<%BANNER%>

Conducting Inference on Ripley's K-Function of Spatial Point Patterns with Applications

MISSING IMAGE

Material Information

Title:
Conducting Inference on Ripley's K-Function of Spatial Point Patterns with Applications
Physical Description:
1 online resource (170 p.)
Language:
english
Creator:
Hyman, Michael A
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Interdisciplinary Ecology
Committee Chair:
Young, Linda
Committee Co-Chair:
Staudhammer, Christina L
Committee Members:
Cropper, Wendell P, Jr
Bliznyuk, Nikolay A

Subjects

Subjects / Keywords:
bootstrap -- inference -- pattern -- process -- spatial
Interdisciplinary Ecology -- Dissertations, Academic -- UF
Genre:
Interdisciplinary Ecology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
In many sciences, spatially-referenced data are collected and analyzed. In some cases, these data represent the locations of a set of events recorded over an area. Collections of these data are referred to as point patterns and the underlying distributions determining these data are point processes. Specifically, when data are collected in 2-dimensional space, the underlying distributions for these data are referred to as spatial point processes. In these cases, analysis is typically focused on the number of points observed and the locations of these events relative to one another.  Many summary functions have been developed to describe the interaction among points of spatial point patterns. Ripley's K-function is a commonly used function to describe the spatial interaction of an observed set of points at a range of distances. The statistical properties of this function and its estimators are unknown for most cases. Thus, empirical methods are typically used to to conduct inference on the K-function of an observed point pattern.  In this work, we propose several new inferential methods to conduct inference on Ripley's K-function. A new method of setting confidence intervals for Ripley's K-function is proposed using a bootstrap technique for spatial point patterns. The proposed method accounts for the intensity and interaction among points in the pattern and adjusts the bootstrap sample accordingly. Confidence intervals are estimated using the quantiles from bootstrap estimates of the K-function. The variance of the proposed bootstrap estimator more closely approximates the variance of the estimator of the K-function than for many current methods of bootstrapping spatial point patterns. A simulation study is conducted to compare this new method to current methods of interval estimation. The percent coverage of the resulting confidence intervals for the K-function and confidence interval widths are determined for the proposed method and current bootstrap methods using point processes with different intensities and interactions among points.  A hypothesis test used to compare the K-function across multiple observed patterns is proposed. The purpose of the proposed test is to compare the K-function from single realizations of spatial point processes. Here, two test statistics are proposed using different methods to account for the heteroskedasticity of the K-function at larger distances. A permutation test is used to calculate $p$-values for the tests. A simulation study compares the proposed test to an existing permutation test using processes with varying intensities and interactions among points. The size of the proposed tests are better controlled when testing patterns with different intensities.  The proposed methods for conducting inference on Ripley's K-function are applied to several point patterns recorded at the Joseph W. Jones Ecological Research Center. These patterns represent the locations of longleaf pine trees on plots with different understory composition and different harvesting schemes. The interaction of adult trees and juvenile pine trees is assessed for plots with different treatments and/or forest characteristics. Conclusions drawn from this analysis are helpful for management and conservation efforts of longleaf pine forests.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Michael A Hyman.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Young, Linda.
Local:
Co-adviser: Staudhammer, Christina L.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045916:00001

MISSING IMAGE

Material Information

Title:
Conducting Inference on Ripley's K-Function of Spatial Point Patterns with Applications
Physical Description:
1 online resource (170 p.)
Language:
english
Creator:
Hyman, Michael A
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Interdisciplinary Ecology
Committee Chair:
Young, Linda
Committee Co-Chair:
Staudhammer, Christina L
Committee Members:
Cropper, Wendell P, Jr
Bliznyuk, Nikolay A

Subjects

Subjects / Keywords:
bootstrap -- inference -- pattern -- process -- spatial
Interdisciplinary Ecology -- Dissertations, Academic -- UF
Genre:
Interdisciplinary Ecology thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
In many sciences, spatially-referenced data are collected and analyzed. In some cases, these data represent the locations of a set of events recorded over an area. Collections of these data are referred to as point patterns and the underlying distributions determining these data are point processes. Specifically, when data are collected in 2-dimensional space, the underlying distributions for these data are referred to as spatial point processes. In these cases, analysis is typically focused on the number of points observed and the locations of these events relative to one another.  Many summary functions have been developed to describe the interaction among points of spatial point patterns. Ripley's K-function is a commonly used function to describe the spatial interaction of an observed set of points at a range of distances. The statistical properties of this function and its estimators are unknown for most cases. Thus, empirical methods are typically used to to conduct inference on the K-function of an observed point pattern.  In this work, we propose several new inferential methods to conduct inference on Ripley's K-function. A new method of setting confidence intervals for Ripley's K-function is proposed using a bootstrap technique for spatial point patterns. The proposed method accounts for the intensity and interaction among points in the pattern and adjusts the bootstrap sample accordingly. Confidence intervals are estimated using the quantiles from bootstrap estimates of the K-function. The variance of the proposed bootstrap estimator more closely approximates the variance of the estimator of the K-function than for many current methods of bootstrapping spatial point patterns. A simulation study is conducted to compare this new method to current methods of interval estimation. The percent coverage of the resulting confidence intervals for the K-function and confidence interval widths are determined for the proposed method and current bootstrap methods using point processes with different intensities and interactions among points.  A hypothesis test used to compare the K-function across multiple observed patterns is proposed. The purpose of the proposed test is to compare the K-function from single realizations of spatial point processes. Here, two test statistics are proposed using different methods to account for the heteroskedasticity of the K-function at larger distances. A permutation test is used to calculate $p$-values for the tests. A simulation study compares the proposed test to an existing permutation test using processes with varying intensities and interactions among points. The size of the proposed tests are better controlled when testing patterns with different intensities.  The proposed methods for conducting inference on Ripley's K-function are applied to several point patterns recorded at the Joseph W. Jones Ecological Research Center. These patterns represent the locations of longleaf pine trees on plots with different understory composition and different harvesting schemes. The interaction of adult trees and juvenile pine trees is assessed for plots with different treatments and/or forest characteristics. Conclusions drawn from this analysis are helpful for management and conservation efforts of longleaf pine forests.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Michael A Hyman.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Young, Linda.
Local:
Co-adviser: Staudhammer, Christina L.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045916:00001


This item has the following downloads:


Full Text

PAGE 1

CONDUCTINGINFERENCEONRIPLEY'SK-FUNCTIONOFSPATIALPOINTPROCESSESWITHAPPLICATIONSByMICHAELALLENHYMANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

PAGE 2

c2013MichaelAllenHyman 2

PAGE 3

TomylovingfamilyandfriendsTherstlawofecologyisthateverythingisrelatedtoeverythingelse.-BarryCommoner 3

PAGE 4

ACKNOWLEDGMENTS Firstandforemost,Iwouldliketoacknowledgemyadvisor,Dr.LindaJ.Young.Dr.Young,youweretopersontoconvincemetoachievethisdegreefouryearsagoandthepersontopushmeeverystepofthewaytoseeitnished.Throughoutthepastfouryears,youhavedevotedanenormousamountofyourfreetimeintohelpingandteachingme.YouhavealsograntedmemanyincredibleopportunitiesandIthankyouforeveryoneofthem.YouhavegoneaboveandbeyondtheroleofanadvisorandIamforevergrateful.Ihavelearnedsomuchaboutstatisticsandlifeandalsohavehadalotoffunworkingwithyou.Youhavetaughtmetostaycalmwhenthingsdon'tworkoutandhowtoidentifytheproblem,whetheritisbigorsmall.Thankyouforeverythingyouhavedone.IwouldalsoliketoacknowledgeallofmyprofessorsattheUniversityofFlorida.Specically,IwouldliketothankDr.GeorgeCasellaandDr.NikolayBlitznukforgoingoutoftheirwaytohelpstudentsinIFASstatistics.IwouldalsoliketoacknowledgeDr.ChristinaStaudhammerforintroducingmetoappliedstatisticalresearchandDr.MihaiGiurcanuforalwaysndingthetimetohelpmewhenIneededit.Nikolay,thankyoufortakingtheinitiativetostarttheRworkshopthispastyear.NotonlyhaveIbecomeabetterprogrammer,butIwouldnothavenishedthesimulationsforthisdissertationwithoutlearningtousetheHPC.Christie,beforebecomingoneofyourstudentsIhadspenttwoyearsinclassroomslearningthetheoreticalcomponentsofstatistics.WorkingontherstapplicationsinspiredaloveinthesubjectthatbeforeIhadonlyappreciatedandgavemethemotivationtocontinuelearning.Thankyouforstickingbyasmyco-advisordespitethedistance.Dr.Casella,youhaveinspiredsomanypeoplewithyourpassionforstatisticsandloveoflife.Despitebeingincrediblybusy,youalwaysfoundtimetohelpstudents,nomatterhowsmalltheproblem.YouareatrueinspirationandIamsohappytohavehadtheopportunitytoknowyouandlearnfromyou. 4

PAGE 5

Finally,Iwouldliketoacknowledgemyfamilyandfriends.TomyMomandDad,thankyouforconvincingme(ormakingme)gotograduateschool.Youhaveshownyourloveandsupportinsomanyways.ThankyouforalwaysbeingtherewhenIneededyou.TomysisterCaseyandbrother-in-lawElliot,thankyoufortheendlesssupportandencouragement.Tomygirlfriend,Whitney,thankyouforyourloveandsupportoverthepastfouryears.Iwouldnothavebeenabletodothiswithoutyouworkingbymyside.Finally,I'dliketoacknowledgeallofmyfriendsintheSNREandstatisticsdepartment.ThankyouEmilyandDanforalwaysbeingtherewhenahappyhourwasneeded.NateandKenny,thankyouforbeingsuchgoodfriendsandmakingthesepastfewyearssomuchfun.Iwouldnothavebeenabletocompletethisdegreewithoutyouguyswalkingtheroadaheadofme.Thankyouforallyourhelp. 5

PAGE 6

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 12 CHAPTER 1BACKGROUNDANDMOTIVATION ........................ 14 1.1Introduction ................................... 14 1.2PointProcessSummary ............................ 15 1.3InferenceonPointProcesses-CondenceIntervals ............ 22 1.4InferenceonPointProcesses-HypothesisTesting ............. 34 1.5Objectives .................................... 43 2CONFIDENCEINTERVALSFORRIPLEY'SK-FUNCTION ........... 47 2.1Introduction ................................... 47 2.2BootstrappingaSpatialPointPattern ..................... 48 2.3NetworkResamplingtoConstructCondenceIntervalsofK(r) ...... 52 2.4UnbiasednessofBootstrappedEstimatorofK(r) .............. 55 2.5SimulationStudy ................................ 58 2.6Results ..................................... 60 2.7Discussion ................................... 65 3HYPOTHESISTESTINGFORRIPLEY'SK-FUNCTION ............. 89 3.1Introduction ................................... 89 3.2Hahn's(2012)StudentizedPermutationTest ................. 93 3.3ProposedTestStatistic1 ........................... 96 3.4ProposedTestStatistic2 ........................... 100 3.5SimulationStudy ................................ 101 3.6Results ..................................... 104 3.7Discussion ................................... 108 4APPLICATIONOFMETHODSWITHJOSEPHW.JONESECOLOGICALRESEARCHCENTERDATA .................................... 131 4.1Introduction ................................... 131 4.2JosephW.JonesEcologicalResearchCenter ................ 132 4.3ExploratoryDataAnalysis ........................... 136 4.4EstimationofCondenceIntervalsfortheK-Function ............ 139 4.5HypothesisTestingoftheK-Function ..................... 142 6

PAGE 7

4.5.1Hypothesis1 .............................. 146 4.5.2Hypothesis2 .............................. 147 4.5.3Hypothesis3 .............................. 147 4.6Discussion ................................... 148 5FUTUREWORK ................................... 157 5.1EstimatingCondenceIntervalsfortheK-function ............. 157 5.2AppropriateNumberofNetworkstoResample ............... 159 5.3ExtensiontoInhomogeneousSpatialPointProcesses ........... 160 5.4BayesianMethodsofInference ........................ 160 5.5HypothesisTestingfortheK-function ..................... 162 REFERENCES ....................................... 164 BIOGRAPHICALSKETCH ................................ 170 7

PAGE 8

LISTOFTABLES Table page 2-1Theprocessesandtheirrespectiveparametersusedinthesimulationstudy. 68 4-1NumberofeachtreeclassicationobservedineachplotoftheJoseph.W.JonesEcologicalResearchCenter. ........................ 150 4-2EstimatedDeviationfromstationarityforeachpatternobservedateachplot.TheunitofmeasurementforeachPlotis1meterx1meter. .......... 150 4-3P-valuesfortestsofHypothesis1usingadultandjuveniletreesinallplots. 150 4-4P-valuesfortestsofHypothesis2usingadulttreesinPlot1(wiregrassunderstory)andPlot2(old-eldplot). ............................ 150 4-5P-valuesfortestsofHypothesis2usingjuvenilepinetreesinPlot1(wiregrassunderstory)andPlot2(old-eldplot). ...................... 150 4-6P-valuesfortestsofHypothesis3usingjuvenilepinetreesinPlot1(singletreeharvesting)andPlot3(control-noharvesting). .............. 151 8

PAGE 9

LISTOFFIGURES Figure page 1-1SamplepatternsandtheirresultingKandLfunctions .............. 45 1-2Exampleoftilingmethod .............................. 46 1-3ExampleofLohandStein'smarkedpointmethod ................. 46 2-1Exampleoftoriodalwrapping ............................ 69 2-2ExampleofLohandStein'smarkedpointmethod ................. 70 2-3Dendrogramofnetworkingmethod ......................... 71 2-4Realizationsofpointpatternsforcondenceintervalsimulation ......... 71 2-5PercentcoveragesforPoissonpatternswithintensity=100 ............ 72 2-6PercentcoveragesforPoissonpatternswithintensity=250 ............ 72 2-7PercentcoveragesforPoissonpatternswithintensity=500 ............ 73 2-8Percentcoveragesforsoftcorepatternswithintensity=100 ............ 73 2-9Percentcoveragesforsoftcorepatternswithintensity=250 ............ 74 2-10Percentcoveragesforsoftcorepatternswithintensity=500 ............ 74 2-11PercentcoveragesforMaternclusteredpatternswithintensity=100 ....... 75 2-12PercentcoveragesforMaternclusteredpatternswithintensity=250 ....... 76 2-13PercentcoveragesforMaternclusteredpatternswithintensity=250 ....... 77 2-14PercentcoveragesforMaternclusteredpatternswithintensity=500 ....... 78 2-15CondenceintervalwidthsforPoissonpatternswithintensity=100 ....... 79 2-16CondenceintervalwidthsforPoissonpatternswithintensity=250 ....... 80 2-17CondenceintervalwidthsforPoissonpatternswithintensity=500 ....... 81 2-18Condenceintervalwidthsforsoftcorepatternswithintensity=100 ....... 82 2-19Condenceintervalwidthsforsoftcorepatternswithintensity=250 ....... 83 2-20Condenceintervalwidthsforsoftcorepatternswithintensity=500 ....... 84 2-21CondenceintervalwidthsforMaternclusteredpatternswithintensity=100 .. 85 2-22CondenceintervalwidthsforMaternclusteredpatternswithintensity=250 .. 86 9

PAGE 10

2-23CondenceintervalwidthsforMaternclusteredpatternswithintensity=250 .. 87 2-24CondenceintervalwidthsforMaternclusteredpatternswithintensity=500 .. 88 3-1SizesoftestsforPoissonpointpatternsofvaryingintensities .......... 111 3-2Sizesoftestsforsoftcorepointpatternsofvaryingintensities .......... 112 3-3SizesoftestsforMaternpointpatterns1ofvaryingintensities ......... 113 3-4SizesoftestsforMaternpointpatterns2ofvaryingintensities ......... 114 3-5Sizesoftestsforhardcorepointpatternsofvaryingintensities ......... 115 3-6SizesoftestsforPoissonpointpatternswhenpatternshavedifferentintensities 116 3-7PowersoftestscomparingMatern1patternsandPoissonpatternsofvaryingintensities ....................................... 117 3-8PowersoftestscomparingMatern2patternsandPoissonpatternsofvaryingintensities ....................................... 118 3-9PowersoftestscomparingsoftcorepatternsandPoissonpatternsofvaryingintensities ....................................... 119 3-10PowersoftestscomparinghardcorepatternsandPoissonpatternsofvaryingintensities ....................................... 120 3-11SizesoftestsforPoissonpointpatternsusingdifferentnumbersofquadrats .. 121 3-12SizesoftestsforMatern1pointpatternsusingdifferentnumbersofquadrats 122 3-13SizesoftestsforMatern2pointpatternsusingdifferentnumbersofquadrats 123 3-14Sizesoftestsforsoftcorepointpatternsusingdifferentnumbersofquadrats 124 3-15Sizesoftestsforhardcorepointpatternsusingdifferentnumbersofquadrats 125 3-16SizesoftestsforPoissonpointpatternswithdifferentintensitiesanddifferentnumbersofquadrats ................................. 126 3-17PowersoftestscomparingMatern1patternsandPoissonpatternsusingdifferentnumbersofquadrats ................................. 127 3-18PowersoftestscomparingMatern2patternsandPoissonpatternsusingdifferentnumbersofquadrats ................................. 128 3-19PowersoftestscomparingsoftcorepatternsandPoissonpatternsusingdifferentnumbersofquadrats ................................. 129 3-20PowersoftestscomparinghardcorepatternsandPoissonpatternsusingdifferentnumbersofquadrats ................................. 130 10

PAGE 11

4-1LocationsoftreesinthreeplotsoftheJosephW.JonesResearchCenter ... 151 4-2EstimatedK-functionsfrompatternsoftreesinthreeplots ............ 152 4-3Cross-Kfunctionsofadultandjuvenilepinetrees ................. 153 4-4Condenceintervalsforadulttrees,Plot1 ..................... 154 4-5Condenceintervalsforjuvenilepinetrees,Plot1 ................ 154 4-6Condenceintervalsforadulttrees,Plot2 ..................... 155 4-7Condenceintervalsforjuvenilepinetrees,Plot2 ................ 155 4-8Condenceintervalsforadulttrees,Plot3 ..................... 156 4-9Condenceintervalsforjuvenilepinetrees,Plot3 ................ 156 11

PAGE 12

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCONDUCTINGINFERENCEONRIPLEY'SK-FUNCTIONOFSPATIALPOINTPROCESSESWITHAPPLICATIONSByMichaelAllenHymanAugust2013Chair:LindaJ.YoungMajor:InterdisciplinaryEcologyInmanysciences,spatially-referenceddataarecollectedandanalyzed.Insomecases,thesedatarepresentthelocationsofasetofeventsrecordedoveranarea.Collectionsofthesedataarereferredtoaspointpatternsandtheunderlyingdistributionsdeterminingthesedataarepointprocesses.Specically,whendataarecollectedin2-dimensionalspace,theunderlyingdistributionsforthesedataarereferredtoasspatialpointprocesses.Inthesecases,analysisistypicallyfocusedonthenumberofpointsobservedandthelocationsoftheseeventsrelativetooneanother.Manysummaryfunctionshavebeendevelopedtodescribetheinteractionamongpointsofspatialpointpatterns.Ripley'sK-functionisacommonlyusedfunctiontodescribethespatialinteractionofanobservedsetofpointsatarangeofdistances.Thestatisticalpropertiesofthisfunctionanditsestimatorsareunknownformostcases.Thus,empiricalmethodsaretypicallyusedtotoconductinferenceontheK-functionofanobservedpointpattern.Inthiswork,weproposeseveralnewinferentialmethodstoconductinferenceonRipley'sK-function.AnewmethodofsettingcondenceintervalsforRipley'sK-functionisproposedusingabootstraptechniqueforspatialpointpatterns.Theproposedmethodaccountsfortheintensityandinteractionamongpointsinthepatternandadjuststhebootstrapsampleaccordingly.CondenceintervalsareestimatedusingthequantilesfrombootstrapestimatesoftheK-function.Thevarianceoftheproposedbootstrap 12

PAGE 13

estimatormorecloselyapproximatesthevarianceoftheestimatoroftheK-functionthanformanycurrentmethodsofbootstrappingspatialpointpatterns.Asimulationstudyisconductedtocomparethisnewmethodtocurrentmethodsofintervalestimation.ThepercentcoverageoftheresultingcondenceintervalsfortheK-functionandcondenceintervalwidthsaredeterminedfortheproposedmethodandcurrentbootstrapmethodsusingpointprocesseswithdifferentintensitiesandinteractionsamongpoints.AhypothesistestusedtocomparetheK-functionacrossmultipleobservedpatternsisproposed.ThepurposeoftheproposedtestistocomparetheK-functionfromsinglerealizationsofspatialpointprocesses.Here,twoteststatisticsareproposedusingdifferentmethodstoaccountfortheheteroskedasticityoftheK-functionatlargerdistances.Apermutationtestisusedtocalculatep-valuesforthetests.Asimulationstudycomparestheproposedtesttoanexistingpermutationtestusingprocesseswithvaryingintensitiesandinteractionsamongpoints.Thesizeoftheproposedtestsarebettercontrolledwhentestingpatternswithdifferentintensities.TheproposedmethodsforconductinginferenceonRipley'sK-functionareappliedtoseveralpointpatternsrecordedattheJosephW.JonesEcologicalResearchCenter.Thesepatternsrepresentthelocationsoflongleafpinetreesonplotswithdifferentunderstorycompositionanddifferentharvestingschemes.Theinteractionofadulttreesandjuvenilepinetreesisassessedforplotswithdifferenttreatmentsand/orforestcharacteristics.Conclusionsdrawnfromthisanalysisarehelpfulformanagementandconservationeffortsoflongleafpineforests. 13

