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Ecological
Indicators
jou rn al h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / e c o l i n d
Novel
application
of
a
quantitative
spatial
comparison
tool
to
species
distribution
data
Esther
L.
Jones
a,b,∗,
Luke
Rendell
a,
Enrico
Pirotta
c,
Jed
A.
Long
daSeaMammalResearchUnit,ScottishOceansInstitute,UniversityofStAndrews,StAndrewsKY168LB,UnitedKingdom
bCentreforResearchintoEcologicalandEnvironmentalModelling,TheObservatory,BuchananGardens,UniversityofStAndrews,StAndrewsKY169LZ, UnitedKingdom
cSchoolofMathematics,WashingtonStateUniversity,Vancouver98686,WA,USA
dDepartmentofGeographyandSustainableDevelopment,SchoolofGeographyandGeosciences,IrvineBuilding,UniversityofStAndrews,StAndrews,Fife, ScotlandKY169AL,UnitedKingdom
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received5January2016
Receivedinrevisedform11May2016 Accepted31May2016
Keywords:
Edgeeffects Mapcomparison Movingwindow Spermwhale SSIMindex Uncertainty
a
b
s
t
r
a
c
t
Comparinggeographicallyreferencedmapshasbecomeanimportantaspectofspatialecology(e.g. assessingchangeindistributionovertime).Whilsthumansareadeptatrecognisingandextracting struc-turefrommaps(i.e.identifyingspatialpatterns),quantifyingthesestructurescanbedifficult.Here,we showhowtheStructuralSimilarity(SSIM)index,aspatialcomparisonmethodadaptedfromtechniques developedincomputersciencetodeterminethequalityofimagecompression,canbeusedtoextract additionalinformationfromspatialecologicaldata.WeenhancetheSSIMindextoincorporate uncer-taintyfromtheunderlyingspatialmodels,andprovideasoftwarealgorithmtocorrectforinternaledge effectssothatlossofspatialinformationfromthemapcomparisonislimited.TheSSIMindexusesa spatially-localwindowtocalculatestatisticsbasedonlocalmean,variance,andcovariancebetweenthe mapsbeingcompared.AnumberofstatisticscanbecalculatedusingtheSSIMindex,rangingfromasingle summarystatistictoquantifysimilaritiesbetweentwomaps,tomapsofsimilaritiesinmean,variance, andcovariancethatcanprovideadditionalinsightintounderlyingbiologicalprocesses.Wedemonstrate theapplicabilityoftheSSIMapproachusingacasestudyofspermwhalesintheMediterraneanSeaand identifyareaswherelocal-scaledifferencesinspace-usebetweengroupsandsingletonwhalesoccur. Weshowhownovelinsightsintospatialstructurecanbeextracted,whichcouldnotbeobtainedby visualinspectionorcell-by-cellsubtraction.Asanapproach,SSIMisapplicabletoabroadrangeofspatial ecologicaldata,providinganovel,implementabletoolformapcomparison.
©2016TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Ecologicalsystemstypicallyexhibitspatialheterogeneity aris-ingfromunderlyingprocessesthatinfluencespeciesoccurrence, abundance,anddiversity.Characterisingspatialheterogeneity,and changestoit,areessentialtounderstandingthestructureof ecolog-icalsystems(FortinandDale,2005).Spatialecologicaldatarange fromspatiallydiscreteeventsorindividuals,representedasbasic plotsoflocationsinspacereferencedbyapoint(e.g.vegetation assemblagesingeographicalspace,Penttinenetal.,1992),to dis-tributionsofspeciesacrosshabitats,characterisedbycontinuous densitymaps(McKinneyetal.,2012).Geographicallyreferenced mapsareaneffectivewaytoconveycomplexspatialinformation
∗Correspondingauthorat:SeaMammalResearchUnit,ScottishOceansInstitute, UniversityofStAndrews,StAndrewsKY168LB,UnitedKingdom.
E-mailaddress:[email protected](E.L.Jones).
because the human visual system excels at recognising struc-tureinthesefamiliarandintuitivelyreadimages.However,visual interpretationofspatialpatternsinsuchmapsissubjective(Da
Silva-Buttkusetal.,2009),which canbefurthercomplicatedby
thecharacteristicsofthemappeddata,suchasscale(e.g.grain andextent) andtheparticularcartographicrepresentationused (e.g.projection,colour,symbology)(MacEachren,1995).Therefore, methodshavemovedtowardsobjectivelyquantifyingthepatterns observed inmappeddata toproduceconsistentand repeatable analyses(FortinandDale,2005).
