedone

MostoftheworkneedstobedoneinGIS

Surveyshapefiles,covariatesetc

ImporttoR,runsmodels,showsoutput

github.com/dill/ltdesigntester

Settingupasurveysimulation

DensityGridinpolygonofstudyarea

Eitherspecifysimplegradientoruseothertoolstomakecomplexdensity

Densityasgrid

DesignGenerateusingGIS/Distance

Exporttoshapefile

DetectionfunctionFunctionalform(half-normal,hazard-rate)

Parameters(scale,shape)

Truncation

(Covariatesviamultiplefunctions,morelater)

SpecificationtosimulationGeneratemultiplerealizations

Analyseeachwithamanymodels

Differentspatial,H-T

Compareresults

TestmodelsSpatialsmoothers

thinplatespline,bs="tp"(Wood,2003)

thinplatesplinewithshrinkage,bs="ts"(Marraetal.,2011)

Duchonspline,bs="ds", m=c(1, 0.5)(Milleretal.,2014)

tensorofthinplatespline(w/andw/orotatedcovariates)

Stratifiedestimates

Horvitz-Thompson(w/andw/ocovariates)

stratifiedHorvitz-Thompson(w/andw/ocovariates)

Comparingperformance

ImportantcaveatsNomodelchecking

Dependenton“good”detectionspecs

Nogroupsizemodel

No oravailabilityg(0)

Quickexamplecodelibrary(ltdesigntester)# setup a simulationmy_sim <- build_sim(design_path="path/to/shp", dsurf=density_surface_matrix, n_grid_x=dsurf_dim_x, n_grid_y=dsurf_dim_y, n_pop=true_N, df=detection_function_specs, region="path/to/shp")# run it!res <- do_sim(nsim=number_of_sims, scenario=my_sim, pred_dat=prediction_data_frame, ...)

Wemadeabigdealaboutweatherearlier...Wecanaddcovariatestoo(aweebitunwieldyatthemoment)

Buildmultipledetectionfunctions/simsinlist()Covariatesvaryaccordingto:

logitfunctionE-W(cansetpars,2state)

setvaluesinsegmentdata(alreadyobserved)

Iamgoingtostoptalkingverysoon

SummaryWecantestmultiple(simple)scenarios

Assumptionofsimplegradients

Modelslikelywon'tworkfordifficultstuffiftheydon'tworkforsimplethings

Whatwillwork/whatwon't

Simplesummaryplots

Betterthantherest good≠

PartV

ModelcheckingforDSMs

ModelcheckingCountdistribution

Basiscomplexity

Model(term)selection

Sensitivity

Observedvs.expected

Cross-validation(replicability)

(Plusalltheusualstufffordetectionfunctions!)

CountdistributionsResponseisacount

Often,it'smostlyzero

Aggregationsoccuratscalessmallerthanspatialmodel

Wantresponsedistributionthatdealswiththat

Couldmess-upvarianceifignored

Linkedtosegmenting

Flexiblemean-variancerelationship

Negativebinomial

Estimate

Isquadraticrelationshipa“strong”assumption?

SimilartoPoisson:

Var (count) =(count) + κ(count)2

κ

Var (count) = (count)

Tweediedistribution

Commondistributionsaresub-cases:

Poisson

Gamma

inverse-Gaussian

Weareinterestedin

(here)

Var (count) = ϕ(count)q

q = 1 ⇒q = 2 ⇒q = 3 ⇒

1 < q < 2

q = 1.2, 1.3, … , 1.9

BasiscomplexityBefore:

Howbigshouldkbe?“Bigenough”Penaltytakescareoftherest

?gam.checkgivesusefuloutput(alsoresidualchecksetc)

s(x) = (x)∑ Kk=1 βkbk

gam.checktextoutputgam.check(dsm_env_tw)

Method: REML Optimizer: outer newtonfull convergence after 8 iterations.Gradient range [-3.139726e-08,2.036272e-08](score 375.9503 & scale 4.316452).Hessian positive definite, eigenvalue range [0.5725432,298.5906].Model rank = 28 / 28

Basis dimension (k) checking results. Low p-value (k-index<1) mayindicate that k is too low, especially if edf is close to k'.

k' edf k-index p-values(Depth) 9.000 4.049 0.814 0.36s(NPP) 9.000 2.846 0.779 0.04s(SST) 9.000 4.916 0.771 0.04

Tobler'sfirstlawofgeography“Everythingisrelatedtoeverythingelse,butnearthingsaremorerelatedthandistantthings”

Tobler(1970)

ImplicationsofTobler'slaw

Whatcanwedoaboutthis?Carefulinclusionofterms

Testforsensitivity(lotsofmodels)

Fitmodelsusingrobustcriteria(REML)