PAGE 14

CHAPTER1BACKGROUNDANDMOTIVATION 1.1IntroductionSpatially-referenceddataarecommonlyobservedinecologyaswellasotherdisciplines.Ingeneral,spatialdatapresentschallengesinanalysisduetospatialcorrelationamongtheobservations.Thetwobasictypesofspatialdataaregeostatisticalandpointprocesses.Ingeostatistics,geographically-referenced,quantitativerandomvariablesareobservedatasetoflocations.Inferenceisconductedonthedistributionofthisquantitativevariable,adjustingforthespatialcovariancestructure.Apointprocessisanunderlyingprocessgivingrisetospatially-referencedevents,andaspatialpointpatternisarealizationofthatprocess.Thedistributionofallpossiblerealizationsisapointprocessdistribution.Forpointprocesses,theeventsthemselvesaretheobservations,andinterestisinthenumberofpointslocatedinanareaandthepositionofthepointsrelativetooneanother.Thefocusofthisworkisinferentialproceduresforspatialpointprocesses.Aspatialpointpatternmayberecordedasamarkedpointprocess,inwhichacategoricalorquantitativemarkisassociatedwitheachofthepoints.Thismarkmightrepresentthespeciesordiameterofatreeobservedataparticularlocation.Markedpointprocessescanhelpassesstheinteractionamongseveralclassicationsofpoints.Apointpatternisobservedind-dimensionalspace,wheredistypically1,2,or3dimensionsformostapplications.Thelocationsofcellularphonetowersinastateisanexampleofa2-dimensionalspatialpointpattern.Similarly,thelocationsofcellnucleionapieceofbraintissueorthelocationsofgalaxyclustersintheobservableuniverseareexamplesof2-dimensionaland3-dimensionalpointpatterns,respectively,observedatvastlydifferentscales.Inecology,pointprocessescanrepresentthelocationsofsomespeciesofinterestorthelocationsatwhichaneventthataffectstheecosystemoccurs.Examplesincludethelocationsofnestingsitesofendangeredspeciesorthelocations 14

PAGE 15

ofaparticularinvasiveplantspeciesrecordedoveranareaofspace.AspecicexampleisgiveninChapter4wherethemarkedpointpatternscontainthelocationsoftreesinseveralplotsoflandintheJosephW.JonesEcologicalResearchCenter.Themarksareanageclassication,labelingeachtreeaseitheradultorjuvenile.Aswiththesedata,spatially-referencedecologicaldataisrecordedind=2dimensions.Fortheremainderofthisdissertation,apointprocesswillrepresentthegeneralprocessinsomed-dimensionalspace,andaspatialpointprocesswillspecicallyrefertoapointprocessin2-dimensions.Recently,ecologicalmodelingmethodshavebeenproposedthatutilizepointprocessdistributions( IllianandBurslem 2007 ; Illianetal. 2009 ; WartonandShephard 2010 ).However,manyofthestatisticalpropertiesofthedistributionsthemselvesareunknown,andconductinginferenceischallenging.Summarystatisticshavebeendevelopedtoassessthequantitiesofinterestforpointprocesses(i.e.,themeannumberofpointsperunitarea,theinteractionamongpoints,etc.).Thestatisticaldistributionsofthesestatisticsaregenerallyunknown,exceptforthesimplestcases.Therefore,empiricalmethodsofconductinginferencehavebeenproposedtoanalyzeobservedpointpatterns.Thepurposeofthisdissertationistoexpanduponthesemethodsandtoproposenewmethodsforconductinginferenceonspatialpointprocesses.InChapter4,weapplythesenewandsomeexistingmethodstoseveralrecordedspatialpointpatternstodemonstratehowtheycanbeusedtoinformmanagementdecisionsinforestry. 1.2PointProcessSummaryAspatialprocessisasetofeventsZoccurringatlocationssinasetX,asubsetofRd;thatisfZ(s):s2XRdg.Forspatialpointprocesses,XisasubsetofR2.Inpointprocesstheory,theexactlocationsofthepointsfs1,s2,...,sngaretypicallylostinfavorofmoreconvenientnotation.LetNbeacountingmeasureonX.ForeachBorelsetA,N(A)representsthenumberofeventsonA.ThusN(A)=f0,1,2,...gforall 15

PAGE 16

A2X,whereXrepresentstheBorel-eldofXR2.IfN(A)isknownforeverysetA2X,thenthisisequivalenttoknowingalllocationsofeventsfs1,...sng.IfthecountingmeasureN(A)islocallynite,thenN(A)<1forallboundedsetsA2X( Cressie 1991 ).Herewealsomaketheassumptionthatpointprocessesaresimple.Thatis,theprobabilityofobservingmorethanonepointatagivenlocationis0.Thisassumptionoftenholdsinecologicalapplicationandviolationscanmakeinterpretationandsomeinferentialmethodsdifcult.Intheabovenotation,Xcanbethoughtofasasurfaceoverwhichthestatisticaldistributionisfound.ThisstatisticaldistributionisdenedbythecountingmeasureN.Foranyobserved,boundedregionAofthissurface,thenumberofpointsN(A)observedinAisarandomvariablewithaprobabilityassignedtoeachpossiblevalue(N(A)=0,1,2,....).Notatingapointprocessinthiswayallowsustoassignprobabilitydistributionstoaprocessobservedonabounded,closedregionandtoconductbasicinferenceonpointpatterns.Similartoquantitativedistributions,pointprocessescanbedenedbytheirmoments.Therst-ordermomentofapointprocessisreferredtoastherst-orderintensityoftheprocess(oftenreferredtoastheintensity)andisanalogoustothemeanofaquantitativedistribution.Formally,ifdrepresentsaLebesguemeasureofD2RdandN(ds)representsacountingmeasureontheinnitesimallysmallBorelsetdsDcenteredatpoints,then (s)=limds!0E[N(ds)] d(ds).(1)Inthecasewhered=2,d(A)representstheareaoftheBorelsetAD,andifd=3,d(A)representsthevolumeoftheballforAD.Iftheprocessisstationary,then(s)isconstantforalllocationssandtheintensity()ofapointprocessistheexpectednumberofpointsperunitarea, =E[#ofpointsperunitarea].(1) 16

PAGE 17

Iftheprocessisheterogeneous,then(s)representstheintensityoftheprocessatlocationsX.Theintensitymaybeafunctionofasetofcovariateswhosevaluesarerecordedateachlocations.Inthiscase,theaverageintensityoveranareaAcanbedeterminedby A=1 jAjZA(s)ds(1)Undertheassumptionofstationarity,anunbiasedestimatoroftheintensityofaprocessis ^=n jAj,(1)wherenistheobservednumberofpointsinregionA.HerewewillletjAjrefertotheareaofaboundedregionin2-dimensionalspace.Stationary,isotropicpointprocessescanbecategorizedasoneofthreeclasses:completelyrandom,spatiallyclustered,orregular.Apointprocessisacompletelyrandompattern,andissaidtoexhibitcompletespatialrandomnessorCSRif1)theaveragenumberofpointsperunitareaisconstantovertheentireareaAand2)thenumberofeventsintwonon-overlappingBorelsets,A1andA2,areindependentofoneanother.Inthiscase,thenumberofeventsinAisPoissondistributed( SchabenbergerandGotway 2005 ).Acompletelyspatiallyrandom(CSR)processisalsoknownasahomogeneousPoissonpointprocess.UnderCSR,eacheventhasaconstantprobabilityofoccurringatanylocationintheregion( Cressie 1991 ; Illianetal. 2008 ).Moreformally,thecountingmeasuredenedonasetAisdistributedas P(N(A)=n)=e)]TJ /F14 7.97 Tf 6.58 0 Td[((A)((A))n n!;(1)where(A)representsthevolumeofAind-dimensionalspace(theareaifd=2)andrepresentsthemeannumberofpointsperunitarea(constantatalllocations).Forspatialpointprocesses,therstnullhypothesistestedisgenerallythatapatternexhibitsCSR.IfapatternexhibitsCSR,pointsaredispersedrandomlyandaresearchercandolittlemoretosummarizeorexplainthem.Ifthisnullhypothesisisrejected,a 17

PAGE 18

researchercanconcludethatthereisheterogeneityorinteractionamongeventsandmaychoosetoinvestigatetheintensityfunctionorinterpointdependenceinthepattern( SchabenbergerandGotway 2005 ).AbinomialpointprocessiscloselyrelatedtoahomogeneousPoissonprocessandresultsfromaPoissonprocessconditionalonnpointsbeingobserved.Inthiscase,ifapatternisarealizationofahomogeneousPoissonprocesswithnpointsobservedonA,thenthenumberofpointsinanysubsetAsubAisdistributedasabinomialrandomvariablewithameanequaltonjAsubj jAj.AlthoughthehomogeneousPoissonprocessisacommonbenchmarkdistributiontotestallobservedrealizationsagainst,itisrarelyseeninpractice.Pointsrecordedindisjointregionsmightindeedbeindependentofoneanother,yetthedensityatwhichthepointsoccurmaynotbehomogeneous.Inothercases,aconstantdensitymightbeobservedacrossasamplearea,buttheeventsarenotindependentofoneanother.InteractionsmightexistamongpointsmakingitmoreorlessprobablethatanotherpointislocatednearbyrelativetoCSR.GeneralclassicationsforstationarypatternsthatdepartfromCSRareclusteredandregular.Inregularpatterns,theprobabilityofobservingapointnearanarbitrarypointinthepatternissmaller,andthustheexpectednumberofpointswithinagivenradiusofanarbitrarypointisalsosmaller,thanitisundertheassumptionofCSR.Anexampleofthismightincludethelocationsofnestingsitesofaspeciesthattendtoavoidothersduetocompetition.Forclusteredpatterns,theprobabilityofobservingpointsthatarenearbyoneanotherisgreaterthanunderCSR.ThustheexpectationofthenumberofpointswithinagivenradiusofaparticularpointisgreaterthanunderCSR.Treesmaybeclusteredatearlystagesofforestdevelopmentbecauseparenttreesdropseedsfromwhichseedlingsgrow.However,treescompeteforthesameresourcesand,aftersometime,mayhavearegularpattern.Thesecond-orderintensityofapointprocessdescribestheinteractionorrelativepositionamongpoints.Ifsiandsjaretwopointsind-dimensionalspaceanddsianddsj 18

PAGE 19

theinnitesimallysmallballscenteredatthesepoints,thenthesecond-orderintensityofapointprocessis 2(si,sj)=limd(dsi)!0,d(dsj)!0E[N(dsi)N(dsj)] d(dsi)d(dsj).(1)Interpretationofthesecondorderintensityisdifcultandsimplermeasurestoassessthedependencyamongpointsaredesired.Thus,RipleyintroducedtheuseoftheK-function(alsoknownasthereducedsecondmomentmeasure)toassessthesecond-ordercharacteristicsofstationary,isotropicprocesses( Ripley 1976 ).TheK-functionisafunctionofdistancer: K(r)=2 2Zr0x2(x)dx.(1)Forasimpleprocess,K(r)representsthenumberofpointslessthandistancerfromanarbitrarypointintheprocessandthus, K(r)=E[N(s0,r)jpointats0] .(1)Themomentsofpointprocessescanbeusedtodenesomeofthetypicalassumptionsthataremadewhenconductinginferenceonpointprocesses.Apointprocessisconsideredhomogeneousiftherst-ordermoment(theintensity)isconstantoverspace.Aprocessisconsideredstationaryiftheprocessisinvarianttotranslations( SchabenbergerandGotway 2005 ).Aprocessiscalledisotropicifitisinvarianttorotationsaroundapoint(thesecond-ordermomentdependsonlyonthedistancebetweentwoevents)( SchabenbergerandGotway 2005 ).Together,K(r)anddenetherstandsecondmomentsofastationaryandisotropicprocess( Stoyanetal. 1995 ).However,justasthemeanandcovarianceoftworandomvariablesdonotprovideacompletedescriptionoftheirbivariatedistribution,therstandsecondorderintensitymeasuresgiveanincompletedescriptionofapointprocess( BaddeleyandSilverman 1984 ). 19

PAGE 20

TheK-functionhasseveralpropertiesthatmakeitthemostcommonfunctionforassessingthesecond-orderpropertiesofanobservedpattern.BecauseK(r)representstheaveragenumberofpointswithindistancerfromanarbitrarypointinthepattern,itiseasilyinterpretable.Thesecond-orderintensityforaprocesscanbederivedifitsK-functionisknown( SchabenbergerandGotway 2005 ).TheK-functionisinvarianttotheintensityofaprocesssothesecond-ordercharacteristicsofpatternscanbecompareddespitedifferencesinthenumbersofpoints( Baddeleyetal. 2000 ).TheK-functioncanbeusedtoobservetheinteractionamongpointsatarangeofdistances( SchabenbergerandGotway 2005 ).Thiscanbeusefulforprocessesthatexhibitoneparticulartypeofinteractionatsmalldistancesandadifferentformofinteractionatlargerdistances.Finally,theK-functionisinvarianttoobservationsmissingcompletelyatrandom( SchabenbergerandGotway 2005 ).ThetheoreticalvalueforK(r)underCSRisafunctionofr,K(r)=r2,theareaofthecircleofradiusr: K(r)=E[#ofpoints
PAGE 21

Thus,inthecaseofahomogeneousPoissonprocess,L(r)=r.Figure 1-1 showsrealizationsofthreespatialpointprocesses:aCSRpattern,aclusteredpattern,andaregularpattern.ThecorrespondingKandLfunctionsarealsoshownforeachofthepatterns.ToestimateK(r),theterm2d(A)K(r)istypicallyestimatedanddividedbyanestimatorof2d(A).Anaiveestimatorof2d(A)K(r)isPx2APy6=xIfjjy)]TJ /F4 11.955 Tf 11.95 0 Td[(xjjrgwhereIrepresentstheindicatorfunctionandjj.jjrepresentsEuclideandistance.However,thisestimatorisbiasedlow.ThebiasisduetopointslyingoutsidetheboundariesofregionAthatarenotobserved,butarestillwithindistancerofapointinthepattern.Thatis,ifpointzisunobservedbecauseitliesoutsideofAbutjjx)]TJ /F4 11.955 Tf 12.95 0 Td[(zjjr,thenthispointpairisnotincludedinthesummation.Thiscanmakesubstantialdifferencesforvaluesofrthatarelargerelativetod(A).MultiplemethodscanbeusedtoobtainmoreaccurateorunbiasedestimatorsofK(r)despitebeingunabletoobservepointsoutsideoftheregion'sboundary.Mostmethodsassignaweightw(x,y)toeachpairofpoints(x,y).Ripley'sisotropicedgecorrection( Ripley 1988 )isacommonedgecorrectionweightthatisusefulundertheassumptionsofstationarityandisotropy.LetAbethe2-dimensionregioninwhicharealizationofaprocessisobserved.Letx,y2AbepointsobservedinA.IfCrepresentsthecircumferenceofthecirclecenteredatpointxandpassingdirectlythroughpointy,andCsrepresentsthelengthofCthatliesinsideA,thenweightw(x,y)=1 Cs=C.Thatis,w(x,y)isthereciprocaloftheproportionofthecircumferenceofthecirclecenteredatxandpassingthroughythatliesinsideoftheregionA.Thisimpliesthat,ifapointfelldirectlyonthestraightboundaryofanopenset,anycirclewithanarbitraryradiusrcenteredatthispointwouldfallhalfinsidetheregion.Therefore,anypointfallingwithindistancerofthispointwouldbeweightedbytwo,becauseitwouldbeequallylikelytohaveobservedanotherpointoutsideoftheboundary.Calculatingthisweightforeach 21

PAGE 22

pointpairobservedinA,anestimatorofK(r)is ^K(r)=jAj n2Xx2AXy6=xw(x,y)I(jx)]TJ /F17 11.955 Tf 11.96 0 Td[(yj2.Inaddition,otherestimatorsandedgecorrectionweightshavebeensuggestedthatresultinanunbiasedestimateofK(r)( Cressie 1991 ; Ohser 1983 ).Ripley'sweightsareeasilycalculatedforarectangularorcircularwindow;however,otherweightsmaybemoreappropriateformorecomplexwindows( Ohser 1983 ). 1.3InferenceonPointProcesses-CondenceIntervalsForspatialpointpatterns,inferencemustbeconductedontheinformationprovidedbythepointsthemselves.Becauseinterpointdistancesaresomeofthemostinformativeanddeningcharacteristicsofaparticularpointprocess,thesummarystatisticsusedtodescribeapattern,suchastheK-function,tendtobefunctionsofdistance.Second-orderanalysisofpointprocessesisarelativelyunexploredareaofstatistics,somanyoftheoriginalmethodsarethemostcommonlyusedtechniquestoday.Exactdistributionsofthecommonestimatorsandthestatisticalpropertiesoftheprocessesareunknown,exceptunderCSRorothersimpleprocesses.Thus,workonspatialpointprocessestendstobeempiricalinnature.Herewereviewthepreviousworkintheareaofintervalestimationandhypothesistestingforpointprocessesorforaspecicparameterofapointprocess.Notethat,althoughtheseworksarerelated,theymayhavedifferentobjectives.Forinstance,theK-functioncanbeidenticalfor 22

PAGE 23

processeshavingdifferentsecond-orderstructures( Baddeleyetal. 2000 ; BaddeleyandSilverman 1984 ).ThustestingtheequivalenceofK(r)formultipleprocessesisnotequivalenttotestingwhetherthesecond-orderintensityoftheprocessesisequal.However,testingtheequivalenceoftheK-functioncanresultinvaluableinformationaboutthesecond-orderstructuresoftheprocesses.Themotivationforthefollowingworksrangefromcondenceintervalcalculationandhypothesistestingundermodelspecication,totestingprocessequivalence.Ripley'sdevelopmentoftheK-function( Ripley 1976 )allowedresearcherstointerpretthespatialdependenceinherentinapointpattern.MonteCarloapproachesarethefoundationforthemostwidelyappliedinferentialmethodsforK(r)andothersummarystatisticsforpointprocesses( Barnard 1963 ; BesagandDiggle 1977 ; Chiu 2007 ; Hope 1968 ; Koen 1991 ).Mostcommonly,anobservedstatisticiscomparedtothedistributionofthestatisticundertheassumptionofCSR.Supposeanobservedpointpatterncontainsnpoints.TodetermineasimulationenvelopeforaparameterundertheassumptionofCSR,npointsaresimulatedrandomlyinaniteregionA,andtheestimateofthatparameteriscalculated.ThisisreplicatedBtimesresultinginasimulationenvelope.Forexample,tondanintervalfortheK-functionunderCSR,BhomogeneousPoissonpatternsaresimulatedand^Ki(r)fori=1,...,Bcalculated.Thenthe100%simulationenvelopeisgivenby ^Kl(r)=mini=1,...,Bf^Ki(r)gand^Ku(r)=maxi=1,...,Bf^Ki(r)g.(1)TotestwhethertheobservedpatternisfromaCSRprocess,theobservedK-functioniscomparedtothissimulatedenvelope.Manymethodsofcomparisoncanbeused,andCressie( Cressie 1991 )suggestsateststatisticoftheform TS=Z10^K(r)1=2)]TJ /F11 11.955 Tf 11.95 0 Td[(1=2r2dr(1) 23