The comparison of two (or more)geographically referenced mapsaimstocharacterisedifferencesinspatialheterogeneityand structure,andcalculatedefinedspatialmetricsbetweenthem.The problemofmapcomparison(Jacquez,1995)hasbeenstudiedfor decadesbygeographers(Tobler,1965),aswellasecologists(Levine etal.,2009).Therearemanyecologicalapplicationswheremap comparisoncanleadtonewinsights.Ecologicaldataoftenhave intrinsicpropertiesthatmakethemchallengingtocompare
spa-http://dx.doi.org/10.1016/j.ecolind.2016.05.051
tially: datatend tobe continuous-valued (e.g.spatially explicit modelpredictions)andhaveunderlyingspatialdependencies(e.g. neighbouringcellsarenotindependent).However,therearefew establishedspatialcomparisontechniquesdocumentedinthe eco-logicalliteraturedirectlyrelatingtothetypeofproblemsoutlined above,asavailablemethodsgenerallyonlyaddressoneorotherof theseproperties.
Inrecentyears,emphasishasbeenplacedoncomparisonsof mapped categorical data (Hagen-Zanker and Lajoie, 2008) and methodsforassessingspatialstructureinmapsofcontinuous val-ueddataorspatiallyexplicitmodelpredictionsonaregularspatial latticeremainlimited inboth scopeand sophistication(
Hagen-Zanker,2006a).Cell-by-cellcomparisonsandnon-spatiallyexplicit
indexesweightedbygridcellarewidelyusedinremotesensing,but donotaccountforspatialdependenciesbetweencells(Horn,1966;
Leitãoetal.,2011).Likewise,Moran’sIorGeary’sCtests(Cliffand
Ord,1970)assessspatialautocorrelationbutprovidesingleindices acrossspace,whichdonotretainlocationalinformation.Metrics usedtoinvestigatenichesimilaritybetweenspeciesdistributions predictedwithEnvironmentalNicheModelsalsolosespatial infor-mationtogiveasinglemeasureofoverlaporequivalency(Warren etal.,2008).Overlapindicesandtestsforspatialautocorrelation measureonlyoneformofspatialstructureinthedata,and this maynotbesufficientfortheecologicalquestionbeingposed.
AStructuralSimilarityIndex(SSIMindex)wasproposed origi-nallybyWangetal.(2004)forcomparingcompressiontechniques usedindigitalimaging(e.g.JPEGcompression).Theindexusesa spatially-localmovingwindowtogenerateindependent compo-nentsrelatingtolocalsimilaritiesinthemean,variance,andspatial correlationbetweenthetwomapsbeingcompared.SSIMcanassess continuousdataandsimultaneouslyconsiderslocalmagnitudeand spatialstructure,makingitsuitabletobeadaptedforthe applica-tionofcomparingspatialecologicaldata.Mapcomparisonmethods toecologicalproblemsshouldallowuncertaintyassociatedwith thedataormodelpredictionstobeincludedinthemap compar-isontoaidinterpretation.Ecologicalmapsoftenhaveuncertainty estimatesassociatedwitheachgridcellwhenvaluesareobtained usingspatiallyexplicitpredictivemodels(Rocchinietal.,2011),and theseshouldbeincorporatedinamapcomparisonapproach. Addi-tionally,localstatisticssuchastheSSIMindexaresusceptibleto edgeeffectsarisingfromtheuseofaspatiallylocalneighbourhood
(Boots,2002).Edgeeffects(i.e.theinclusionofnullareasoutside
thestudy)areexacerbatedbyirregularlyshapedboundariescaused byarbitrarilyshapedadministrativeunitsorgeographicalfeatures (e.g.islands).Thesemayormaynotinfluencethespatialprocess understudy.Ecologicalprocessesoftenchangeonornear bound-aries(Wiensetal.,1985),forexample,theboundaryoftheAntarctic CircumpolarCurrent affects the surroundingmarine ecosystem
(Tynan,1998),andsotheseareascanbeofspecificinterest.
There-fore,weproposetwoenhancementstotheSSIMindextoaddress commonissuesfacedinspatialecologicalanalysisby incorporat-inguncertaintyassociatedwiththeunderlyingdataintothemap comparison,andcorrectingforedgeeffects.Wedemonstrateuse oftheSSIMmethodologyandourenhancementsbyapplyingthem toacasestudytocomparehabitatpreferencebygroupsand sin-gletonsofspermwhales(Physetermacrocephalus,Linneaus1758) intheMediterraneanSea(Pirottaetal.,2011).
2. Methods
2.1. Mapcomparison
Considertwocontinuous valuedmaps(Aand B)each repre-sentedas regular grids. For each cell, a localneighbourhood is definedby(n)neighbouringspatialunitsgivenaweighting(w).
Thesizeoftheneighbourhoodisuser-defined,hasalowerlimitof 3×3cells andcantakeanynon-evenvalue.Wangetal.(2004)
proposedtheuseof a(circular)Gaussian weightingfunctionof
w=
wi|i=1,2...,nwherewiisobtainedfromaGaussian
ker-nelcentredonthefocalcell.Thestandarddeviation,=n/3,is normalisedsothatn
i=1wi=1.