Testforconcurvity(mgcv::concurvity,dsm::vis.concurvity)

Termselection(approximate) values(Marra&Wood,2012)

pathdependenceissues

shrinkagemethods(Marra&Wood,2011)

ecological-leveltermselection

whichbiomassmeasure?

includespatialsmoothornot?

p

Observedvs.expectedDiagnostic–compareobservedvs.expectedcounts

Comparefordifferentcovariate/aggregations

Innextdsm,obs_exp()doesthisGoingbacktothoseroughPOWERmodels…

> obs_exp(b, "beaufort") 1 2 34Observed 3.00000 10.00000 80.00000Expected 6.97715 12.42649 83.03773

> obs_exp(b_nc, "beaufort") 1 2 34Observed 3.000000 10.00000 80.00000Expected 8.478759 17.00705 73.23535

Cross-validationHowwelldoesthemodelreproducewhatwesaw?

Leaveoutonearea,re-fitmodel,predicttonewdata

Wenger&Olden(2012)havegoodspatialexamples

Cross-validationexample

Cross-validationexample

Iamgoingtostoptalkingverysoon

2(ormore)-stagemodelsNot“cool”(statistically),but…

Multi-stagemodelsarehandy!

Understandandcheckeachpart

Splityourmodellingeffortsamongstpeople

ConclusionsThismethodologyisgeneral

Bears,birds,beercans,LochNessmonsters…

Modelsareflexible!

Linearthings,smooththings,randomeffectthings(andmore)

IfyouknowGLMs,youcangetstartedwithDSMs

Maturetheoreticalbasis,stilllotstodo

Activeusercommunity,activesoftwaredevelopment

Resources

distancesampling.org/R/

distancesampling.org/workshops/duke-spatial-2015/

Thanks!Slidesw/referencesavailableatconverged.yt

ReferencesFewster,R.M.,Buckland,S.T.,Burnham,K.P.,Borchers,D.L.,Jupp,P.E.,Laake,J.L.,etal.(2009)EstimatingtheEncounterRateVariancein

DistanceSampling.Biometrics,65,225–236.

Fewster,R.M.(2011),VarianceEstimationforSystematicDesignsinSpatialSurveys.Biometrics,67:1518–1531.

Hedley,S.L.,&Buckland,S.T.(2004).Spatialmodelsforlinetransectsampling.JournalofAgricultural,Biological,andEnvironmental

Statistics,9(2).

Marques,T.A.,Thomas,L.,Fancy,S.G.,&Buckland,S.T.(2007).Improvingestimatesofbirddensityusingmultiple-covariatedistance

sampling.TheAuk,124(4).

Marra,G.,&Wood,S.N.(2011).Practicalvariableselectionforgeneralizedadditivemodels.ComputationalStatisticsandDataAnalysis,

55(7).

Marra,G.,&Wood,S.N.(2012).CoveragePropertiesofConfidenceIntervalsforGeneralizedAdditiveModelComponents.Scandinavian

JournalofStatistics,39(1).

Wenger,S.J.andOlden,J.D.(2012)Assessingtransferabilityofecologicalmodels:anunderappreciatedaspectofstatisticalvalidation.

MethodsinEcologyandEvolution,3,260–267.

Handyawkwardquestionanswers

Don'tthrowawayyourresiduals!

gam.check

rqgam.check(DunnandSmyth,1996)

PenaltymatrixForeach calculatethepenalty

Penaltyisafunctionof

calculatedonce

smoothingparameter( )dictatesinfluence

bk

βλ SββT

Sλ

Howwigglyarethings?Wecansetbasiscomplexityor“size”( )

Maximumwigglyness

Smoothshaveeffectivedegreesoffreedom(EDF)

EDF<

Set “largeenough”

k

kk

Let'stalkaboutdetectability

Detectability

Distancesampling“Fittothehistogram”

Model:

Calculatetheaverageprobabilityofdetection:

ℙ [animal detected | animal at distance y] = g(y; θ)

= g(y; )dyp̂ 1w ∫

w

0θ̂

Distancesampling(extensions)Covariatesthataffectdetectability(Marquesetal,2007)

Perceptionbias( )(Burtetal,2014)

Availabilitybias(Borchersetal,2013)

Detectionfunctionformulations(MillerandThomas,2015)

Measurementerror(Marques,2004)

FigurefromMarquesetal(2007)

g(0) < 1

That'snotreallyhowtheoceanworks...

Availability

WecanonlyseewhalesatthesurfaceWhatproportionofthetimearetheythere?

Acoustics

Tags(DTAGsetc)

Behaviouralstudies

Fixedcorrectionto ?

ModelviafancyMarkovmodels(Borchersetal,2013)

PicturefromUniversityofStAndrewsLibrarySpecialCollections

p̂