PAGE 24

wherethisstatisticiscalculatedfortheobservedpatternandforeachofthesimulatedpatternsfromaCSRprocess.AnobservedteststatisticthatisgreaterthananycalculatedfromCSRwouldimplydeviationfromCSR,eitherfromclusteringorregularizationoracombinationofclusteringandregularization.Testsofthisnaturecanbeusedinecologytotestforpatterntransference(suchastheobservedlocationsofmigratoryanimalsoverdifferentregions)andspace-timeinteractions(suchastestingforcontagionintheobservedlocationsofaparticularevent)byspecifyinganullmodelexhibitingtheabsenceoftheseinteractionsandcomparingtheobservedandsimulatedK-functionsfromthenullmodel( BesagandDiggle 1977 ).MonteCarlomethodsfortestinganullhypothesishaveseveraladvantages.Anapproximationofthedistributionoftheteststatisticisnotnecessaryandthusthep-valuesareexactinthatsense( SchabenbergerandGotway 2005 ).Theyarealsoexibleinthatthenullhypothesiscaneasilybeadaptedtotestcomplexpointprocessdistributions.However,manycriticalchoicesarelefttotheresearcher,suchasthenumberofsimulationstoperformandthedistancetowhichtheteststatisticistobeevaluated.Diggleandothers( Diggle 1977 1979 ; HoandChiu 2006 )exploredthesechoicesaswellasadditionalteststatistics.MonteCarlomethodsarebenecialundertheassumptionofaspeciednullmodel.Methodsofmodelttingandparameterestimationexistforpointprocessdata.However,variationinrealizationsfromaparametricpointprocessmodelcanbegreat,andthepowerandcondenceoftherespectivehypothesistestsandcondenceintervalsarecontingentontheaccuracyofthettedmodels( LohandStein 2004 ).Modelttingtypicallyinvolvestheassumptionofaspecictypeofprocess,andwrongmodelspecicationcanresultinpoorperformanceofsimulationenvelopesforbothcondenceintervalestimationandhypothesistesting.Often,condenceintervalsfortheK-functionaredesiredforanobservedpointpatternwithoutspecicationofanullprocess.Replicatedpatternsfromaprocessaretypicallynotavailabletoestimatethe 24

PAGE 25

standarderrorofK(r).Asanalternative,bootstrapmethodsfordependentdatahavebeenappliedtopointpatterns( DavisonandHinkley 1997 ; LohandStein 2004 )toestimatecondenceintervalsfortheK-function,aswellasotherparameters.ThesimplestwaytoobtaincondenceintervalsfortheK-functionofapointpatternobservedoverasquareregion,calledthesplittingmethod,istodividetheregionintoNcongruentsubregionsandtocalculateNseparateestimatesof^K(r)( LohandStein 2004 ).AssumingthattheNestimatesareindependentandapproximatelynormallydistributed,anestimateforthevarianceof^K(r)canbeobtainedandthe100(1)]TJ /F11 11.955 Tf 12.36 0 Td[()%condenceintervalcanbecomputedby ^K(r)tN)]TJ /F9 7.97 Tf 6.59 0 Td[(1,=2s ^Varf^Ki(r)g N(1)where^Varf^Ki(r)gisthesamplevarianceof^K1(r),^K2(r),...,^KN(r),and^K(r)istheoverallestimateofK(r)( LohandStein 2004 ).Forpatternswithsufcientnumbersofpoints,thismethodofcondenceintervalcalculationcreatesfairlyaccurateintervals.However,theseintervalscanbewideduetobeingcalculatedfromasmallnumberofsamples(e.g.,thequadrats).Anobviouslimitationisinthedistancerangethatintervalscanbedetermined.Foraunitpatterndividedintoquadratsofarea0.25,thelengthofeachquadratedgeis0.5.CalculationsofK(r)fordistancegreaterthan0.25begintohighlyweighttheedgecorrectionbecauseahighproportionofanycircleofthatradiuswouldfalloutsideofeachquadrat.Atthesedistances,thetypeofedgecorrectionusedtoestimateK(r)hasamuchlargerinuenceontheestimatesoftheK-functionthanatshorterdistances.Issueswithsmallsamplesizes,dependenceamongquadrats,andnon-normalsamplesfromquadrats,encounteredwhencalculatingcondenceintervalsbydividingapatternintomultipleindependentsamples,ledtothedevelopmentofresamplingmethodsforpointpatterns.Hall(1985)extendedblockbootstrappingmethodsusedtoestimatethestatisticalcharacteristicsoftimeseriesdatatospatialBooleanmodels 25

PAGE 26

intwodimensions( Hall 1985 ).DavisonandHinkley(1997)gaveamoregeneralexplanationofthismethod,referringtoitastileresampling( DavisonandHinkley 1997 )andreferredtohereastiling.Inthismethod,thegoalistocreatenewpatternsthatmaintainthespatialdependenceoftheobservedpattern.IftheobservedregionAR2ispartitionedintoNdisjointtiles,A1,A2,...,AN,thestatisticofinterestcanbedenedasT=t(A1,A2,...,AN).ThenaresampledpatterniscreatedbytakingrandomsamplesoftheNdisjointtiles,A1,A2,...,AN,andabootstrappedestimateofthestatisticofinterestiscalculatedfromthisnewlycreatedpatternT=t(A1,A2,...,AN).Acommonvariationistousemoving,overlappingtilesbysettingAj=Uj+AjwhereUjisarandomvector( DavisonandHinkley 1997 ).PolitisandRomano(1992)suggestusingtoroidalwrappingbeforeresamplingsuchthattilesareallowedtofalloutsideoftheboundaryand,inthiscase,arewrappedaroundtotheoppositesideofthewindow( PolitisandRomano 1992 ).Thiscanbeaccomplishedbycreatinganewregionwhichisa3x3gridoftheobservedregion(assumingthattheobservedregionisrectangular).Thetilesthencontainidenticalpointsasiftoroidalwrappingisused.Thishelpsavoidbiascreatedbyundersamplingpointsneartheboundariesoftheobservedregion( DavisonandHinkley 1997 ).ByperformingBresamplesandcalculating^K1(r),...,^K2(r),...,^KB(r)withtheorderedestimatesofthestatisticofinterestfrombootstrapping,a100(1)]TJ /F11 11.955 Tf 12.25 0 Td[()%condenceintervalcanbecreatedforK(r)usingtheformula: h2^K(r))]TJ /F5 11.955 Tf 16.1 2.66 Td[(^K(B+1)(1)]TJ /F14 7.97 Tf 6.58 0 Td[(=2)(r),2^K(r))]TJ /F5 11.955 Tf 16.1 2.66 Td[(^K(B+1)(=2)(r)i(1)( DavisonandHinkley 1997 ).Thiscreatesanequal-tailedcondenceintervalforK(r).Itispossiblethatothermethodsofcondenceintervalestimationmightbemoreappropriate.However,interesthereisinthemethodofbootstrappingfromthepattern.Figure 1-2 showsanexampleofthecreationofanarticialrealizationfromtheprocessusingthetilemethod.Forfurtherinformationonthestatisticalpropertiesusingthis 26

PAGE 27

resamplingmethod,seeHall(1985)fora2-dimensionalspatialcaseandKunsch(1989)fora1-dimensionaltimeseriesanalysis( Hall 1985 ; Kunsch 1989 ).Theprimaryproblemwithcreatingbootstrappedpatternsusingtilingisthatwhentilesarerearrangedtoproducethenewpattern,pointsthatarenotoriginallypositionedtogethercanbeplacedincloseproximity.UndertheassumptionofCSR,theplacementofpointswithinthebootstrapsamplesisrandomandtheresultingpatternsshouldstillfollowthesamedistribution.However,iftheprocesshasobviousspatialdependence,constructionofanewpatternmightviolatetheinterpointdependencestructureintheprocess.Ifthespatialdependencebetweenpointsisrelativelyshort-ranged,andtheblocksarelargeenoughtoadequatelycapturethespatialdependencebetweenpoints,consistentresultscanbeobtained( DavisonandHinkley 1997 ).However,ifthespatialdependenceislongerranged,thismethodfailsatmaintainingtheoriginaldistributionoftheobservedpattern.Lahiri(1993)showsthatputtingindependentresampledblockstogetherdestroysthelong-rangedependenceoftheoriginalobservations( Lahiri 1993 ).Asimpleexampleisthatofahardcoreinhibited(regular)process.Inthiscase,theprobabilityofpointsfallingwithinacertainradiusofinteractionr0ofoneanotheris0.Thusfordistanceslessthanr0,theresultingK-functionisK(r)=0forr
PAGE 28

areformedwithtilesofsizejAj=N,allowingforoverlappingbetweentilesandtoroidalwrappingaroundA.Letxijandxik,j,k=1,...,nirepresenttwodistinctpointsinsubregionAi.ThentheestimateofK(r)canbecalculatedusing ^K(r)=jAj PniPniNXi=1niXj6=iwAi(xij,xik)I(jjxij)]TJ /F17 11.955 Tf 11.95 0 Td[(xijjj
PAGE 29

Thersttwomomentsoftheprocess,p1(),andp2(1,2),aredenedas p1()=limh!0P(X((,+h])>0) h (1) andp2(1,2)=limh1!0,h2!0P(X((1,1+h1])>0,X((2,2+h2])>0) h1h2, (1) respectively.Similartotheestimatorsoftherstandsecond-orderintensities,thersttwomomentsareestimatedfromanobservedpatternontheinterval(0,T]usingthefollowingequations( Brillinger 1975 ): ^p1=1 TX((0,T]) (1) (1) ^p2()=1 hTXxif#ofpointsin(xi+,xi++h)gwhere=j2)]TJ /F11 11.955 Tf 9.38 0 Td[(1jandhisawindow,orbinwidthparameter.Underseveralassumptionsregardinghigher-ordermomentsoftheprocessdenedbyBrillinger( Brillinger 1975 1978 ),theseestimatorsareapproximatelyasymptoticnormalasTincreasestoinnity.However,theasymptoticvariancefor^p1isdifculttoestimate,andtheasymptoticvarianceof^p2isapoorapproximationtothetruevarianceoftheestimator,evenforlargeintervals(largevaluesofT)( BraunandKulperger 1998 ).Thus,anewblockbootstrapapproachprovidesmoreaccuratecondenceintervals( BraunandKulperger 1998 ).BraunandKulpergerrstdescribeanapproachsimilartothetilemethoddescribedinDavisonandHinkley(1997).LetXrepresentapointprocessofwhichapatternXisobservedonaninterval(0,T].LetArepresentthesetofpoints(locations) 29

PAGE 30

ton(0,T],suchthatx(t)=1.Thenthepointpatternxcanbebootstrappedusingthefollowingmethod: 1. Takebtobesomepositiveinteger.Foreachintegeri=1,...,b,generateuniformvariatesU1,U2,...,Ubontheinterval(0,T)]TJ /F4 11.955 Tf 11.95 0 Td[(T=b]. 2. Forj=1,2,...b,seteachofthefollowing: a)Aj=(Uj,Uj+T=b]\A b)Aj=Aj)]TJ /F4 11.955 Tf 11.96 0 Td[(Uj+(j)]TJ /F9 7.97 Tf 6.58 0 Td[(1)T b c)Xj(.)=jAj\.j. 3. SetX=Pbj=1Xj.Essentially,brandomlyselectedblocksoflengthT=baretakenfromtheobservedpattern.ThepointsoccurringintheselectedblocksarerepositionedtocreateanarticialrealizationofX.Theauthorsprovidesomeasymptoticresultsforthedistributionoftheestimatoroftherstmomentfromequation 1 ( BraunandKulperger 1998 ).AswiththeHallandDavisonandHinkleymethods,theBraunandKulpergerbootstrapmethodfailstocapturethesecond-ordercharacteristicsoftheprocess.Inthesamepaper,theauthorssuggestanotherbootstrapmethodforestimatingthesecond-orderpropertiesofapointprocess,whichtheylabelthemarkedpointprocessmethod.Againusingtheirnotationforthe1-dimensionalcase,thesecond-orderpropertiesofthepointprocess,()(analogoustoK(r))areestimatedforthe1-dimensionregion(0,T].Foreachobservedpointx2(0,T],amarkissetequaltothenumberofpointsintheinterval(x+,x++h]foraxedvalueh.Theestimateofthesecondorderintensity()isgivenbytheequation: ^()=1 hTXxif#ofpointsin(xi+,xi++h]g.(1)Thefollowingtheoremisofferedbytheauthors. 30

PAGE 31

THEOREM:SupposethepointprocessXhasniteandintegrablefourthmomentdensitiesinthesenseofBrillinger(1975).Foragivenh,conditionalontheobservedpointprocessXon[0,T], p hT)]TJ /F5 11.955 Tf 7.73 -8.17 Td[(^2())]TJ /F4 11.955 Tf 11.96 0 Td[(E(2()jX))N)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(0,22(1)asT!1and2=2()+O(h).Forsmallh,thelimitingvariance22isapproximatelythelimitingvariance^2()( BraunandKulperger 1998 ).Thus,BraunandKulperger(1998)showthat,forsmallvaluesofh,thebootstrapestimatesofthesecond-ordermomentgiveapproximatelythecorrectdistributionasymptotically( BraunandKulperger 1998 ).Itisstillunclearhowcloseofanapproximationthebootstrapestimatorsaretothetruesecond-ordermoment.InasimulationstudyusingaMaternclusteredprocess,useofthemarkedbootstrapapproachledtomoreaccuratenominallevelcondenceintervalsthantheirblockedbootstrapapproach.Resultsimprovedforsmallervaluesofh.Itisalsounknownwhethertheseresultsholdindimensionsgreaterthan1.LohandStein(2004)provideanadaptationoftheBraunandKulperger(1998)markedpointmethodfor2-dimensionalpointpatterns,referredtohereasthemarkedpointmethodormarking.ForeachpointxinanobservedpatternXinregionA,amarkmx(r)isassignedfordistancer.ThemarkequalsthesumofallweightswA(x,y)forpointsywithindistancerofpointx.Thus, mx(r)=Xy:y6=xwA(x,y)Ifjjy)]TJ /F17 11.955 Tf 11.96 0 Td[(xjj
PAGE 32

markassociatedwithpointxwouldbemx(r)=wA(x,y)+wA(x,z).Thewrappedsquarerepresentstheresamplestrategy.So,whereaspointviswithindistancerofpointxintheresampledsubregion,itisnotrecordedinthemarkgiventox.Theestimateof2jAjK(r)isobtainedbyaddingupthemarksofallpointsincludedintheresample.Therefore,ifsubregionsAi,i=1,2,...,N,areusedtoresamplepointswheresubregionAicontainspointsxij,j=1,2,...,ni,eachxijhasamarkmij(r)=Py:y6=xwA(xij,y)Ify2A:jjy)]TJ /F17 11.955 Tf 11.96 0 Td[(xijjjrg.ThentheestimateofK(r)is ^K(r)=a Pni(Pni)NXi=1niXj=1mij.(1)CondenceintervalsareproducedusingEquation 1 ( LohandStein 2004 ).Themarkedpointmethodhasseveraladvantagesoverothermethodsofbootstrappingapointprocess.First,articialpointpairsarenotcreated,avoidingthebiasresultingfromrecreatingnewpatternsfromsubsamples.Also,edgecorrectionweightsarecalculatedusingtheentireobservedregion,minimizingtheirinuenceontheresultingbootstrapestimationsofK(r).Someinformationfromoutsidethesubsampledregionsisretainedintheresamples,preservingsomeofthespatialinformationthatislostinothermethods.Finally,themarkedpointmethodobtainsacomputationaladvantageasedgecorrectionweightsandmarksareonlycalculatedonceandthenusedrepeatedlyintheresamples.Inpreviouslyintroducedmethods,edgecorrectionweightsmustberecalculatedforeveryresampledsubregionoreverynewpatterncreatedfromresampledsubregions( LohandStein 2004 ).Themajorityoftheresamplingmethodsandvariousadaptationsdescribedearlierhadonlybeenappliedtovariousrandomspatialprocesses(notnecessarilypointprocesses)or1-dimensionalproblemspriorto2004.LohandStein(2004)comparefourofthemostprominentmethodsforcreatingcondenceintervalsfortheK-functionof2-dimensionalspatialpointpatterns:theseincludethesplittingmethod,thetilingmethod,thesubsettingmethod,andthemarkedpointmethod.Eachofthesevarious 32

PAGE 33

methodsisappliedtothreedifferentpatterns:ahomogeneousPoissonpointprocess,aNeymann-Scottclusteredprocess(calledaMaternclustereldbyStoyanandStoyan(1994)( StoyanandStoyan 1994 )),andasoft-coreprocess( LohandStein 2004 ).Thesoftcoreprocessisregularandhasareduced,butpositiveprobabilityofpointsbeingobservedwithinaradiusofinteractionofapointinthepattern( LohandStein 2004 ).Anaverageintensityofapproximately250pointsisusedforallthreeprocessesgeneratedonaunitsquare.Foreachmethodofcalculatingintervals,threetilesizesareusedtoresamplethepatterns.Thepercentcoverageofnominal95%condenceintervalsisdeterminedinasimulationof1000realizationsofeachprocess;thatis,thepercentageofintervalsthatcontainthetruevalueofK(r)fromtheunderlyingpointprocessisdeterminedforarangeofrvalues.LohandStein(2004)alsoinvestigatethewidthsofnormalizedcondenceintervalsforK(r)andvariousotheraspectsrelatedtoresamplingpointpatterns,includingshapesofthetilesandtheeffectsoftoroidalwrapping.Ofthefourmethodsexamined,noneproduceperfectlyaccuratenominalcondenceintervalsacrossalltypesofpointprocesses.Splittingwithlargequadrats(quadratswithasidelengthequalto0.5units)producetheclosestintervalstothenominal95%levelacrossallprocesses.However,thismethodcreatesestimatesofvarianceofK(r)basedononlyfourobservations,andthecondenceintervalscanbecometoowideduetosmallsamplesizeswhenusinglargequadrats.Theempiricalcondencelevellowerswhenusingsmallerquadrats.Usingsplittingwithlargequadrats(sidelengthof0.5units),clusteredpatternsproducecoveragebelowthenominallevel(approximately90%).Quadratsofsmallersizessignicantlyreducethecoverageoftheintervalsforclusteredpatterns(coverageofapproximately50%-80%,dependingonquadratsize).Formarkingandsubsetting,percentcoverageisalsolowerthanthenominallevelforclusteredpatterns(approximately80%-85%)andvariesbasedontilesizeandprocess.ForregularpatternsandPoissonpatterns,intervalsaretoowideatlargerdistancescausingcoveragetobegreaterthanthenominal95%level.Meanintervalwidthofthe 33

PAGE 34

normalizedcondenceintervalsappearssmallestatallconsidereddistancesforthemarkedpointmethodacrossallprocesseswithtilesizesof0.5squareunits.Widthsvaryusingtilesofdifferentsizesbasedontheprocessanddistanceanalyzed.LohandStein(2004)useprocessesexhibitingCSR,clustering,andregularitybutdidnotdeterminehowtheintensitiesoftheprocessesorrangeofspatialinteractionbetweenpointsaffecttheresultsofeachmethodforobtainingcondenceintervals. 1.4InferenceonPointProcesses-HypothesisTestingHypothesistestsofanobservedspatialpatternagainstanullmodelusingMonteCarlomethodsarequitecommon.Thesetestsdonotcomparepatternstooneanother,butcompareapatterntoattedorchosenmodel,whichisthensimulated.Earlyteststocomparetheunderlyingprocessesfrommultiplepatternsrelyonreplicatepatternsthattheresearcherassumesarefromthesameprocess.Diggle(1979)discussedtheuseofnumerousgoodness-of-tstatisticstodeterminewhetherapatternisarealizationfromCSR.Someofthesestatisticswerelaterusedinthedevelopmentofteststatistics.TheteststatisticsexploredusetheK-functionandnearestneighbordistributionstodeterminewhetherapatternistherealizationofaspecicprocess( Diggle 1979 ).Thesestatisticsstillrequireanullmodelspecication,andsimulationtousedtestthenullhypothesis.Thenullmodelisspeciedandtheparameterofinterest(e.g.theKorGfunctions)iscalculatedfortheobservedpatternandthenullprocess.Ateststatisticiscalculatedbythedistancetheobservedsummarystatisticisfromthetheoreticalvalueunderthenullhypothesis.Thevariationoftheteststatisticisdeterminedunderthenullhypothesisbysimulatingpatternsfromthenullmodelandcalculatingtheteststatisticforthesesimulations.Thentheobservedteststatisticiscomparedtothedistributionfromthenullhypothesis( Diggle 1979 ).ThemethodsusedbyDiggletotmodelsusinggoodness-of-tstatisticsaresimilartotheteststatisticslaterdevelopedtocomparereplicatedandsinglepatternsfrommultipleprocesses.Diggle(1979)discussedminimizationofaparticularfunction(he 34