Theindexiiteratesthroughallncellswithineachlocalregion toproducemeansandvariancesforeachmapaswellascovariance betweenthetwogriddedmaps.
a=
n
i=1
wiai (1)
2a=
n
i=1
wi(ai−a)2 (2)
ab=
n
i=1
wi(ai−a)(bi−b) (3)
a, b2,and ab represent spatially local measuresof mean,
varianceandcovariance,computedforeachcell,whereaiandbi
representthevaluesin cellifor mapsAandBrespectively.The threecomponentsoftheSSIMmethodarethencalculatedfrom thesestatistics,givingspatiallylocalmeasuresofsimilarityinthe mean,variance,andcovarianceofthetwomaps.
SIM(A,B)= 2ab+c1
2
a+2b+c1
(4)
SIV(A,B)= 2ab+c2
2
a+b2+c2
(5)
SIP(A,B)= ab+c3
ab+c3
(6)
Thestatisticsare namedSimilarityin Mean(SIM),Similarity inVariance(SIV),andSimilarityinPattern(SIP)ofspatial covari-ance,sothattheycanbeinterpretedintuitivelyinecologicalterms
(Table1).Constants(c1–c3)areusedinequations(4)–(6)toaid
sta-bilitywhenthedenominatorsoftheequations,sumofthesquared
means
2a+2b
,sumof thesquaredvariances
2a+b2
,and productofthestandarddeviations (ab) arecloseto0.
Follow-ingguidelinesproposedbyWangetal.(2004),theconstantscan beestimatedheuristicallyfromtherangeofvaluesofthe underly-ingmapsbeingcompared(R)togetherwithk1=0.01andk2=0.03.
Therefore,c1=(k1R)2,c2=(k2R)2,andc3=c2/2.
Anoverall measurefor comparison canbe computedasthe productofallthreecomponents.
SSIM(A,B)=[SIM(A,B)]˛·[SIV(A,B)]ˇ·[SIP(A,B)] (7)
Table1
DescriptionoflocalstatisticscalculatedinSSIMindex,usingpairsofimages(mapsAandB)todemonstratehighandlowsimilarity.Theinterpretationcolumnprovidesa generalecologicalinterpretationofeachmetricusingthecaseexampleofonespeciesinMapAandadifferentspeciesinMapB.
Index Description Bounds Interpretation MapA MapB
Similarityinmeans
(SIM)
Ratiooftwicethe
productofthelocal
meanstotheir
summedsquares.
(0,1) 0=MapAhashighvalues;mapBlowvalues.
Themeansaredissimilar(e.g.specieshave
differentlocalabundances).
1=BothmapsAandBhavesimilarlyhigh(or
low)values(e.g.specieshavesimilarlocal
abundances).
Similarityinvariance
(SIV)
Ratiooftwicethe
productofthelocal
standarddeviationsto
theirsummed
variances.
(0,1) 0=MapAhashighvariance;mapBlow
variance.Thevariancesaredissimilar(e.g.one
speciesisspatiallyclustered,theotherhasa
homogeneousdistribution).
1=BothmapsAandBhavesimilarlyhigh(or
low)variance(e.g.bothspecieshavesimilar
degreesofspatialclustering,orbothhave
homogeneouslocaldistributions).
Similarityinpattern
(SIP)ofspatial
covariance
Ratioofthelocal
covariancetothe
productofthelocal
standarddeviations.
(−1,1) −1=MapAhashighvaluesinsomecells;Map
Bhashighvaluesinalternatecells.Spatial
correlationisnegative(e.g.speciesexhibit
spatialpartitioning).
0=MapAandBexhibitnospatialcorrelation
(e.g.speciesdistributionsareindependent).
1=MapAandBhavehighandlowvaluesin
thesamecells.Spatialcorrelationispositive
(e.g.speciesareusingthesameresources,or
havepredator-preyinteractions).
formofecologicalinteraction,suchaspredation,areoccurring.The
meansofeachmetriccanbecalculatedtoproducesummary
statis-tics(SIM,SIV ,SIP)ifrequired.ThemeanofSSIM(SSIM)willprovide
anoverall metricof mapcomparison, capturingthesimilarities
betweenmeans,variances,andcovarianceinasinglevalue.
Themeanandvarianceofeachgridcellintheunderlyingmaps
areresampledtogenerateaseriesofrealisations(N).SSIMstatistics
arecalculatedforeachsetofrealisations(1···N)ofthetwomaps
beingcompared.Avariance-adjustedmeasureofSSIMiscalculated
bytakingthemeanofeachstatisticovertheresultingcomparisons.
Upperandlower95%confidencelimitsofthestatisticscanbe
cal-culatedfromthemeanandvarianceofthesampledcomparisons.
