Methods Ecol Evol 201891679ndash1691 wileyonlinelibrarycomjournalmee3emsp|emsp1679

Received21January2018emsp |emsp Accepted26April2018DOI 1011112041-210X13027

R E S E A R C H A R T I C L E

Statistical inference for home range overlap

Kevin Winner1 emsp|emspMichael J Noonan2 3 emsp|emspChristen H Fleming2 3emsp|emsp Kirk A Olson4emsp|emspThomas Mueller2 5 6emsp|emspDaniel Sheldon1 7emsp|emspJustin M Calabrese2 3

1CollegeofInformationandComputerSciencesUniversityofMassachusettsAmherstAmherstMassachusetts2SmithsonianConservationBiologyInstituteNationalZoologicalParkFrontRoyalVirginia3DepartmentofBiologyUniversityofMarylandCollegeParkMaryland4WildlifeConservationSocietyMongoliaProgramUlaanbaatarMongolia5SenckenbergBiodiversityandClimateResearchCentreSenckenbergGesellschaftfuumlrNaturforschungFrankfurt(Main)Germany6DepartmentofBiologicalSciencesGoetheUniversityFrankfurt(Main)Germanyand7DepartmentofComputerScienceMountHolyokeCollegeSouthHadleyMassachusetts

Correspondence JustinMCalabreseSmithsonianConservationBiologyInstituteNationalZoologicalPark1500RemountRdFrontRoyalVA22630 Emailcalabresejsiedu

Funding informationUSNSFAdvancesinBiologicalInformaticsprogramGrantAwardNumberABI-1458748SmithsonianInstitutionCGPSgrantRobertBoschFoundation

HandlingEditorKylieScales

Abstract1 DespitetheroutinenatureofestimatingoverlappingspaceuseinecologicalresearchtodatenoformalinferentialframeworkforhomerangeoverlaphasbeenavailabletoecologistsPartofthisissueisduetotheinherentdifficultyofcomparingtheesti-matedhomerangesthatunderpinoverlapacrossindividualsstudiessitesspeciesandtimesAsoverlapiscalculatedconditionallyonapairofhomerangeestimatesbiasesintheseestimateswillpropagateintobiasesinoverlapestimatesFurthercom-poundingtheissueofcomparabilityinhomerangeestimatorsisthehistoricallackofconfidenceintervalsonoverlapestimatesThismeansthatitisnotcurrentlypossibletodetermineifasetofoverlapvaluesisstatisticallydifferentfromoneanother

2 Asasolutionwedevelopthefirstrigorousinferentialframeworkforhomerangeover-lapOurframeworkisbasedontheautocorrelated-Kerneldensityestimation(AKDE)familyofhomerangeestimatorswhichcorrectforbiasesduetoautocorrelationsmalleffectivesamplesizeandirregularsamplingintimeCollectivelytheseadvancesallowAKDEestimatestovalidlybecomparedevenwhensamplingstrategiesdifferWethencoupletheAKDEestimateswithanovelbias-correctedBhattacharyyacoefficient(BC)toquantifyoverlapFinallywepropagateuncertaintyintheAKDEestimatesthroughtooverlapandthusareabletoputconfidenceintervalsontheBCpointestimate

3 Using simulateddatawedemonstratehowour inferential frameworkprovidesaccurateoverlapestimatesandreasonablecoverageofthetrueoverlapevenatsmallsamplesizesWhenappliedtoempiricaldatawefoundthatbuildinganin-teractionnetworkforMongoliangazellesProcapra gutturosabasedonallpossibletiesvsonlythosetieswithstatisticalsupportsubstantiallyinfluencedthenet-workrsquospropertiesandanypotentialbiologicalinferencesderivedfromit

4 OurinferentialframeworkpermitsresearcherstocalculateoverlapestimatesthatcanvalidlybecomparedacrossstudiessitesspeciesandtimesandtestwhetherobserveddifferencesarestatisticallymeaningfulThismethodisavailableviatheRpackagectmm

K E Y W O R D S

autocorrelated-KerneldensityestimationanimalmovementautocorrelationBhattacharyyacoefficientctmmKerneldensityestimate

copy2018TheAuthorsMethodsinEcologyandEvolutionpublishedbyJohnWileyampSonsLtdonbehalfoftheBritishEcologicalSocietyThisarticlehasbeencontributedtobyUSGovernmentemployeesandtheirworkisinthepublicdomainintheUSA

1680emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

1emsp |emspINTRODUC TION

Ecologistshavelongbeeninterestedinpatternsanddriversofanimalspaceuse(BrownampOrians1970Burt1943Jetz2004)Decisionsonwhatareastooccupycaninfluencefitnessthroughawiderangeof pathways such as foraging efficiency (MitchellampPowell 2012)orpredatorndashpreydynamics(MitchellampLima2002)andevendriveevolutionarytrajectories(LukasampClutton-Brock2013)Relatedtothis is the question of overlapping space use between individualsandorpopulationsQuantifyingoverlapcanprovideaninformativemetric for testinghypotheseson interspecificcompetition (BergerampGese2007)territoriality(GrantChapmanampRichardson1992)and mating systems (Powell 1979) Furthermore overlap can beusedtounderpinanalysesofsocialnetworkstructure(Fregravereetal2010) and contact rates with implications for disease transmis-sion(DoughertySeidelCarlsonSpiegelampGetz2018SanchezampHudgens2015)Trendsinoverlappingspaceusearealsoroutinelyusedindeterminingallometricscalinglaws(Grantetal1992Jetz2004)Therapidincreaseinboththeavailabilityandqualityoftrack-ingdatainrecentyears(KaysCrofootJetzampWikelski2015)hasmadetheconceptofhomerange(HR)overlapincreasinglyrelevantEcologistsarenowinapositiontoaddressoverlap-relatedquestionsforalargernumberofspeciesandindividualsinmoreecosystemsandwithmoreaccuratedatathaneverbefore

Despitetheseadvancesaformal inferentialframeworkforHRoverlap is still lackingOverlap is typically quantified by the first- estimatingHRsfromtrackingdataandthenapplyinganoverlapmet-rictotherangeestimates (FiebergampKochanny2005MillspaughGitzenKernohan LarsonampClay 2004)Awide rangeofoverlapmetricshavebeenproposed in the literature spanning thegamutfrom ad hocindicestomoreformalmeasuresThesedifferentmet-rics have contrasting properties and can produce highly differentoverlapestimatesonthesamedata(seeFiebergampKochanny2005Millspaughetal 2004) Further compounding this problem is theinherent difficulty of comparing the estimatedHRs that underpinoverlapacrossstudiessitesspeciesandtimes(FlemingampCalabrese2017)ThereisbroadagreementintheliteraturethatHRestimatesbasedondifferentsamplingstrategiesaredifficult tocompareastheymaybeexposedtodifferentdegreesofbias(FiebergampBoumlrger2012Flemingetal2018Frairetal2010)Moresubtlyeveniden-tical sampling strategies can still produce differentially biasedHRestimates if theunderlyingparametersofmovementdifferamongindividualsinthecomparison(FlemingampCalabrese2017)Asover-lapiscalculatedconditionallyonapairofHRestimatesbiasesintheHRestimateswillpropagateintobiasesinoverlapestimates(FiebergampKochanny2005)ItfollowsthenthatdifferentialbiasesinHRes-timates amongdifferent groupsof interestwill tend to propagateintodifferentialbiasesinoverlapestimatesrenderingcomparisonsdifficulttointerpretandpotentiallyunreliable

Additionallynoneoftheoverlapmetricsofwhichweareawarecome equipped with confidence intervals to quantify the uncer-taintyintheestimatesThismeansthatitiscurrentlynotpossibletodetermineifasetofoverlapvaluesisstatisticallydifferentfromone

anotherorfromareferencevalueofinterestToseethisconsideracasewhereonewishestocomparetwooverlapestimatesfromtwopairsof individuals035and055 Ifthe95confidenceintervalsforeachestimatearedisjointthenwemayconcludethatthetwopairs have significantly different measures of overlap If the 95confidenceintervalsarenotdisjointthenthepointestimatesmaynot be significantly different In otherwords without confidenceintervals one cannotproperly interpretdifferencesbetweenesti-mates(Pawitan2001)

Herewedevelopthefirst inferential frameworkforHRover-lap by building on previous work in quantifying overlap (FiebergampKochanny 2005) and by leveraging recent advances inHR es-timation (FlemingampCalabrese2017FlemingFaganetal2015Flemingetal2018)WebaseourapproachontheBhattacharyyacoefficient(BCBhattacharyya1943alsocalledtheBhattacharyyaaffinity) which has a formal basis as a measure of similarity be-tween twoprobabilitydistributionsand is straightforward tocal-culate and interpret (Fieberg amp Kochanny 2005)We couple theBC with autocorrelated-Kernel density estimation (AKDE) as ageneralHRestimator(FlemingampCalabrese2017)BasingoverlapestimationonAKDEhastwoprimaryadvantagesFirstAKDEcor-rectsforbiasduetoautocorrelation(FlemingFaganetal2015)ordinary small-sample-size bias (FlemingampCalabrese 2017) andtemporalsamplingbias(Flemingetal2018)ThenetresultisthatAKDEHRestimatescanvalidlybecomparedacrossstudiessitesspeciesandtimesevenwhensamplingstrategiesandunderlyingmovementparametersdiffer(FlemingampCalabrese2017Flemingetal2018Noonanetal in review)Second theerrorpropaga-tiontechniquesusedtodevelopconfidenceintervalsonAKDEareaestimates(FlemingampCalabrese2017)canbeextendedtooverlapestimationallowingustodevelopconfidenceintervalsforoverlapestimates Inadditionoverlapestimatescanexhibitnegativebias(FiebergampKochanny2005)wherepartofthisproblemistheresultofsmall-sample-sizebiasintheBC(DjouadiampSnorrason1990)Asasolutionwederiveanapproximatefirst-orderbiascorrectiontotheBC

Weuseacombinationofsimulatedandempiricaldatatodemon-stratethepowerofourinferentialframeworkFirstbasedonsim-ulations we study the bias in BC estimates as a function of theamountofautocorrelation in thedataandof theeffectivesamplesizebothincaseswheretheunderlyingHRestimatorsaccountforthese biases (AKDE) and where they do not (conventional KDEWorton1989)WeuseasimilarapproachtoquantifytherealizedcoverageofourconfidenceintervalsWethenshowhowourframe-workcanbeusedtoaccuratelyestimateoverlapevenwhenindivid-ualsexhibiteddifferentmovementstrategiesandorweresubjecttocompletelydifferentsamplingdesignswhereasconventionalmeth-odsfailFinallyweshowhowourapproachcanbeusedinldquodown-streamrdquoapplicationsthatdependonoverlapSpecificallywebuildaninteractionnetwork(WeyBlumsteinShenampJordaacuten2008)forMongoliangazellesProcapra gutturosawhereedgesareestablishedonlybetweenindividualswhoseoverlapestimatesreceivedstatisti-calsupport

emspensp emsp | emsp1681Methods in Ecology and Evolu13onWINNER Et al

2emsp |emspMATERIAL S AND METHODS

Ourinferentialframeworkconsistsofbias-correctedHRestimatesabias-correctedBCestimatorandconfidence intervalsontheBCpointestimateWedescribeeachoftheseelementsinturnWethendescribehowourframeworkcanbeusedinpracticeviathectmm R package by extending the workflow for HR analysis described inCalabrese Fleming andGurarie (2016)or through theweb-basedgraphical user interface at ctmmshinyappsioctmmweb (DongFlemingampCalabrese2017)

21emsp|emspHome range estimation

Ataminimum calculatingoverlap requiresapairofHRestimates(FiebergampKochanny2005Millspaughetal2004)MoregenerallycomparisonsofoverlapamongdifferentgroupsspeciesplacesortimesmayalsobeofinterestNonethelessasoverlapestimatesareconditionalonestimatedHRsthoseunderlyingHRestimatesmustbedirectly comparableacross thedifferentgroups the researcherwishes to evaluate Unfortunately HR estimates are subject to anumberofbiasesanddifferencesineithersamplingscheduleunder-lyingmovementparametersorbothcanexposedifferentdatasets to different degrees of bias (Fieberg amp Boumlrger 2012 Fleming ampCalabrese2017)Datasetscharacterizedbyoneofmoreof theseformsofbiaswhicharethenorminpracticecanthusrendercom-parisonofHRestimatesacrossgroupsofinteresthighlymisleadingThepropagationofdifferentiallybiasedHRestimatesintodifferen-tiallybiasedoverlapestimateshasbeenakeyimpedimenttothede-velopmentofareliableinferentialframeworkforHRoverlap

Indecreasingorderofimportancethethreemainsourcesofbiasin HR estimation are unmodeled autocorrelation (Fleming Faganetal 2015) small effective sample sizes (Fleming amp Calabrese2017) and temporally biased sampling (Fleming etal 2018) ThemagnitudeofthenegativebiasinHRestimatesthatresultsfromas-sumingthedataisindependentandidenticallydistributed(IID)whenin fact theyareautocorrelatedcanbearbitrarily large (FlemingampCalabrese2017)AllelsebeingequalthebiaswillincreasewiththestrengthofautocorrelationinthedataIncontrastsmallsamplesizebiaswillbeestimator-specificandwilltendtobeofsmallermagni-tudethanautocorrelation-relatedbiasformodernGPSdataForex-ampleKDEsbasedontheconventionalGaussianreferencefunction(GRF)approximationtendtooverestimateHRareasatsmallsamplesize(FlemingampCalabrese2017)Temporallybiasedsamplingoccurswhensometimesareoversampledwhileothersareundersampled(Frairetal2010)whichcanproducedatathatarenotrepresenta-tiveoftheindividualsspaceuse(Flemingetal2018)Biasduetononrepresentative sampling in timewill tend to increasewith thedegreeofunevennessinthesamplingschedule

ThesethreesourcesofbiasmustbemitigatedtovalidlycompareHRestimatesandbyextensiontovalidlycompareoverlapestimatesWe now describeHR estimationmethods thatwhen used in com-bination largelycorrects thesebiasesAutocorrelated-KDE isagen-eralization of the GRF-KDE (Fleming Fagan etal 2015) The core

advanceinAKDEisthattheoptimizationofthesmoothingbandwidthσBexplicitlyaccountsforautocorrelationinthedataSpecificallyanautocorrelated movement model is used to represent the autocor-relationstructureofthedatainthebandwidthoptimization(Flemingetal 2014c Fleming Subaşi amp Calabrese 2015) Model selection(detailedbelow)canbeusedtoarriveatanappropriatemodelforthedatarsquos autocorrelation structure (Calabrese etal 2016) When thedataexhibitnoautocorrelationtheIIDmodelwouldbeselectedandAKDEconditionalontheIIDmodelisexactlyequivalenttothewell-knownGRF-KDERecently FlemingandCalabrese (2017)derivedasmall-sample-sizearea-basedcorrectionthatmitigatesthetendencyofKDEsbasedontheGRFapproximation includingAKDE toover-smooththedataFinally (Flemingetal2018)developedanoptimalweightingscheme termedldquowAKDErdquo that leverages theautocorrela-tion structure of the data to appropriately upweight undersampledtimes and downweight oversampled times When used in concertthese innovations result inmore accurateHRestimates that are di-rectlycomparableacrossgroupsofinterestAtechnicalintroductiontotheseestimatorsisprovidedinSupportingInformationAppendixA1

22emsp|emspThe Bhattacharyya coefficient

Therearemanydifferentmeasureswhichquantifytherelativesimi-larity (overlap) or dissimilarity (distance) of two probability distri-butionsWhileboth typesofmetrics canbeused todescribe thedegreeofsharedspaceusebetweenindividualsmeasuresofover-lapareusedmorecommonly inbiologicalcontexts thanmeasuresofdistance(butseeKranstauberSmollaampSafi2016)Intheircom-parativeanalysisofoverlapmetricsFiebergandKochanny (2005)concludedthattheBCandVolumeofIntersectionstatistic(VIalsoknown as the overlap coefficient Inman amp Bradley 1989) werethemostrobustoverlapestimatorsWhilethesetwovalidchoicesexistwesuggestthatforinferentialpurposesanoverlapestimatorshouldsatisfythefollowingcriteria

(i) Statistical validity An appropriate overlap estimator shouldbebasedonanestablishedmeasureofstatisticaldistanceordivergencethatsatisfiesrelatedmathematicalproperties

(ii) Geometric interpretabilityForuniformdistributionsoverlapshouldbeproportionaltotheareaofintersection

(iii) ObjectivityOverlapshouldnotdependonad hocparameterssuch as particular isopleths (eg 95or50)or discretizeddistributions

(iv) Computational efficiencyComputingtheoverlapoftwodis-tributionsshouldscaleefficientlywiththesamplesizeandex-tentofbothdistributions

(v) Asymptotic consistency An overlap estimator should con-vergetothetrueoverlapinthelargesamplesizelimit

(vi) Minimal bias An overlap estimator should have good small-sample-sizebehaviour

(vii) Quantifiable uncertaintyOverlapisanestimatederivedfromdata and shouldbeaccompaniedby ameasureof the confi-denceinthatestimate(Pawitan2001)

emspensp emsp | emsp1681Methods in Ecology and Evolu13onWINNER Et al

2emsp |emspMATERIAL S AND METHODS

Ourinferentialframeworkconsistsofbias-correctedHRestimatesabias-correctedBCestimatorandconfidence intervalsontheBCpointestimateWedescribeeachoftheseelementsinturnWethendescribehowourframeworkcanbeusedinpracticeviathectmm R package by extending the workflow for HR analysis described inCalabrese Fleming andGurarie (2016)or through theweb-basedgraphical user interface at ctmmshinyappsioctmmweb (DongFlemingampCalabrese2017)