PAGE 35

citesspecicallythelogtransformationoftheK-function),integratedoveradistanceforparameterestimation( Diggle 1979 ).Hisrecommendationstoaccountforanintervalofdistancesinateststatistichavebeenexploredandveriedthroughfurtherstudies( Diggle 1979 ; Diggleetal. 1991 ).Doguwa(1989)developedseveralstatisticsforpointprocesscomparison( Doguwa 1989 ).ThestatisticswerefunctionsoftheK-functionscalculatedfrommultiplepatterns.However,Doguwashowedthatthestatisticshedevelopedcouldbeusedtodifferentiateamongrealizationsfromdifferentprocesses,eventhosewiththesameK-function.Diggleetal.(1991)developedoneoftherstmodel-freeteststotesttheequivalenceoftheK-functionbycomparingtheinterpointinteractioninreplicatedpointpatterns.Thisandotherearlytestscomparingmultiplepatternsutilizedreplicatedpatternsthatareassumedtoberealizationsfromthesamepointprocessanduntilveryrecently,methodstocomparetwoormorepatternsdirectlydidnotexist( Diggleetal. 1991 2000 ).ThetestinDiggleetal.(1991)comparestheintensitiesandspatialclusteringinobservedpatternsfromreplicatedspatialpointpatterns.Diggleetal.(1991)usedthetesttocomparepatternscreatedbypyramidalneuronsonbraintissueofpatientsdiagnosedaseitherhealthy,schizoaffective,orschizophrenic.Theobjectivewastoassessdifferencesamongthesethreegroupsinboththeintensitiesandcellulararrangement.Thepurposeofthespatialpatternanalysiswastwo-fold: 1. DetermineifthepatternfromeachsubjectexhibitedCSR. 2. Ifthepatternswerenotspatiallyrandom,determineifdifferencesamonggroupsweresignicantafteradjustingfordifferencesinintensity.TherstquestionwasassessedusingMonteCarlomethodstotesttheG-function.Theauthorsconcludedthatthesmall-scalespatialregularitiesforallgroupsweresignicantlydifferenttothatexpectedfromCSR.Toanswerwhetherpatternsfromdifferentgroupsweresignicantlydifferentfromoneanother,analysisofRipley's 35

PAGE 36

K-functionwasused.TheestimatedK-functionfromeachpattern(^Kij(r))wascalculated,andthegroupspecicmeanK-functionwasdeterminedforeachgroup: Ki(r)=miXj=1wij^Kij(r),i=1,...,g,(1)wherewij=nij=niandni=Pmij=1nij.Heremiisthenumberofobservedpatternsingroupi,nijisthenumberofobservedpointsinthejthpatternofgroupi,andgisthetotalnumberofgroupsbeingcompared.Similarly,anoverallmeanfunctioniscalculated, K(r)=1 ngXi=1niKi(r),i=1,...,g.(1)TomeasurethedifferenceoftheK-functionamonggroups,astatisticwascreatedbyintegratingoveragivendistancer0, Dg=gXi=1Zr00q Ki(r))]TJ /F13 11.955 Tf 11.95 14.98 Td[(q K(r)2.(1)Forthisanalysis,r0=0.25.Diggle(1991)claimsthatDgislooselyanalogoustoaresidualsumofsquaresinaconventionalone-wayANOVA( Diggleetal. 1991 ).BecauseananalyticalformofthedistributionofDgisnotknown,empiricalmethodsmustbeappliedtotestamonggroupdifferences.Diggleetal.(2000)comparetheteststatistictoitsempiricaldistributionapproximatedfromabootstrapmethodandarandomizationtest.Therandomizationtestpermutes^Kij(r)acrossgroupsandcalculatestheteststatisticDgusingthenewgroups.Theauthorsconcludedthatabootstraptestprocedureisamorepowerfulapproachthanusingarandomizationtest.Thebootstrapprocedureforthistestisasfollows: 1. TheresidualK-functionforapatternisdenedas ^Rij(r)=p nijh^Kij(r))]TJ /F5 11.955 Tf 13.74 2.65 Td[(Ki(r)i.(1)Here^Rij(r)expressesthevariationobservedfortheK-functionsofeachgroupfromthegroupmean.UnderthenullhypothesisofequivalenceinKi(r)foralli, 36

PAGE 37

theseareapproximatelyexchangeablebecausethesamplingvarianceofeach^Kij(r)isproportionalto1 nij( Diggleetal. 1991 ). 2. BootstrapsamplesofKij(r),^Kij,areobtainedforallgroupsi,byusingthegroupmeanK-functionsKi(r),andtheresidualK-functiondistributioncalculatedforeachgroup: ^Kij=K(r)+1 p nij^Rij(r).(1)Here^Rijareobtainedbydrawingfromtheempiricaldistributionof^Rij(r)randomlyandwithreplacement.Thenumberofbootstrapsampleswithineachgroupiskeptconsistentwiththenumberofpatternsobservedfromeachgroup. 3. TheteststatisticDgiscalculatedforalargenumberofbootstrapsamples,providinganempiricalapproximationtothedistributionofDgunderthenullhypothesis.Theobservedteststatisticiscomparedtothisdistributiontoobtainap-valueforthistest.Diggleetal.(1991)useasimulationstudytoexplorethepowerandsizeofthisbootstraptestfordifferentpointprocessdistributions.Theyfoundthatthebootstrapproceduregivesaslightlyconservativetestandismorepowerfulinatwo-groupcasethanwhenappliedtothreeormoregroups.ThepowerofthetestcomparingaregularprocesstoCSRdependedonhowextremetheregularityoftheprocesswas(i.e.,howmuchthepatterndepartedfromCSR).Diggleetal.(2000)comparedthebootstraptestdescribedinDiggleetal.(1991)toaparametricmethodofcomparingreplicatedspatialpointpatternsfromdifferentgroups.Theparametricmethodassumesaspecictypeofmodel(e.g.,aPoissonpointprocessframework)andcomparesthepseudo-likelihoodoftheobservedpatternstothatunderthenullhypothesisthattheobservedinteractionamongpointsisthesameacrossgroups.IfXisacongurationofnpointsobservedina2-dimensionalregionA,representedasX=fxi2A:i=1,...,ng,ajointdensityofthesenpointcanbedenedwithinaPoissonpointprocessframework,allowingforinteractionusinganinteractionfunctionofdistance.ForaPoissonmodel,thejointdensityofthepointpatternXwith 37

PAGE 38

nobservedpointscanbewrittenas, f(X)=C)]TJ /F9 7.97 Tf 6.59 0 Td[(1nexp()]TJ /F8 7.97 Tf 17.65 14.95 Td[(nXi=1Xj>i(jjxi)]TJ /F4 11.955 Tf 11.96 0 Td[(xjjj;)).(1)isaparameterdeningtheintensityoftheprocess.Largervaluesofresultingreaterlikelihoodofobservingnpointsinthepattern.representsapair-potentialfunction,afunctionofdistancebetweentwoeventswhichdependsonthesetofparameters.Typically,(r)isadecreasingfunctionofdistanceandhasavalueequaltozeroforsomemaximumradiusofinteractionamongpoints.Positivevaluesof(r)indicateregularitybetweentwopoints( Cressie 1991 ).Thus,theinterpointdistancebetweenallpairsofpointsisincludedinthejointdensityofthepattern.Cisanormalizingconstantdependingontheparameterandthefunction.InthecaseofahomogeneousPoissonprocesswithnointeraction,expf(r)g=1foralldistancesr.Ife(r)representstheinteractionfunctionexpf)]TJ /F5 11.955 Tf 15.27 0 Td[((r)g,theninterestistypicallyininferencefortheparametersofe(r)asopposedtotheintensityoftheobservedpatterns(i.e.,theinteractionobservedinthepatterns).ThusthejointdensityofthelocationsofX,conditionalonobservingnpoints,areconsidered,andthejointdensitycanbewrittenas f(X)=cn()Yi.Thismodelisusedin( Diggle 1986 ; Diggleetal. 1994 ).Largervaluesofcorrespondtohigherlevelsofrepulsiveinteraction. 38

PAGE 39

Thesecondmodelis e(r)=(r=r1r>.Inthiscase,therangeofinteractionisstilldenedby,butthereisdiscontinuityat=0. Thelastmodelhastheinteractionfunction e(r)=1)]TJ /F5 11.955 Tf 11.95 0 Td[(expf)]TJ /F5 11.955 Tf 15.28 0 Td[((r=)2g.Inthiscase,therangeofinteractionamongpointsisinnitealthoughthedegreeofregularitystillincreaseswith.Fortheparametricapproach,Diggleetal.(2000)usedanedge-correctedpseudo-likelihoodapproachtoestimatetheparameters( Besag 1977 ).Thisisdescribedherewithouttheadditionofedgecorrections.Let(u;X)representtheconditionalintensityofaneventatlocationu62X;thatis,apointthatisnotpartofthepatternX.Thentheconditionaldensityis, (u;X)=f(X[u) f(X).(1)Bysubstitutingthejointdensitiesforthenumeratoranddenominator,itfollowsthat (u;X)=exp()]TJ /F8 7.97 Tf 17.64 14.95 Td[(nXj=1(jju)]TJ /F4 11.955 Tf 11.96 0 Td[(xjjj;))=0(u;X).(1)Here0(u;X)representstheconditionaldensityatpointuwiththeintensityparameterremoved.IfXirepresentsthepointpatternX,excludingpointxi(callthisXi=X)]TJ /F4 11.955 Tf 12.23 0 Td[(xi),thenthepseudo-likelihoodfortheobservedpointpatternXis pl(,;X)=exp)]TJ /F13 11.955 Tf 11.29 16.27 Td[(ZA(u;X)dunYi=1(xi;Xi)(1)( Besag 1977 ; JensonandMoller 1991 ).Here,RA(u;X)duisthetotalintensityoftheobservedregionandtheproductontherightoftheequationisthejointlikelihoodofnobservedpointsinthepattern.Theobjectiveistomaximize 1 withrespecttoandtoobtainmaximumpseudo-likelihoodestimatorsoftheseparameters.Ignoring(becauseinterestliesintheinteractionparameters),thefollowingmaximizationcriterion 39

PAGE 40

canbederivedfor: PL()=nXi=1logf0(xi;Xi)g)]TJ /F4 11.955 Tf 20.59 0 Td[(nlogZA0(u;X)du(1)( Diggleetal. 2000 ).Maximizationofthispseudo-log-likelihoodequationwithrespecttoobtainsthemaximumpseudo-likelihoodestimatorfortheinteractionparameterset,^.Propertiesoftheseestimatorsarediscussedin( Jenson 1993 ; JensonandMoller 1991 )ToextendtheparametricmethodofDiggleetal.(2000)tothecaseofreplicatedpatterns,differentintensityparametersareallowedforacrossthereplicatedpatternsfromwithineachgroupandijaretreatedasnuisanceparameters;thatis,interestisindeterminingdifferencesamongtheinteractionofpointsobservedindifferentgroups.Letplij(,;Xij)representthepseudo-likelihood(Equation 1 )forthejthpatternintheithgroup,whereequation 1 andplij(,;Xij)containthepointsofXandXij,respectively.Notethatijcanbeeliminatedfromthepseudo-likelihoodequationofeachobservedpatternbysubstitutionofEquation 1 intoEquation 1 anddifferentiationwithrespecttoij.Thepooledpseudo-log-likelihoodfunctionforallreplicatedpatternsintheithgroupis PLi()=miXj=1PLij(),(1)wherePLij()isgivenbyEquation 1 forpatternXij,thejthpatternoftheithgroup.Ifacommonvalueoftheinteractionparametersetisassumedforeachofggroups,thentheoveralllog-pseudo-likelihoodis PL()=gXi=1PLi()(1)Totestthenullhypothesisthattheinteractionparametersetisequalforallgroups(i=foralli=1,2...,g)againstthealternativehypothesisthattheinteractionparametersetisdifferentfordifferentgroups(i=jforsomei6=j),thefollowingprocedureisused: 40

PAGE 41

1. Separateparametersiareassumedforeachoftheggroups.Estimateiforeachoftheggroupsandcalculatethemaximizedtotalpseudo-log-likelihoodallowingtheseparameterstovary: PL1=gXi=1PLi(^i).(1) 2. Assumeacommonparameteracrossallgroupsandnd^thatmaximizesEquation 1 andcalculatePL0=PL(^). 3. Theteststatisticfortestingagainstthenullhypothesisofequalityinamonggroupsisthedifferencebetweenthepseudo-log-likelihoodsdeterminedbyallowingtheseparameterstodifferamonggroupsandkeepingthemconstantforallgroups.Thatis,T=PL1)]TJ /F4 11.955 Tf 11.96 0 Td[(PL0. 4. Newpatternsaresimulatedfromthejointdensityusingidenticalnumbersofpointsnijforeachofthepatternsobservedandaconstantvalueofequaltothemaximumpseudo-likelihoodestimateof,underthenullhypothesisthatisequalforallggroups.ThebootstrapteststatisticTiscalculatedforeachofthesimulatedsetsofpatterns.ThisisrepeatedalargenumberoftimesandtheobservedstatisticTiscomparedtotherankedvaluesofTtodeterminethep-valueofthetest.TheDiggleetal.(1991)bootstraptestandtheDiggleetal.(2000)parametrictestwerecomparedinasimulationstudytestingthenullhypothesisofequivalenceintheinteractionamongdifferentgroupsformultiplerealizationofdifferentprocesses.Asonemightexpect,Diggleetal.(2000)foundthattheparametrictestperformedbetterthanthenonparametrictestwhenaccuratemodelspecicationhadbeenmade.Whenthemodelformwasmisspecied,thenonparametricteststatisticoftheform 1 wasmoresuccessfulinsizeandpower.Tothispoint,testsdesignedtocomparetheunderlyingdistributionsofobservedpointpatternshaveeitherrequiredspecicationofanullmodelormultipleobservedpatternsthatareassumedtobefromthesameprocess.Unlesstestingagainstaspecicmodel(suchasCSR),modelspecicationcanleadtoerrorduetochoosingthewrongmodel.Also,itisoftennotknownwhetherdifferentobservedpatternsaretheresultofthesamepointprocessdistribution.Generallyonlyasinglepatternisrecordedattwoormoredisjointareas,andaresearcher'sgoalistodeterminewhether 41

PAGE 42

thesepatternsarerealizationsfromthesameprocess.In2012,Hahndevelopedtherststatisticaltesttotestwhethermultiplepatternsaretherealizationofthesamepointprocessdistribution( Hahn 2012 ).Hahn(2012)extendedtheresultsofDiggleetal.(1991,2000)todevelopatesttocomparetheinteractionobservedintwoormoreindependentpointpatterns.TocompareppatternsobservedinwindowsA1,A2,...,Ap,forp2,themethodcanbesummarizedasfollows: 1. ForeachwindowAi,i=1,2,...,p,generateasetfAijg,j=1,...,mi,ofpairwisedisjointquadrats,ofatleastroughlythesamesizeandshape. 2. CalculatetheempiricalK-functionsforeachsetfAijg,named^Kij.OnlyquadratsAijwithatleastnminpointsareincluded.Thiswillpossiblydecreasethesamplesizemi. 3. CalculatetheteststatisticTorTwhere T=X1i
PAGE 43

Thesepermutedvaluesareorderedandtheobservedteststatisticisrankedamongthemtodeterminethep-valueassociatedwiththetest.Hahn(2012)conductedalargesimulationusing6differentprocesses(ahomogeneousPoissonprocess,3clusteredprocesses,and2inhibitedprocesseswithvaryingdegreesofpoint-to-pointinteraction).Resultsshowedthatunderthenullhypothesis,thetestisslightlyliberalforstronglyclusteredpatternsandconservativeforinhibitedpatterns.Thelevelofthetestalsovarieswiththeparameterr0andquadratnumberandsize.UnderthenullhypothesisofCSR,thepowerofthetest(usingvaluer0=0.15)isbelowthenominallevelandincreaseswiththeintensityofthepattern.Usingfewer,largerquadratsincreasedthepowerofthetestandproducedmoreaccuratelevelsforrelativelysmallintensities.Hahn(2012)appliedthepermutationtesttopointpatternsobservedfromcapillaryprolesonhealthyandcancerousprostatetissue.Twopatternsweredividedinto3x3quadratseach.Teststatisticswerecalculatedandcomparedtorandompermutationsofthequadratsusingdifferentvaluesofr0.p-valuesweredeterminedandshowntobeminimizedatapproximatelythevaluesuggestedbyRipley(1979)(r0=1.25=p ).Hahn'spermutationtestwassuccessfulincomparingcancerousandhealthytissueusingpointpatternmethodology. 1.5ObjectivesInthiswork,theK-functionforstationary,isotropicprocessesisthekeyfocus.Themotivationistoperformmoreaccurateinferenceonthisspecicprocessparameter.Chapter2'sprimaryfocusisthedevelopmentofcondenceintervalsfortheK-function.WeextendtheanalysisperformedbyLohandStein(2004)tostudytheeffectsofintensityandthedegreeofclusteringontheaccuracyofseveralbootstrapproceduresforpointprocesses.Wealsointroduceanewmethodofbootstrappingfromapointpatterntocalculatecondenceintervals.Wecomparethesemethodsusingasimulation 43

PAGE 44

studywiththegoalofbeingabletorecommendamethodofintervalcalculationfortheK-functiontoecologicalresearchersworkingwithspatialpointpatterns.Chapter3focusesonhypothesistestsdesignedtotesttheequivalenceoftheK-functionfrommultiplepointpatterns;thatis,anewprocedureisdescribedtotestthenullhypothesis H0:K1(r)=K2(r)=...=Kg(r)forg2.(1)ItisimportantheretonotethedistinctionbetweentestingtheequivalenceoftheK-functionformultiplepatternsandtestingwhethermultiplepatternsaretrulytheresultsofthesameprocesses.Inthiswork,wedeveloptheformer.Inecologicalapplications,werarelydealwithpointdatathathavethesamenumberofpointsandtheunderlyingmotivationofecologicalandbiologicalstudiesiswhetherthebiologicalprocessdemonstratedinoneobservedpatternisthesameasthatdemonstratedinanother.Therefore,ourfocusisonthesecond-orderintensityofstationarypointprocesses.Ifthenullhypothesisisrejected,itcanbeconcludedthattwoormorepatternshavedifferentpoint-to-pointinteractionsandaretheresultsoftwodifferentprocesses.InChapter3,weagainuseasimulationstudytodeterminethesizeandpowerofourtestusingprocesseswithdifferentintensitiesandlevelsofinteraction.WecomparetheseresultstoHahn's(2012)studentizedpermutationtestwiththegoalofrecommendingthebesttestforecologicalresearcherstouseinpracticetocomparetheapparentinteractionobservedinpointpatterns.InChapter4,ananalysisofadatasetisconductedusingdatafromtheJosephW.JonesEcologicalResearchCenter.Thesedataconsistofseveralplotsinwhichthepositionofseveralclassesoftreesarerecorded.Themethodsdescribedinthisdissertationareappliedaftercheckingthenecessaryassumptions.Thepurposeofthisapplicationistoinformecologicalresearchersonhowtoapplythedescribedmethods,andhowtheycanbeusedformanagementpurposes. 44

PAGE 45

Figure1-1. SamplepatternsandtheirresultingKandLfunctions:TheleftcolumnisaCSRpattern,thecentercolumnisaclusteredpattern,andtherightcolumnisaregularpattern.ThesecondrowshowstheK-functioncalculatedfromeachusingRipley'sisotropicestimator.ThethirdrowshowstheL-functioncalculatedfromeach,atransformationoftheK-function.RedlinesindicatefunctionsforaCSRprocess. 45

PAGE 46

Figure1-2. Anexampleofbootstrappingapointpatternusingthetilingmethod. Figure1-3. ExampleofLohandStein'smarkedpointmethod:Fordistancer,pointsyandzcontributetopointx'smarkthroughtheirweightsassociatedwithx.Pointvdoesnotcontributedtothemarkmx(r).Herepointsx,y,andvareresampled(blockbootstrapwithtoroidalwrapping)butpointzisnot.However,someoftheinformationfromzhasbeenresampledthroughitsmarkontheotherpoints.Figuretakenfrom( LohandStein 2004 ) 46

PAGE 47

CHAPTER2CONFIDENCEINTERVALSFORRIPLEY'SK-FUNCTION 2.1IntroductionRipley'sK-function,aparameterofapointprocess,isusedtointerpretitssecond-orderintensity.Forasimpleprocess,K(r)representsthenumberofpointslessthandistancerfromanarbitrarypointintheprocessandthus, K(r)=E[N(s0,r)jpointats0] .(2)AnestimatorofK(r)is ^K(r)=jAj n2Xx2AXy6=xw(x,y)I(jjx)]TJ /F17 11.955 Tf 11.96 0 Td[(yjj
PAGE 48

spatialpointprocessesforthepurposeofestablishingcondenceintervalsfortheK-function.Theapproachestheyconsideredaswellasrelatedmethodsofresamplingpointpatternsarereviewed.Anewbootstrapmethodisalsoproposed.ThenotationhereisthatofLohandStein(2004). 2.2BootstrappingaSpatialPointPatternPerhapsthesimplestmethodofcreatingacondenceintervalforK(r)ofaspatialpointprocessisthesplittingmethod,whichisnotabootstrapapproach.ThesplittingmethoddividestheregionAassociatedwiththeobservedpatternintoNcongruentsubregions.Then^K(r)iscalculatedseparatelyovereachsubregion.Foraspecicvaluer,NestimatesofK(r):^K1(r),^K2(r),...,^KN(r),areobtained.AssumingtheNestimatesareindependentandGaussiandistributed,a100(1)]TJ /F11 11.955 Tf 12.07 0 Td[()%condenceintervalis ^K(r)tN)]TJ /F9 7.97 Tf 6.59 0 Td[(1,=2vuut ^Var^Ki(r) N(2)where^Var(^Ki(r))isthesamplevariancecalculatedfrom^K1(r),^K2(r),...,^KN(r)andtN)]TJ /F9 7.97 Tf 6.58 0 Td[(1,=2isthe=2percentileofthetdistributionwithN)]TJ /F5 11.955 Tf 11.96 0 Td[(1degreesoffreedom( LohandStein 2004 ).Thismethodhastheobviousadvantagethatbootstrappingisnotneeded,andintervalsarecalculatedonlywiththeavailabledata.However,relativelylargequadratsizestendtoresultinthemostaccurateintervalsleadingtofewobservationsbeingusedintheircalculation,resultinginwideintervalsinsomecases( LohandStein 2004 ).Forinstance,quadratswithasidelengthof0.5unitsonlyallowfourobservationstobeusedifthepatternisobservedontheunitsquare.Splittingalsohaslimitationsinthedistancetowhichintervalsareaccurate.BecauseestimatesofK(r)arecalculatedoneachquadrat,theareasinwhichtheestimatesareformedaresmaller.Thismeansthat,atlargerdistances,edge-correctionweightshaveagreaterinuenceontheestimatesof 48