Tocorrectforedgeeffects,areflectionalgorithmisimplemented
togeneratesyntheticbuffersandensurethespatialextentofthe
mapcomparisonispreserved(AppendixAinSupplementary).
2.2. Casestudy:spermwhalesintheMediterranean
2.2.1. Introduction
IntheMediterranean,asmallpopulationofspermwhales
per-sist.Spermwhalesshowsexuallydimorphicbehaviourasadults:
malesbecomeincreasinglysolitaryastheymature,andsegregate
fromlong-termsocialunitsofadultfemalesandtheiroffspring,
exceptingshorttermassociationsformatingpurposes(Whitehead,
2003).It is unclear what drives this segregationand
hypothe-sesinclude:groupsoffemalesoutcompetingsolitarymaleswhen exploitingmid-watersquidpatches,malesandfemaleshaving dif-ferentdietaryandhencehabitatpreferences,orhighermalegrowth ratesthat requirewider searchareastolocatehighprey
densi-ties(Whitehead,2003).Understandinghabitatuseinareaswhere
Fig.1. Predictedprobabilityofoccurrenceforbothsocialassemblages:(a)groups;(b)singletons(c)varianceingrouppresence;and(d)varianceinsingletonpresence.(a) and(b)aremodifiedfromPirottaetal.(2011).GlobalSelf-consistent,Hierarchical,High-resolutionGeographyDatabase(GSHHG)shorelinedatafromNOAAwereused, availabletodownloadfromhttp://www.ngdc.noaa.gov/mgg/shorelines/gshhs.htmlhttp://www.ngdc.noaa.gov/mgg/shorelines/gshhs.html.
providenovelbiologicalinsightsnotreadilyapparentfromvisual assessmentsalone.
2.2.2. Data
Thestudyareawaslocatedfrom38to41◦Nand0.5–5◦E,centred ontheislandsof Ibiza,Mallorcaand Menorca(Fig.1). Informa-tiononspermwhaleoccurrencewascollectedduringdedicated summerresearchcruisescoveringthewatersaroundtheBalearic archipelago.Each cruise lasted for approximatelya month and wasrepeatedover6consecutiveyears(2003–2008).Whaleswere locatedand tracked acoustically from theirecholocation clicks, andanencounterwasdefinedasaperiodofcontinuous acous-ticcontactwithoneormoreanimals.Pirottaetal.(2011)useda GeneralisedAdditiveModelling(GAM)approachtomodelsperm whaleoccurrenceasafunctionofseveralenvironmentaland tem-poralpredictors,combinedwithGeneralisedEstimatingEquations (GEEs) to account for autocorrelation in the residuals. Further detailsontheenvironmentaldatasetsandanalyticalapproachcan befoundinPirottaetal.(2011)andaresummarisedhere:Separate analyseswerecarriedoutforsingletonsandgroupstodetermine whetherhabitatpreferencewascharacterisedbydifferentextrinsic drivers.Thefinalmodelforspermwhalegroupsincludedlatitude, longitude,weeklyseasurface temperature(SST)and slope gra-dient.Forsingletons, latitude, longitude,year,monthlySST and slopeaspectwereretainedbymodelselection.Theauthorsnoted
qualitativelydifferentspatialpatternsemergingforthetwosocial assemblagesinthefinalpredictionmaps,quantitativelysupported byaninverserelationshipwithSST,andsuggestedthatthesemight betheresultoffine-scalehabitatsegregation.
2.2.3. Analysis
Thepredictedprobabilityofpresenceofgroupsandsingletons, andcorrespondingestimatesofvarianceweremappedataspatial resolutionof 2nauticalmiles (NM)ona regulargrid.To calcu-lateSSIMstatistics,thesizeofthelocalneighbourhoodforboth mapsshouldbedefinedbytakingthenatureoftheunderlyingdata andecologicalprocessinquestionintoaccount.Lewisetal.(2007)
examinedthenearest-neighbourdistancesbetweenspermwhales intheMediterraneanSeausingasimilaracousticsurveyapproach
todatausedinPirottaetal.(2011)andfoundthatanimalsdefined
asbelongingtoa‘cluster’mostlyhadanupperlimitof2.7NMof perpendiculardistancebetweenthem,whereasdispersed (single-ton)animalswereseparatedbydistancesbeyondthisthreshold. Thesizeofthelocalneighbourhoodwasdefinedina3×3(n=9) cellwindow(6NMx6NM),suchthattheedgeofthewindowwas atleast2NM(1gridcell)fromanyanimalsencounteredinthe cen-trecell.AcircularGaussianweightingkernel(w={wi|i=1,2...,9})
Fig.2.Mapcomparisonbetweenthepredictedprobabilityofoccurrencesofgroupandsingletonspermwhales.Areflectionalgorithmwasusedtocounteractinternaledge effects:(a)SimilarityInMeans(0–1);(b)SimilarityInVariance(0–1);(c)SimilarityInPatternofspatialcovariance(−1to1);and(d)StructuralSimilarityindex(−1to1).