21emsp|emspHome range estimation

Ataminimum calculatingoverlap requiresapairofHRestimates(FiebergampKochanny2005Millspaughetal2004)MoregenerallycomparisonsofoverlapamongdifferentgroupsspeciesplacesortimesmayalsobeofinterestNonethelessasoverlapestimatesareconditionalonestimatedHRsthoseunderlyingHRestimatesmustbedirectly comparableacross thedifferentgroups the researcherwishes to evaluate Unfortunately HR estimates are subject to anumberofbiasesanddifferencesineithersamplingscheduleunder-lyingmovementparametersorbothcanexposedifferentdatasets to different degrees of bias (Fieberg amp Boumlrger 2012 Fleming ampCalabrese2017)Datasetscharacterizedbyoneofmoreof theseformsofbiaswhicharethenorminpracticecanthusrendercom-parisonofHRestimatesacrossgroupsofinteresthighlymisleadingThepropagationofdifferentiallybiasedHRestimatesintodifferen-tiallybiasedoverlapestimateshasbeenakeyimpedimenttothede-velopmentofareliableinferentialframeworkforHRoverlap

Indecreasingorderofimportancethethreemainsourcesofbiasin HR estimation are unmodeled autocorrelation (Fleming Faganetal 2015) small effective sample sizes (Fleming amp Calabrese2017) and temporally biased sampling (Fleming etal 2018) ThemagnitudeofthenegativebiasinHRestimatesthatresultsfromas-sumingthedataisindependentandidenticallydistributed(IID)whenin fact theyareautocorrelatedcanbearbitrarily large (FlemingampCalabrese2017)AllelsebeingequalthebiaswillincreasewiththestrengthofautocorrelationinthedataIncontrastsmallsamplesizebiaswillbeestimator-specificandwilltendtobeofsmallermagni-tudethanautocorrelation-relatedbiasformodernGPSdataForex-ampleKDEsbasedontheconventionalGaussianreferencefunction(GRF)approximationtendtooverestimateHRareasatsmallsamplesize(FlemingampCalabrese2017)Temporallybiasedsamplingoccurswhensometimesareoversampledwhileothersareundersampled(Frairetal2010)whichcanproducedatathatarenotrepresenta-tiveoftheindividualsspaceuse(Flemingetal2018)Biasduetononrepresentative sampling in timewill tend to increasewith thedegreeofunevennessinthesamplingschedule

ThesethreesourcesofbiasmustbemitigatedtovalidlycompareHRestimatesandbyextensiontovalidlycompareoverlapestimatesWe now describeHR estimationmethods thatwhen used in com-bination largelycorrects thesebiasesAutocorrelated-KDE isagen-eralization of the GRF-KDE (Fleming Fagan etal 2015) The core

advanceinAKDEisthattheoptimizationofthesmoothingbandwidthσBexplicitlyaccountsforautocorrelationinthedataSpecificallyanautocorrelated movement model is used to represent the autocor-relationstructureofthedatainthebandwidthoptimization(Flemingetal 2014c Fleming Subaşi amp Calabrese 2015) Model selection(detailedbelow)canbeusedtoarriveatanappropriatemodelforthedatarsquos autocorrelation structure (Calabrese etal 2016) When thedataexhibitnoautocorrelationtheIIDmodelwouldbeselectedandAKDEconditionalontheIIDmodelisexactlyequivalenttothewell-knownGRF-KDERecently FlemingandCalabrese (2017)derivedasmall-sample-sizearea-basedcorrectionthatmitigatesthetendencyofKDEsbasedontheGRFapproximation includingAKDE toover-smooththedataFinally (Flemingetal2018)developedanoptimalweightingscheme termedldquowAKDErdquo that leverages theautocorrela-tion structure of the data to appropriately upweight undersampledtimes and downweight oversampled times When used in concertthese innovations result inmore accurateHRestimates that are di-rectlycomparableacrossgroupsofinterestAtechnicalintroductiontotheseestimatorsisprovidedinSupportingInformationAppendixA1

22emsp|emspThe Bhattacharyya coefficient

Therearemanydifferentmeasureswhichquantifytherelativesimi-larity (overlap) or dissimilarity (distance) of two probability distri-butionsWhileboth typesofmetrics canbeused todescribe thedegreeofsharedspaceusebetweenindividualsmeasuresofover-lapareusedmorecommonly inbiologicalcontexts thanmeasuresofdistance(butseeKranstauberSmollaampSafi2016)Intheircom-parativeanalysisofoverlapmetricsFiebergandKochanny (2005)concludedthattheBCandVolumeofIntersectionstatistic(VIalsoknown as the overlap coefficient Inman amp Bradley 1989) werethemostrobustoverlapestimatorsWhilethesetwovalidchoicesexistwesuggestthatforinferentialpurposesanoverlapestimatorshouldsatisfythefollowingcriteria

(i) Statistical validity An appropriate overlap estimator shouldbebasedonanestablishedmeasureofstatisticaldistanceordivergencethatsatisfiesrelatedmathematicalproperties

(ii) Geometric interpretabilityForuniformdistributionsoverlapshouldbeproportionaltotheareaofintersection

(iii) ObjectivityOverlapshouldnotdependonad hocparameterssuch as particular isopleths (eg 95or50)or discretizeddistributions

(iv) Computational efficiencyComputingtheoverlapoftwodis-tributionsshouldscaleefficientlywiththesamplesizeandex-tentofbothdistributions

(v) Asymptotic consistency An overlap estimator should con-vergetothetrueoverlapinthelargesamplesizelimit

(vi) Minimal bias An overlap estimator should have good small-sample-sizebehaviour

(vii) Quantifiable uncertaintyOverlapisanestimatederivedfromdata and shouldbeaccompaniedby ameasureof the confi-denceinthatestimate(Pawitan2001)

1682emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

TheBC(Bhattacharyya1943)isasolidbasisforinferenceonHRoverlapbecauseitsatisfiescriteria(i)ndash(v)andhastheadditionalbenefitofbeingwellknowntotheecologicalcommunity(FiebergampKochanny2005)AlthoughtheVIalsomeetsthesecriteria(FiebergampKochanny2005) approximating confidence intervals on theVI for the caseofunequalvariancespresentsseveredifficulties(ReiserampFaraggi1999)ConsequentlywebaseourapproachontheBCTheBCbetweentwocontinuousdistributionsp1 and p2isgivenby

TheBC is thusa functionof theproductof thetwodistributionsranging from 0leBCle1 with BC=0 only when p1 and p2 have nosharedsupportandBC=1onlywhenp1=p2Wenowturnourattentiontocriteria (vi)and(vii)andderiveaconfidence intervalapproximationandbiascorrectionthatallowtheBCtosatisfytheseadditionalcriteria

221emsp|emspConfidence intervals for the BC

WhenmeasuringtheoverlapoftwoHRstheBCasgivenaboveisapointestimateoftheoverlapbetweenthetwodistributionsbutdoesnot captureanyofouruncertainty in theHRestimationprocedureToaddressthislimitationwederiveconfidenceintervalsfortheBCintheGaussianreferencefunction(GRF)approximationAKDEsfirststep involves fitting stochastic movement models (Fleming Faganetal2015)toestimatethemeanandcovarianceparameters

where r(t)=(x(t)y(t)) denotes the individuals location In theGRFapproximationtheindividualspatialdensityestimatesaregivenby

andsotheBCbetweenGaussiandensityestimatesresolvesto

in termsof thearithmeticandgeometricmeansof thecovariancematrices

andtheMahalanobisdistance(Mahalanobis1936)betweenthetwodistributions

The closely related Bhattacharyya distance BD=minuslogBCBhattacharyya(1946)isdefined

whichhereresolvesto

Term-by-termallcomponentsoftheBDarenonnegativewiththefirstsetoftermsinvolvingtheMahalanobisdistancebeingzeroonlyfor identical mean locations and the second set of terms invok-ingtheAM-GMinequalitybeingzeroonlyfor identicalcovariancematrices

Firstwepropagateuncertaintyinthemeanandcovariancepa-rameters intouncertainty inBD via thedeltamethod (Cox 2005)toobtainVAR[BD]SecondasanimprovementoverasymptoticallynormalCIsandastheBDroughlytakestheformofasquaredis-tancewe approximate theBD statistic as being chi-squaredwithdegreesoffreedomequalto

inaccordancewiththechi-squarevarianceformulaWethentrans-formtheBDCIsbackintoBCCIsviaBC=exp(minusBD)FinallyforthekerneldensityBCCIsweapplythesameχ2approximation(9)butwiththeAKDEpointestimatefortheBDandtheGRFestimateforVAR[BD]

222emsp|emspBias correction for the BC

As noted by Fieberg andKochanny (2005) overlap is likely to benegativelybiasedatsmallsamplesizesInadditiontonegativebiasesinHRestimationdrivenbyunmodeledautocorrelationpartofthisproblem is the result of small-sample-size bias in theBC (DjouadiampSnorrason1990)whichisacommonpropertyofasymptoticallyconsistentestimators(Basu1956)AsasolutionherewederiveanapproximatebiascorrectionfortheBD

whichwewillalsoapplytotheAKDEBDpointestimateEvenifthe two distributions are Gaussian the BD plug-in estimatormdashwhichcalculatestheBDdirectlybyassumingthatthedensityes-timates are truemdashis severely biased This bias correctionwill beexact in the case of IID processes of equal variance which isknown tobe solvable (DjouadiampSnorrason1990) but approxi-matelygeneralizedforthemovementprocessesweconsiderandverifiedwith simulation (Supporting Information Appendix A2)Mostofthebiasisduetothefactthatuncertaintyinthecentroidstranslates strictly intopositiveBDeven if the twodistributionsare identicalFirstweaddressthis largestsourceofbiasbyde-composingthemeanestimates intotheirexpectationvaluesand(mean-zero)error

whereuponwecanexpressthefirstexpectedBDterm

(1)BC(p1p2)=

+infin

intminusinfin

+infin

intminusinfin

radic

p1(xy)p2(xy) dx dy

(2)= ⟨r(t)⟩ = COV[r(t)r(t)]

(3)p(r)=eminus

1

2(rminus)T

minus1(rminus)

radic

det (2)

(4)BC=

radic

det

(

GM

AM

)

eminus

1

4MD

2

(5)AM=1+2

2 GM=

radic

12

(6)MD=

radic

(1 minus 2)T AM

minus1(1 minus 2)

(7)BD=minus log BC 0le BD ltinfin

(8)BD=

1

8MD

2+1

2tr log

(

AM

GM

)

(9)DOF=2 BD

2

VAR[BD]

(10)

BD=1

8

(

1 minus 2

)Tminus1

(

1 minus 2

)

+1

2log det minus

1

4log det 1minus

1

4log det 2

(11)equiv 1

2

(

1+ 2

)

(12)=+ ⟨⟩=0 COV[]= COV[]

(13)

1 minus 2

Tminus1

1 minus 2

= tr

1 minus 2

1 minus 2T

minus1

+

1 minus 2

T⟨minus1

⟩

1 minus 2

+⋯

emspensp emsp | emsp1683Methods in Ecology and Evolu13onWINNER Et al

plustermslikeminus1thatweignorebecauseξismeanzeroandas-ymptoticallyuncorrelatedwithNextwenotetheapproximation

which is exact formany stationary processes (eg Fleming etal2014c)withaproportionalityconstantequaltotheeffectivesamplesizeofthemeanThereforewehave

whenthetwocovariancesaresimilarallowingustohereignorethebiasesinminus1 Wenotethat ingeneralthistermrelatedtohome-rangecentroiduncertaintyisbyfarthelargestsourceofbiasinBDestimation Furthermore if the two movement process are inde-pendentofeachotherthenwehave

For the remaining terms of the plug-in BD estimator we requiresomedistributionalassumptionsonthecovarianceestimates12 and Wetake1 and 2tobeWishartdistributed(Wishart1928)whereeffectivesamplesizesN1 and N2areestimatedwiththepa-rameters (FlemingampCalabrese2017)For theaveragecovariance we construct aWelchndashSatterthwaite (Satterthwaite 1946) likeapproximationthatisexactforequalcovariancesIf were χ2dis-tributedtheordinaryWelchndashSatterthwaiteapproximationwouldfixitsdegreesoffreedomviatherelationshipbetweenitsvarianceandthatof itsconstituentsHowever ismatrixvaluedandhasmanyvariancesWechoosetoconservethetracevariancewhichisbothadditiveandrotationallyinvariant

Nexttheexpectedinverseestimatematrixresolvesto

and so we clamp our effective sample size estimates to Ngedim(σ)+2whichisthesmallestdiscretenumberofIIDlocationswithwhichonecanestimateproperlyBelowthisvaluetheestimateislikelynotapproximatelyWishartdistributedandNislikelynotwellestimatedSobyclampingNweeffectivelyclampourbiascorrec-tionNexttheexpectedlog-determinanttermsresolveto

intermsofthemultivariatedigammafunctionψdFinallyasBDge0wedebiastheplug-inestimatorbydividingby

alargenumberratherthanbysubtractingalargenumber

whichisthesametofirstorderThisservesasafirst-orderbiascor-rectiontoboththeBDandtheBC

23emsp|emspWorkflow

Theresultingcenterpieceofour inferential framework isabias-cor-rected BC estimate with confidence intervals that is comparableacrossstudiesTogettothatpointtheusermustfirstproceedthroughaworkflowdesignedtoproducethebestpossibleestimatesfromtheirdatabutwarnwhensuchananalysisisinappropriateThisworkflowbuildsonthatdescribedinCalabreseetal(2016)forHRanalysis

ThefirststepisensuringthatthedataathandareappropriateforHRanalysiswhichmeansthattheremustbeclearevidenceofrange-residencyDatafromnon-range-resident individualsorfromrange-residentintervalsthatwereonlybrieflytrackedmaynotsat-isfythiscriterionWhenthedatadonotshowevidenceofrange-res-idency HR estimation is not appropriate (Calabrese etal 2016FlemingampCalabrese2017)whichimpliesthatHRoverlapanalysisis also not appropriateWe therefore strongly recommend start-ingwithvisualverificationofrange-residencyviavariogramanalysis(Fleming etal 2014b) Specifically the variogramof a range-resi-dentindividualshouldshowaclearasymptote

Oncerange-residencyhasbeenverified thenextstep is to fita series of range-residentmovementmodels to the data such asthe IID Ornstein-Uhlenbeck (OU Uhlenbeck amp Ornstein 1930)andOU-Foraging (OUF Fleming etal 2014bc) processesModelselection should thenbeemployed to identify thebestmodel forthe data (Fleming etal 2014c Fleming Subaşi etal 2015) Theselectedmodelshouldthenbevisuallycomparedtothevariogramtoensurethatthemodel iscapturingthekeyfeatures inthedataModelsthatfailtoconvergeorthatdonotprovideareasonablefittothedataareanotherindicationthatHRanalysismaybeinappro-priate(Calabreseetal2016)

With a fitted selected movement model in hand AKDE HRestimatescanthenbecalculatedandthesecanbeusedtoobtainBC estimates andCIs These overlap estimatesmay either be thefinal product of the analysis or be used in subsequent analysesImportantlytheconfidenceintervalsattachedtoeachBCestimatecanbestraightforwardlypropagatedintoderivedquantitiessuchasthemeanoverlapwithinagroupwhichcanfacilitatetestinghypoth-eses on similarity or differences among groups of interestWhiletheworkflowwe describe involves several steps thectmm pack-ageandgraphicaluserinterface(Dongetal2017)streamlinethisprocedureAfullexampleoftheworkflowisshowninSupportingInformationAppendixB

24emsp|emspSimulation study

ToexaminethestatisticalpropertiesoftheBCandthecoverageofourCIswesimulatedtrackingdatawithvariablesamplingdurations

(14)COV[]prop

(15)tr⟨

(

1 minus 2) (

1 minus 2)T

minus1⟩

asymp tr[

COV[1 minus 2]minus1]

(16)COV[1minus2]=COV[1]+COV[2]

(17)tr VAR[]=1

4tr VAR

[

1

]

+1

4tr VAR

[

2

]

(18)tr diag()2

N=tr diag(1)

2

4N1

+tr diag(2)

2

4N2

(19)N=

4 tr diag()2

tr diag(1)2

N1

+tr diag(2)2

N2

(20)⟨minus1⟩=

N

N minus dim() minus 1

(21)⟨log det ⟩= log det+120595dim()(N∕2)minus log (N∕2)dim()

(22)rarr

(

+ BIAS[]

)

= minusBIAS[]+(

Nminus2)

1684emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

and frequencies Datawere simulated based on pairs of both IIDprocessesandOUFprocesses(Flemingetal2014bc)parameter-izedsuchthatthetrueoverlapbetweenthesepairswasfixedat05SimulatingfromanOUFprocessgeneratesrelocationsthatfeatureautocorrelated positions and velocities aswell as restricted spaceuseandarerepresentativeofmodernGPStrackingdatacommonlyusedinHRanalyses(FlemingampCalabrese2017)

Importantlythetimescaleoverwhichautocorrelationinpositiondecaysτp(alsotermedtheHRcrossingtimeCalabreseetal2016)is a key parameter for HR estimation (Noonan etal in review)Formallyτpcanbequantifiedfromthedataasthetimescaleoverwhichanindividualspositionalautocorrelationdecaysbyafactorof1

eanditsmovementprocessrevertstothemeanlocation(FlemingampCalabrese2017FlemingFaganetal2015)Thedurationoftheobservationperiod(T)inrelationtoτpwillthusdictatetheeffectivesamplesize(ne)ofadatasetvia

which may be interpreted as the approximate number of rangecrossingsthatoccurredduringthesamplingperiodWetailoredoursimulationsaccordingtotheirrelativeeffectsonneThesewere