PAGE 49

K(r).Theassumptionsofindependentandapproximatelynormalobservationsamongquadratsmaynotholdwithsomepatterns.ThelimitationsofthesplittingmethodhaveledresearcherstoexploreothermethodsofsettingcondenceintervalsforK(r).Thetilingmethodresamplesarectangularwindowbyplacingblocksortilesofagivenareaovertheregioninwhichthepatternisobserved(( Hall 1985 ; Kunsch 1989 ; LiuandSingh 1992 )demonstratetileresamplinginonedimension).Thetilescreatesubregionswithsubpatternsconsistingofthepointscontainedwithineachsubregion.ForawindowAofareajAj,Nsubregionsaresampled,eachofsizejAj=N.ThenthesubregionsarearrangedinapredeterminedwaytocreateanewpatterninawindowofareajAj,whichhasthesamedimensionsasA.ThenewpatternA`doesnotnecessarilyhavethesamenumberofpointsasAsothepointsofA`arereferredtoasxi`,i=1,2,...,n`.Wecanthencalculatetheestimate^K(r)fromthenewpatternA`usingEquation 2 ^K(r)=jAj n`2n`Xi=1n`Xj=1,j6=iw(x`i,x`j)I(jjx`i)]TJ /F17 11.955 Tf 11.96 0 Td[(x`jjjr).(2)ThisprocessisrepeatedBtimes.Undertheassumptionthatthedistributionof^K(r))]TJ /F5 11.955 Tf -450.61 -21.25 Td[(^K(r)issimilartothatof^K(r))]TJ /F4 11.955 Tf 11.15 0 Td[(K(r),a100(1)]TJ /F11 11.955 Tf 11.15 0 Td[()%intervalforK(r)iscreatedusingthebootstrapinterval h2^K(r))]TJ /F5 11.955 Tf 13.73 2.65 Td[(^K(B+1)(1)]TJ /F14 7.97 Tf 6.59 0 Td[(=2)(r),2^K(r))]TJ /F5 11.955 Tf 13.73 2.65 Td[(^K(B+1)=2(r)i.(2)Severalvariationsofthetilingmethodexist.Originally,tileswerenon-overlappingandconsistentwithdividingtheobservedpatternintoNcongruentsubregions( DavisonandHinkley 1997 ).Allowingtilestooverlap,butlimitingthemtobecontainedintheregion,resultsinundersamplingpointsneartheboundariesoftheregion( DavisonandHinkley 1997 ).Toriodalwrappingisavariationusedtoavoidthisproblem( DavisonandHinkley 1997 ).Toriodalwrappingcanbevisualizedastheoriginalpatternreplicated 49

PAGE 50

ina3x3gridasshowninFigure 2-1 .Thetilesarethenrandomlyplacedinthecenterwindowofthegridbutallowedtooverlaptheboundaries.Theobjectiveoftilingistocreatenewpatternsthathaveanintensityandspatialstructuresimilartothatoftheobservedpattern.Bootstrapsamplesarearticialrealizationsoftheunderlyingprocessoftheobservedpattern.Althoughpointsthatareinthesamesubregionarexedrelativetooneanother,therelativepositionofpointsindifferentsubregionschanges.Asaconsequence,pointsclosetoaboundaryofasubregionmaybeplacedneareachotherinabootstrapsamplewhentheyarenotinfactclosetoeachotherintheobservedpattern.Changingtherelativepositionofthepointsmayalterthesecond-ordercharacteristicsoftheprocessandcreatebiasintheestimatesofK(r),especiallyatsmalldistances( Lahiri 1993 ; LohandStein 2004 ).Onesimpleexampleofhowtheinteractionamongpointscanbechangedisahardcorepattern,inwhichpointscannotfallwithinaxedradiusofinteractionofeachother;thatis,P(K(r)=0)=1forr
PAGE 51

weightsarecalculatedusingthesamemethodofedgecorrectionusedtocalculated^K(r).TheresultingbootstrappedestimateofK(r)is ^K(r)=jAj PNi=1ni2NXi=1niXj6=kwAi(xij,xik)I(jjxij)]TJ /F17 11.955 Tf 11.95 0 Td[(xikjjr).(2)Subsettingreducestheproblemofviolatingthesecond-orderstructureoftheprocessbecausesubregionsarenotrearranged.Somearticialpointpairsarestillproducediftoriodalwrappingisusedinthesampling;however,farfeweroccurthanduringtiling( LohandStein 2004 ).Similartosplitting,thedistancetowhichintervalscanaccuratelybecalculatedislimited.Becauseestimatesarebasedonthesmallersubregions,edgecorrectionweightsagaininuencetheresampledestimatorsatrelativelylargedistances( LohandStein 2004 ).Thus,eitheralargerwindowisrequiredfortheoriginalsample,orinterestshouldbeinintervalcalculationforshortdistancesrelativetothesizeofthewindow.LohandStein(2004)alsoadaptedthemarkedpointmethodfromBraunandKulperger(1998)forspatialpointprocesses.Foraparticulardistancer,eachpointxisgivenamark,mx(r),equaltothesumofweightsforallpointswithindistancerofx: mx(r)=Xy:y6=xwA(x,y)Ifjjy)]TJ /F17 11.955 Tf 11.96 0 Td[(xjj
PAGE 52

toresamplepointswheresubregionAicontainspointsxij,j=1,2,...,ni,eachxijhasamarkmij(r)=Py:y6=xwA(xij,y)Ify2A:jjy)]TJ /F17 11.955 Tf 11.96 0 Td[(xijjjrg.ThentheestimateofK(r)is ^K(r)=a Pni(Pni)NXi=1niXj=1mij(r).(2)CondenceintervalsareproducedusingEquation 2 ( LohandStein 2004 ).ThepositionsofpointsintheresampledblocksarenotrecordedbecauseestimationoftheK-functionisthesumofthemarksbeingresampled.Thus,marksareonlycalculatedoncepriortoresamplingandthenresampledtoestimateK(r).Markinghasseveraladvantagesasamethodofbootstrappingaspatialpointpattern.Nonewpointpairsareproducedthatviolatethesecond-orderstructureofthepointprocess.Eveniftoriodalwrappingisusedwhenbootstrapping,theblocksareonlyusedtoresamplethepointsofthepatternandarenottreatedassamplesoftheprocess.Instead,thepointsthatareresampledareusedtocalculateanewestimateofK(r)bycombiningtheirassignedmarks.Thepatternisnotbrokenintosmallersubregionsaswithsplittingandsubsetting.ThusedgecorrectionweightsdonotinuencethebootstrappedestimatorsanymorethantheyinuencethetotalestimateofK(r).Finally,markingprovidesacomputationaladvantagetotilingandsubsettingastheedgecorrectionweightsandmarksonlyneedtobecalculatedonce.Whenthepointsareresampled,theyretainthesamemarksasintheoriginalpattern. 2.3NetworkResamplingtoConstructCondenceIntervalsofK(r)Here,anewmethodofresamplingapointpatterntoobtaincondenceintervalsforK(r),referredtoasthenetworkmethodornetworking,isproposedasfollows: 1. Atuningparameterischosenbasedoninformationfromtheobservedpattern.Thevalueofchosenforaspecicpatternistheresponseinthelinearfunction =11 ^+2DA+3DA ^(2) 52

PAGE 53

where DA=1 r0Zr00)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(L(t))]TJ /F11 11.955 Tf 11.95 0 Td[(t2dtandL(t)=r K(t) .(2)isameasureofthepattern'sdeparturefromCSR.Here,theL-function,atransformationoftheK-function,isusedbecausetheL-functionhasareducedvariancecomparedtoK(r),asdistanceincreases.Thevaluer0ischosenasthedistancewheremaximumdeviationfromtheliner2occurs.Forpatternsexhibitingclusteringofpoints,DAisapositivevalue.Forpatternsexhibitinginhibition,DAisnegative.Thettedcoefcientsinthelinearequationaredescribedlater. 2. ThenpointsthatareobservedinwindowAareclusteredintonetworks.Ifthedistancebetweentwopointsislessthan,thenthesepointsbelongtothesamenetwork.Nnetworksarecreatedfromthenpoints.Thus,theinterpointdistancebetweenanytwopointsindisjointnetworksisgreaterthan.Figure 2-3 showsthedendrogramofasamplepatterndividedintonetworksbasedon(thehorizontallineintheplot). 3. Edgecorrectionweightsarecalculatedforallpointsinthepattern,withoutconsiderationofnetworks.Fori=1,...,N,letnetworkihavenipointswithinit.Thenthetermyi(r)istheaveragenumberofpointswithindistancerofanarbitrarypointintheithnetwork.Thatis, yi(r)=1 niniXk=1nXj6=kw(xik,xij)I(jjxik)]TJ /F17 11.955 Tf 11.95 0 Td[(xijjjr).(2) 4. BootstrapMethod:Abootstrapsampleisdrawnbyrandomlyselectingnetworkswithprobabilityproportionaltosize,thatispi=ni nfornetworki.Letn(i)mbethenumberofpointsintheithnetwork(networki)thatisresampledonthemthdrawfromthepopulationofNnetworks.SamplingcontinueswithreplacementuntilMnetworksareresampledsuchthatPMm=1n(i)m=n)]TJ /F5 11.955 Tf 12.12 0 Td[(1=2n,wherenisthemeannumberofpointsinarandomlysamplednetworkusingtheweightedsamplingprobabilities. 5. TheK-functionisestimatedfromabootstrapsampleusingtheHansen-Hurwitzestimator: ^K(r)=jAj n1 MMXm=1y(i)m(r),(2)wherey(i)m(r)isthevalueofyi(r)fromtheithnetworkofthepopulationthatissampledonthemthdraw( Lohr 2010 ). 53

PAGE 54

6. Bbootstrapsamplesaredrawn.Foragivendistancer,a100(1)]TJ /F11 11.955 Tf 12.29 0 Td[()%bootstrapcondenceintervalforK(r)is h2^K(r))]TJ /F5 11.955 Tf 16.1 2.66 Td[(^K(B+1)(1)]TJ /F14 7.97 Tf 6.58 0 Td[(=2)(r),2^K(r))]TJ /F5 11.955 Tf 16.1 2.66 Td[(^K(B+1)(=2)(r)i,(2)where^K(r)istheestimationofK(r)fromEquation 2 .Thetuningparameterwaschosensothat95%condenceintervalsprovideapproximately95%coverageforK(r)forr0.5.Itsvaluewasdeterminedforover40processesofvariousintensitiesanddegreesofclustering.CoveragewasbasedonB=1000bootstrapsamplesfromeachpatternandatleast200patternsfromeachprocess.Alinearmodel =0+11 +2DA+3DA +e,forr00.5(2)wastusingtheseselectedvaluesof.Forpatternswherecoveragevariedwithdistance,thevalueofwasselectedas =argmin1 0.5Z0.50(0.95)]TJ /F4 11.955 Tf 11.95 0 Td[(PC(r))2dr(2)wherePC(r)isthepercentcoverageoftheresultingintervalsusingthenetworkradiusparameteratdistancer.TheR2fromthelinearttotheoptimizedvaluesofis0.87.Conceptually,thevalueofadjuststhenetworksizetoaccountformoreorlessvariationintheprocessduetoclusteringorlackthereof.TheK-functionofclusteredprocessestendstobemorevariablethantheK-functionofCSRorregularprocesses.Thusforclusteredpatterns,thevalueofthenetworkradiusisgreaterthanthatofCSRorregularpatternsresultinginfewernetworks.Theclustersretaintheobservedvariabilityin^K(r)forroftheobservedspatialpointpattern.Forpatternsexhibitingregularity,pointshaveasmallerprobabilityoflyingwithinaradiusofinteractionfromotherpoints.Inthiscase,thevariabilityinthenumberofpointswithinadistancerofanarbitrarypointinthepatternissmallersotheK-functionislessvariable.Asa 54

PAGE 55

consequence,thenetworkradiusissmallerthanthatofCSRorclusteredprocesses,allowingpointstobesampledinsmallerclustersorasindividuals.If^K(r)isunbiasedforK(r),thenthebootstrapestimatesofK(r)arealsounbiased,conditionalon.SomeofthepreviousmethodsofcalculatingcondenceintervalsmayalsoproduceunbiasedbootstrapestimatesoftheK-function;however,others,suchasthetilemethod,donot.BecausethevarianceofestimatorsoftheK-functionaregenerallyunknown,thevarianceofthenetworkbootstrapestimatorisunknown.Percentcoveragefromsimulationsprovidesanindirectmeasureofaccuracyoftheestimatedvariance.Inthefollowingsection,itisshownthat,given,theestimatoroftheK-functionusingnetworkresamplingisunbiasedfortheoverallestimatorofK(r). 2.4UnbiasednessofBootstrappedEstimatorofK(r)TheHansen-HurwitzestimatorforclustersamplingwithunequalprobabilitiesandwithreplacementisusedtoestimateK(r)fromthebootstrapsamples.Theapplicationofthisestimator,conditionalontheNnetworksformedforaparticularvalueof,willbeshowntobeunbiasedforknownintensity.RecallthatourestimatorofK(r)is ^K(r)=1 1 nnXi=1nXj6=iw(xi,xj)I(jjxi)]TJ /F17 11.955 Tf 11.96 0 Td[(xjjj
PAGE 56

whereTi(r)representsthenumberofallpointsinthebootstrapsamplethatarewithindistancerofpointsintheithnetwork.BootstrapsamplesareusedtoestimateT(r).EachbootstrapsamplehasMnetworks,butMisnotxedanddependson,aswellastherandomselectionprocess.SupposethatMnetworksarerandomlyselectedwithreplacementandwithprobabilityproportionaltosize;thatis,pi=ni=n,fori=1,2,...,N.ThenusingtheHansen-Hurwitzestimator,anunbiasedestimateofT(r)is ^T(r)=1 MMXm=1T(i)m(r) p(i)m.(2)HereT(i)m(r)isthenumberofpointsinthebootstrapsamplewithindistancerfromanypointintheithpopulationnetworkthatisresampledonthemthdrawfromthepopulationofNnetworks.p(i)m=ni=nistheprobabilityofselectingtheithnetworkfromthepopulation,whereniisthenumberofpointsintheithpopulationnetworkthatisselectedonthemthdrawfromthepopulationofNnetworks.Thisestimatorcanberewrittenas ^T(r)=1 MNXi=1QiTi(r) pi(2)whereQirepresentsthenumberoftimesnetworkiisdrawnintheMnetworkssampled.Thus,Qi=0,1,2...,MandPNi=1Qi=M( Lohr 2010 ). 56

PAGE 57

Referringto 2 ,itiseasytoseethatE[QijM]=Mpiandthus^T(r)isunbiasedforT(r)givenM: Eh^T(r)jMi=E"1 MNXi=1QiTi(r) pijM# (2) =1 MNXi=1E[QijM]Ti(r) pi (2) =1 MNXi=1MpiTi(r) pi (2) =1 MNXi=1MTi(r) (2) =NXi=1Ti(r) (2) =T(r). (2) (2) Thus,conditionalonresamplingMnetworks,thebootstrapestimator^T(r)isunbiasedforT(r).Then Eh^T(r)i=EhEh^T(r)jMii (2) =E[T(r)] (2) =E"nXi=1nXj6=iw(xi,xj)I(jjxi)]TJ /F17 11.955 Tf 11.96 0 Td[(xjjj
PAGE 58

Thus,ifisknown, ^K(r)=1 1 n^T(r) (2) =1 1 n1 MMXm=1T(i)m(r) p(i)m (2) =1 1 n1 MMXm=1T(i)m(r) n(i)m=n (2) =1 1 MMXm=1T(i)m(r) n(i)m. (2) (2) Notethatyi=Ti(r) niandsotheestimator ^K(r)=1 1 MMXm=1y(i)m(r)(2)isunbiasedforK(r)forknown.ReplacingwithitsestimatorintroducesbiasintheestimationofK(r)( Cressie 1991 ). 2.5SimulationStudyAsimulationstudywasconductedtocomparethemethodsofcalculatingcondenceintervalsforRipley'sK-function.Threedifferenttypesofspatialpointprocesseswereusedinthissimulation:ahomogeneousPoissonpointprocess,aNeymann-Scottclustereld,andasoftcoreregularpattern.TheNeymann-Scottprocess(referredtoasaMaternclustereldby( StoyanandStoyan 1994 ))hasaxednumberofparentpoints,,generatedwithintheunitsquare.EachparentpointhasaPoissonnumberofdaughterpoints,withmean,generatedwithinaradiusofaroundthem.Theobservedpatternistheunionofalldaughterpoints.LetbetheexpectationofthenumberofpointsinaNeymann-Scottprocess.Becausetheprobabilityofadaughterpointfallingoutsideofthewindowisgreaterthan0andonlypointscontainedinthewindowareobserved,theexpectednumberofobservedpointsfortheNeymann-Scottprocessislessthan.Restrictingpointsto 58

PAGE 59

fallinsidethewindowchangestheunderlyingprocess.AnexampleofarealizationofaNeymann-Scottprocesswith25parentpoints,ameanof10daughterpointsperparent,andaradiusofinteractionof0.1unitsisdisplayedinFigure 2-4 Thesoftcoreprocessisaninhibitedprocessinwhichpointshaveareduced,butpositiveprobabilityoflyingwithintheradiusofinteractionfromoneanother.ThisprocessissimulatedbyrstgeneratingahomogeneousPoissonpatternwithanintensitygreaterthanthatofthedesiredsoftcoreprocess.Foreachpoint,aradius`isgeneratedbasedonagivenprobabilitydistribution.Themaximumvalueofthedistributionistheradiusofinteractionintheprocess.Eachpointisalsoassignedarandommarkmwhichisauniform[0,1]randomvariable.Apointinthepatternisdeletedifatleastoneotherpointiscloserthanitsassignedradius`andtheotherpointhasasmallermarkm`.Thenumberofpointsoriginallysimulatedandthedistributionoftheradiiforeachpointareadjustedtohaveadesiredexpectationforthenumberofpointsandtheradiusofinteraction.ThesoftcorepatterndisplayedinFigure 2-4 hasaradiusofinteractionof0.05unitsandanintensityofapproximately250points.ThewindowofthesimulatedpatternswastheunitsquareinR2.Foreachtypeofprocess,parametervalueswerechangedtovarythemeanintensityandradiusofinteraction.Foreachtypeofprocess,patternswithmeanintensitiesapproximatelyequalto100,250,and500pointsweresimulated.FortheMaternclusteredprocesswithmeanintensityequalto250points,tworadiiofinteractionwereused.TheradiusofinteractionforatightlyclusteredMaternprocesswas0.05units.Theradiusforamoredispersedclusteredprocesswas0.1units.PatterninformationforeachsimulatedprocessisdisplayedinTable 2-1 .Foreachtypeofprocessandeachintensity,1000patternsweresimulated.95%condenceintervalsforK(r)werecalculatedusingthesplitting,tiling,subsetting,marking,andnetworkingmethodsfordistances0.01r0.25atintervalsof0.01units.Onethousandbootstrapsampleswereselectedfromeachpattern.Formethodsusingblockbootstrapping,blocksofsidelengths0.25and 59