fromthecasestudy(Section2.2.4andAppendixBin Supplemen-tary).Uncertaintyfromtheunderlyingdatawasincludedinthe mapcomparisonsusingparametricbootstrapping.Samplesfroma multivariatenormaldistributionweregeneratedusingmodel coef-ficientsandeachcovariancematrixtoproduce500realisationsof modelcoefficientsforthegroupandsingletonmodels.Thesewere usedtopredict500setsofprobabilitiesforthegroupand single-tonmodels.SSIMstatisticswerecalculatedforeachpairofmaps generatedfrombootstrappeddata.Meanandvarianceofpredicted probabilitiesineach gridcellweretakenforeachstatistic(SIM, SIV,SIP,andSSIM).Allanalysiswasconductedusingthestatistical softwarepackageR(RCoreTeam,2014),andcodeanddataused forcalculatingtheSSIMindexcanbeobtainedfromAppendicesC andDinSupplementaryrespectively.
2.2.4. Sensitivitytesting
Setsofsensitivitytestswereconductedtodemonstratehow varyingspecific(user-defined)parameterscouldpotentiallyaffect resultsofthemapcomparisonanalysisforthespermwhaledata: (1)AcircularGaussianweightingkernelwasappliedtothelocal neighbourhoodwindowvs.noweighting;(2)thesizeofthelocal neighbourhoodwasvaried,using3×3,5×5,and7×7gridcells; and(3)areflectionalgorithmtocorrectforedgeeffectswasapplied vs.noedgecorrection.ForGaussianweightingtests,thesizeofthe localneighbourhood(w)wassetat3×3gridcellsandthereflection algorithmwasimplemented.Forlocalneighbourhoodtests, Gaus-sianweightingandthereflectionalgorithmwereapplied.Foredge
effectstests,thesizeofthelocalneighbourhood(w)wassetat3×3 gridcellsandGaussianweightingwasapplied.Inalltests,=n/3 andonlymeanvaluesfromtheunderlyingmapsbeingcompared wereused.SSIMstatisticswerecalculatedforeachsetoftestsand meansandvariancesofeachstatistic(SIM,SIV,SIP,SSIM)were cal-culatedtoprovidesummarystatistics.Welchtwo-samplet-tests wereusedtocomparetheSSIMstatisticforeachsetoftests.
3. Results
Fig.3. Mapcomparisonbetweenthepredictedprobabilityofoccurrenceofgroupandsingletonspermwhalesusingbootstrappeduncertaintyfromthehabitatpreference modelsofbothsocialassemblages.Areflectionalgorithmwasusedtocounteractinternaledgeeffects:(a)SimilarityInMeans(0–1);(b)SimilarityInVariance(0–1);(c) SimilarityInPatternofspatialcovariance(−1to1);and(d)StructuralSimilarityindex(−1to1).
probabilityofoccurrence,andvaluescloseto0(red)showareas wherethevarianceisdifferent.Forexample,theareatothenorth ofMenorcashowsatransitionzonewheregroupsandsingletons areusingspacedifferently—groupshaveheterogeneous,sporadic spaceuse(i.e.highvariance),singletonsareutilisingspacein a consistent,homogeneousway(i.e.lowvariance).Fig.2cshowsthe similarityinpatterns(SIP)ofspatialcovariancebetweenthemaps. TheSIPmetricisthemostdifficulttocapturethroughvisual com-parisonofhabitatusebetweengroupsandsingletons(c.f.Figs.1a andb).Valuescloseto1(yellow)denotelocalregionswherethe spatialstructure betweenpredictedprobabilityofoccurrenceof groupsandsingletonsissimilar,meaninggridcellswithrelative highandlowvarianceareinthesamelocationsineachunderlying map.Underlyingmechanismsofdirectcompetitionforresources couldbeoccurring,for example tothenorth,east and west of Menorca,andnorthandwestofMallorca.Valuescloseto−1(red) indicateareaswherelocalspatialstructureisdissimilar,suggesting spatialpartitioningmaybeoccurring(northofMenorcainthe tran-sitionzonediscussedpreviously,andthesouthernedgeofthestudy area).Fig.2dshowsSSIM,whichistheproductoftheotherthree statistics.DifferencesinspatialstructuredetectedinSIV(Fig.2b) andSIP (Fig.2c)atthesouthernedgeof thestudyarearemain apparentintheSSIMindex.Somespatial structuralsimilarities seenthroughoutFigs.2a–ctothenorth-westandeastofMallorca, andsouth-eastofFormenteraarealsoretainedinSSIM.Themean
valueofSSIMwascalculated(SSIM=0.22),showingpositivespatial structurebetweentheunderlyingmaps.