(i) Sampling duration Observations were recorded eight timesday andwemanipulated sampling duration (ranging from1 to4096days in a doubling series) For OUF simulations the HRcrossingtimewassetto1dayandthevelocityautocorrelationtimescaleto15ofadayNotablythisparameterizationwassuchthatinthesesimulationsthesamplingdurationindaysexhibiteda11relationshipwithne

(ii)Sampling frequency Here the sampling durationwas fixed at32days and we manipulated the sampling frequency (rangingfrom1to1024fixesdayinadoublingseries)AgainfortheOUFprocessHRcrossingtimewassetto1dayandthevelocityauto-correlationtimescaleto15ofadayThefixedsamplingdurationinthesesimulationsresultedinnebeingfixedat32irrespectiveofvariationinthesamplingfrequency

We then compared the accuracy of the underlying HR es-timates the accuracyof the estimatedoverlap and the realizedcoverageoftheconfidenceintervalsResultswereaveragedover1000simulationspermanipulationThecomputationswerecon-ductedontheSmithsonian InstitutionHighPerformanceCluster(SIHPC)

25emsp|emspEmpirical study

We demonstrate the functionality of this method using GPSdata from Mongolian gazelles Mongolian gazelles are medium-sized herbivores that cross their ranges on seasonal timescales(Flemingetal2014bc)Positionaldatafor36Mongoliangazellewere collected inMongolias EasternSteppebetween2007and2011 (Fleming etal 2014a) Both variogram analysis (Fleming

etal 2014c) and model selection (Calabrese etal 2016) wereused to confirm that there was evidence of range-residency inthedataFromthesediagnosticchecks13individualsshowednosigns of range-resident behaviour and we restricted our analy-sestothe23range-residentindividualsHRestimationwasthencarried out using KDE and AKDE as described aboveWe thencomputedallpairwiseBCsplusmn95CIson theKDEandAKDEes-timatesNotablythelongHRcrossingtimescales(x = 1115daysrange=80ndash4432) and comparatively short tracking durations(x = 3810daysrange=672ndash7550)hereproducedameanne of 61(range=07ndash246)Thisisaregimewherethenegativebiasofconventional KDE is known to have serious implications forHRestimatesonautocorrelateddata(FlemingampCalabrese2017)

251emsp|emspDownstream analyses

To further highlight the utility of these confidence intervalswe used the estimated overlap to quantify the edges of a spa-tial interaction network (Wey etal 2008) As point estimateswereaccompaniedbyCIswewereabletosubsetedgesintotwocategories

(i) SupportedWell-supportededgeswereidentifiedascaseswheretwoindividualsexhibitedoverlappingspaceusewithaminimumCIthatwasgreaterthan001mdashthatistherewasa95certaintythattheoverlapwasge001

(ii)Unsupported Unsupported edges were identified as caseswherethepointestimatesuggestedoverlappingspaceusebutwith aminimumCI thatwas less than 001mdashthat is therewasinsufficientevidencetobecertainthattheoverlapdifferedsig-nificantlyfrom0

Wethenquantifiedanumberofcommonlyuseddiagnostics(ienetwork density mean path length and closeness centrality Weyetal2008)toinvestigatehowthesemightdifferwhenthenetworkwasbasedonlyonstatisticallysupportededgesvstheinclusionofun-supportededges

All analyses were conducted in the R environment (R CoreTeam2016)usingthemethodsimplementedinthepackagectmm (Calabreseetal2016)

3emsp |emspRESULTS

31emsp|emspSimulation results

311emsp|emspAsymptotic properties of the BCss

SimulationsrevealedthatforIIDdatabothAKDEandKDEHResti-matesprovidedidenticalresultsandwererelativelyunbiasedexceptatverysmallsamplesizes(Figure1a)Theresultingoverlapwasalsoidentical between estimators and increasing the number of fixesbyeitherincreasingthesamplingduration(Figure1b)orfrequency(Figure1e)hadtheexpectedeffectofincreasingtheaccuracyofthe

(23)neasympT

τp

emspensp emsp | emsp1685Methods in Ecology and Evolu13onWINNER Et al

overlapestimateanddecreasingtheuncertaintyNotablytheCIsontheBCoffered reasonable coverageof the trueoverlapacross allsamplingregimesalbeitwithsomepersistentnegativebiasatlargesamplesizes(Figure1cf)ThiswastheresultofbiasintheBCdecay-ingtooslowlyrelativetothevariance(seeSupportingInformationAppendixA3)

For autocorrelated data in contrast AKDE 95HR estimateswere generally accurate across the range of sample durations(Figure2a)andfrequencies(Figure2d)wesimulatedwhereasKDEHRestimateswereseverelybiasedforallbutthelargestdatasetsAsaresultwhiletheestimatedoverlapbetweenAKDEandKDEes-timatesbothconvergedtothetruthassamplingdurationincreased(Figure2b)asymptoticconsistencyforKDEestimateswasseverelydelayedFurthermoreincreasingthesamplingfrequencyincreasedthenegativebias inoverlapestimatesderived fromKDEbut ap-propriately did not influence overlap estimates based on AKDE(Figure2e)

The coverage of 95 CIs for the KDE-derived overlap esti-mateswas severely biased under all of the scenarioswe tested(Figure2cf) In contrast the coverageofCIson theAKDEesti-matesconsistentlyprovidedclosetonominalcoverageofthetrueoverlap

312emsp|emspComparability of estimates

Ourbaselinesimulationstudycontrolledtheeffectofthemovementparametersbyassumingtheindividualsexhibitedidenticalmovementstrategies andwere sampled at the exact same timesUnder theseconditionstheimprovedaccuracyofAKDEHRsestimatesresultedinmoreaccurateoverlapestimateswith95CIsthatprovidedclosetonominalcoverage(Figure3a)Therearerealisticcomplicationstoourbasicsimulationstrategyhoweverincludingcaseswhereindividualsaresubjecttothesamesamplingdesignbutexhibitdifferentmove-mentstrategiesandcaseswherebothmovementstrategiesandsam-plingdesignsdifferImportantlywefoundthatAKDE-basedoverlapstillprovidedreasonablecoverageforbothofthesecases(Figure3ce)IncontrastbecauseofthedifferentialbiasinKDEHRestimatestheestimatedoverlapdifferedsubstantiallybetweeneachof thesesce-nariosandineverycasefailedtoprovidecoverageofthetruevalue(Figure3bdf)

32emsp|emspEmpirical case study

ConsistentwithoursimulatedfindingsofnegativebiasinKDEHRandBCestimatesatmidtolowneonautocorrelateddataempirical

F I G U R E 1 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedIIDdataaswellasthecoverageoftheCIs(panelscandf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIs

04

08

12

16

1 4 16 64 256 1024 4096

Sampling duration (days)

95

are

a es

timat

es

AKDEKDE

(a)

000

025

050

075

100

1 4 16 64 256 1024 4096

Sampling duration (days)

Bha

ttach

arry

a co

effi

cien

t

(b)

00

02

04

06

08

10

1 4 16 64 256 1024 4096

Sampling duration (days)

Cov

erag

e

(c)

04

08

12

16

1 4 16 64 256 1024

Sampling frequency (fixesday)

95

are

a es

timat

es

(d)

000

025

050

075

100

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Sampling frequency (fixesday)

Bha

ttach

arry

a co

effi

cien

t(e)

00

02

04

06

08

10

1 4 16 64 256 1024

Sampling frequency (fixesday)C

over

age

(f)

1686emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

AKDEHR estimates were larger than KDE estimates for all pairs(Figure4a)Medianpairwiseoverlapbetweenthe276pairsofindi-vidualswas066(95CI058ndash076)whentheoverlapwasestimatedfromAKDEHRestimatesbutfivefoldlowerwhenestimatedfromKDEestimates(median=01395CI006ndash022)

TheseverenegativebiasofKDE-derivedoverlapwaspersistentacrossall individualsThiscanbe illustrated ina specificexamplewhere theKDEHR estimates resulted in an estimated overlap of002(95CI001ndash003)whereastheAKDEHRsresultedinanover-lapof080(95CI022ndash099)Visualinspectionoftherangeesti-matesfortheseindividualsrevealedsubstantialnegativebiasintheKDEHRwhereastheAKDEHRwaslargerwithappropriatelywideCIsconsideringthesmallne of c4foreachHRestimate(Figure4bc)

321emsp|emspDownstream analyses

Astheseoverlapestimateswereaccompaniedbyconfidenceinter-vals theuncertaintycanbeused to informdownstreamanalysesFor instance a spatial network analysis based on the estimatedoverlaprevealed461edgesofvariablestrength(Figure5)Ofthese275werewell supportedwhereas186hadnostatistical supportWe found that basing the network off of all possible edges vsonlythoseedgeswithstatisticalsupport influenceditsproperties

andanypotentialbiologicalinferencesthatwouldbederivedfromit For instance network density was reduced from 086 to 063whentheanalysiswasrestrictedtoonlythewell-supportededgesFurthermore onlyutilizing statistically supportededges increasedthemeanpath length from113 to139 Interestinglydespitede-creasingdensityandincreasingthemeanpathlengthconstructingthenetworkbasedononlywell-supportededgesresultedinatwo-fold increase in theclosenesscentralitycompared to thenetworkconstructedwithbothsupportedandunsupportededges(045vs023respectively)

4emsp |emspDISCUSSION

Despite the routine nature of estimating overlapping space use(eg Berger amp Gese 2007 Dougherty etal 2018 Fregravere etal2010SanchezampHudgens2015)thereexistsnoformalinferentialframeworkforthisanalysisThis is largelyduetothe inherentdif-ficultiesassociatedwithHRestimation(FiebergampBoumlrger2012)andexacerbated by the historical lack of CIs on bothHR and overlapestimatesAsasolutionwehavedemonstratedhowAKDEHResti-mates(FlemingampCalabrese2017FlemingFaganetal2015)canserveasareliablefoundationonwhichtobasestatisticalinference

F I G U R E 2 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedautocorrelatedtrackingdataandthecoverageoftheCIs(candf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIsNotablyconvergencetothetruthwasmuchslowerforKDEandthecoverageofKDEsCIswasfarfromappropriateinallcases

1 4 16 64 256 1024 4096

00

05

10

15

1 4 16 64 256 1024 4096

neffective

Sampling duration (days)

95

are

a es

timat

es AKDEKDE

(a)

1 4 16 64 256 1024 4096

000

025

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075

100

1 4 16 64 256 1024 4096

neffective

Sampling duration (days)

Bha

ttach

arry

a co

effi

cien

t

(b)

1 4 16 64 256 1024 4096

00

02

04

06

08

10

1 4 16 64 256 1024 4096

neffective

Sampling duration (days)

Cov

erag

e

(c)

32 32 32 32 32

04

08

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16

1 4 16 64 256

neffective

Sampling frequency (fixesday)

95

are

a es

timat

es

(d)

32 32 32 32 32

000

025

050

075

100

1 4 16 64 256

neffective

Sampling frequency (fixesday)

Bha

ttach

arry

a co

effi

cien

t(e)

32 32 32 32 32

00

02

04

06

08

10

1 4 16 64 256

neffective

Sampling frequency (fixesday)

Cov

erag

e

(f)

emspensp emsp | emsp1687Methods in Ecology and Evolu13onWINNER Et al

F I G U R E 3 emsp HRandoverlapestimatesfortwosimulatedindividualswithatrueoverlapof050Inallpanelsthedashedcirclesdepictthetrue95areasthesolidblacklinestheestimated95areasandthegreylinesthe95CIsontheareaestimatesInthefirstrowrelocationsweresimulatedfromOUFmodelswithidenticalmovementparametersandsamplingtimesInthesecondrowsamplingwasheldconsistentbuttheindividualplottedinyellowhadaHRcrossingtimeof1weekvs1dayfortheindividualinredInthethirdrowmovementagaindifferedbetweenindividualsbutheretheindividualinyellowwassampledonceevery30minvsonceevery3hrsfortheindividualinredNotehowinallcasesAKDE-basedoverlapestimateswererelativelyconsistentandprovidedcoverageofthetrueoverlapwhereasKDE-basedoverlapestimatesvariedsubstantiallyandconsistentlyfailedtoprovidecoverageofthetruth

Overlap 048 95 CIs 032 ndash 066

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(a) AKDEOverlap 04 95 CIs 036 ndash 044

ndash1000

ndash500

0

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Y (

met

ers)

(b) KDE

Overlap 047 95 CIs 029 ndash 067

ndash1000

ndash500

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Y (

met

ers)

(c)

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(d)

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Overlap 035 95 CIs 031 ndash 039

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ndash500

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Y (

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(f)

F I G U R E 4 emsp (a)TherelationshipbetweenpairwiseestimatesoftheBCforMongoliangazellecomputedfromKDEandAKDEHRestimatesThedashed11linedepictsparitybetweentheseNotehowallcasesfallabovethislinehighlightinghowAKDE-derivedBCsuggestsmoreoverlapthanKDE-derivedBCAnexampleofthisdiscrepancyisdepictedin(b)withAKDEBCsuggestingextensiveoverlap080(022ndash099)whereasin(c)thenegativebiasinKDEpropagatestoproduceabiasedestimateoftheoverlap002(001ndash003)Cruciallywitheffectivesamplesizesofc4foreachHRestimatetheCIsapproximatedfromtheAKDEestimateswereappropriatelywidevsKDEsdeceivinglynarrowCIs

000

025

050

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100

000 025 050 075 100KDE overlap

AK

DE

ove

rlap

(a)

ndash3e+05

ndash2e+05

ndash1e+05

0e+00

1e+05

2e+05

ndash3e+05 ndash2e+05 ndash1e+05 0e+00 1e+05 2e+05

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Y (

met

ers)

(b)

ndash3e+05

ndash2e+05

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ndash3e+05 ndash2e+05 ndash1e+05 0e+00 1e+05 2e+05

X (meters)

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met

ers)

(c)

1688emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

Inadditionwehave implementedasmall-sample-sizebiascorrec-tionfortheBCandderivedwell-behavedapproximateCIsonthepoint estimateCollectively theseadvancespermit researchers toaccuratelyquantifyHRoverlapevenwhensamplingstrategiesandunderlyingmovementparametersdifferamonggroupsbeingcom-pared and testwhether anyobserveddifferencesare statisticallymeaningful

41emsp|emspHome range and overlap estimation an intrinsic relationship

Acrucialcomponentofanystatisticalinferenceishavingcompara-blemeasuresonwhichtobaseanalysesOverlapistypicallycondi-tionalonHRestimates(FiebergampKochanny2005Millspaughetal2004)whichare themselvesestimated fromanimal trackingdataAs overlap estimation relies on at least three separate estimates(twoHRestimatesandtheiroverlap)itfollowsthatthisanalysisisparticularlyvulnerabletoissuesofestimatorbiasAccurateHResti-mationisadeceptivelychallengingproblemhoweverasautocorre-lation(FlemingFaganetal2015)small-sample-sizebias(FlemingampCalabrese2017)andsamplingirregularities(Flemingetal2018Frairetal2010)willsignificantlyinfluenceanystatisticalanalysesappliedtoanimaltrackingdataMoresubtlyevenidenticalsamplingstrategiescanstillproducedifferentiallybiasedHRestimatesiftheunderlying parameters of movement differ markedly between in-dividuals (FlemingampCalabrese2017Noonanetal inreview)Asthesearenearlyubiquitousaspectsof animal trackingdata accu-rateoverlapestimationrequiresstatisticalmethodsthatcanhandlethesecomplicationswithoutintroducingartifactualdifferencesduepurelytoestimatorbias

Inthisrespectoursimulationstudyrevealedthatforautocor-related data KDE regularly underestimated HR sizes (Fleming amp

Calabrese2017Noonanetalinreview)andthisnegativebiaswasdirectlypropagated tooverlapestimatesForKDE theamountofdatarequiredtoachieveanaccuratemeasureofoverlapwasverylargeandmostempiricalcasesarelikelytounderestimatethetrueoverlap (FiebergampKochanny2005) In contrastAKDEHRswerelarger but significantly more accurate which translated to moreaccurateoverlapestimatesCruciallywhenwevariedthesamplingdesignandmovementstrategiesbetweenthe individualswewerecomparingAKDE-basedestimatesprovidedreliablecoverageofthetrueoverlapwhereasthiswasnotthecaseforKDEConsistentwiththe resultsofour simulation study empiricalAKDEHRs fromau-tocorrelatedMongoliangazelleGPSdatawerecatwiceaslargeasKDEestimatesThisresultedinthemedianpairwiseoverlapbeingfivefoldlargerwhenbasedonAKDEvsKDEHadananalysisbeenbased on the biasedKDE estimates onewould have erroneouslyconcludedthattherewaslittlespatialoverlapinthissystemwhereastheresultsbasedonAKDEsmorerigorousestimatesrevealedtheseindividualsactuallyexhibitedextensiveoverlapAlthoughtheseem-piricalestimatescouldnotbecomparedtoatruthasperoursimula-tionsthisfindingisalsoconsistentwitharecentanalysisbyNoonanetal (in review) Ina large-scale comparative studyencompassing369 individualsacross30species they found thatAKDE95HRestimates consistently included c 95 of holdout observationswhereasKDEestimatesincludedc92athighne(gt256)butonlyc75atlowneThismeansAKDEslargerestimatesareaccuratewhile those producedby conventionalKDEon the samedata areconsistentlyandoftengrosslytoosmallThenetresultisthatAKDEprovides a solid foundation for estimating overlap under realisticsampling regimes resulting inaccurateoverlapestimates that canvalidlybecomparedacrossstudies