PAGE 60

0.5unitswereresampled.Foreachmethodofcalculatingcondenceintervals,thepercentcoverageandmeancondenceintervalwidthweredeterminedforeachpointprocess.Foreachmethod,thepercentcoverageoftheresulting95%condenceintervalforK(r)wasevaluated.Theestimatedpercentcoverageisthepercentageofthe1000patternswhoseintervalscontainthetruetheoreticalK(r)valuesatagivendistancer.ForthePoissoncase,thetheoreticalvalueofK(r)isr2.ThetheoreticalvalueoftheMaternclusterprocessis K(r)=r2+h(r 2rmax) ,(2)where h(z)=2+1 (8z2)]TJ /F5 11.955 Tf 11.95 0 Td[(4)cos)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z))]TJ /F5 11.955 Tf 11.96 0 Td[(2sin)]TJ /F9 7.97 Tf 6.59 0 Td[(1(z)+4zp (1)]TJ /F4 11.955 Tf 11.96 0 Td[(z2)3)]TJ /F5 11.955 Tf 11.95 0 Td[(6zp 1)]TJ /F4 11.955 Tf 11.95 0 Td[(z2(2)forz1andh(z)=1forz>1( MollerandWaagpetersen 2003 ; StoyanandStoyan 1994 ).ThemeanK-functionfrom10,000simulatedpatternsisusedforthetheoreticalvalueofthesoftcoreprocessasthetrueK-functionisunknown.Percentcoverageisevaluatedtoadistanceof0.25whereapplicable.Forsplittingandsubsettingusingsmallblocksizes,evaluationisonlycarriedouttothemaximumdistancesuchthatedgecorrectionmethodsdonotencompassmultiplecornersofthewindow(adistanceofapproximately0.5timesthesubregion'ssidelength),therebyreducingtheroleofedgecorrectionweightsasmuchaspossible.Thus,ifsubregionsofsidelength0.25areused,evaluationofK(r)isonlyconductedtoadistanceof0.12. 2.6ResultsThepercentcoverageforeachbootstrapmethodoneachprocessisdisplayedinFigures 2-5 2-14 .Figures 2-5 2-7 2-8 2-10 ,and 2-11 2-14 showthecoverageforPoissonprocesses,softcoreprocesses,andMaternclusterprocesses,respectively.Panelsontheleftprovidethecoverageofintervalscalculatedusingtilesofsidelength0.5unitstoresampleandpanelsontherightshowcoveragefromintervalsusingaside 60

PAGE 61

lengthof0.25unitsinresampling.Becausethenetworksareformedandresampledinthenetworkingmethod,thecoverageisthesameintheleftandrightpanels.Thenominal95%condenceintervalsbasedonthesplittingmethodhavecoverageclosestto95%formostoftheprocessesevaluated.ForthePoissonprocess,coverageisaccurateforallintensitiesandbothtilesizesusedtocalculatethevarianceofK(r).Forlargerintensities,coverageisaccurateforregularprocessesaswell.Incontrast,condenceintervalcoverageofthesoftcorepatternwithanaverageintensityof100pointsisextremelylow(approximately0%).Forclusteredprocesseswithasmallintensity,percentcoveragevarieswithdistancebutmaintainsatleast80%formostdistancesifthetilesusedforresamplingarelarger(sidelengthof0.5units).Coverageismoreaccurateandstableforgreaterintensitieswhenusinglargertilestoresample.Usingsmallertiles,theintervalsdonotaccuratelyaccountforthevariationintheprocess,leadingtocondenceintervalsthataretoonarrow.Tilingresultsinpercentcoveragefurthestfromthenominallevel.CoverageforPoissonprocessesisbetween80%and90%forallintensities,anddecreasessteadilyasdistanceincreases.Forclusteredpatterns,coverageislowerthan80%andalsodecreasessteadilyasdistanceincreases.Coverageismarginallygreaterforpatternsthataremoretightlyclustered.Percentcoveragealsoincreasesastheintensityoftheprocessincreases.Atlargerdistances,coverageforclusteredprocesseswithsmallintensitiesisverylow(lessthan50%forr0.18for=100and250).Moreaccuratecoverageisobtainedforclusteredpatternsusinglargerblockstoresamplethepattern.Percentcoverageforregularpatternsisbelowthenominallevel(70%-85%)andisconstantoveralldistances.Thepercentcoverageincreasesnoticeablyasintensityoftheprocessincreases.Tilesizehaslittleeffectonthecoverageoftheresultingcondenceintervals.Subsettingproducesaccurateresultsforprocessesthatarenotclustered.Formostprocesses,coveragewassignicantlyimprovedbyusingsmalltilestoresamplethe 61

PAGE 62

pattern.ForPoissonandsoftcoreprocesses,smalltilesresultincoveragethatisclosetothenominallevel.However,usinglargertilesresultsinpercentcoveragebetween80%and90%atmostdistances.Fortheseprocesses,theintensityoftheprocesshaslittleimpactonthecoverageofthecondenceintervals.Forclusteredprocesseswithasmallintensity,percentcoverageisfarfromthenominallevelatsmalldistancesandincreaseswithdistance,reachingamaximumcoverageofapproximately85%.Usingsmallerresampletiles,thiscoverageisslightlyimproved.Forprocesseswithgreaterintensitiesandasmallerradiusofinteraction,percentcoverageisinitiallygreaterthanthenominallevelandsteadilydecreaseswithdistance.Atlargedistances,coveragefallsbelow80%usinglargeresampletiles.Usingsmallresampletiles,coverageisclosetothenominallevel,butalsodecreaseswithdistance.Formoredispersedclusteredprocesseswithgreaterintensities,coverageislowerthanthenominallevel(80%-90%usinglargeresampletilesand90%-95%usingsmallresampletiles).Usingthemarkedpointmethod,coverageofcondenceintervalsformostoftheprocessesincreaseswithdistanceandobtainsapercentcoverageofover95%aftersomedistance.ForthePoissonprocess,coverageisinitiallylowatsmalldistances(approximately75%forintensity=100and90%forgreaterintensities)andincreasesto100%atlargerdistances.Asintensityincreases,thedistanceatwhich100%coverageisobtaineddecreases(approximatelyr=0.19forintensity=250andr=0.10forintensity=500).Usingsmallertilestoresamplethepatternsappearstoleadtoslightlymoreaccuratecoverageatmostdistances,butthisimprovementissmall.Resultsaresimilarforinhibitedprocesseswithcoveragereaching100%atapproximatelythesamevaluesaswiththePoissonprocess.Again,itappearsthatsmallertilesslightlyimprovetheaccuracyofthecondenceintervals.Forclusteredprocesses,percentcoverageislowforsmallintensities(reachingamaximumofapproximately70%).Forclusteredprocesseswithgreaterintensities,themarkedpointmethodperformsfairlywell.Using 62

PAGE 63

smallertilestoresample,percentcoveragewasbetween90%and100%fortheseprocesses.Thecoverageofthenetworkingmethodvariesbasedontheprocessanddistancebeingevaluated.ForPoissonprocesses,thebestresultsareobtainedatsmallintensities(=100)withapercentcoveragebetween90%and95%foralldistancesrlargerthan0.03.Forallintensities,percentcoverageisbetween90%and95%for0.05r0.15.Thecoveragedecreasessteadilywithrforr0.10forpatternswithaveragesof250and500points.Similarresultsareobtainedforinhibitedprocesseswithcoveragebeingaccurateforsmallintensitiesanddecreasingastheintensityincreases.CoverageforaMaternprocesswithasmallintensityislow(approximately70%formostdistances)andcomparabletotheresultsfromthemarkedpointmethod.Forintensitiesof250points,thecoverageisstilllowerthanthenominallevel,butmaintainsafairlysteadycoveragebetween85%and90%.Theradiusofinteractiondoesnotappeartohaveagreatimpactonthecoverageoftheresultingintervals.Percentcoverageisfairlyaccurateforclusteredpatternswithalargeintensity(=500).Thepercentcoverageofmostofthemethodsvarieswiththeintensityofthepatternstosomedegree.Thesplittingmethodhasthemostvolatilereactionsbasedonintensitywithcoverageforaninhibitedpatternwithsmallintensityequalingapproximately0.Thecoverageforthesplittingmethodonclusteredpatternswithsmallintensityhasfairlyhighvariationwithdistanceaswell.However,withthesetwoexceptions,thepercentcoverageofthismethodisgood.Coverageofintervalsusingtilingimprovesastheintensityofthepatternincreases.Theperformanceofthemarkedpointmethodandthesubsettingmethodarefairlyconsistentwithintensityasthepercentcoverageincreasessteadilywithdistanceformostprocessesatallintensities.Therateatwhichthepercentcoverageincreasesseemstoincreaseasintensityincreases.Forclusteredpatterns,resultsfromthesubsettingmethodarepoorforsmallintensities.Theeffectsofintensityontheperformanceofthenetworkingmethodvaries 63

PAGE 64

basedonthetypeofprocess.ForsoftcoreandPoissonprocesses,thismethodseemstoperformbestatsmallintensities.However,forclusteredprocesses,thepercentcoverageoftheintervalsincreaseswithintensityoftheprocess.Surprisingly,thedegreeofclusteringfortheMaternprocesswithanintensityof250pointshadlittleeffectonthecoverageoftheintervals.ThemeanwidthsofthecondenceintervalsfromthebootstrapmethodsaredisplayedinFigures 2-15 2-24 .TocomparethevarianceofeachbootstrapestimatorforK(r)tothevarianceof^K,1000patternsfromeachprocessweresimulatedandthewidthsofthe95%simulationenvelopefortheK-functionfromthesesimulationsplotted.WiththeexceptionoftheMaternprocesswithhighintensity,intervalscreatedusingthenetworkingmethodaresubstantiallynarrowerthantheothermethods,especiallyatthelargerdistancesevaluated.FortheMaternprocesseswithgreaterintensities(=250and500),thecondenceintervalwidthsarecomparableforallmethods.However,thewidthsoftheseintervalsarewiderthanthosebasedonthesimulatedpatterns.Forallprocesses,markingandsplittingcreatedintervalsthataresignicantlywiderthanthevariationofK(r)estimatedfromsimulatedpatterns.ForthePoissonandsoftcoreprocessesathighintensities(=500),thewidthsofintervalscreatedusingthenetworkingmethodarecomparabletothewidthsoftheintervalsbasedonsimulations,indicatingthattheseintervalsgiveagoodapproximationofthevarianceoftheK-functionfortheseprocesses.ThepercentcoveragewasalsodeterminedforeachbootstrapmethodforaPoissonprocessandaMaternclusteredprocesswithrelativelysmallintensities(=25).Forthesplittingmethod,smallresampleblocksfailedtoaccountforthevariationintheprocessandcoveragewaslowerthanthenominallevel.Usinglargeresampleblocks,coveragewasclosetothenominallevel,althoughdroppingtoabout80%afterdistancesof0.1unitsfortheclusteredprocess.Themarkedpointmethodhadsimilarperformancetoprocesseswithgreatintensities.ForthePoissonprocess,estimatedcoveragesteadily 64

PAGE 65

increasedfrom80%atsmalldistancesto100%forr0.15.FortheMaternprocess,estimatedcoveragereachedamaximumof90%atadistanceof0.1unitsanddroppedtoasteadyvaluejustbelow80%.Thenetworkingmethoddidnotperformnearlyaswell,withcoverageofthenominal95%intervalsbetween60%and70%foraPoissonprocessandmuchlowerforaclusteredprocess.Itisinterestingtonotethatthepercentcoverageforthesplittingmethodandmarkedmethodusinglargerresampletilesismaximizedatapproximatelyadistanceequaltotheradiusofinteractionfortheclusteredprocess. 2.7DiscussionBootstrappingmethodstoconstructcondenceintervalsforRipley'sK-functionofpointprocesseshavevaryingresults.Thepercentcoverageofthecondenceintervalsconstructedusingthetilingmethodarefarthestfromthenominallevel.Splittingprovidesaccuratenominal95%condenceintervalsforthemajorityofprocessesandmaintainsaccuratecoverageoverarangeofdistances.However,whensplittingfailstogiveaccuratecoverage,thecoverageisfarfromthenominallevel.Forthesoftcoreprocesswithanaverageintensityof100points,coverageisapproximately0%.Markingresultedinintervalsthataretoowideinmostcases.Formostprocesses,thepercentcoverageislowerthanthenominallevelatshortdistancesandsteadilyincreasesto100%.Usingsmallerresamplingtiles(sidelength=0.25units)increasestheaccuracyofintervals,butonlymarginally.Networkingcreatesintervalswithstablecoveragethattendtobebelowthenominallevelformostoftheprocesses.Forclusteredprocesses,coverageismoreaccurateasintensityincreases.ForthePoissonandinhibitedprocesses,theaccuracyofthecoveragedecreasesasintensityincreases.Formostprocesses,splittingandmarkingcreateintervalsthatarewiderthanthesimulationenvelopeforK(r)determinedbysimulatingtheprocessandestimatingK(r).Networkingcreatesintervalsthatarewiderthanthesimulatedintervals;however,thewidthbecomesclosertotheenvelopewidthasintensityincreasesforPoissonand 65

PAGE 66

inhibitedprocesses.Thewidthsofthecondenceintervalsforclusteredprocessesarecomparableacrossallmethods.Splittinghastheadvantagethatitiseasilyimplemented.Italsoproducesintervalsthatareaccurateinmostcases.However,theintervalsarewiderthanthevariationfromtheprocessesthemselves.Thisislikelyduetobeingbasedonsmallernumbersofobservations.Thechoiceoftilesizeusedtoresamplethepatternvariesbasedontheprocess.Insomecases,largerresampletilesresultedinintervalswithmoreaccuratecoverage.However,inothercases,smallertilesizesresultedinmoreaccuratecoveragefromtheintervals.Thiscreatesachoicefortheresearcherastowhichisthemostappropriateblocksizefortheobservedpattern.Markingproducesintervalswhosecoverageismoreaccuratethansomeoftheothermethods.However,thepercentcoveragechangesbasedonthedistancebeingevaluatedandreaches100%forthemajorityofprocessesassessed,indicatingthatcondenceintervalsaretoowide.However,theaccuracyusingmarkingisnotasvolatileassplittingand,ifconservativeintervalsareofinterest,thisisaviablemethodforcalculatingcondenceintervals.Thepercentcoverageusingthenetworkingmethodisaccurateinmostscenarios.Generally,thecoverageisbelowthenominallevel.Thecoveragealsovariesbasedontheintensityoftheprocessandthedirectionofthisvariationdependsontheprocess.Forclusteredpatterns,highlyaccurateresultsareobtainedathighintensitiesandcoverageclearlyimprovesasintensityincreases.ForPoissonandregularpatterns,themostaccuratecoverageisobtainedatsmallintensities.Forsparsepatterns,intervalsusingnetworkinghavepoorcoverage.However,formostprocesses,theintervalscreatedusingnetworkingarebyfarthenarrowestandstillproducefairlyaccuratecoverage,providingthebestapproximationofthevarianceoftheK-functionfromtheunderlyingprocess. 66

PAGE 67

Foranecologistworkingwithpointpatterndata,thechoiceofmethodtocalculatecondenceintervalsforasummaryfunctionoftheunderlyingprocesslikelydependsontheobjectiveoftheresearcher.Thesplittingmethodistheeasiesttoimplementandproducesaccurateintervalsforamajorityofprocesses.However,thecondenceintervalstendtobewideand,forsomeprocesses,theintervalsfailtocapturethetheoreticalvalueofthefunctionofinterest.Thus,abootstrappingapproachhasmerit.Ifconservativeresultsarepreferredsothattheresearcheriscondentthattruevaluesliewithinthisinterval,themarkingmethodshouldbeused.Ifnarrowintervalsaredesiredandtheresearchercanaffordtobeslightlyliberalincalculatingcondenceintervals,thenetworkingmethodisrecommended.Ifinterestisinthevariationofthefunctionfromtheunderlyingprocess,theintervalsfromnetworkingprovidemoreaccurateestimatesofthis,especiallyasintensityoftheprocessincreases. 67

PAGE 68

Table2-1. Theprocessesandtheirrespectiveparametersusedinthesimulationstudy. ProcessIntensityRadiusofInteractionParentPointsDaughterPoints Poisson100NANANA250NANANA500NANANAMatern1000.110102500.125102500.0525105000.15010Softcore1000.1NANA2500.05NANA5000.02NANA 68

PAGE 69

Figure2-1. Exampleoftoriodalwrapping:ThisgureexplainstoriodalwrappingusedforresamplingbyreplicatingthewindowAina3x3grid.TilescanbedrawnanywhereinAandallowedtooverlapintoneighboringpatternsduringresampling. 69

PAGE 70

Figure2-2. ExampleofLohandStein'smarkedpointmethod:Fordistancer,pointsyandzcontributetopointx'smarkthroughtheirweightsassociatedwithx.Pointvdoesnotcontributedtothemarkmx(r).Herepointsx,y,andvareresampled(blockbootstrappingwithtoroidalwrapping),butpointzisnot.However,someoftheinformationfromzhasbeenresampledthroughitsmarkontheotherpoints.Figuretakenfrom( LohandStein 2004 ) 70

PAGE 71

Figure2-3. Dendrogramofnetworkingmethod:Thedendrogramofasamplepattern,separatedintonetworksusinghierarchicalclustering.Thehorizontallinerepresentsthevalueof,thenetworkradius.Pointsthatareconnectedbelowthislinebelongtothesamenetwork. Figure2-4. RealizationsfromahomogeneousPoissonpointprocess,aNeymann-Scottclusteredprocess,andasoftcoreinhibitedprocess.Eachprocesshasanintensityofapproximately250points. 71

PAGE 72

Figure2-5. Percentcoveragesof95%condenceintervalsforK(r)ofPoissonpatternswithintensity=100:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. Figure2-6. Percentcoveragesof95%condenceintervalsforK(r)forPoissonpatternswithintensity=250:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. 72

PAGE 73

Figure2-7. Percentcoveragesof95%condenceintervalsforK(r)forPoissonpatternswithintensity=500:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. Figure2-8. Percentcoveragesof95%condenceintervalsforK(r)forsoftcorepatternswithintensity=100:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. 73

PAGE 74

Figure2-9. Percentcoveragesof95%condenceintervalsforK(r)forsoftcorepatternswithintensity=250:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. Figure2-10. Percentcoveragesof95%condenceintervalsforK(r)forsoftcorepatternswithintensity=500:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. 74

PAGE 75

Figure2-11. Percentcoveragesof95%condenceintervalsforK(r)forMaternclusteredpatternswithintensity=100:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. 75

PAGE 76

Figure2-12. Percentcoveragesof95%condenceintervalsforK(r)forMaternclusteredpatternswithintensity=250:Thisprocesshasaradiusofinteractionof0.05units.Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. 76

PAGE 77

Figure2-13. Percentcoveragesof95%condenceintervalsforK(r)forMaternclusteredpatternswithintensity=250:Thisprocesshasaradiusofinteractionof0.1units.Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. 77

PAGE 78

Figure2-14. Percentcoveragesof95%condenceintervalsforK(r)forMaternclusteredpatternswithintensity=500:Theleftpanelindicatesintervalscalculatedusingtilesofsidelength0.5unitstoresamplethepattern.Therightpanelindicatesintervalscalculatedusingtilesofsidelength0.25unitstoresamplethepattern. 78

PAGE 79

Figure2-15. CondenceintervalwidthsforPoissonpatternswithintensity=100:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 79

PAGE 80

Figure2-16. CondenceintervalwidthsforPoissonpatternswithintensity=250:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 80

PAGE 81

Figure2-17. CondenceintervalwidthsforPoissonpatternswithintensity=500:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 81

PAGE 82

Figure2-18. Condenceintervalwidthsforsoftcorepatternswithintensity=100:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 82

PAGE 83

Figure2-19. Condenceintervalwidthsforsoftcorepatternswithintensity=250:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 83

PAGE 84

Figure2-20. Condenceintervalwidthsforsoftcorepatternswithintensity=500:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 84

PAGE 85

Figure2-21. CondenceintervalwidthsforMaternclusteredpatternswithintensity=100:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 85

PAGE 86

Figure2-22. CondenceintervalwidthsforMaternclusteredpatternswithintensity=250:Thesepatternshavearadiusofinteractionof0.05units.Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 86

PAGE 87

Figure2-23. CondenceintervalwidthsforMaternclusteredpatternswithintensity=250:Thesepatternshavearadiusofinteractionof0.1units.Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 87

PAGE 88

Figure2-24. CondenceintervalwidthsforMaternclusteredpatternswithintensity=500:Theblacklineindicatesthewidthofthe95%intervalbasedon1000simulatedpatternsfromtheprocess.Theleftpanelindicatesintervalsbasedtilesofsidelength=0.5unitsforresampling.Therightpanelindicatesintervalscalculatedwithtilesofsidelength0.25unitsusedtoresamplethepattern. 88