Results incorporatinguncertainty from theunderlyingmaps intothecomparisoncalculationareprovidedinFig.3,andshow similarinferences to those in Fig.2, although each of thefour comparisonmetricsexhibitlessextremevalues.Anareaof par-ticularinterestis southof Mallorca(Fig.3c), whereSIP isclose to−1(red),characterisingdifferentspatialpatternsinhabitatuse betweengroupsandsingletons.Fig.4focusesonthisarea,whichis situatedoverthecontinentalslopeandhaspreviouslybeen iden-tifiedasafeedinggroundforspermwhales(GannierandPraca,
2007;Gannieretal.,2002).AlthoughFig.4ashowsthatbothsocial
assemblageshavesimilar(high)habitatpreference(SIMiscloseto 1),thereisstrongnegativeSIPinspecificareas(Fig.4c), indicat-inglocal-scalespatialpartitioningbetweengroupsandsingletons. Thesepatternsoccurmostlyalongbathymetriccontoursatdepths rangingbetween1000to2000m.Thespatialstructurecanbeseen inSSIM(Fig.4d).
Table 2 Results determining sensitivity of the SSIM index when user-defined parameters were varied: applying Gaussian weighting to the local neighbourhood vs. no weighting; varying the size of the local neighbourhood; and implementing the reflection algorithm vs. not countering for edge effects. Sensitivity analysis Parameters SIM SIV SIP SSIM t-test on SSIM Mean Low 95%CI Upp 95%CI Mean Low 95%CI Upp 95%CI Mean Low 95%CI Upp 95%CI Mean Low 95%CI Upp 95%CI t; p -value Weighting Gaussian weighting 0.432 − 0.183 1.048 0.741 0.169 1.313 0.654 − 0.127 1.434 0.218 − 0.293 0.729 0.06; 0.95 No Gaussian weighting 0.432 − 0.183 1.048 0.740 0.167 1.313 0.652 − 0.130 1.435 0.218 − 0.294 0.729 Local neighbourhood 3 × 3 cells 0.432 − 0.183 1.048 0.741 0.169 1.313 0.654 − 0.127 1.434 0.218 − 0.293 0.729 4.14; 0.00 5 × 5 cells 0.433 − 0.174 1.041 0.706 0.104 1.308 0.567 − 0.229 1.362 0.191 − 0.284 0.666 7 × 7 cells 0.435 − 0.169 1.039 0.684 0.063 1.306 0.515 − 0.284 1.314 0.177 − 0.279 0.633 2.34; 0.02 a Edge effects Reflection algorithm 0.432 − 0.183 1.048 0.741 0.169 1.313 0.654 − 0.127 1.434 0.218 − 0.293 0.729 0.79; 0.43 No reflection algorithm 0.435 − 0.177 1.047 0.734 0.155 1.314 0.639 − 0.150 1.427 0.212 − 0.287 0.712 a 5 × 5 Window vs. 7 × 7 were compared to give this t -test result.
whencomparing5×5and7×7tests,thereis asignificant dif-ferencebetweenSSIM(t=2.34,p-value=0.02).Whenthereflection algorithmwasnotapplied,thevalueofSSIMwasnotaffected sig-nificantlybutthere wasa reductioninthespatialextentofthe mapcomparison(asvaluesforedgecellscouldnotbecalculated) (AppendixBinSupplementary).
4. Discussion
Wehavedescribedanapproachtoobjectivelycomparespatial patternsbetweentwocontinuousvaluedmaps.Weenhancedthe originalSSIMindex(Wangetal.,2004)byincorporating uncer-taintyfromunderlyingmapsintothecomparisoncalculationand correcting for edge effects. Application of the SSIM approach, includingourenhancements,wasdemonstratedwithacasestudy using spermwhale distributiondata in theMediterranean Sea. Quantitativemapcomparisontoolsarecurrentlylimitedintheir extentandapplicationintheecologicalliterature(Hagen-Zanker,
2006b;Robertson etal.,2014), possiblybecauseecologicaldata
havecharacteristicpropertiessuchascontinuousvaluesand inher-ent spatialdependencies that make quantifyingthe underlying spatialstructure betweengeographically referencedmaps chal-lenging.Aswellasaccountingforthesecharacteristics,theSSIM indexhasseveralkeyadvantagesmakingitidealforbroader ecolog-icalapplications.First,themethodologycanbeeasilyimplemented regardlessofthepredictionorestimationmethodusedtoobtain the underlying maps. For instance, a useful application of the methodwouldbetocomparetwomapswheredifferent statisti-calmethodswereusedtoaddresssimilarquestions.Second,the SSIMindex producesa number of underlyingstatistics,as well asanoverallmeasureofsimilarityinspatialstructure. By com-paringlocalmeans,variances,andcovariancebetweenunderlying maps,differentaspectsofspatialpatternsarecharacterised, poten-tiallyprovidinginsightintounderlyingprocessesthatdrivethese patterns.Finally,thesizeofthelocalneighbourhoodinthemap comparisoncalculationisuser-defined.Priorknowledgeofspatial scaleofthedatacanbeusedtoinformthemapcomparisonanalysis, providingmoremeaningfulresults.