AsdescribedaboveafundamentalcomponentofestimatingHRoverlapishavingcomparablemeasuresonwhichtobaseanalyses

F I G U R E 5 emsp Figuredepicting(a)theGPSlocationsfor23MongoliangazelletrackedinMongoliasEasternSteppe(b)anetworkdiagramwithedgeweightsbasedonoverlapvaluesand(c)anexamplecaseoftwoHRestimateswherethepointestimateoftheoverlapsuggestsaconnectionbuttheCIsontheestimatessuggestthatconnectionmightnotbestatisticallysignificantThedashedlinesin(b)depictpairswherethepointestimatesuggestsaconnectionbutwithCIsthatinclude001andthusmaynotbestatisticallysignificantThetransparencyofthelinesisproportionaltothepointestimateoftheBCTheconnectiondepictedinredontheright-handsideof(b)correspondstothepairin(c)

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

emspensp emsp | emsp1683Methods in Ecology and Evolu13onWINNER Et al

plustermslikeminus1thatweignorebecauseξismeanzeroandas-ymptoticallyuncorrelatedwithNextwenotetheapproximation

which is exact formany stationary processes (eg Fleming etal2014c)withaproportionalityconstantequaltotheeffectivesamplesizeofthemeanThereforewehave

whenthetwocovariancesaresimilarallowingustohereignorethebiasesinminus1 Wenotethat ingeneralthistermrelatedtohome-rangecentroiduncertaintyisbyfarthelargestsourceofbiasinBDestimation Furthermore if the two movement process are inde-pendentofeachotherthenwehave

For the remaining terms of the plug-in BD estimator we requiresomedistributionalassumptionsonthecovarianceestimates12 and Wetake1 and 2tobeWishartdistributed(Wishart1928)whereeffectivesamplesizesN1 and N2areestimatedwiththepa-rameters (FlemingampCalabrese2017)For theaveragecovariance we construct aWelchndashSatterthwaite (Satterthwaite 1946) likeapproximationthatisexactforequalcovariancesIf were χ2dis-tributedtheordinaryWelchndashSatterthwaiteapproximationwouldfixitsdegreesoffreedomviatherelationshipbetweenitsvarianceandthatof itsconstituentsHowever ismatrixvaluedandhasmanyvariancesWechoosetoconservethetracevariancewhichisbothadditiveandrotationallyinvariant

Nexttheexpectedinverseestimatematrixresolvesto

and so we clamp our effective sample size estimates to Ngedim(σ)+2whichisthesmallestdiscretenumberofIIDlocationswithwhichonecanestimateproperlyBelowthisvaluetheestimateislikelynotapproximatelyWishartdistributedandNislikelynotwellestimatedSobyclampingNweeffectivelyclampourbiascorrec-tionNexttheexpectedlog-determinanttermsresolveto

intermsofthemultivariatedigammafunctionψdFinallyasBDge0wedebiastheplug-inestimatorbydividingby

alargenumberratherthanbysubtractingalargenumber

whichisthesametofirstorderThisservesasafirst-orderbiascor-rectiontoboththeBDandtheBC

23emsp|emspWorkflow

Theresultingcenterpieceofour inferential framework isabias-cor-rected BC estimate with confidence intervals that is comparableacrossstudiesTogettothatpointtheusermustfirstproceedthroughaworkflowdesignedtoproducethebestpossibleestimatesfromtheirdatabutwarnwhensuchananalysisisinappropriateThisworkflowbuildsonthatdescribedinCalabreseetal(2016)forHRanalysis

ThefirststepisensuringthatthedataathandareappropriateforHRanalysiswhichmeansthattheremustbeclearevidenceofrange-residencyDatafromnon-range-resident individualsorfromrange-residentintervalsthatwereonlybrieflytrackedmaynotsat-isfythiscriterionWhenthedatadonotshowevidenceofrange-res-idency HR estimation is not appropriate (Calabrese etal 2016FlemingampCalabrese2017)whichimpliesthatHRoverlapanalysisis also not appropriateWe therefore strongly recommend start-ingwithvisualverificationofrange-residencyviavariogramanalysis(Fleming etal 2014b) Specifically the variogramof a range-resi-dentindividualshouldshowaclearasymptote

Oncerange-residencyhasbeenverified thenextstep is to fita series of range-residentmovementmodels to the data such asthe IID Ornstein-Uhlenbeck (OU Uhlenbeck amp Ornstein 1930)andOU-Foraging (OUF Fleming etal 2014bc) processesModelselection should thenbeemployed to identify thebestmodel forthe data (Fleming etal 2014c Fleming Subaşi etal 2015) Theselectedmodelshouldthenbevisuallycomparedtothevariogramtoensurethatthemodel iscapturingthekeyfeatures inthedataModelsthatfailtoconvergeorthatdonotprovideareasonablefittothedataareanotherindicationthatHRanalysismaybeinappro-priate(Calabreseetal2016)

With a fitted selected movement model in hand AKDE HRestimatescanthenbecalculatedandthesecanbeusedtoobtainBC estimates andCIs These overlap estimatesmay either be thefinal product of the analysis or be used in subsequent analysesImportantlytheconfidenceintervalsattachedtoeachBCestimatecanbestraightforwardlypropagatedintoderivedquantitiessuchasthemeanoverlapwithinagroupwhichcanfacilitatetestinghypoth-eses on similarity or differences among groups of interestWhiletheworkflowwe describe involves several steps thectmm pack-ageandgraphicaluserinterface(Dongetal2017)streamlinethisprocedureAfullexampleoftheworkflowisshowninSupportingInformationAppendixB

24emsp|emspSimulation study

ToexaminethestatisticalpropertiesoftheBCandthecoverageofourCIswesimulatedtrackingdatawithvariablesamplingdurations

(14)COV[]prop

(15)tr⟨

(

1 minus 2) (

1 minus 2)T

minus1⟩

asymp tr[

COV[1 minus 2]minus1]

(16)COV[1minus2]=COV[1]+COV[2]

(17)tr VAR[]=1

4tr VAR

[

1

]

+1

4tr VAR

[

2

]

(18)tr diag()2

N=tr diag(1)

2

4N1

+tr diag(2)

2

4N2

(19)N=

4 tr diag()2

tr diag(1)2

N1

+tr diag(2)2

N2

(20)⟨minus1⟩=

N

N minus dim() minus 1

(21)⟨log det ⟩= log det+120595dim()(N∕2)minus log (N∕2)dim()

(22)rarr

(

+ BIAS[]

)

= minusBIAS[]+(

Nminus2)

1684emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

and frequencies Datawere simulated based on pairs of both IIDprocessesandOUFprocesses(Flemingetal2014bc)parameter-izedsuchthatthetrueoverlapbetweenthesepairswasfixedat05SimulatingfromanOUFprocessgeneratesrelocationsthatfeatureautocorrelated positions and velocities aswell as restricted spaceuseandarerepresentativeofmodernGPStrackingdatacommonlyusedinHRanalyses(FlemingampCalabrese2017)

Importantlythetimescaleoverwhichautocorrelationinpositiondecaysτp(alsotermedtheHRcrossingtimeCalabreseetal2016)is a key parameter for HR estimation (Noonan etal in review)Formallyτpcanbequantifiedfromthedataasthetimescaleoverwhichanindividualspositionalautocorrelationdecaysbyafactorof1

eanditsmovementprocessrevertstothemeanlocation(FlemingampCalabrese2017FlemingFaganetal2015)Thedurationoftheobservationperiod(T)inrelationtoτpwillthusdictatetheeffectivesamplesize(ne)ofadatasetvia

which may be interpreted as the approximate number of rangecrossingsthatoccurredduringthesamplingperiodWetailoredoursimulationsaccordingtotheirrelativeeffectsonneThesewere

(i) Sampling duration Observations were recorded eight timesday andwemanipulated sampling duration (ranging from1 to4096days in a doubling series) For OUF simulations the HRcrossingtimewassetto1dayandthevelocityautocorrelationtimescaleto15ofadayNotablythisparameterizationwassuchthatinthesesimulationsthesamplingdurationindaysexhibiteda11relationshipwithne

(ii)Sampling frequency Here the sampling durationwas fixed at32days and we manipulated the sampling frequency (rangingfrom1to1024fixesdayinadoublingseries)AgainfortheOUFprocessHRcrossingtimewassetto1dayandthevelocityauto-correlationtimescaleto15ofadayThefixedsamplingdurationinthesesimulationsresultedinnebeingfixedat32irrespectiveofvariationinthesamplingfrequency

We then compared the accuracy of the underlying HR es-timates the accuracyof the estimatedoverlap and the realizedcoverageoftheconfidenceintervalsResultswereaveragedover1000simulationspermanipulationThecomputationswerecon-ductedontheSmithsonian InstitutionHighPerformanceCluster(SIHPC)

25emsp|emspEmpirical study

We demonstrate the functionality of this method using GPSdata from Mongolian gazelles Mongolian gazelles are medium-sized herbivores that cross their ranges on seasonal timescales(Flemingetal2014bc)Positionaldatafor36Mongoliangazellewere collected inMongolias EasternSteppebetween2007and2011 (Fleming etal 2014a) Both variogram analysis (Fleming

etal 2014c) and model selection (Calabrese etal 2016) wereused to confirm that there was evidence of range-residency inthedataFromthesediagnosticchecks13individualsshowednosigns of range-resident behaviour and we restricted our analy-sestothe23range-residentindividualsHRestimationwasthencarried out using KDE and AKDE as described aboveWe thencomputedallpairwiseBCsplusmn95CIson theKDEandAKDEes-timatesNotablythelongHRcrossingtimescales(x = 1115daysrange=80ndash4432) and comparatively short tracking durations(x = 3810daysrange=672ndash7550)hereproducedameanne of 61(range=07ndash246)Thisisaregimewherethenegativebiasofconventional KDE is known to have serious implications forHRestimatesonautocorrelateddata(FlemingampCalabrese2017)

251emsp|emspDownstream analyses

To further highlight the utility of these confidence intervalswe used the estimated overlap to quantify the edges of a spa-tial interaction network (Wey etal 2008) As point estimateswereaccompaniedbyCIswewereabletosubsetedgesintotwocategories

(i) SupportedWell-supportededgeswereidentifiedascaseswheretwoindividualsexhibitedoverlappingspaceusewithaminimumCIthatwasgreaterthan001mdashthatistherewasa95certaintythattheoverlapwasge001

(ii)Unsupported Unsupported edges were identified as caseswherethepointestimatesuggestedoverlappingspaceusebutwith aminimumCI thatwas less than 001mdashthat is therewasinsufficientevidencetobecertainthattheoverlapdifferedsig-nificantlyfrom0

Wethenquantifiedanumberofcommonlyuseddiagnostics(ienetwork density mean path length and closeness centrality Weyetal2008)toinvestigatehowthesemightdifferwhenthenetworkwasbasedonlyonstatisticallysupportededgesvstheinclusionofun-supportededges

All analyses were conducted in the R environment (R CoreTeam2016)usingthemethodsimplementedinthepackagectmm (Calabreseetal2016)

3emsp |emspRESULTS

31emsp|emspSimulation results

311emsp|emspAsymptotic properties of the BCss

SimulationsrevealedthatforIIDdatabothAKDEandKDEHResti-matesprovidedidenticalresultsandwererelativelyunbiasedexceptatverysmallsamplesizes(Figure1a)Theresultingoverlapwasalsoidentical between estimators and increasing the number of fixesbyeitherincreasingthesamplingduration(Figure1b)orfrequency(Figure1e)hadtheexpectedeffectofincreasingtheaccuracyofthe

(23)neasympT

τp

emspensp emsp | emsp1685Methods in Ecology and Evolu13onWINNER Et al

overlapestimateanddecreasingtheuncertaintyNotablytheCIsontheBCoffered reasonable coverageof the trueoverlapacross allsamplingregimesalbeitwithsomepersistentnegativebiasatlargesamplesizes(Figure1cf)ThiswastheresultofbiasintheBCdecay-ingtooslowlyrelativetothevariance(seeSupportingInformationAppendixA3)

For autocorrelated data in contrast AKDE 95HR estimateswere generally accurate across the range of sample durations(Figure2a)andfrequencies(Figure2d)wesimulatedwhereasKDEHRestimateswereseverelybiasedforallbutthelargestdatasetsAsaresultwhiletheestimatedoverlapbetweenAKDEandKDEes-timatesbothconvergedtothetruthassamplingdurationincreased(Figure2b)asymptoticconsistencyforKDEestimateswasseverelydelayedFurthermoreincreasingthesamplingfrequencyincreasedthenegativebias inoverlapestimatesderived fromKDEbut ap-propriately did not influence overlap estimates based on AKDE(Figure2e)

The coverage of 95 CIs for the KDE-derived overlap esti-mateswas severely biased under all of the scenarioswe tested(Figure2cf) In contrast the coverageofCIson theAKDEesti-matesconsistentlyprovidedclosetonominalcoverageofthetrueoverlap

312emsp|emspComparability of estimates

Ourbaselinesimulationstudycontrolledtheeffectofthemovementparametersbyassumingtheindividualsexhibitedidenticalmovementstrategies andwere sampled at the exact same timesUnder theseconditionstheimprovedaccuracyofAKDEHRsestimatesresultedinmoreaccurateoverlapestimateswith95CIsthatprovidedclosetonominalcoverage(Figure3a)Therearerealisticcomplicationstoourbasicsimulationstrategyhoweverincludingcaseswhereindividualsaresubjecttothesamesamplingdesignbutexhibitdifferentmove-mentstrategiesandcaseswherebothmovementstrategiesandsam-plingdesignsdifferImportantlywefoundthatAKDE-basedoverlapstillprovidedreasonablecoverageforbothofthesecases(Figure3ce)IncontrastbecauseofthedifferentialbiasinKDEHRestimatestheestimatedoverlapdifferedsubstantiallybetweeneachof thesesce-nariosandineverycasefailedtoprovidecoverageofthetruevalue(Figure3bdf)

32emsp|emspEmpirical case study

ConsistentwithoursimulatedfindingsofnegativebiasinKDEHRandBCestimatesatmidtolowneonautocorrelateddataempirical

F I G U R E 1 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedIIDdataaswellasthecoverageoftheCIs(panelscandf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIs

04

08

12

16

1 4 16 64 256 1024 4096

Sampling duration (days)

95

are

a es

timat

es

AKDEKDE

(a)

000

025

050

075

100

1 4 16 64 256 1024 4096

Sampling duration (days)

Bha

ttach

arry

a co

effi

cien

t

(b)

00

02

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06

08

10

1 4 16 64 256 1024 4096

Sampling duration (days)

Cov

erag

e

(c)

04

08

12

16

1 4 16 64 256 1024

Sampling frequency (fixesday)

95

are

a es

timat

es

(d)

000

025

050

075

100

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Sampling frequency (fixesday)

Bha

ttach

arry

a co

effi

cien

t(e)

00

02

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08

10

1 4 16 64 256 1024

Sampling frequency (fixesday)C

over

age

(f)

1686emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

AKDEHR estimates were larger than KDE estimates for all pairs(Figure4a)Medianpairwiseoverlapbetweenthe276pairsofindi-vidualswas066(95CI058ndash076)whentheoverlapwasestimatedfromAKDEHRestimatesbutfivefoldlowerwhenestimatedfromKDEestimates(median=01395CI006ndash022)

TheseverenegativebiasofKDE-derivedoverlapwaspersistentacrossall individualsThiscanbe illustrated ina specificexamplewhere theKDEHR estimates resulted in an estimated overlap of002(95CI001ndash003)whereastheAKDEHRsresultedinanover-lapof080(95CI022ndash099)Visualinspectionoftherangeesti-matesfortheseindividualsrevealedsubstantialnegativebiasintheKDEHRwhereastheAKDEHRwaslargerwithappropriatelywideCIsconsideringthesmallne of c4foreachHRestimate(Figure4bc)

321emsp|emspDownstream analyses

Astheseoverlapestimateswereaccompaniedbyconfidenceinter-vals theuncertaintycanbeused to informdownstreamanalysesFor instance a spatial network analysis based on the estimatedoverlaprevealed461edgesofvariablestrength(Figure5)Ofthese275werewell supportedwhereas186hadnostatistical supportWe found that basing the network off of all possible edges vsonlythoseedgeswithstatisticalsupport influenceditsproperties

andanypotentialbiologicalinferencesthatwouldbederivedfromit For instance network density was reduced from 086 to 063whentheanalysiswasrestrictedtoonlythewell-supportededgesFurthermore onlyutilizing statistically supportededges increasedthemeanpath length from113 to139 Interestinglydespitede-creasingdensityandincreasingthemeanpathlengthconstructingthenetworkbasedononlywell-supportededgesresultedinatwo-fold increase in theclosenesscentralitycompared to thenetworkconstructedwithbothsupportedandunsupportededges(045vs023respectively)

4emsp |emspDISCUSSION

Despite the routine nature of estimating overlapping space use(eg Berger amp Gese 2007 Dougherty etal 2018 Fregravere etal2010SanchezampHudgens2015)thereexistsnoformalinferentialframeworkforthisanalysisThis is largelyduetothe inherentdif-ficultiesassociatedwithHRestimation(FiebergampBoumlrger2012)andexacerbated by the historical lack of CIs on bothHR and overlapestimatesAsasolutionwehavedemonstratedhowAKDEHResti-mates(FlemingampCalabrese2017FlemingFaganetal2015)canserveasareliablefoundationonwhichtobasestatisticalinference

F I G U R E 2 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedautocorrelatedtrackingdataandthecoverageoftheCIs(candf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIsNotablyconvergencetothetruthwasmuchslowerforKDEandthecoverageofKDEsCIswasfarfromappropriateinallcases