PAGE 89

CHAPTER3HYPOTHESISTESTINGFORRIPLEY'SK-FUNCTION 3.1IntroductionScientistsareofteninterestedinwhetherthesamebiologicalprocessesgaverisetopointpatternsobservedatdifferentlocationsand/ortimes.Althoughthisquestioncannotbefullyanswered,testingtheequivalenceofthesecond-orderstructureoftwoormoreobservedspatialpointpatternsassesseswhethertheinteractionamongeventsfromthesepatternscouldbethesame.Understandinghowtheinteractionamongeventschangesforpointpatternswithdifferentmanagementschemes,differentenvironmentalcharacteristics,orrecordedatdifferentpointsintimecanhelpresearchersunderstandthebiologicalprocessesandhowtheychangeorreacttoexternalcovariates.ComparisonoftheestimatedK-functionscanhelpidentifydifferencesinthesecond-orderstructureofmultipleobservedspatialpointpatternsandwhethertheinteractionamongobservedpointsissimilar.AsdescribedinChapter2,Ripley'sK-functionandtheclosely-relatedL-functionareusedtosummarizethesecond-orderstructureofapointpattern.Undertheassumptionsofstationarityandisotropy,theK-functionforaprocessrepresentstheexpectednumberofpointswithinadistancerofanarbitrarypointintheprocess,standardizedbytheprocessintensity.AnumberofhypothesistestsofthespatialinteractionamongpointsinanobservedpatternhavebeendevelopedusingK(r)oranothersummaryfunctiontocalculateateststatistic.However,mostofthesearedesignedtotestthehypothesisthatapatternistherealizationofaspeciednullprocessusingMonteCarlomethods( BesagandDiggle 1977 ; Diggle 1979 ; HoandChiu 2006 ; Ripley 1979 ).Here,interestisinconductinginferenceonRipley'sK-functionfortwoormoreobservedspatialpointpatterns.BecauseRipley'sK-functionisstandardizedbytheintensityoftheprocess,theK-functionforaprocessisindependentoftheintensity, 89

PAGE 90

assumingothercharacteristicsoftheprocessesarethesame( Baddeleyetal. 2000 ).Forinstance,twohomogeneousPoissonprocesseswithdifferentintensitieshavethesametheoreticalK-function,thoughthevarianceoftheestimatorsaredifferent.Likewise,twoMaternclusteredprocesseswiththesamenumberofparentpointsandanidenticalradiusofinteractionhaveanidenticaltheoreticalK-function,regardlessofthenumberofdaughterpointsperparent.However,changingthenumberofparentpointsalterstheprocessandresultsinadifferentK-function.Pointprocesseswithdifferentsecond-ordermomentscanhaveidenticalK-functionsinsomecases( BaddeleyandSilverman 1984 ).Thus,claricationofthenullhypothesisiscriticalforstatisticalinferenceonspatialpointprocesses.Inthiswork,teststhatcomparetheK-functionsfrommultipleobservedpatternsareconsidered.Throughthese,thesecond-ordermomentsofpointpatternscanbeassessed.AsinDiggleetal.(1991),theinterestisindeterminingwhethertheobservedpatternsdiffersignicantlyfromgrouptogroupafteradjustingforthedifferencesinintensity.Inthiscase,noreplicatedpatternswithingroupsareobservedandanalysisisconductedongroupsofsizeone.Thusifgpatternsareobserved,wewishtotestthenullhypothesis H0:K1(r)=K2(r)=...=Kg(r)forg2andforalldistancesrr0.(3)versusthealternativehypothesisthatKi(r)isnotequaltotheotherK-functionsforatleastonepatterni.Formalinferenceonparametersofpointprocesses,suchastheK-function,isdifcultasthedistributionsoftheirestimatorsaregenerallyunknown.LohandStein(2004)studiedintervalestimationandvarianceapproximationfortheK-function(Chapter2).Also,thedevelopmentofteststatisticsthatfollowknowndistributionsisdifcultduetochallengeswithnon-normalityanddependenceofobservations.Thus,withoutfurtherknowledgeofthedistributionoftheestimators,parametric 90

PAGE 91

hypothesistestingisdifcult,leadingmostteststoemployMonteCarlomethodsorsimilartechniques( Diggleetal. 1991 2000 ; Hahn 2012 ).Challengesalsoarisebecausemostparametersdetailingthesecond-orderstructureofpointprocessesarefunctionsofdistance.Thisleadstothequestionofwhetheranalysisshouldbeconductedatindividualdistancesorcombinedoversomerangeofdistances.Ifinterestisintheanalysisataparticulardistance(i.e.,thenumberofpointsthatliewithinagivenradius),testscanbeconstructedtoassessthis.However,ifinterestisincomparisonsatmultipledistancesorofafunctionasawhole,thechoiceofthespecicdistanceorrangeofdistancesandthetestprovidingthegreatestpowerwhilecontrollingthesizedependsonthealternativehypothesis( Hahn 2012 ).Mostauthorsrecommendcomparingthefunctionsacrossthedistancesofinterestbyestablishingsomedistancemeasurement.Letf1andf2representtheobservedfunctionsoftwoobservedpatternsortheempiricalfunctionandtheoreticalfunctionfromapatternandanullprocess.Thentwocommonmeasuresofdistancebetweenf1andf2arethesupremumdistance d1(f1,f2;r0)=suprr0jf1(r))]TJ /F4 11.955 Tf 11.95 0 Td[(f2(r)j(3)andtheL2distance d2(f1,f2;r0)=Zrr0(f1(r))]TJ /F4 11.955 Tf 11.95 0 Td[(f2(r))2dr(3)( Hahn 2012 ).Inthecaseoftheproposednullhypothesis,f1andf2representtheKorLfunctionsfromtwoobservedpatterns.Noticethatthemeasuresof 3 and 3 ofthedistancebetweenthefunctionf1andf2dependonthechoiceofr0.Ripley(1979)suggestedthevaluer0=1.25=p ()toprovidepowerfultests,whereistheintensityoftheprocess( Ripley 1979 ).HoandChiu(2006)investigatedthechoiceofr0,concludingthatadaptedestimatorsfortheintensityoftheprocesscanbeusedtoimprovethepowerofthetests( HoandChiu 2006 ).HoandChiu(2009)suggestedusingdistance-basedweightingfunctionsinstead 91

PAGE 92

ofequallyweightingalldistancesrasinEquation 3 ( HoandChiu 2009 ).ThisadjustsfortheheteroscedasticityintheK-functionand,toalesserextent,intheL-function.Severaltestshavebeenproposedtocomparefunctionsorparametersdescribingthespatialinteractionamongeventsinreplicatedpatternsfromdifferentgroups.Specically,Diggleetal.(1991)andDiggleetal.(2000)presenttwoteststocomparethesecond-orderstructureofseveralgroupsfromwhichmultiplepatternshavebeenobserved( Diggleetal. 1991 2000 ).Diggleetal.(1991)developedateststatisticcomparingthemeanK-functionacrossdifferentgroupswhenreplicatedpatternsareobservedfromeachgroup.P-valuesareobtainedbybootstrappingtheK-functionfromtheobserveddistributionofK-functionsofthepatternsandcomparingtheobservedteststatistictothosecalculatedfromthesebootstraps( Diggleetal. 1991 ).Diggleetal.(2000)comparedthistesttooneusingthepseudo-likelihoodsoftheobservedpatterns.InDiggleetal.(2000),weakmodelassumptionsaremade,andthespatialinteractionisdenedasafunctionofdistance.Parametersofthisfunctionareestimatedusingmaximumpseudo-likelihoodunderthenullhypothesisthattheseparametersareequalacrossgroupsandthealternativehypothesisthattheseparameterscandifferamonggroups.Ateststatisticiscalculatedfromthedifferenceinthepseudo-likelihoodsunderthealternativeandnullhypothesis.Theteststatisticiscomparedtotheempiricaldistributionofthestatisticcalculatedfromsimulatedpatternsunderthenullhypothesestoobtainap-valueforthetest( Diggleetal. 2000 ).Sometimesasmallnumberofspatialpatternsisobserved.Theunderlyingprocessofeachisunknownandnoreplicatesareavailable.Yetatestoftheequalityofthesecond-orderstructure,orspecically,equalityoftheK-functionsamongthepatternsisofinterest.Hahn(2012)developedastudentizedpermutationtesttocomparetwoormorepatterns,usingsubsamplesfromthesepatternstoestimatethevarianceofK(r)foreachprocess.TheteststatisticusedinthistestisbasedonDiggleetal.(1991),inwhichgroupmeansfortheK-functionrecordedfromquadratsofdisjointpatternsare 92

PAGE 93

compared,adjustingforthevarianceintheseestimates.Hahn'sanalysisfoundthattheMonteCarlomethodsusedbyDiggleetal.(1991,2000)produceliberaltestswhenthenumberofreplicatepatternspergroupissmall.Becausethenumberofdisjointquadratsusedtoobtainvarianceestimatesfortheteststatisticislimited,Hahnusedapermutationtestsothatthesizeofthetestisexact( Hahn 2012 ).Givenapermutationofthequadratsfromtheobservedpatterns,theteststatisticiscalculatedforthisnewpermutation.Theobservedteststatisticiscomparedtothedistributioncalculatedfromthepermutationstoobtainap-value.Becausethenumberofpermutationscanmakethecomputationsforapermutationtestinfeasible,Hahn(2012)alsoconsideredconductingthetestbasedonaprespeciednumberofrandompermutations.Here,twonewtestsaredevelopedtocomparetheK-functionsoftwoormorespatialpointpatterns.ThemarkedpointmethodofBraunandKulperger(1998)thatwasadaptedtospatialpointpatternsbyLohandStein(2004)isadoptedhere.K-functionsarecomparedbythesumsofsquareddeviationofthefunctioncalculatedoverquadratsofpatternsunderthenullhypothesisthattheK-functionsfordifferentpatternsareequalandthealternativehypothesisthattheK-functionforatleastoneofthepatternsisdifferent.TheheteroscedasticityoftheK-functionisadjustedforintwoways.Inthersttest,theL-function,atransformationoftheK-function,isusedtoreducetheheteroscedasticityintheestimateofK(r).Inthesecondtest,thesquareddeviationfromthemeanK-functionsisadjustedbytheapproximatedvarianceof^K(r)foraPoissonprocess.Foreachtest,theobservedteststatisticiscomparedtothedistributioncalculatedfromanumberofrandompermutationsofthequadrats.AsimulationstudycomparesthesizeandpoweroftheseteststothosedevelopedbyHahn(2012). 3.2Hahn's(2012)StudentizedPermutationTestHahn(2012)introducedthersthypothesistesttocomparetheK-functionsofsinglerealizationsofmultiplepointprocesses.Heproposedtwoteststatistics,whichare 93

PAGE 94

extensionsoftheDiggleetal.(1991)teststatisticforcomparingreplicatedpatternsfromdifferentgroups.Insteadofmultiplepatternsbeingobservedfromeachgroup,eachpatternistreatedasthegroupandthepatternsaredividedintomiquadrats,whereirepresentsoneofgobservedpatternsfori=1,...,gandg2.Foreachquadratofeachpattern,K(r)isestimated.ThegroupmeansaredeterminedbyaveragingtheestimatedK-functionsfromallquadratsofapattern.TheteststatisticdevelopedbyHahn(2012)isthesquareddifferenceinthemeanK-functionsfromdifferentpatterns,adjustedbythesumofthevarianceof^K(r)fromeachpattern.SincetheK-functionisafunctionofdistance,thisfunctionisintegratedtoapredetermineddistancer0.Ifmorethan2patternsareobserved(g>2),thisintegratedfunctionissummedforallcombinationsofpatterns.Thus,ifg=3,thestatisticiscalculatedforpatterns1and2,patterns2and3,andpatterns1and3,andsummedtoobtaintheteststatisticforHahn'stest.Asimilarteststatisticisalsocalculatedusingasmoothedestimationofthevarianceof^K(r)foreachpattern.P-valuesforHahn'stestareobtainedusingapermutationtest.Forapermutation,allquadratsareassignedtooneoftheobservedpatternsandthenumberofquadratsassignedtoeachpatternisequaltothenumberofthatpattern'squadratsinthecalculationoftheteststatistic.Theteststatisticiscalculatedforthisnewpermutation.Thisisdoneforeitherallpermutations(ifthenumberofquadratsissmall)orforapredenednumberofrandompermutations(ifthenumberoftotalquadratsislarge).Underthenullhypothesis,quadratsareexchangeable,providinganempiricaldistributionoftheteststatistic.Theobservedteststatisticisthencomparedtotheempiricaldistributiontoobtainap-value,withlargevaluesoftheteststatisticbeingassociatedwithsmallp-values.Specically,forHahn's(2012)test,eachpatterni,fori=1,...,g,isdividedintomidisjointquadrats.Foreachquadratjinpatterni,Kij(r)iscalculatedfromthepointscontainedintherespectivequadratandusingthequadratboundariesasthewindowfor 94

PAGE 95

theestimateofK(r).TheteststatisticisthencalculatedfromtheestimatesofK(r)fromthequadrats T=X1i
PAGE 96

todeterminethep-valueassociatedwiththetest.Iftheteststatisticislargecomparedtothosecalculatedfromthepermutedquadrats,thep-valuefromthetestissmall.Thenumberofpermutationsincreasesrapidlywiththenumberofpatternsornumberofquadratsperpattern.Toillustrate,fortwopatterns,eachpartitionedintoninequadrats(patternsaredividedinto3x3gridssuchthatm1=m2=9),thereare)]TJ /F9 7.97 Tf 5.48 -4.38 Td[(189=2=24,310permutationsforwhichtocalculatetheteststatistic.Forscenarioswithalargernumberofpermutations,Hahnusesp-valuesdeterminedfrom4000randompermutations.Hahnconductedalargesimulationstudytoexploretheempiricalsizeandpoweroftheseteststatistics.Theeffectsofquadratnumber,size,patternintensity,anddegreeofclusteringwereassessed.Thepowerofthetestsfordifferentvaluesofr0,thedistancetowhichtheteststatisticiscalculated,wasalsoexplored. 3.3ProposedTestStatistic1Hahn's(2012)teststatistichashighpowerwhentwopatternsarerealizationsofverydifferentprocesses.However,whenprocessesaresimilar,thepowerofHahn'stestfallsbelow0.4atmostdistances.EstimationofK(r)foreachquadratcausesedgecorrectionweightstohavealargeinuenceonresults,especiallyforsmallquadrats.Thismayhavealargeeffectonthetestifquadratsizesorshapesvarywithpattern.TwonewstatisticalteststocomparetheK-functionsestimatedfromtwoormoreobservedspatialpointpatternsareproposed.AmarkedpointmethodisusedintheestimationofK(r)toreducetheinuenceofedgecorrectionweightsforsmallquadrats( BraunandKulperger 1998 ; LohandStein 2004 ).Thetwotestsaresimilar,butrelyondifferentmethodstoaccountfortheheteroscedasticityoftheK-functionasdistanceincreases.TherstisbasedonthetransformationoftheK-function,theL-function.ThesecondmethodusesaweightingfunctionsimilartothatofHoandChiu(2009).Theproposedteststatisticismotivatedbytestscomparingfullversusreducedmodelsinmultiplelinearregression.AswithLohandStein's(2004)markedpointmethodforintervalcalculation,marksareassignedtoeachpointintheobserved 96

PAGE 97

patterns.Eachmarkequalsthenumberofotherpointswithindistancer(weightedforedgecorrections)ofthatparticularpoint.PatternsaredividedintoquadratsandtheK-functionsareestimatedforeachquadratbysummingthemarksforallpointswithinthequadrat.ToreduceheteroscedasticityoftheK-function,estimatesofK(r)aretransformedtotheL-function.Becausethisisamonotonetransformation,useoftheL-functiondoesnotaffectthehypothesisorresultsofthetest.Asdescribedindetaillater,thesumofsquaresofdeviationsfordistanceriscalculatedbyusingthedifferencebetweenestimatesofL(r)foreachquadrattothemeanL-functionsunderthenullhypothesisthattheL-functionsfromdifferentpatternsareequal,andunderthealternativethattheL-functionfromatleastonepatternisdifferent.Theteststatisticisanadjustedratioofthesesumsofsquares,takingintoaccountthedegreesoffreedomusedforestimationofthemeanL-functions.SimilartoHahn's(2012)test,arandomizationtestisusedtoobtainp-valuesfortheteststatistics.Quadratsarerandomlypermuted,andtheteststatisticiscalculatedforthepermutation.Thisisrepeatedforapredenednumberofrandompermutations(i.e.,permutationsarenotnecessarilyunique)andtheobservedteststatisticiscomparedtotheempiricaldistributiongeneratedfromthepermutations.Similartestscanbeconstructedusingallpermutationsorapredenednumberofuniquepermutations.Morerigorously,assumethatgpatternsareobservedoverdisjointwindows.SimilartothemarkedpointmethodintroducedbyLohandStein(2004),marksareassignedtoeachpoint,foreachdistancer,representingthesumofallweightsforpointswithindistancerofthepoint.Thatis,forpointx, mx(r)=Xy:y6=xw(x,y)Ifjjy)]TJ /F17 11.955 Tf 11.95 0 Td[(xjj
PAGE 98

quadratsandanestimatorofKij(r)iscalculatedby ^Kij(r)=jAijj n2ijXx2Qijmx.(3)whereQijrepresentsquadratjinpatterniandjAijjrepresenttheareaofthequadrat,andnijrepresentstherespectivenumberofpoints.ToaccountfortheincreasingvarianceoftheK-functionasdistanceincreases,^Kij(r)istransformedto^Lij(r).Recall ^Lij(r)=s ^Kij(r) .(3)TheestimatorofLi(r)foreachpatternistheaverageoftheestimatedL-functionsfromeachquadrat Li(r)=1 mimiXj=1^Lij(r).(3)AnestimateofthemeanL-functionfromallpatternsisalsocalculatedinasimilarmannerbyaveraging^Lij(r)overallmiquadratsinallgpatterns L(r)=1 Pgi=1migXi=1miXj=1^Lij(r).(3)NotethatthisisdifferentfromthetypicalestimatorofL(r).ThiscanbemoreeasilyseenusingtheK-function.ThetypicalestimatorofK(r)is ^Ki(r)=jAij n2iXx2AXy6=xw(x,y)I(jx)]TJ /F17 11.955 Tf 11.96 0 Td[(yj
PAGE 99

Equalityholdsin 3 ifnijisequalforallj.Thus,whenthepatternshavedifferentintensitiestheestimatorK(r)equallyweightsthequadratsinmultiplepatterns.Theprobabilityofanegativeteststatisticispositivewhenaweightedaverageofthe^Ki(r)isused.ThesumsofsquaresarecalculatedusingtheestimatedL-functionsfromeachquadrat,averagedwithinapatternandaveragedoverpatternsforaparticulardistancer.SSNullisthesumofsquarescalculatedunderthenullhypothesisofacommonK-function,andthereforeL-functionforallpatterns.SSAltisthesumofsquaresunderthealternativehypothesisL-functions.Thatis, SSNull(r)=gXi=1miXj=1^Lij(r))]TJ /F5 11.955 Tf 12.21 2.66 Td[(L(r)2(3)and SSAlt(r)=gXi=1miXj=1^Lij(r))]TJ /F5 11.955 Tf 12.2 2.65 Td[(Li(r)2.(3)Thesesumsofsquaresaretheintegratedtoapredenedvalueofr0, SSNullr0=Zr00SSNull(r)dr(3)and SSAltr0=Zr00SSAlt(r)dr.(3)Finally,theteststatisticiscalculatedas TS=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(SSNullr0)]TJ /F4 11.955 Tf 11.95 0 Td[(SSAltr0=(g)]TJ /F5 11.955 Tf 11.96 0 Td[(1) SSAltr0=(Pmi)]TJ /F4 11.955 Tf 11.96 0 Td[(g).(3)Althoughmoreinvestigationneedstobeconductedonthedistributionoftheteststatistic,itdoesnotfollowthetypicalF-distributionasinlinearregressionorANOVA.Thus,arandomizationtestisusedtodeterminethep-valuesassociatedwiththeteststatistic.SimilartoHahn(2012),quadratsarerandomlypermutedandtheteststatisticiscalculated.Thisisrepeatedforapredenednumberofrandompermutations. 99

PAGE 100

Permutationsaredrawnwithreplacementandarenotnecessarilyunique.Ineachpermutation,quadratsareonlyusedonceandthenumberofquadratsassignedtopatterniismi.Therankedpositionoftheteststatisticfromtheoriginalpatternsintheempiricaldistributionofteststatisticscalculatedfromthepermutationsservesasthep-valueforthetest.Iftheteststatisticcalculatedfromtheobservedpatternsislargerelativetothosefromthepermutedquadrats,thep-valueissmall. 3.4ProposedTestStatistic2ThepreviousteststatisticusestheL-function,avariancestabilizingtransformationoftheK-function,tocopewiththeissuethatthevarianceoftheestimateoftheK-functionincreasesasdistanceincreases.Asaconsequence,theweightfordeviationsatsmalldistancesaremoreequaltothedeviationsatlargerdistanceswhencalculatingtheintegratedsumsofsquares.Here,asimilarteststatisticisproposedusingtheK-functionandaweightingfunction,similartothatproposedinHoandChiu(2009),toadjustforheteroscedasticity.Again,itisassumedthatgpatternsareobservedoverdisjointregionsandthemarksmx(r)aredeterminedforallpointsinthepatterns.TheK-functionfromeachquadratisestimatedasinEquation 4 .TheestimatorofKi(r)iscalculatedforeachpatternbyaveragingtheK-functionsfromeachquadrat Ki(r)=1 mimiXj=1^Kij(r).(3)AnestimateoftheK-functionfromthegpatternsisalsocalculatedastheaverageoftheestimatesofK(r)fromallquadratsinallpatterns: K(r)=1 Pgi=1migXi=1miXj=1^Kij(r).(3) 100