Dependentonthesizeofthelocalneighbourhood,edgeeffects occurwhencomparingmapsbecausenon-valuedcellsbeyondthe boundaryofthestudyareaareincluded.Toensurethemap com-parisonproducedthesamespatialextentastheunderlyingmaps,a reflectionalgorithmwaschosentocorrectforedgeeffectsbecause ofitsabilitytodealwithcomplexedgesandeaseof implementa-tion.Thealgorithmreflectedknowndataalongedgestoextrapolate outsideofthestudyarea.Alimitationofthismethodisthatitcan emphasisefine-scaleorlocalpatternsinareaswhereitis imple-mented,andsocareshouldbetakenwheninterpretingresultsclose toedgesinthestudyarea.
Thedefinitionofspatiallylocalneighbourhoodsandtheeffects oftheirsizehavebeenwellstudied(Chefaoui,2014;Longetal.,
2010;Zurlinietal.,2007).Inecology,localneighbourhoodsizemust
ecolog-Fig.4. Mapcomparisonbetweenthepredictedprobabilityofoccurrenceofgroupandsingletonspermwhalesusingbootstrappeduncertaintyfromthehabitatpreference modelsofbothsocialassemblages.Areflectionalgorithmwasusedtocounteractinternaledgeeffects.FocusingontheareaofinterestsouthofMallorca:(a)SimilarityIn Means(0–1);(b)SimilarityInVariance(0–1);(c)SimilarityInPatternofspatialcovariance(−1to1);and(d)StructuralSimilarityindex(−1to1).
ically.WerecommenduseofalocalneighbourhoodintheSSIM indexthatbearsrelationto:(1)thescaleofunderlyingdata(e.g. movementofanimals)representedinthemapsbeingcompared, (2)ecologicalquestionsbeinginvestigatedthroughmap compar-isons,and,(3)scale-dependentpatternsofunderlyingecological processesbeinginvestigated.
4.1. Casestudy
Themapcomparisonshoweddifferencesinspaceusebetween groupsandsingletonsperm whales.Theareasouth ofMallorca wasalsoidentifiedbyPirottaetal.(2011)asimportanttoboth social assemblages.Data samplingeffortwasgreatesthere and thereforegroupsandsingletonmodelssufferedlessfromsampling biasthaninotherareas.Inthisarea,bothsocialassemblageshad highpredictedprobabilityofoccurrence.Statisticaldifferencesin patternsofspace-usewereassociatedwithfine-scalefeaturesat depthsbetween1000and2000m,andtheprobabilityof occur-renceforbothsocialassemblageswaspreviouslyfoundtobedriven bybathymetricfeatures(Pirottaetal.,2011).Althoughthesexof animalsincludedinthestudywasnotverified,singletonsshowed divingbehaviourtypicalof solitarymales,andgroups ofsperm whalesaregenerallyassociatedwithadultfemalesandtheir imma-tureoffspring(Drouotetal.,2004;Whitehead,2003).Whitehead
(2003)suggestedthatreducedforagingsuccessformalesinareas
wherebothsocialassemblagesexistmaybearesultofresource
competition.Ourresultsrevealthatgroupsandsingletonsdo inter-actspatiallyinsomemutuallyexclusiveway.Thishasimplications forbothunderstandinglocalspaceuse,andinformingmore gen-eralhypothesesabouttheevolutionofextremebehaviouralsexual dimorphisminspermwhales(WhiteheadandWeilgart,2000).The resultsprovidea specific targetareasothat efficientresources canbeputinto studyingsexualsegregationof groupsand sin-gletons.Hypothesescouldbeinvestigatedtodeterminewhether patternsofmutuallyexclusive(presumed)foragingofgroupsand individualsshowstableresourcepartitioning(inwhichcaseboth socialassemblagesmaybeforagingoptimally),orwhether pat-ternsaretheresultofonesocialassemblagebeingoutcompeted andforcedtoutilisesub-optimalhabitat.Sensitivitytestsindicated thatcomparisonresultswereaffectedbyneighbourhoodsize,and anyinterpretationshouldtakeaccountofthis.