1 4 16 64 256 1024 4096

00

05

10

15

1 4 16 64 256 1024 4096

neffective

Sampling duration (days)

95

are

a es

timat

es AKDEKDE

(a)

1 4 16 64 256 1024 4096

000

025

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1 4 16 64 256 1024 4096

neffective

Sampling duration (days)

Bha

ttach

arry

a co

effi

cien

t

(b)

1 4 16 64 256 1024 4096

00

02

04

06

08

10

1 4 16 64 256 1024 4096

neffective

Sampling duration (days)

Cov

erag

e

(c)

32 32 32 32 32

04

08

12

16

1 4 16 64 256

neffective

Sampling frequency (fixesday)

95

are

a es

timat

es

(d)

32 32 32 32 32

000

025

050

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100

1 4 16 64 256

neffective

Sampling frequency (fixesday)

Bha

ttach

arry

a co

effi

cien

t(e)

32 32 32 32 32

00

02

04

06

08

10

1 4 16 64 256

neffective

Sampling frequency (fixesday)

Cov

erag

e

(f)

emspensp emsp | emsp1687Methods in Ecology and Evolu13onWINNER Et al

F I G U R E 3 emsp HRandoverlapestimatesfortwosimulatedindividualswithatrueoverlapof050Inallpanelsthedashedcirclesdepictthetrue95areasthesolidblacklinestheestimated95areasandthegreylinesthe95CIsontheareaestimatesInthefirstrowrelocationsweresimulatedfromOUFmodelswithidenticalmovementparametersandsamplingtimesInthesecondrowsamplingwasheldconsistentbuttheindividualplottedinyellowhadaHRcrossingtimeof1weekvs1dayfortheindividualinredInthethirdrowmovementagaindifferedbetweenindividualsbutheretheindividualinyellowwassampledonceevery30minvsonceevery3hrsfortheindividualinredNotehowinallcasesAKDE-basedoverlapestimateswererelativelyconsistentandprovidedcoverageofthetrueoverlapwhereasKDE-basedoverlapestimatesvariedsubstantiallyandconsistentlyfailedtoprovidecoverageofthetruth

Overlap 048 95 CIs 032 ndash 066

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(a) AKDEOverlap 04 95 CIs 036 ndash 044

ndash1000

ndash500

0

500

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1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(b) KDE

Overlap 047 95 CIs 029 ndash 067

ndash1000

ndash500

0

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1500

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Y (

met

ers)

(c)

Overlap 031 95 CIs 026 ndash 035

ndash1000

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500

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met

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(d)

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(e)

Overlap 035 95 CIs 031 ndash 039

ndash1000

ndash500

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ndash1000 ndash500 0 500 1000 1500

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Y (

met

ers)

(f)

F I G U R E 4 emsp (a)TherelationshipbetweenpairwiseestimatesoftheBCforMongoliangazellecomputedfromKDEandAKDEHRestimatesThedashed11linedepictsparitybetweentheseNotehowallcasesfallabovethislinehighlightinghowAKDE-derivedBCsuggestsmoreoverlapthanKDE-derivedBCAnexampleofthisdiscrepancyisdepictedin(b)withAKDEBCsuggestingextensiveoverlap080(022ndash099)whereasin(c)thenegativebiasinKDEpropagatestoproduceabiasedestimateoftheoverlap002(001ndash003)Cruciallywitheffectivesamplesizesofc4foreachHRestimatetheCIsapproximatedfromtheAKDEestimateswereappropriatelywidevsKDEsdeceivinglynarrowCIs

000

025

050

075

100

000 025 050 075 100KDE overlap

AK

DE

ove

rlap

(a)

ndash3e+05

ndash2e+05

ndash1e+05

0e+00

1e+05

2e+05

ndash3e+05 ndash2e+05 ndash1e+05 0e+00 1e+05 2e+05

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Y (

met

ers)

(b)

ndash3e+05

ndash2e+05

ndash1e+05

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2e+05

ndash3e+05 ndash2e+05 ndash1e+05 0e+00 1e+05 2e+05

X (meters)

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met

ers)

(c)

1688emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

Inadditionwehave implementedasmall-sample-sizebiascorrec-tionfortheBCandderivedwell-behavedapproximateCIsonthepoint estimateCollectively theseadvancespermit researchers toaccuratelyquantifyHRoverlapevenwhensamplingstrategiesandunderlyingmovementparametersdifferamonggroupsbeingcom-pared and testwhether anyobserveddifferencesare statisticallymeaningful

41emsp|emspHome range and overlap estimation an intrinsic relationship

Acrucialcomponentofanystatisticalinferenceishavingcompara-blemeasuresonwhichtobaseanalysesOverlapistypicallycondi-tionalonHRestimates(FiebergampKochanny2005Millspaughetal2004)whichare themselvesestimated fromanimal trackingdataAs overlap estimation relies on at least three separate estimates(twoHRestimatesandtheiroverlap)itfollowsthatthisanalysisisparticularlyvulnerabletoissuesofestimatorbiasAccurateHResti-mationisadeceptivelychallengingproblemhoweverasautocorre-lation(FlemingFaganetal2015)small-sample-sizebias(FlemingampCalabrese2017)andsamplingirregularities(Flemingetal2018Frairetal2010)willsignificantlyinfluenceanystatisticalanalysesappliedtoanimaltrackingdataMoresubtlyevenidenticalsamplingstrategiescanstillproducedifferentiallybiasedHRestimatesiftheunderlying parameters of movement differ markedly between in-dividuals (FlemingampCalabrese2017Noonanetal inreview)Asthesearenearlyubiquitousaspectsof animal trackingdata accu-rateoverlapestimationrequiresstatisticalmethodsthatcanhandlethesecomplicationswithoutintroducingartifactualdifferencesduepurelytoestimatorbias

Inthisrespectoursimulationstudyrevealedthatforautocor-related data KDE regularly underestimated HR sizes (Fleming amp

Calabrese2017Noonanetalinreview)andthisnegativebiaswasdirectlypropagated tooverlapestimatesForKDE theamountofdatarequiredtoachieveanaccuratemeasureofoverlapwasverylargeandmostempiricalcasesarelikelytounderestimatethetrueoverlap (FiebergampKochanny2005) In contrastAKDEHRswerelarger but significantly more accurate which translated to moreaccurateoverlapestimatesCruciallywhenwevariedthesamplingdesignandmovementstrategiesbetweenthe individualswewerecomparingAKDE-basedestimatesprovidedreliablecoverageofthetrueoverlapwhereasthiswasnotthecaseforKDEConsistentwiththe resultsofour simulation study empiricalAKDEHRs fromau-tocorrelatedMongoliangazelleGPSdatawerecatwiceaslargeasKDEestimatesThisresultedinthemedianpairwiseoverlapbeingfivefoldlargerwhenbasedonAKDEvsKDEHadananalysisbeenbased on the biasedKDE estimates onewould have erroneouslyconcludedthattherewaslittlespatialoverlapinthissystemwhereastheresultsbasedonAKDEsmorerigorousestimatesrevealedtheseindividualsactuallyexhibitedextensiveoverlapAlthoughtheseem-piricalestimatescouldnotbecomparedtoatruthasperoursimula-tionsthisfindingisalsoconsistentwitharecentanalysisbyNoonanetal (in review) Ina large-scale comparative studyencompassing369 individualsacross30species they found thatAKDE95HRestimates consistently included c 95 of holdout observationswhereasKDEestimatesincludedc92athighne(gt256)butonlyc75atlowneThismeansAKDEslargerestimatesareaccuratewhile those producedby conventionalKDEon the samedata areconsistentlyandoftengrosslytoosmallThenetresultisthatAKDEprovides a solid foundation for estimating overlap under realisticsampling regimes resulting inaccurateoverlapestimates that canvalidlybecomparedacrossstudies

AsdescribedaboveafundamentalcomponentofestimatingHRoverlapishavingcomparablemeasuresonwhichtobaseanalyses

F I G U R E 5 emsp Figuredepicting(a)theGPSlocationsfor23MongoliangazelletrackedinMongoliasEasternSteppe(b)anetworkdiagramwithedgeweightsbasedonoverlapvaluesand(c)anexamplecaseoftwoHRestimateswherethepointestimateoftheoverlapsuggestsaconnectionbuttheCIsontheestimatessuggestthatconnectionmightnotbestatisticallysignificantThedashedlinesin(b)depictpairswherethepointestimatesuggestsaconnectionbutwithCIsthatinclude001andthusmaynotbestatisticallysignificantThetransparencyofthelinesisproportionaltothepointestimateoftheBCTheconnectiondepictedinredontheright-handsideof(b)correspondstothepairin(c)

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

1684emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

and frequencies Datawere simulated based on pairs of both IIDprocessesandOUFprocesses(Flemingetal2014bc)parameter-izedsuchthatthetrueoverlapbetweenthesepairswasfixedat05SimulatingfromanOUFprocessgeneratesrelocationsthatfeatureautocorrelated positions and velocities aswell as restricted spaceuseandarerepresentativeofmodernGPStrackingdatacommonlyusedinHRanalyses(FlemingampCalabrese2017)

Importantlythetimescaleoverwhichautocorrelationinpositiondecaysτp(alsotermedtheHRcrossingtimeCalabreseetal2016)is a key parameter for HR estimation (Noonan etal in review)Formallyτpcanbequantifiedfromthedataasthetimescaleoverwhichanindividualspositionalautocorrelationdecaysbyafactorof1

eanditsmovementprocessrevertstothemeanlocation(FlemingampCalabrese2017FlemingFaganetal2015)Thedurationoftheobservationperiod(T)inrelationtoτpwillthusdictatetheeffectivesamplesize(ne)ofadatasetvia

which may be interpreted as the approximate number of rangecrossingsthatoccurredduringthesamplingperiodWetailoredoursimulationsaccordingtotheirrelativeeffectsonneThesewere

(i) Sampling duration Observations were recorded eight timesday andwemanipulated sampling duration (ranging from1 to4096days in a doubling series) For OUF simulations the HRcrossingtimewassetto1dayandthevelocityautocorrelationtimescaleto15ofadayNotablythisparameterizationwassuchthatinthesesimulationsthesamplingdurationindaysexhibiteda11relationshipwithne

(ii)Sampling frequency Here the sampling durationwas fixed at32days and we manipulated the sampling frequency (rangingfrom1to1024fixesdayinadoublingseries)AgainfortheOUFprocessHRcrossingtimewassetto1dayandthevelocityauto-correlationtimescaleto15ofadayThefixedsamplingdurationinthesesimulationsresultedinnebeingfixedat32irrespectiveofvariationinthesamplingfrequency

We then compared the accuracy of the underlying HR es-timates the accuracyof the estimatedoverlap and the realizedcoverageoftheconfidenceintervalsResultswereaveragedover1000simulationspermanipulationThecomputationswerecon-ductedontheSmithsonian InstitutionHighPerformanceCluster(SIHPC)

25emsp|emspEmpirical study

We demonstrate the functionality of this method using GPSdata from Mongolian gazelles Mongolian gazelles are medium-sized herbivores that cross their ranges on seasonal timescales(Flemingetal2014bc)Positionaldatafor36Mongoliangazellewere collected inMongolias EasternSteppebetween2007and2011 (Fleming etal 2014a) Both variogram analysis (Fleming

etal 2014c) and model selection (Calabrese etal 2016) wereused to confirm that there was evidence of range-residency inthedataFromthesediagnosticchecks13individualsshowednosigns of range-resident behaviour and we restricted our analy-sestothe23range-residentindividualsHRestimationwasthencarried out using KDE and AKDE as described aboveWe thencomputedallpairwiseBCsplusmn95CIson theKDEandAKDEes-timatesNotablythelongHRcrossingtimescales(x = 1115daysrange=80ndash4432) and comparatively short tracking durations(x = 3810daysrange=672ndash7550)hereproducedameanne of 61(range=07ndash246)Thisisaregimewherethenegativebiasofconventional KDE is known to have serious implications forHRestimatesonautocorrelateddata(FlemingampCalabrese2017)

251emsp|emspDownstream analyses

To further highlight the utility of these confidence intervalswe used the estimated overlap to quantify the edges of a spa-tial interaction network (Wey etal 2008) As point estimateswereaccompaniedbyCIswewereabletosubsetedgesintotwocategories

(i) SupportedWell-supportededgeswereidentifiedascaseswheretwoindividualsexhibitedoverlappingspaceusewithaminimumCIthatwasgreaterthan001mdashthatistherewasa95certaintythattheoverlapwasge001

(ii)Unsupported Unsupported edges were identified as caseswherethepointestimatesuggestedoverlappingspaceusebutwith aminimumCI thatwas less than 001mdashthat is therewasinsufficientevidencetobecertainthattheoverlapdifferedsig-nificantlyfrom0

Wethenquantifiedanumberofcommonlyuseddiagnostics(ienetwork density mean path length and closeness centrality Weyetal2008)toinvestigatehowthesemightdifferwhenthenetworkwasbasedonlyonstatisticallysupportededgesvstheinclusionofun-supportededges

All analyses were conducted in the R environment (R CoreTeam2016)usingthemethodsimplementedinthepackagectmm (Calabreseetal2016)

3emsp |emspRESULTS

31emsp|emspSimulation results

311emsp|emspAsymptotic properties of the BCss

SimulationsrevealedthatforIIDdatabothAKDEandKDEHResti-matesprovidedidenticalresultsandwererelativelyunbiasedexceptatverysmallsamplesizes(Figure1a)Theresultingoverlapwasalsoidentical between estimators and increasing the number of fixesbyeitherincreasingthesamplingduration(Figure1b)orfrequency(Figure1e)hadtheexpectedeffectofincreasingtheaccuracyofthe

(23)neasympT

τp

emspensp emsp | emsp1685Methods in Ecology and Evolu13onWINNER Et al

overlapestimateanddecreasingtheuncertaintyNotablytheCIsontheBCoffered reasonable coverageof the trueoverlapacross allsamplingregimesalbeitwithsomepersistentnegativebiasatlargesamplesizes(Figure1cf)ThiswastheresultofbiasintheBCdecay-ingtooslowlyrelativetothevariance(seeSupportingInformationAppendixA3)

For autocorrelated data in contrast AKDE 95HR estimateswere generally accurate across the range of sample durations(Figure2a)andfrequencies(Figure2d)wesimulatedwhereasKDEHRestimateswereseverelybiasedforallbutthelargestdatasetsAsaresultwhiletheestimatedoverlapbetweenAKDEandKDEes-timatesbothconvergedtothetruthassamplingdurationincreased(Figure2b)asymptoticconsistencyforKDEestimateswasseverelydelayedFurthermoreincreasingthesamplingfrequencyincreasedthenegativebias inoverlapestimatesderived fromKDEbut ap-propriately did not influence overlap estimates based on AKDE(Figure2e)

The coverage of 95 CIs for the KDE-derived overlap esti-mateswas severely biased under all of the scenarioswe tested(Figure2cf) In contrast the coverageofCIson theAKDEesti-matesconsistentlyprovidedclosetonominalcoverageofthetrueoverlap

312emsp|emspComparability of estimates

Ourbaselinesimulationstudycontrolledtheeffectofthemovementparametersbyassumingtheindividualsexhibitedidenticalmovementstrategies andwere sampled at the exact same timesUnder theseconditionstheimprovedaccuracyofAKDEHRsestimatesresultedinmoreaccurateoverlapestimateswith95CIsthatprovidedclosetonominalcoverage(Figure3a)Therearerealisticcomplicationstoourbasicsimulationstrategyhoweverincludingcaseswhereindividualsaresubjecttothesamesamplingdesignbutexhibitdifferentmove-mentstrategiesandcaseswherebothmovementstrategiesandsam-plingdesignsdifferImportantlywefoundthatAKDE-basedoverlapstillprovidedreasonablecoverageforbothofthesecases(Figure3ce)IncontrastbecauseofthedifferentialbiasinKDEHRestimatestheestimatedoverlapdifferedsubstantiallybetweeneachof thesesce-nariosandineverycasefailedtoprovidecoverageofthetruevalue(Figure3bdf)

32emsp|emspEmpirical case study

ConsistentwithoursimulatedfindingsofnegativebiasinKDEHRandBCestimatesatmidtolowneonautocorrelateddataempirical

F I G U R E 1 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedIIDdataaswellasthecoverageoftheCIs(panelscandf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIs

04

08

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1 4 16 64 256 1024 4096

Sampling duration (days)

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1686emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

AKDEHR estimates were larger than KDE estimates for all pairs(Figure4a)Medianpairwiseoverlapbetweenthe276pairsofindi-vidualswas066(95CI058ndash076)whentheoverlapwasestimatedfromAKDEHRestimatesbutfivefoldlowerwhenestimatedfromKDEestimates(median=01395CI006ndash022)

TheseverenegativebiasofKDE-derivedoverlapwaspersistentacrossall individualsThiscanbe illustrated ina specificexamplewhere theKDEHR estimates resulted in an estimated overlap of002(95CI001ndash003)whereastheAKDEHRsresultedinanover-lapof080(95CI022ndash099)Visualinspectionoftherangeesti-matesfortheseindividualsrevealedsubstantialnegativebiasintheKDEHRwhereastheAKDEHRwaslargerwithappropriatelywideCIsconsideringthesmallne of c4foreachHRestimate(Figure4bc)

321emsp|emspDownstream analyses

Astheseoverlapestimateswereaccompaniedbyconfidenceinter-vals theuncertaintycanbeused to informdownstreamanalysesFor instance a spatial network analysis based on the estimatedoverlaprevealed461edgesofvariablestrength(Figure5)Ofthese275werewell supportedwhereas186hadnostatistical supportWe found that basing the network off of all possible edges vsonlythoseedgeswithstatisticalsupport influenceditsproperties

andanypotentialbiologicalinferencesthatwouldbederivedfromit For instance network density was reduced from 086 to 063whentheanalysiswasrestrictedtoonlythewell-supportededgesFurthermore onlyutilizing statistically supportededges increasedthemeanpath length from113 to139 Interestinglydespitede-creasingdensityandincreasingthemeanpathlengthconstructingthenetworkbasedononlywell-supportededgesresultedinatwo-fold increase in theclosenesscentralitycompared to thenetworkconstructedwithbothsupportedandunsupportededges(045vs023respectively)

4emsp |emspDISCUSSION

Despite the routine nature of estimating overlapping space use(eg Berger amp Gese 2007 Dougherty etal 2018 Fregravere etal2010SanchezampHudgens2015)thereexistsnoformalinferentialframeworkforthisanalysisThis is largelyduetothe inherentdif-ficultiesassociatedwithHRestimation(FiebergampBoumlrger2012)andexacerbated by the historical lack of CIs on bothHR and overlapestimatesAsasolutionwehavedemonstratedhowAKDEHResti-mates(FlemingampCalabrese2017FlemingFaganetal2015)canserveasareliablefoundationonwhichtobasestatisticalinference

F I G U R E 2 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedautocorrelatedtrackingdataandthecoverageoftheCIs(candf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIsNotablyconvergencetothetruthwasmuchslowerforKDEandthecoverageofKDEsCIswasfarfromappropriateinallcases

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emspensp emsp | emsp1687Methods in Ecology and Evolu13onWINNER Et al

F I G U R E 3 emsp HRandoverlapestimatesfortwosimulatedindividualswithatrueoverlapof050Inallpanelsthedashedcirclesdepictthetrue95areasthesolidblacklinestheestimated95areasandthegreylinesthe95CIsontheareaestimatesInthefirstrowrelocationsweresimulatedfromOUFmodelswithidenticalmovementparametersandsamplingtimesInthesecondrowsamplingwasheldconsistentbuttheindividualplottedinyellowhadaHRcrossingtimeof1weekvs1dayfortheindividualinredInthethirdrowmovementagaindifferedbetweenindividualsbutheretheindividualinyellowwassampledonceevery30minvsonceevery3hrsfortheindividualinredNotehowinallcasesAKDE-basedoverlapestimateswererelativelyconsistentandprovidedcoverageofthetrueoverlapwhereasKDE-basedoverlapestimatesvariedsubstantiallyandconsistentlyfailedtoprovidecoverageofthetruth

Overlap 048 95 CIs 032 ndash 066

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F I G U R E 4 emsp (a)TherelationshipbetweenpairwiseestimatesoftheBCforMongoliangazellecomputedfromKDEandAKDEHRestimatesThedashed11linedepictsparitybetweentheseNotehowallcasesfallabovethislinehighlightinghowAKDE-derivedBCsuggestsmoreoverlapthanKDE-derivedBCAnexampleofthisdiscrepancyisdepictedin(b)withAKDEBCsuggestingextensiveoverlap080(022ndash099)whereasin(c)thenegativebiasinKDEpropagatestoproduceabiasedestimateoftheoverlap002(001ndash003)Cruciallywitheffectivesamplesizesofc4foreachHRestimatetheCIsapproximatedfromtheAKDEestimateswereappropriatelywidevsKDEsdeceivinglynarrowCIs

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1688emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

Inadditionwehave implementedasmall-sample-sizebiascorrec-tionfortheBCandderivedwell-behavedapproximateCIsonthepoint estimateCollectively theseadvancespermit researchers toaccuratelyquantifyHRoverlapevenwhensamplingstrategiesandunderlyingmovementparametersdifferamonggroupsbeingcom-pared and testwhether anyobserveddifferencesare statisticallymeaningful

41emsp|emspHome range and overlap estimation an intrinsic relationship

Acrucialcomponentofanystatisticalinferenceishavingcompara-blemeasuresonwhichtobaseanalysesOverlapistypicallycondi-tionalonHRestimates(FiebergampKochanny2005Millspaughetal2004)whichare themselvesestimated fromanimal trackingdataAs overlap estimation relies on at least three separate estimates(twoHRestimatesandtheiroverlap)itfollowsthatthisanalysisisparticularlyvulnerabletoissuesofestimatorbiasAccurateHResti-mationisadeceptivelychallengingproblemhoweverasautocorre-lation(FlemingFaganetal2015)small-sample-sizebias(FlemingampCalabrese2017)andsamplingirregularities(Flemingetal2018Frairetal2010)willsignificantlyinfluenceanystatisticalanalysesappliedtoanimaltrackingdataMoresubtlyevenidenticalsamplingstrategiescanstillproducedifferentiallybiasedHRestimatesiftheunderlying parameters of movement differ markedly between in-dividuals (FlemingampCalabrese2017Noonanetal inreview)Asthesearenearlyubiquitousaspectsof animal trackingdata accu-rateoverlapestimationrequiresstatisticalmethodsthatcanhandlethesecomplicationswithoutintroducingartifactualdifferencesduepurelytoestimatorbias

Inthisrespectoursimulationstudyrevealedthatforautocor-related data KDE regularly underestimated HR sizes (Fleming amp

Calabrese2017Noonanetalinreview)andthisnegativebiaswasdirectlypropagated tooverlapestimatesForKDE theamountofdatarequiredtoachieveanaccuratemeasureofoverlapwasverylargeandmostempiricalcasesarelikelytounderestimatethetrueoverlap (FiebergampKochanny2005) In contrastAKDEHRswerelarger but significantly more accurate which translated to moreaccurateoverlapestimatesCruciallywhenwevariedthesamplingdesignandmovementstrategiesbetweenthe individualswewerecomparingAKDE-basedestimatesprovidedreliablecoverageofthetrueoverlapwhereasthiswasnotthecaseforKDEConsistentwiththe resultsofour simulation study empiricalAKDEHRs fromau-tocorrelatedMongoliangazelleGPSdatawerecatwiceaslargeasKDEestimatesThisresultedinthemedianpairwiseoverlapbeingfivefoldlargerwhenbasedonAKDEvsKDEHadananalysisbeenbased on the biasedKDE estimates onewould have erroneouslyconcludedthattherewaslittlespatialoverlapinthissystemwhereastheresultsbasedonAKDEsmorerigorousestimatesrevealedtheseindividualsactuallyexhibitedextensiveoverlapAlthoughtheseem-piricalestimatescouldnotbecomparedtoatruthasperoursimula-tionsthisfindingisalsoconsistentwitharecentanalysisbyNoonanetal (in review) Ina large-scale comparative studyencompassing369 individualsacross30species they found thatAKDE95HRestimates consistently included c 95 of holdout observationswhereasKDEestimatesincludedc92athighne(gt256)butonlyc75atlowneThismeansAKDEslargerestimatesareaccuratewhile those producedby conventionalKDEon the samedata areconsistentlyandoftengrosslytoosmallThenetresultisthatAKDEprovides a solid foundation for estimating overlap under realisticsampling regimes resulting inaccurateoverlapestimates that canvalidlybecomparedacrossstudies

AsdescribedaboveafundamentalcomponentofestimatingHRoverlapishavingcomparablemeasuresonwhichtobaseanalyses

F I G U R E 5 emsp Figuredepicting(a)theGPSlocationsfor23MongoliangazelletrackedinMongoliasEasternSteppe(b)anetworkdiagramwithedgeweightsbasedonoverlapvaluesand(c)anexamplecaseoftwoHRestimateswherethepointestimateoftheoverlapsuggestsaconnectionbuttheCIsontheestimatessuggestthatconnectionmightnotbestatisticallysignificantThedashedlinesin(b)depictpairswherethepointestimatesuggestsaconnectionbutwithCIsthatinclude001andthusmaynotbestatisticallysignificantThetransparencyofthelinesisproportionaltothepointestimateoftheBCTheconnectiondepictedinredontheright-handsideof(b)correspondstothepairin(c)

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

emspensp emsp | emsp1685Methods in Ecology and Evolu13onWINNER Et al

overlapestimateanddecreasingtheuncertaintyNotablytheCIsontheBCoffered reasonable coverageof the trueoverlapacross allsamplingregimesalbeitwithsomepersistentnegativebiasatlargesamplesizes(Figure1cf)ThiswastheresultofbiasintheBCdecay-ingtooslowlyrelativetothevariance(seeSupportingInformationAppendixA3)

For autocorrelated data in contrast AKDE 95HR estimateswere generally accurate across the range of sample durations(Figure2a)andfrequencies(Figure2d)wesimulatedwhereasKDEHRestimateswereseverelybiasedforallbutthelargestdatasetsAsaresultwhiletheestimatedoverlapbetweenAKDEandKDEes-timatesbothconvergedtothetruthassamplingdurationincreased(Figure2b)asymptoticconsistencyforKDEestimateswasseverelydelayedFurthermoreincreasingthesamplingfrequencyincreasedthenegativebias inoverlapestimatesderived fromKDEbut ap-propriately did not influence overlap estimates based on AKDE(Figure2e)

The coverage of 95 CIs for the KDE-derived overlap esti-mateswas severely biased under all of the scenarioswe tested(Figure2cf) In contrast the coverageofCIson theAKDEesti-matesconsistentlyprovidedclosetonominalcoverageofthetrueoverlap

312emsp|emspComparability of estimates

Ourbaselinesimulationstudycontrolledtheeffectofthemovementparametersbyassumingtheindividualsexhibitedidenticalmovementstrategies andwere sampled at the exact same timesUnder theseconditionstheimprovedaccuracyofAKDEHRsestimatesresultedinmoreaccurateoverlapestimateswith95CIsthatprovidedclosetonominalcoverage(Figure3a)Therearerealisticcomplicationstoourbasicsimulationstrategyhoweverincludingcaseswhereindividualsaresubjecttothesamesamplingdesignbutexhibitdifferentmove-mentstrategiesandcaseswherebothmovementstrategiesandsam-plingdesignsdifferImportantlywefoundthatAKDE-basedoverlapstillprovidedreasonablecoverageforbothofthesecases(Figure3ce)IncontrastbecauseofthedifferentialbiasinKDEHRestimatestheestimatedoverlapdifferedsubstantiallybetweeneachof thesesce-nariosandineverycasefailedtoprovidecoverageofthetruevalue(Figure3bdf)

32emsp|emspEmpirical case study

ConsistentwithoursimulatedfindingsofnegativebiasinKDEHRandBCestimatesatmidtolowneonautocorrelateddataempirical

F I G U R E 1 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedIIDdataaswellasthecoverageoftheCIs(panelscandf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIs

04

08

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1686emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

AKDEHR estimates were larger than KDE estimates for all pairs(Figure4a)Medianpairwiseoverlapbetweenthe276pairsofindi-vidualswas066(95CI058ndash076)whentheoverlapwasestimatedfromAKDEHRestimatesbutfivefoldlowerwhenestimatedfromKDEestimates(median=01395CI006ndash022)

TheseverenegativebiasofKDE-derivedoverlapwaspersistentacrossall individualsThiscanbe illustrated ina specificexamplewhere theKDEHR estimates resulted in an estimated overlap of002(95CI001ndash003)whereastheAKDEHRsresultedinanover-lapof080(95CI022ndash099)Visualinspectionoftherangeesti-matesfortheseindividualsrevealedsubstantialnegativebiasintheKDEHRwhereastheAKDEHRwaslargerwithappropriatelywideCIsconsideringthesmallne of c4foreachHRestimate(Figure4bc)

321emsp|emspDownstream analyses

Astheseoverlapestimateswereaccompaniedbyconfidenceinter-vals theuncertaintycanbeused to informdownstreamanalysesFor instance a spatial network analysis based on the estimatedoverlaprevealed461edgesofvariablestrength(Figure5)Ofthese275werewell supportedwhereas186hadnostatistical supportWe found that basing the network off of all possible edges vsonlythoseedgeswithstatisticalsupport influenceditsproperties

andanypotentialbiologicalinferencesthatwouldbederivedfromit For instance network density was reduced from 086 to 063whentheanalysiswasrestrictedtoonlythewell-supportededgesFurthermore onlyutilizing statistically supportededges increasedthemeanpath length from113 to139 Interestinglydespitede-creasingdensityandincreasingthemeanpathlengthconstructingthenetworkbasedononlywell-supportededgesresultedinatwo-fold increase in theclosenesscentralitycompared to thenetworkconstructedwithbothsupportedandunsupportededges(045vs023respectively)

4emsp |emspDISCUSSION

Despite the routine nature of estimating overlapping space use(eg Berger amp Gese 2007 Dougherty etal 2018 Fregravere etal2010SanchezampHudgens2015)thereexistsnoformalinferentialframeworkforthisanalysisThis is largelyduetothe inherentdif-ficultiesassociatedwithHRestimation(FiebergampBoumlrger2012)andexacerbated by the historical lack of CIs on bothHR and overlapestimatesAsasolutionwehavedemonstratedhowAKDEHResti-mates(FlemingampCalabrese2017FlemingFaganetal2015)canserveasareliablefoundationonwhichtobasestatisticalinference

F I G U R E 2 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedautocorrelatedtrackingdataandthecoverageoftheCIs(candf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIsNotablyconvergencetothetruthwasmuchslowerforKDEandthecoverageofKDEsCIswasfarfromappropriateinallcases

1 4 16 64 256 1024 4096

00

05

10

15

1 4 16 64 256 1024 4096

neffective

Sampling duration (days)

95

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(a)

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000

025

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neffective

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ttach

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a co

effi

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t

(b)

1 4 16 64 256 1024 4096

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neffective

Sampling duration (days)

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(c)

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(f)

emspensp emsp | emsp1687Methods in Ecology and Evolu13onWINNER Et al

F I G U R E 3 emsp HRandoverlapestimatesfortwosimulatedindividualswithatrueoverlapof050Inallpanelsthedashedcirclesdepictthetrue95areasthesolidblacklinestheestimated95areasandthegreylinesthe95CIsontheareaestimatesInthefirstrowrelocationsweresimulatedfromOUFmodelswithidenticalmovementparametersandsamplingtimesInthesecondrowsamplingwasheldconsistentbuttheindividualplottedinyellowhadaHRcrossingtimeof1weekvs1dayfortheindividualinredInthethirdrowmovementagaindifferedbetweenindividualsbutheretheindividualinyellowwassampledonceevery30minvsonceevery3hrsfortheindividualinredNotehowinallcasesAKDE-basedoverlapestimateswererelativelyconsistentandprovidedcoverageofthetrueoverlapwhereasKDE-basedoverlapestimatesvariedsubstantiallyandconsistentlyfailedtoprovidecoverageofthetruth

Overlap 048 95 CIs 032 ndash 066

ndash1000

ndash500

0

500

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F I G U R E 4 emsp (a)TherelationshipbetweenpairwiseestimatesoftheBCforMongoliangazellecomputedfromKDEandAKDEHRestimatesThedashed11linedepictsparitybetweentheseNotehowallcasesfallabovethislinehighlightinghowAKDE-derivedBCsuggestsmoreoverlapthanKDE-derivedBCAnexampleofthisdiscrepancyisdepictedin(b)withAKDEBCsuggestingextensiveoverlap080(022ndash099)whereasin(c)thenegativebiasinKDEpropagatestoproduceabiasedestimateoftheoverlap002(001ndash003)Cruciallywitheffectivesamplesizesofc4foreachHRestimatetheCIsapproximatedfromtheAKDEestimateswereappropriatelywidevsKDEsdeceivinglynarrowCIs

000

025

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000 025 050 075 100KDE overlap

AK

DE

ove

rlap

(a)

ndash3e+05

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1688emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

Inadditionwehave implementedasmall-sample-sizebiascorrec-tionfortheBCandderivedwell-behavedapproximateCIsonthepoint estimateCollectively theseadvancespermit researchers toaccuratelyquantifyHRoverlapevenwhensamplingstrategiesandunderlyingmovementparametersdifferamonggroupsbeingcom-pared and testwhether anyobserveddifferencesare statisticallymeaningful

41emsp|emspHome range and overlap estimation an intrinsic relationship

Acrucialcomponentofanystatisticalinferenceishavingcompara-blemeasuresonwhichtobaseanalysesOverlapistypicallycondi-tionalonHRestimates(FiebergampKochanny2005Millspaughetal2004)whichare themselvesestimated fromanimal trackingdataAs overlap estimation relies on at least three separate estimates(twoHRestimatesandtheiroverlap)itfollowsthatthisanalysisisparticularlyvulnerabletoissuesofestimatorbiasAccurateHResti-mationisadeceptivelychallengingproblemhoweverasautocorre-lation(FlemingFaganetal2015)small-sample-sizebias(FlemingampCalabrese2017)andsamplingirregularities(Flemingetal2018Frairetal2010)willsignificantlyinfluenceanystatisticalanalysesappliedtoanimaltrackingdataMoresubtlyevenidenticalsamplingstrategiescanstillproducedifferentiallybiasedHRestimatesiftheunderlying parameters of movement differ markedly between in-dividuals (FlemingampCalabrese2017Noonanetal inreview)Asthesearenearlyubiquitousaspectsof animal trackingdata accu-rateoverlapestimationrequiresstatisticalmethodsthatcanhandlethesecomplicationswithoutintroducingartifactualdifferencesduepurelytoestimatorbias

Inthisrespectoursimulationstudyrevealedthatforautocor-related data KDE regularly underestimated HR sizes (Fleming amp