4.2. Broaderapplicationsandfurtherdevelopment
space-use. There are many applications to spatial ecology problems suchasidentifyingareasofconflictbetweenanthropogenic activ-itiesandwildlife:depredationondomesticlivestockandfarmed speciesbyapexpredators(Berlandetal.,2008;Rippleetal.,2014;
Suryawanshietal.,2013).Animportantapplicationisthe
assess-mentofchangeindistributionbetweenandwithinspecies,suchas comparingdensitymapsobtainedusingdifferentmethods(Bailey etal.,2014),assessingcompetitionandspatialsegregationbetween species (Suryawanshiet al.,2013; Wilson, 2010), and seasonal changesindistribution(Millspaughetal.,2015).
TheMarineStrategyFrameworkDirectiveusesan ecosystem-basedapproachtomanagementofanthropogenicactivitieswithin themarineenvironment(Oleninetal.,2010).Underthis frame-work,ecosystems are assessed through a set of environmental abundanceanddistributionindicatorstodetermineconservation status. Spatially-explicit indicators such as biodiversity indices (speciesrichness and diversity)present mean values over time
(Piroddietal.,2015).TheSSIMindexandenhancementspresented
herecanbeusedtoelevatetheseindicatorstoaspatio-temporal contextandassessbiodiversityovertime.Whenusedin conjunc-tionwithabundanceestimates,thesecanfurtherinformthespatial managementprocess.
Themethodologycouldbedevelopedfurther.Currently,spatial resolutionandextentofthemapsbeingcomparedmustbe regu-larlyspacedandidentical,andthecasestudyusedtodemonstrate the methodology benefitted from having these characteristics. However,comparinganimaldistributions(e.g.usinglinetransect datafromanimal sightingssurveys) mayresult invarying spa-tialresolutionandextentbecausesamplingeffortandsurveyarea canchangeovertime. AdaptingSSIMmethodologytoallowfor mapswithnon-regularlatticesandpoint-processpatternstobe compared,wouldbebeneficialfor effectiveanalyses.Forlonger time-series(Baileyetal.,2014)ormultiplespeciescomparisons
(Wilson,2010),mapcomparisonfunctionalitycouldbeextended
tocomparemorethantwomapsatonce,eithersequentially,or throughpair-wisecomparisons.
5. Conclusions
TheSSIMindexandenhancementspresentedhereoffera com-prehensivetooltoobjectivelycomparespatiallyexplicitecological datawithinan implementableframework. Anadvantageofthe SSIMindexisthatdifferentaspectsofspatialcomparisoncanbe investigated:mapsofSIM,SIV,andSIP(relatingtosimilaritiesin localmeans,variances,andcovariance,respectively)canbe calcu-latedtorevealspatialpatternsthatcannotbeseenthroughvisual inspectionoftheunderlyingmaps.TheSSIMmetricsummarises SIM,SIV,and SIP intoone mapbecausesummary statistics are oftenrequiredtocondenseinformation.Thiscanbefurther sum-marisedbycalculatingthemeanoverSSIMtogiveasinglevalue representingsimilaritybetweentheunderlyingmaps.
WepresentedenhancementstotheSSIMindexby incorporat-inguncertaintyfromtheunderlyingmapsandcorrectingforedge effectssothatthemethodologycanbebroadlyappliedtomany typesofspatialecologicaldata.Usinganecologicalcasestudyto comparegroupsandsingletonsspermwhaledistributioninthe MediterraneanSea,wedemonstratedthepresenceoflocal-scale spatialstructurethatcouldnotbedetectedeithervisuallyorusing mapsubtractiontechniques.Wefoundthatintheseareaswhere (presumed)foragingwastakingplace,singletonsand groupsof whaleswerespatiallymutuallyexclusive.Thisenabledusto rec-ommendthatfuturebehaviouralstudiesfocusingoninteractions betweensingletonsandgroups ofwhales whilstforaging could mosteffectivelybecarriedout intheareasofinterest wehave identified.
Acknowledgements
E.L.J. was funded under Scottish Government grant
MMSS001/11. Sperm whale data were collected with support from One World Wildlife, the Natural Environment Research Council(NER/I/S/2002/00632),Whaleand DolphinConservation (WDC), and J.M. Brotons of the Balearic Government Office of FisheriesManagement.L.R.wassupportedbytheMASTSpooling initiative,fundedbytheScottishFundingCouncil(HR09011)and contributinginstitutionsandtheirsupportaregratefully acknowl-edged.ThankyoutoJ.B.Illian,G.D.Ruxton,andtwoanonymous reviewersforreviewinganearlierversionofthemanuscriptand providingvaluablefeedback.Thismanuscriptpartiallyfulfilsthe PhDsubmissionofE.L.J,part-fundedbyCREEM,UniversityofSt Andrews.
Appendices. SupplementaryMaterial
Supplementarydataassociatedwiththisarticlecanbefound, intheonlineversion,athttp://dx.doi.org/10.1016/j.ecolind.2016.
05.051.
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