Calabrese2017Noonanetalinreview)andthisnegativebiaswasdirectlypropagated tooverlapestimatesForKDE theamountofdatarequiredtoachieveanaccuratemeasureofoverlapwasverylargeandmostempiricalcasesarelikelytounderestimatethetrueoverlap (FiebergampKochanny2005) In contrastAKDEHRswerelarger but significantly more accurate which translated to moreaccurateoverlapestimatesCruciallywhenwevariedthesamplingdesignandmovementstrategiesbetweenthe individualswewerecomparingAKDE-basedestimatesprovidedreliablecoverageofthetrueoverlapwhereasthiswasnotthecaseforKDEConsistentwiththe resultsofour simulation study empiricalAKDEHRs fromau-tocorrelatedMongoliangazelleGPSdatawerecatwiceaslargeasKDEestimatesThisresultedinthemedianpairwiseoverlapbeingfivefoldlargerwhenbasedonAKDEvsKDEHadananalysisbeenbased on the biasedKDE estimates onewould have erroneouslyconcludedthattherewaslittlespatialoverlapinthissystemwhereastheresultsbasedonAKDEsmorerigorousestimatesrevealedtheseindividualsactuallyexhibitedextensiveoverlapAlthoughtheseem-piricalestimatescouldnotbecomparedtoatruthasperoursimula-tionsthisfindingisalsoconsistentwitharecentanalysisbyNoonanetal (in review) Ina large-scale comparative studyencompassing369 individualsacross30species they found thatAKDE95HRestimates consistently included c 95 of holdout observationswhereasKDEestimatesincludedc92athighne(gt256)butonlyc75atlowneThismeansAKDEslargerestimatesareaccuratewhile those producedby conventionalKDEon the samedata areconsistentlyandoftengrosslytoosmallThenetresultisthatAKDEprovides a solid foundation for estimating overlap under realisticsampling regimes resulting inaccurateoverlapestimates that canvalidlybecomparedacrossstudies

AsdescribedaboveafundamentalcomponentofestimatingHRoverlapishavingcomparablemeasuresonwhichtobaseanalyses

F I G U R E 5 emsp Figuredepicting(a)theGPSlocationsfor23MongoliangazelletrackedinMongoliasEasternSteppe(b)anetworkdiagramwithedgeweightsbasedonoverlapvaluesand(c)anexamplecaseoftwoHRestimateswherethepointestimateoftheoverlapsuggestsaconnectionbuttheCIsontheestimatessuggestthatconnectionmightnotbestatisticallysignificantThedashedlinesin(b)depictpairswherethepointestimatesuggestsaconnectionbutwithCIsthatinclude001andthusmaynotbestatisticallysignificantThetransparencyofthelinesisproportionaltothepointestimateoftheBCTheconnectiondepictedinredontheright-handsideof(b)correspondstothepairin(c)

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

1686emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

AKDEHR estimates were larger than KDE estimates for all pairs(Figure4a)Medianpairwiseoverlapbetweenthe276pairsofindi-vidualswas066(95CI058ndash076)whentheoverlapwasestimatedfromAKDEHRestimatesbutfivefoldlowerwhenestimatedfromKDEestimates(median=01395CI006ndash022)

TheseverenegativebiasofKDE-derivedoverlapwaspersistentacrossall individualsThiscanbe illustrated ina specificexamplewhere theKDEHR estimates resulted in an estimated overlap of002(95CI001ndash003)whereastheAKDEHRsresultedinanover-lapof080(95CI022ndash099)Visualinspectionoftherangeesti-matesfortheseindividualsrevealedsubstantialnegativebiasintheKDEHRwhereastheAKDEHRwaslargerwithappropriatelywideCIsconsideringthesmallne of c4foreachHRestimate(Figure4bc)

321emsp|emspDownstream analyses

Astheseoverlapestimateswereaccompaniedbyconfidenceinter-vals theuncertaintycanbeused to informdownstreamanalysesFor instance a spatial network analysis based on the estimatedoverlaprevealed461edgesofvariablestrength(Figure5)Ofthese275werewell supportedwhereas186hadnostatistical supportWe found that basing the network off of all possible edges vsonlythoseedgeswithstatisticalsupport influenceditsproperties

andanypotentialbiologicalinferencesthatwouldbederivedfromit For instance network density was reduced from 086 to 063whentheanalysiswasrestrictedtoonlythewell-supportededgesFurthermore onlyutilizing statistically supportededges increasedthemeanpath length from113 to139 Interestinglydespitede-creasingdensityandincreasingthemeanpathlengthconstructingthenetworkbasedononlywell-supportededgesresultedinatwo-fold increase in theclosenesscentralitycompared to thenetworkconstructedwithbothsupportedandunsupportededges(045vs023respectively)

4emsp |emspDISCUSSION

Despite the routine nature of estimating overlapping space use(eg Berger amp Gese 2007 Dougherty etal 2018 Fregravere etal2010SanchezampHudgens2015)thereexistsnoformalinferentialframeworkforthisanalysisThis is largelyduetothe inherentdif-ficultiesassociatedwithHRestimation(FiebergampBoumlrger2012)andexacerbated by the historical lack of CIs on bothHR and overlapestimatesAsasolutionwehavedemonstratedhowAKDEHResti-mates(FlemingampCalabrese2017FlemingFaganetal2015)canserveasareliablefoundationonwhichtobasestatisticalinference

F I G U R E 2 emsp TheasymptoticpropertiesofKDEandAKDEHRestimators(aandd)andtheBC(bande)forsimulatedautocorrelatedtrackingdataandthecoverageoftheCIs(candf)asafunctionofsamplingduration(toprow)andfrequency(bottomrow)Inallpanelsthedashedhorizontallinesdepictthetruththesolidlinethemeanpointestimateandtheshadedregionsthe95CIsNotablyconvergencetothetruthwasmuchslowerforKDEandthecoverageofKDEsCIswasfarfromappropriateinallcases

1 4 16 64 256 1024 4096

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05

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Sampling duration (days)

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emspensp emsp | emsp1687Methods in Ecology and Evolu13onWINNER Et al

F I G U R E 3 emsp HRandoverlapestimatesfortwosimulatedindividualswithatrueoverlapof050Inallpanelsthedashedcirclesdepictthetrue95areasthesolidblacklinestheestimated95areasandthegreylinesthe95CIsontheareaestimatesInthefirstrowrelocationsweresimulatedfromOUFmodelswithidenticalmovementparametersandsamplingtimesInthesecondrowsamplingwasheldconsistentbuttheindividualplottedinyellowhadaHRcrossingtimeof1weekvs1dayfortheindividualinredInthethirdrowmovementagaindifferedbetweenindividualsbutheretheindividualinyellowwassampledonceevery30minvsonceevery3hrsfortheindividualinredNotehowinallcasesAKDE-basedoverlapestimateswererelativelyconsistentandprovidedcoverageofthetrueoverlapwhereasKDE-basedoverlapestimatesvariedsubstantiallyandconsistentlyfailedtoprovidecoverageofthetruth

Overlap 048 95 CIs 032 ndash 066

ndash1000

ndash500

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ers)

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(f)

F I G U R E 4 emsp (a)TherelationshipbetweenpairwiseestimatesoftheBCforMongoliangazellecomputedfromKDEandAKDEHRestimatesThedashed11linedepictsparitybetweentheseNotehowallcasesfallabovethislinehighlightinghowAKDE-derivedBCsuggestsmoreoverlapthanKDE-derivedBCAnexampleofthisdiscrepancyisdepictedin(b)withAKDEBCsuggestingextensiveoverlap080(022ndash099)whereasin(c)thenegativebiasinKDEpropagatestoproduceabiasedestimateoftheoverlap002(001ndash003)Cruciallywitheffectivesamplesizesofc4foreachHRestimatetheCIsapproximatedfromtheAKDEestimateswereappropriatelywidevsKDEsdeceivinglynarrowCIs

000

025

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000 025 050 075 100KDE overlap

AK

DE

ove

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(a)

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X (meters)

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(c)

1688emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

Inadditionwehave implementedasmall-sample-sizebiascorrec-tionfortheBCandderivedwell-behavedapproximateCIsonthepoint estimateCollectively theseadvancespermit researchers toaccuratelyquantifyHRoverlapevenwhensamplingstrategiesandunderlyingmovementparametersdifferamonggroupsbeingcom-pared and testwhether anyobserveddifferencesare statisticallymeaningful

41emsp|emspHome range and overlap estimation an intrinsic relationship

Acrucialcomponentofanystatisticalinferenceishavingcompara-blemeasuresonwhichtobaseanalysesOverlapistypicallycondi-tionalonHRestimates(FiebergampKochanny2005Millspaughetal2004)whichare themselvesestimated fromanimal trackingdataAs overlap estimation relies on at least three separate estimates(twoHRestimatesandtheiroverlap)itfollowsthatthisanalysisisparticularlyvulnerabletoissuesofestimatorbiasAccurateHResti-mationisadeceptivelychallengingproblemhoweverasautocorre-lation(FlemingFaganetal2015)small-sample-sizebias(FlemingampCalabrese2017)andsamplingirregularities(Flemingetal2018Frairetal2010)willsignificantlyinfluenceanystatisticalanalysesappliedtoanimaltrackingdataMoresubtlyevenidenticalsamplingstrategiescanstillproducedifferentiallybiasedHRestimatesiftheunderlying parameters of movement differ markedly between in-dividuals (FlemingampCalabrese2017Noonanetal inreview)Asthesearenearlyubiquitousaspectsof animal trackingdata accu-rateoverlapestimationrequiresstatisticalmethodsthatcanhandlethesecomplicationswithoutintroducingartifactualdifferencesduepurelytoestimatorbias

Inthisrespectoursimulationstudyrevealedthatforautocor-related data KDE regularly underestimated HR sizes (Fleming amp

Calabrese2017Noonanetalinreview)andthisnegativebiaswasdirectlypropagated tooverlapestimatesForKDE theamountofdatarequiredtoachieveanaccuratemeasureofoverlapwasverylargeandmostempiricalcasesarelikelytounderestimatethetrueoverlap (FiebergampKochanny2005) In contrastAKDEHRswerelarger but significantly more accurate which translated to moreaccurateoverlapestimatesCruciallywhenwevariedthesamplingdesignandmovementstrategiesbetweenthe individualswewerecomparingAKDE-basedestimatesprovidedreliablecoverageofthetrueoverlapwhereasthiswasnotthecaseforKDEConsistentwiththe resultsofour simulation study empiricalAKDEHRs fromau-tocorrelatedMongoliangazelleGPSdatawerecatwiceaslargeasKDEestimatesThisresultedinthemedianpairwiseoverlapbeingfivefoldlargerwhenbasedonAKDEvsKDEHadananalysisbeenbased on the biasedKDE estimates onewould have erroneouslyconcludedthattherewaslittlespatialoverlapinthissystemwhereastheresultsbasedonAKDEsmorerigorousestimatesrevealedtheseindividualsactuallyexhibitedextensiveoverlapAlthoughtheseem-piricalestimatescouldnotbecomparedtoatruthasperoursimula-tionsthisfindingisalsoconsistentwitharecentanalysisbyNoonanetal (in review) Ina large-scale comparative studyencompassing369 individualsacross30species they found thatAKDE95HRestimates consistently included c 95 of holdout observationswhereasKDEestimatesincludedc92athighne(gt256)butonlyc75atlowneThismeansAKDEslargerestimatesareaccuratewhile those producedby conventionalKDEon the samedata areconsistentlyandoftengrosslytoosmallThenetresultisthatAKDEprovides a solid foundation for estimating overlap under realisticsampling regimes resulting inaccurateoverlapestimates that canvalidlybecomparedacrossstudies

AsdescribedaboveafundamentalcomponentofestimatingHRoverlapishavingcomparablemeasuresonwhichtobaseanalyses

F I G U R E 5 emsp Figuredepicting(a)theGPSlocationsfor23MongoliangazelletrackedinMongoliasEasternSteppe(b)anetworkdiagramwithedgeweightsbasedonoverlapvaluesand(c)anexamplecaseoftwoHRestimateswherethepointestimateoftheoverlapsuggestsaconnectionbuttheCIsontheestimatessuggestthatconnectionmightnotbestatisticallysignificantThedashedlinesin(b)depictpairswherethepointestimatesuggestsaconnectionbutwithCIsthatinclude001andthusmaynotbestatisticallysignificantThetransparencyofthelinesisproportionaltothepointestimateoftheBCTheconnectiondepictedinredontheright-handsideof(b)correspondstothepairin(c)

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

emspensp emsp | emsp1687Methods in Ecology and Evolu13onWINNER Et al

F I G U R E 3 emsp HRandoverlapestimatesfortwosimulatedindividualswithatrueoverlapof050Inallpanelsthedashedcirclesdepictthetrue95areasthesolidblacklinestheestimated95areasandthegreylinesthe95CIsontheareaestimatesInthefirstrowrelocationsweresimulatedfromOUFmodelswithidenticalmovementparametersandsamplingtimesInthesecondrowsamplingwasheldconsistentbuttheindividualplottedinyellowhadaHRcrossingtimeof1weekvs1dayfortheindividualinredInthethirdrowmovementagaindifferedbetweenindividualsbutheretheindividualinyellowwassampledonceevery30minvsonceevery3hrsfortheindividualinredNotehowinallcasesAKDE-basedoverlapestimateswererelativelyconsistentandprovidedcoverageofthetrueoverlapwhereasKDE-basedoverlapestimatesvariedsubstantiallyandconsistentlyfailedtoprovidecoverageofthetruth

Overlap 048 95 CIs 032 ndash 066

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(a) AKDEOverlap 04 95 CIs 036 ndash 044

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(b) KDE

Overlap 047 95 CIs 029 ndash 067

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(c)

Overlap 031 95 CIs 026 ndash 035

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(d)

Overlap 046 95 CIs 031 ndash 064

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(e)

Overlap 035 95 CIs 031 ndash 039

ndash1000

ndash500

0

500

1000

1500

ndash1000 ndash500 0 500 1000 1500

X (meters)

Y (

met

ers)

(f)

F I G U R E 4 emsp (a)TherelationshipbetweenpairwiseestimatesoftheBCforMongoliangazellecomputedfromKDEandAKDEHRestimatesThedashed11linedepictsparitybetweentheseNotehowallcasesfallabovethislinehighlightinghowAKDE-derivedBCsuggestsmoreoverlapthanKDE-derivedBCAnexampleofthisdiscrepancyisdepictedin(b)withAKDEBCsuggestingextensiveoverlap080(022ndash099)whereasin(c)thenegativebiasinKDEpropagatestoproduceabiasedestimateoftheoverlap002(001ndash003)Cruciallywitheffectivesamplesizesofc4foreachHRestimatetheCIsapproximatedfromtheAKDEestimateswereappropriatelywidevsKDEsdeceivinglynarrowCIs

000

025

050

075

100

000 025 050 075 100KDE overlap

AK

DE

ove

rlap

(a)

ndash3e+05

ndash2e+05

ndash1e+05

0e+00

1e+05

2e+05

ndash3e+05 ndash2e+05 ndash1e+05 0e+00 1e+05 2e+05

X (meters)

Y (

met

ers)

(b)

ndash3e+05

ndash2e+05

ndash1e+05

0e+00

1e+05

2e+05

ndash3e+05 ndash2e+05 ndash1e+05 0e+00 1e+05 2e+05

X (meters)

Y (

met

ers)

(c)

1688emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

Inadditionwehave implementedasmall-sample-sizebiascorrec-tionfortheBCandderivedwell-behavedapproximateCIsonthepoint estimateCollectively theseadvancespermit researchers toaccuratelyquantifyHRoverlapevenwhensamplingstrategiesandunderlyingmovementparametersdifferamonggroupsbeingcom-pared and testwhether anyobserveddifferencesare statisticallymeaningful

41emsp|emspHome range and overlap estimation an intrinsic relationship

Acrucialcomponentofanystatisticalinferenceishavingcompara-blemeasuresonwhichtobaseanalysesOverlapistypicallycondi-tionalonHRestimates(FiebergampKochanny2005Millspaughetal2004)whichare themselvesestimated fromanimal trackingdataAs overlap estimation relies on at least three separate estimates(twoHRestimatesandtheiroverlap)itfollowsthatthisanalysisisparticularlyvulnerabletoissuesofestimatorbiasAccurateHResti-mationisadeceptivelychallengingproblemhoweverasautocorre-lation(FlemingFaganetal2015)small-sample-sizebias(FlemingampCalabrese2017)andsamplingirregularities(Flemingetal2018Frairetal2010)willsignificantlyinfluenceanystatisticalanalysesappliedtoanimaltrackingdataMoresubtlyevenidenticalsamplingstrategiescanstillproducedifferentiallybiasedHRestimatesiftheunderlying parameters of movement differ markedly between in-dividuals (FlemingampCalabrese2017Noonanetal inreview)Asthesearenearlyubiquitousaspectsof animal trackingdata accu-rateoverlapestimationrequiresstatisticalmethodsthatcanhandlethesecomplicationswithoutintroducingartifactualdifferencesduepurelytoestimatorbias

Inthisrespectoursimulationstudyrevealedthatforautocor-related data KDE regularly underestimated HR sizes (Fleming amp

Calabrese2017Noonanetalinreview)andthisnegativebiaswasdirectlypropagated tooverlapestimatesForKDE theamountofdatarequiredtoachieveanaccuratemeasureofoverlapwasverylargeandmostempiricalcasesarelikelytounderestimatethetrueoverlap (FiebergampKochanny2005) In contrastAKDEHRswerelarger but significantly more accurate which translated to moreaccurateoverlapestimatesCruciallywhenwevariedthesamplingdesignandmovementstrategiesbetweenthe individualswewerecomparingAKDE-basedestimatesprovidedreliablecoverageofthetrueoverlapwhereasthiswasnotthecaseforKDEConsistentwiththe resultsofour simulation study empiricalAKDEHRs fromau-tocorrelatedMongoliangazelleGPSdatawerecatwiceaslargeasKDEestimatesThisresultedinthemedianpairwiseoverlapbeingfivefoldlargerwhenbasedonAKDEvsKDEHadananalysisbeenbased on the biasedKDE estimates onewould have erroneouslyconcludedthattherewaslittlespatialoverlapinthissystemwhereastheresultsbasedonAKDEsmorerigorousestimatesrevealedtheseindividualsactuallyexhibitedextensiveoverlapAlthoughtheseem-piricalestimatescouldnotbecomparedtoatruthasperoursimula-tionsthisfindingisalsoconsistentwitharecentanalysisbyNoonanetal (in review) Ina large-scale comparative studyencompassing369 individualsacross30species they found thatAKDE95HRestimates consistently included c 95 of holdout observationswhereasKDEestimatesincludedc92athighne(gt256)butonlyc75atlowneThismeansAKDEslargerestimatesareaccuratewhile those producedby conventionalKDEon the samedata areconsistentlyandoftengrosslytoosmallThenetresultisthatAKDEprovides a solid foundation for estimating overlap under realisticsampling regimes resulting inaccurateoverlapestimates that canvalidlybecomparedacrossstudies

AsdescribedaboveafundamentalcomponentofestimatingHRoverlapishavingcomparablemeasuresonwhichtobaseanalyses

F I G U R E 5 emsp Figuredepicting(a)theGPSlocationsfor23MongoliangazelletrackedinMongoliasEasternSteppe(b)anetworkdiagramwithedgeweightsbasedonoverlapvaluesand(c)anexamplecaseoftwoHRestimateswherethepointestimateoftheoverlapsuggestsaconnectionbuttheCIsontheestimatessuggestthatconnectionmightnotbestatisticallysignificantThedashedlinesin(b)depictpairswherethepointestimatesuggestsaconnectionbutwithCIsthatinclude001andthusmaynotbestatisticallysignificantThetransparencyofthelinesisproportionaltothepointestimateoftheBCTheconnectiondepictedinredontheright-handsideof(b)correspondstothepairin(c)

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

1688emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

Inadditionwehave implementedasmall-sample-sizebiascorrec-tionfortheBCandderivedwell-behavedapproximateCIsonthepoint estimateCollectively theseadvancespermit researchers toaccuratelyquantifyHRoverlapevenwhensamplingstrategiesandunderlyingmovementparametersdifferamonggroupsbeingcom-pared and testwhether anyobserveddifferencesare statisticallymeaningful

41emsp|emspHome range and overlap estimation an intrinsic relationship

Acrucialcomponentofanystatisticalinferenceishavingcompara-blemeasuresonwhichtobaseanalysesOverlapistypicallycondi-tionalonHRestimates(FiebergampKochanny2005Millspaughetal2004)whichare themselvesestimated fromanimal trackingdataAs overlap estimation relies on at least three separate estimates(twoHRestimatesandtheiroverlap)itfollowsthatthisanalysisisparticularlyvulnerabletoissuesofestimatorbiasAccurateHResti-mationisadeceptivelychallengingproblemhoweverasautocorre-lation(FlemingFaganetal2015)small-sample-sizebias(FlemingampCalabrese2017)andsamplingirregularities(Flemingetal2018Frairetal2010)willsignificantlyinfluenceanystatisticalanalysesappliedtoanimaltrackingdataMoresubtlyevenidenticalsamplingstrategiescanstillproducedifferentiallybiasedHRestimatesiftheunderlying parameters of movement differ markedly between in-dividuals (FlemingampCalabrese2017Noonanetal inreview)Asthesearenearlyubiquitousaspectsof animal trackingdata accu-rateoverlapestimationrequiresstatisticalmethodsthatcanhandlethesecomplicationswithoutintroducingartifactualdifferencesduepurelytoestimatorbias

Inthisrespectoursimulationstudyrevealedthatforautocor-related data KDE regularly underestimated HR sizes (Fleming amp

Calabrese2017Noonanetalinreview)andthisnegativebiaswasdirectlypropagated tooverlapestimatesForKDE theamountofdatarequiredtoachieveanaccuratemeasureofoverlapwasverylargeandmostempiricalcasesarelikelytounderestimatethetrueoverlap (FiebergampKochanny2005) In contrastAKDEHRswerelarger but significantly more accurate which translated to moreaccurateoverlapestimatesCruciallywhenwevariedthesamplingdesignandmovementstrategiesbetweenthe individualswewerecomparingAKDE-basedestimatesprovidedreliablecoverageofthetrueoverlapwhereasthiswasnotthecaseforKDEConsistentwiththe resultsofour simulation study empiricalAKDEHRs fromau-tocorrelatedMongoliangazelleGPSdatawerecatwiceaslargeasKDEestimatesThisresultedinthemedianpairwiseoverlapbeingfivefoldlargerwhenbasedonAKDEvsKDEHadananalysisbeenbased on the biasedKDE estimates onewould have erroneouslyconcludedthattherewaslittlespatialoverlapinthissystemwhereastheresultsbasedonAKDEsmorerigorousestimatesrevealedtheseindividualsactuallyexhibitedextensiveoverlapAlthoughtheseem-piricalestimatescouldnotbecomparedtoatruthasperoursimula-tionsthisfindingisalsoconsistentwitharecentanalysisbyNoonanetal (in review) Ina large-scale comparative studyencompassing369 individualsacross30species they found thatAKDE95HRestimates consistently included c 95 of holdout observationswhereasKDEestimatesincludedc92athighne(gt256)butonlyc75atlowneThismeansAKDEslargerestimatesareaccuratewhile those producedby conventionalKDEon the samedata areconsistentlyandoftengrosslytoosmallThenetresultisthatAKDEprovides a solid foundation for estimating overlap under realisticsampling regimes resulting inaccurateoverlapestimates that canvalidlybecomparedacrossstudies

AsdescribedaboveafundamentalcomponentofestimatingHRoverlapishavingcomparablemeasuresonwhichtobaseanalyses

F I G U R E 5 emsp Figuredepicting(a)theGPSlocationsfor23MongoliangazelletrackedinMongoliasEasternSteppe(b)anetworkdiagramwithedgeweightsbasedonoverlapvaluesand(c)anexamplecaseoftwoHRestimateswherethepointestimateoftheoverlapsuggestsaconnectionbuttheCIsontheestimatessuggestthatconnectionmightnotbestatisticallysignificantThedashedlinesin(b)depictpairswherethepointestimatesuggestsaconnectionbutwithCIsthatinclude001andthusmaynotbestatisticallysignificantThetransparencyofthelinesisproportionaltothepointestimateoftheBCTheconnectiondepictedinredontheright-handsideof(b)correspondstothepairin(c)

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

emspensp emsp | emsp1689Methods in Ecology and Evolu13onWINNER Et al

NotablyinthisstudyweconsiderrangeestimatorsinthesenseofBurt(1943)whichestimatealong-runspaceuseassumingthefocalindividualdoesnotchange itsmovementprocess (FlemingFaganetal2015)ThisincludesKDEsminimumconvexpolygons(MCPMohr1947)andtime-naivelocalconvexhulls(LoCoH)(Getzetal2007)Alsoof interestareoccurrencedistributionestimators suchas theBrownianbridge (HorneGartonKroneamp Lewis 2007) ort-LoCoH(LyonsTurnerampGetz2013)whichquantifyuncertaintyintheanimalslocationduringthesamplingperiodincludingtimesnotsampledCruciallythisuncertaintyvanishesinthelimitwhereboththesamplingintervalandtelemetryerrorapproachzeroAlthoughthesetwomathematicallydistinctclassesofdistributionshavebeenhistorically conflated under the umbrella term of ldquoutilization dis-tributionsrdquo theyhaveverydifferent interpretationsandusecases(FlemingFaganetal2015)Consequentlyoverlapbasedonoccur-renceestimateshasaverydifferentmeaningfromoverlapbasedonrangeestimatesandisbeyondthescopeofthepresentwork

WealsonotethatextendingourbiascorrectionandCIstootherHRestimators suchasMCPLoCoHornon-GRFKDEbandwidthoptimizersisnotatractableproblemFirstourmethodsareexplic-itlybasedontheGRFapproximationsotheyarenotconsistentwithnon-GRFestimatorsSecondtheGRF-basedmethodsimplementedin ctmmaretoourknowledgetheonlyHRestimatorsthatquan-tifyuncertaintyAsanuncertaintyestimateisaprerequisiteforourerrorpropagationtechniquesitwouldnotcurrentlybepossibletoadaptourapproachtootherestimatorsFinallythetargetdistribu-tionsandexpectationvaluesofgeometricmethodssuchasMCPandLoCoHareusuallyunknownwhichmakestheseestimatorsincom-patiblewiththemethodsdevelopedhere

42emsp|emspProperties of the overlap estimator

InadditiontoutilizingreliableHRestimatestheoverlapestimatoritselfshouldhavedesirableproperties(FiebergampKochanny2005)Whileseveralvalidestimatorsexist theBC(Bhattacharyya1943)standsoutbecauseofitsstatisticalvaliditygeometricinterpretabil-itycomputationalefficiencyandasymptoticconsistencyAsnotedbyFiebergandKochanny(2005)howevertheBCispronetoexhib-itingnegativesmall-sample-sizebias (DjouadiampSnorrason1990)Tocorrectforthiswederivedasmall-sample-sizebiascorrectionwhichimprovedtheaccuracyofBCestimates(DjouadiampSnorrason1990)

Furthermore problematic is the historical lack ofCIs on over-lapestimatesOverlapisanestimatederivedfromdataandshouldbeaccompaniedbyameasureof theuncertainty (Pawitan2001)Withoutthisonecannotproperly infer the importanceofagivenestimateAs a solutionwe have derivedCIs on theBCbased onaGRFapproximationUsingsimulateddatawedemonstratedhowthis implementation will provide reasonable coverage of the trueoverlapWenotehoweverthatwhilegenerallywellbehavedtherewassomepersistentnegativebiasinthecoverageoftheseCIsThebiasedcoverage is likely the resultof thebias in theBCpointes-timatedecayingtooslowlyrelativetothevarianceasne increased

(Figure A2) With asymptotically efficient estimators this ratiowoulddecayatarateof1∕

radic

Norbetterwhereashereitincreasesata rateofc

radic

NAs such their coverageshouldbe treatedwithcautionparticularlyat largeneFurthermorebecauseweapproxi-matetheHRsasGaussianwhenestimatinguncertaintytheCIsmayexhibit unintended behaviour when the overlap is dependent onnon-Gaussianfeatures

Despite these limitationswell-behaved CIs for HR overlap is anovel feature and permits true statistical inference on overlap esti-matesForinstancetheseCIscanbeappliedtoareferencevalueofin-terest(egthemeanoverlapbetweenindividualsofthesamespeciesstudiedelsewhere)totestforsignificantdifferencesbetweentheseasopposedtorelyingonad hoccomparisonsAdditionallyifoverlapisbeingusedtoinformsubsequentanalysesCIscanbeusedtoimprovethese For examplewe found that differentiating between the275overlapestimatesthatwerewellsupportedbythedataandthe186thatmayhavebeenartifactualsignificantlyinfluencedthepropertiesof an interactionnetworkofMongolian gazelleWhenbasedon allpossibleedgesthenetworksuggestedalargernumberofedgesbutwithalowclosenesscentralityConverselywhenbasedonlyonedgeswithstatisticalsupportthenetworkdensitydecreasedbutclosenessincreasedThesupportedandunsupportednetworkswouldeachleadtoauniquesetofbiologicalinterpretationswithonlytheformerbeingsupportedbythedata

5emsp |emspCONCLUSION

In conclusion we have developed the first inferential frameworkforHRoverlap tailored for thespecificneedsofecologists that isboth statistically valid and computationally efficient CollectivelythemoreaccurateandcomparableHRestimatesprovidedbyAKDE(Flemingamp Calabrese 2017 Fleming Fagan etal 2015Noonanetal in review) and our novel bias correction andCIs on theBCpermit rigorous overlap estimation This method is now availablevia command line interface through thectmm package (Calabreseetal 2016) or through theweb-based graphical user interface at ctmmshinyappsioctmmweb(Dongetal2017)

ACKNOWLEDG EMENTS

This work was supported by the US NSF Advances in BiologicalInformatics programme (ABI-1458748 to JMC)MJN was sup-portedbyaSmithsonianInstitutionCGPSgrantTMwasfundedbytheRobertBoschFoundation

AUTHORSrsquo CONTRIBUTIONS

KWandMJNcontributedequallytothisworkCHFandJMCconceivedthestudyKWandCHFdevelopedthemethodsKAOandTMcollectedthedataMJNconductedtheanalysesMJNand JMC drafted themanuscriptAll authors contributed to thestudyconceptsandwriting

1690emsp |emsp emspenspMethods in Ecology and Evolu13on WINNER Et al

DATA ACCE SSIBILIT Y

TheMongoliangazelledatausedinthismanuscriptareavailablefromtheDryadDigitalRepositoryhttpsdoiorg105061dryad45157(Flemingetal2014a)

ORCID

Kevin Winner httporcidorg0000-0002-8838-8905

Michael J Noonan httporcidorg0000-0003-4512-0535

Justin M Calabrese httporcidorg0000-0003-0575-6408

R E FE R E N C E S

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Fleming C H amp Calabrese J M (2017) A new kernel density esti-mator for accurate home-range and species-range area estima-tion Methods in Ecology and Evolution 8 571ndash579 httpsdoiorg1011112041-210x12673

FlemingCHCalabreseJMMuellerTOlsonKALeimgruberPampFaganWF(2014a)DatafromFromfine-scaleforagingtohomerangesA semi-variance approach to identifyingmovementmodes

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FlemingCHCalabreseJMMuellerTOlsonKALeimgruberPampFaganWF(2014c)Non-Markovianmaximumlikelihoodestima-tionofautocorrelatedmovementprocessesMethods in Ecology and Evolution5462ndash472httpsdoiorg1011112041-210x12176

FlemingCHFaganWFMuellerTOlsonKALeimgruberPampCalabreseJM(2015)Rigoroushomerangeestimationwithmove-mentdataAnewautocorrelatedkerneldensityestimatorEcology961182ndash1188httpsdoiorg10189014-20101

FlemingCHSubaşiYampCalabreseJM(2015)Maximum-entropyde-scriptionofanimalmovementPhysical Review E Statistical Nonlinear and Soft Matter Physics 91 032107 httpsdoiorg101103physreve91032107

Fleming C H Sheldon D FaganW F Leimgruber P Mueller TNandintsetsegDhellipCalabreseJM(2018)Correctingformissingandirregulardatainhome-rangeestimationEcological Applications1ndash8

Frair J L Fieberg J Hebblewhite M Cagnacci F DeCesare NJ amp Pedrotti L (2010) Resolving issues of imprecise and habi-tat-biased locations in ecological analyses using GPS telemetrydata Philosophical Transactions of the Royal Society of London B Biological Sciences 365 2187ndash2200 httpsdoiorg101098rstb20100084

Fregravere C H Kruumltzen M Mann J Watson-Capps J J Tsai Y JPattersonEMhellipSherwinWB(2010)Homerangeoverlapmatri-linealandbiparentalkinshipdrivefemaleassociationsinbottlenosedolphinsAnimal Behaviour80481ndash486httpsdoiorg101016janbehav201006007

GetzWMFortmann-RoeSCrossPCLyonsAJRyanS JampWilmersCC(2007)LoCoHNonparameterickernelmethodsforconstructinghomerangesandutilizationdistributionsPLoS ONE2e207httpsdoiorg101371journalpone0000207

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InmanHFampBradleyJrEL(1989)TheoverlappingcoefficientasameasureofagreementbetweenprobabilitydistributionsandpointestimationoftheoverlapoftwonormaldensitiesCommunications in StatisticsndashTheory and Methods 18 3851ndash3874 httpsdoiorg10108003610928908830127

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KaysRCrofootMCJetzWampWikelskiM(2015)TerrestrialanimaltrackingasaneyeonlifeandplanetScience348aaa2478ndashaaa2478httpsdoiorg101126scienceaaa2478

Kranstauber B Smolla M amp Safi K (2016) Similarity in spa-tial utilization distributions measured by the earth moverrsquos dis-tance Methods in Ecology and Evolution 8 155ndash160 httpsdoiorg1011112041-210x12649

LukasDampClutton-BrockTH(2013)Theevolutionofsocialmonog-amy in mammals Science 341 526ndash530 httpsdoiorg101126science1238677

LyonsA J TurnerWCampGetzWM (2013)Home rangeplusAspace-time characterization of movement over real landscapesMovement Ecology12httpsdoiorg1011862051-3933-1-2

emspensp emsp | emsp1691Methods in Ecology and Evolu13onWINNER Et al

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MitchellMSampPowellRA(2012)ForagingoptimallyforhomerangesJournal of Mammalogy93917ndash928httpsdoiorg10164411-mamm-s-1571

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NoonanM J TuckerMA FlemingCHAlberts SC Ali AHAltmannJhellipCalabreseJM(2018)WhichhomerangeestimatorshouldIuseAnanalysisofautocorrelationandbiasinhomerangeestimation(inreview)

PawitanY(2001)In all likelihood Statistical modelling and inference using likelihoodOxfordClarendonPress

PowellRA(1979)MustelidspacingpatternsVariationsonathemebyMustelaEthology50153ndash165

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(Urocyon littoralis)Ecology and Evolution52466ndash2477httpsdoiorg101002ece31533

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SUPPORTING INFORMATION

Additional supporting information may be found online in theSupportingInformationsectionattheendofthearticle

How to cite this articleWinnerKNoonanMJFlemingCHetalStatisticalinferenceforhomerangeoverlapMethods Ecol Evol 201891679ndash1691 httpsdoiorg1011112041-210X13027