PAGE 101

Similartothepreviouslyproposedmethod,thesumsofsquaresarecalculatedunderthenullandalternativehypotheses: SSNull(r)=gXi=1miXj=1^Kij(r))]TJ /F5 11.955 Tf 13.73 2.66 Td[(K(r)2(3)and SSAlt(r)=gXi=1miXj=1^Kij(r))]TJ /F5 11.955 Tf 13.73 2.66 Td[(Ki(r)2.(3)Thesesumsofsquaresaremultipliedbytheirrespectiveweightsandintegratedtoapredeterminedvalueofr0, SSNullr0=Zr00w(r)SSNull(r)dr(3)and SSAltr0=Zr00w(r)SSAlt(r)dr.(3)Herew(r)=1 var(KPij(r))isanapproximationofthevarianceoftheestimateofK(r)foraPoissonprocess,conditionalon1 PmiPgi=1nipointsbeingobservedinawindowthesizeofonequadrat(forquadratsofequaldimensions).Theteststatisticiscalculatedforapredeterminednumberofrandompermutationsofthequadrats,withreplacement,andp-valuesarefoundbasedontherankedpositionoftheteststatisticfromtheobservedpatternstothosefromthepermutations. 3.5SimulationStudyAlargesimulationstudywasconductedtoexaminethesizeandpowerofthesehypothesistestsandtocomparethemtoHahn's(2012)studentizedpermutationtest.Tomyknowledge,thesearecurrentlytheonlyavailableteststocomparethesecond-orderstructureofsinglerealizationsfrompointprocessmodels.Inadditiontorecommendingatestforresearchersinterestedincomparingtwoofmorepointpatterns,suggestionsonchoicesforr0andthenumberandsizeofquadratsareexamined.Theresultsforeachtestarealsoexaminedforcomparingquadratsofdifferentshapes. 101

PAGE 102

Toassessthesizeofeachteststatistic,weuseseveraldifferentprocessesatvaryingintensities.ThesizeofeachteststatisticisdeterminedforaPoissonprocess,aMaternclusterprocesswithasmallradiusofinteraction(highdegreeofclustering),aMaternclusterprocesswithalargeradiusofinteraction(smalldegreeofclustering),asoftcoreregularprocess,andahardcoreregularprocess,eachsimulatedontheunitsquare.Foreachprocess,threeintensitiesareused(=100,250,and500).PoissonprocessesofdifferentintensitieshavethesametheoreticalK-function.However,clusteredandsoftcoreprocesseswithdifferentintensitiesdonot.Poissonpatternsofdifferentintensitiesarealsocomparedtooneanothertodeterminethesizeofthetestsforprocessesofdifferentintensities,butwhichhaveidenticalK-functions.ThePoissonprocesswithintensityhasaPoissondistributednumberofpointswithmeanequalto,randomlydistributedintheunitsquarewindow.TheMaternprocesshasaxednumberofparentpoints.Foreachparent,aPoissonnumberofdaughterpointswithameanequaltoisdistributedrandomlywithinaxedradiusofinteraction.Theobservedpatternistheunionofalldaughterpoints.Thesoftcoreprocessisaninhibitedprocessinwhichpointshaveapositiveprobabilityoflyingwithintheradiusofinteraction,,fromoneanother.ThisprocessissimulatedbyrstgeneratingahomogeneousPoissonpatternwithanintensityhigherthanthatofthedesiredsoftcoreprocess.Foreachpoint,aradius,`,isgeneratedbasedonagivenprobabilitydistribution.Themaximumvalueofthedistributionwillbetheradiusofinteractionintheprocess.Eachpointisalsoassignedarandommark,m,uniformon[0,1].Pointsinthepatternaredeletedifthereisatleastoneotherpointlessthanitsassignedradius,`,awayfromitwithasmallermark,m`.Thenumberofpointsoriginallysimulatedandthedistributionoftheradiiforeachpointareadjustedtohaveadesiredintensityandtheradiusofinteractionamongpoints.ThehardcoreprocessissimulatedbysimulatingaPoissonpatternatagivenintensity.Pointsareremovediftheyfallwithinagivenradiusofinteractionofoneanothersuchthattheprobabilityofseeingtwo 102

PAGE 103

pointslessthanrmaxapartfromoneanotheriszero.Theinitialintensityandradiusareadjustedtoachievethedesiredintensityofthehardcoreprocess.Todeterminetheeffectofintensityonthesizeandpowerofeachtest,m1=m2=9quadratsareusedfortwopatternsineachtest.Toassesssize,1000testsaresimulatedforeachprocessandintensitycombinationontheunitsquareunderthenullhypothesis.ThesizeisalsoassessedforPoissonprocesseswithdifferentintensities.TheKand/orLfunctionsareassessedatdistancesbetween0.01and0.25atintervalsof0.01.Thesizeofthetestisdeterminedatarangeofvaluesofr0between0.05and0.25unitsatintervalsof0.05unitsbyintegratingthesumsofsquarestoeachofthesevaluessothattheeffectofthischoiceonthesizeofthetestcanbedeterminedforeachprocess.P-valuesaredeterminedbyrankingtheobservedteststatisticwiththosecalculatedfrom4000randompermutationsofthequadrats,drawnwithreplacement.ThesizeoftestsforeachpatternandintensitycombinationareshowninFigures 3-1 3-6 forlevels=0.01,0.05and0.10.ThepowerofeachteststatisticisalsoevaluatedbycomparingeachprocesstoarealizationofCSRwithsimilarintensity.Eachtestisperformedon1000realizationsoftwopatterns,aPoissonpatternandoneofanotherprocesswithasimilarintensity.Toassesspower,9congruent,disjointquadratsareusedforeachpattern,eachofarea1/9squareunits.TheestimatesofK(r)areassessedatdistancesrangingfrom0.01to0.25unitsatintervalsof0.01units.Thepowerofthetestsisfoundatvaryingvaluesofr0between0.05and0.25unitsatintervalsof0.05units.ThepoweroftestsforeachpatternandintensitycombinationareshowninFigures 3-7 3-10 forlevels=0.01,0.05and0.10.ThesizeandpoweroftheproposedteststatisticsarealsocomparedwhenvaryingthesizeandnumberofthequadratsusedtoestimateK(r)foreachpattern.Becausefortheproposedtests,amarkingsystemisusedtoestimatetheK-function,edgecorrectionweightsarenotdependentonthesizeofthequadrat;thus,onlythenumberofpointsusedtoestimateK(r)foreachquadratischanged.Foralltests,theunit 103

PAGE 104

squareisusedasthewindowforallpatterns.Therefore,theintensityofthepatternsorthetotalnumberofpointsisnotalteredbyusingdifferentnumbersofquadrats.However,thesizeandnumberofquadratsusedaredirectlyrelated.Thesizeandpoweraredeterminedforpatternsdividedinto9(3x3),16(4x4),and25(5x5)quadrats.Forthisanalysis,thesameveprocessesareused.Theaverageintensityofallpatternsis250points,withanexceptionfordeterminingthesizeofatestwithPoissonpatternsofdifferentintensities.Inthatcase,aPoissonpatternwithanintensityof250pointsandaPoissonpatternwithanintensityof100pointsareused.K(r)isestimatedatdistancesrangingfrom0.01to0.25unitsatintervalsof0.01units.Thepowerandsizeofthetestsarefoundatvaryingvaluesofr0between0.05and0.25unitsatintervalsof0.05units.Thepowerofthetestsisdeterminedbycomparingeachnon-PoissonprocesstoaprocesswithCSRatsimilarintensities. 3.6ResultsHahn's(2012)TteststatisticandTteststatistichavesizeapproximatelyequaltothenominallevelformostpatternswithsimilarintensities(Figures 3-1 3-5 ).ForPoissonandregularpatternswithsmallintensities(=100),thesizeofthetestsusingTwasgreaterthanthesizeoftestsusingT.Astheintensityofthesepatternsincreases,thesizeoftestsusingtheseteststatisticsbecomesapproximatelyequal.Forregularpatternswithsmallintensities(=100),testsusingTareclosetothenominallevelwhiletestsusingTareconservative(sizeisapproximately0).Forclusteredpatterns,thetestshaveapproximatelyequalsizeusingbothteststatistics.Thedistancetowhichthetestsareintegratedr0haslittleeffectonthesizeoftestsusingHahn'steststatistics.Forpatternswithlargeintensities(=500),thesizeincreasesslightlyasr0increases.Forpatternswithsmallerintensities,thesizeisconstantforallvaluesofr0.ThesizeoftestsusingHahn'steststatisticisgreaterthanthenominallevelwhencomparingPoissonpatternswithdifferentintensities.Thesizeincreasesasr0increases(Figure 3-6 ).TestsusingThavesmallersizethantestsusingT,althoughstillgreater 104

PAGE 105

thanthenominallevel(approximately0.10-0.30for=0.05).ThesizeofthetestwhenusingTincreasesasthedifferenceintheintensitybetweenpatternsincreases.Althoughthesizeisabovethenominallevel,thedifferenceintheintensitiesbetweentwopatternsbeingcompareddoesnotaffectthesizeofthetestswhenusingT.Todeterminethepowerofthesestatisticaltests,allpatternsarecomparedtoapatternwithCSRandasimilarintensity.ThepoweroftestsusingHahn'steststatisticsvarybasedonthepatternsbeingcomparedtoCSR(Figures 3-7 3-10 ).WhencomparingregularprocessestoCSR,Hahn'stestshavethegreatestpoweratintermediateintensities(=250).Thepowerofthesetestsrangesfrom0.90to1andisconstantforallvaluesofr0.Testsofpatternswithsmallerandlargerintensitiesresultinpowerthatdecreasesasr0increases.Atlevel=0.05,thepowercomparingregularprocessestoCSRis0.60and0.70forTandT,respectively,whenr0=0.05and=100.Forr0=0.25,thepowerisapproximately0.40forbothteststatistics.Thepoweroftestscomparingpatternswithlargeintensitiesbehavesimilarly.Atlevel=0.05,thepowercomparingregularprocessestoCSRisapproximately1forTandT,whenr0=0.05and=500.Thepoweroftestsforr0=0.25isapproximately0.60and0.30forTandTrespectively.ForcomparingclusteredpatternstoCSR,thepowerofHahn'stestsincreasesastheintensityofthepatternsincreases.Thepowerofthesetestsisconstantforallr0.Atlevel=0.05thepoweroftestsusingTandTrangesfrom0.60to0.70when=100andtheradiusofinteractionfortheclusteredpatternissmall(i.e.,thepatternishighlyclustered).Thepowerofbothtestsis1for>100.Whentheclusteredpatternismoredispersed(i.e.,theradiusofinteractionislarger),thepowerdecreasesatallintensities.Thepowerofbothtestsisapproximately0.10,0.70,and0.90,for=100,250,and500,respectivelyandatr0=0.15.Thepowerofthesetestsincreasesslightlyforsmallr0andreachesmaximumpoweratr0=0.15. 105

PAGE 106

ThesizesoftestsusingtheteststatisticcalculatedfromtheL-function(referredtohereastheLteststatistic)andusingaweightfunction(referredtoastheWteststatistic)aresimilartoeachother(Figures 3-1 3-5 ).Inallcases,thesizesofthesetestsvarybasedonr0.WhencomparingPoissonandregularpatternswith500underthenullhypothesis,thesizesofbothtestsareabovethenominallevelforsmallr0.Forr0>0.15,thesizesofthesetestsareconservative(approximately0atalllevels).Thevaluer0atwhichsizereaches0decreasesastheintensityofthepatternsincrease.WhencomparingPoissonandregularpatternswithlargeintensities,thesizesofbothtestsarebelowthenominallevelforallr0,andreach0forr0>0.05.ThesizeoftestsusingtheLteststatisticisslightlylargerthantestsusingWwhencomparingPoissonpatternsandregularpatternsunderthenullhypothesis.ComparedtoHahn'stests,theperformanceoftestsusingtheLandWteststatisticsvarieslesswithchoiceofr0whentestingclusteredpatternsunderthenullhypothesis,althoughthesizeofthetestdecreasesasr0increases.Thesizeofbothtestsisclosetothenominallevelwhen=100,theradiusofinteractioninthepatternsissmall,andr0=0.05.Astheintensityincreases,testsusingtheLandWteststatisticsareconservative(sizeis0.02at=0.05).TestsusingtheLteststatistichavemarginallygreatersizethantestsusingtheWteststatisticwhencomparingthesepatterns.Similarresultsareobtainedwhencomparingclusteredpatternswithlargerradiiofinteraction.Thesizeisconservativefortestscomparingpatternsofallintensities.When=100,thesizesofbothtestsareapproximatelyequaltothemarginallevelwhenr0=0.05.Thesizedecreasesasr0increases(sizeisapproximately0.02whenr0=0.25and=0.05).Whencomparingpatternswithgreaterintensity(=500)underthenullhypothesis,thesizeislessthan0.01at=0.05.ThesizeoftestscomparingPoissonpatternswithdifferentintensitieschangesbasedonthechoiceofr0(Figure 3-6 ).Thedifferenceintheintensitiesoftwopatternsdoesnotaffectthesizeofthetest.Atr0=0.05,thesizesoftestsusingtheLandWtest 106

PAGE 107

statisticsare0.2and0.15respectivelyat=0.05.Thesizeofeachtestreachesthenominallevelatapproximatelyr0=0.10.Thetestsareconservativeforr0>0.10.ThepowersoftestsusingtheLandWteststatisticsdecreaseasr0increasesandastheintensitiesofthepatternsincrease(Figures 3-7 3-10 ).TestscomparingregularpatternstoCSRarepowerfulwhentheintensitiesofthepatternsaresmall(=100)andr00.10(approximately0.80.90forbothtestsat=0.05).Thepowerdecreasesrapidlyforr0>0.10.TestscomparingregularpatternstoCSRwithintensitiesgreaterthan100havehighpoweratr0=0.05.Thepowersofbothtestsdecreaseforlargerchoicesofr0,andbecomeverysmallatr00.15.Therateatwhichthepoweroftestsdecreasesincreasesastheintensitiesofthepatternsincrease.ThepoweroftestsusingtheLteststatisticismarginallylargerthanthepoweroftestsusingtheWteststatisticwhencomparingregularpatternstoCSR.Theeffectsofusingdifferentnumbersofquadratstocalculatetheteststatisticsvarybasedonthepatternsbeingtested.ThesizesofHahn'stestsarenotaffectedbythenumberofquadratsandareapproximatelyequaltothenominallevelforalltestscomparingpatternswithsimilarintensities(Figures 3-11 3-15 ).ThesizesoftheproposedtestsareapproximatelyequalfortestsusingdifferentnumbersofquadratsforPoissonpatternsandregularpatterns.Thesizesofthesetestsarebelowthenominallevelforr00.1.Thesizesoftheproposedtestscomparingclusteredpatternsapproachthenominallevelasthenumberofquadratsusedtocalculatetheteststatisticincreases.Thesetestsareconservativewhenfewerquadratsareused.ThesizesofHahn'stestsandtheproposedtestusingtheLteststatisticvarywhentheintensitiesofthepatternsaredifferent(Figure 3-16 ).ThesizesofHahn'stestsareabovethenominallevelforallvaluesofr0andallchoicesofnumberofquadrats.Thesizesoftestsusingfewerquadratsincreaseasr0increases.Theratesatwhichthesesizesincreasedecreaseasthenumberofquadratsincreases.TestsusingtheteststatisticTand25quadratsforeachpatternhaveaconstantsizeforallr0.Thesizes 107

PAGE 108

oftestsusingTincreaseasr0increases,butthesetestshavesizeswellbelowthatoftestsusingTforallr0.ThesizesoftheproposedtestusingtheLteststatisticincreaseasthenumberofquadratsincreaseswhenthepatternsbeingcomparedhavedifferentintensities.Thesizesofthesetestsareabovethenominallevelforr00.1.Thesetestsareconservativeforr00.15.ThesizesoftheproposedtestusingtheteststatisticWareunaffectedbythenumberofquadratsusedtocalculatedtheteststatisticwhenthepatternsbeingcomparedhavedifferentintensities.ThepowersofalltestscomparingclusteredpatternstopatternswithCSRareaffectedbythenumberofquadratsusedtocalculatetheteststatistics(Figures 3-17 3-20 ).Asthenumberofquadratsincreases,thepoweroftestsusingtheWteststatisticandcomparingclusteredpatternswithsmallradiiofinteractiontopatternswithCSRdecreases.ThepowersofHahn'stestsalsodecreaseasthenumberofquadratsincrease,althoughtoalesserdegree.TestsusingtheLteststatisticarepowerfulbutvarybasedonthechoiceofquadratnumber.Whencomparingclusteredpatternswithradiiofinteractionof0.2unitstopatternswithCSR,thepowersoftestsusingtheLandWteststatisticsaresimilarforallchoicesofthenumberofquadratsusedtocalculatetheteststatistics.ThepowersofHahn'stestsdecreasesignicantlyasthenumberofquadratsincreases.ThepowersofalltestsareunaffectbythenumberofquadratsusedtocalculatetheteststatisticswhencomparingregularpatternstopatternswithCSR. 3.7DiscussionThebestsuitedtestforresearcherscomparingmultiplespatialpointpatternsdependsonthecharacteristicsoftheobservedpatternsandthedistancetowhichtheinteractionamongpointsisevaluated.Inmostcases,theproposedtestshaveasizebelowthenominallevel.Thisresultsindecreasedpowercomparedtothepowerthatresultswhenthesizeisatthenominallevel.Thesetestsbecomemoreconservativeastheintensityofthepatternsbeingcomparedincreases.Thus,testshavelesspowerwhencomparingpatternswithgreaterintensities.ThetestsdevelopedbyHahn(2012) 108

PAGE 109

havesizesthatareapproximatelyequaltothenominallevelwhencomparingprocesseswithsimilarintensities.ThesizesofHahn'stestsareconstantwiththechoiceofr0andtheintensityofthepatternsbeingcompared.Asaresult,Hahn'stestsaremorepowerfulthantheproposedtestsformostoftheprocessescomparedinthissimulation.Thesizeandpoweroftheproposedtestsvaryasthechoiceofr0increases,whentestingPoissonpatternsandregularpatterns.ThesizesoftestscomparingPoissonpatternsofsoftcorepatternsareabovethenominallevelwhen=100andr00.1.Thesizesofthesetestsdecreaseasr0increases,fallingbelowthenominallevelforr0>0.10.SizesoftestscomparingPoissonpatternsorregularpatternswithgreaterintensitiesalsovarywiththechoiceofr0,althoughtheyarebelowthenominallevelforallchoicesofr0.ThepoweroftestscomparingregularpatternstopatternswithCSRdecreasesasr0increases.WhencomparingregularpatternstopatternswithCSR,theproposedtestshavepowergreaterthanorequaltothepowerofHahn'stestswhen=100andr00.15.Asthechoiceofr0increasesortheintensitiesofthepatternsincrease,Hahn'stestsbecomemorepowerful.Thesizesoftheproposedtestscomparingclusteredprocesseshavelessvariationbasedonthechoiceofr0.Theproposedtestsareconservativeforclusteredpatterns,andthesizesofbothtestsdecreaseastheintensitiesofthepatternsincrease.TheproposedtestsandHahn'stestsarepowerfulwhencomparingpatternswithahighdegreeofclusteringtopatternswithCSR.Ifthedegreeofclusteringissmaller,theproposedtestshavegreaterpowerwhen=100,andHahn'stestsaremorepowerfulwhen=500.Themostpowerfultestcomparingpatternswith=250dependsonthechoiceofr0.ThetestsdevelopedbyHahn(2012)havesizesabovethenominallevelwhencomparingpatternswithdifferentintensities.ThetestusingteststatisticThasgreatersizethanthetestusingT,andthesizeincreasesasthedifferencebetweentheintensitiesofthepatternsbeingcomparedincreases.ThetestusingThasasize 109

PAGE 110

abovethenominallevel,althoughthedifferencebetweentheintensitiesofthepatternsbeingcompareddoesnotaffectthesizeofthetest.Thesizesoftheproposedtestsarenotaffectedwhencomparingpatternswithdifferentintensities.Underthenullhypothesis,thesizeoftestscomparingtwoPoissonpatternswithdifferentintensitiesisapproximatelyequaltothenominallevelwhenr0=0.10.Thesizeisabovethenominallevelforr0<0.10,andthetestisconservativeforr0>0.10.Furtherworkisrequiredtodeterminetheappropriatesizeandnumberofquadratsforeachtest.Theappropriatenumberofquadratstouselikelychangesbasedontheintensitiesandinteractionsamongpointsintheobservedpatterns.Theproposedtestshavesizesclosesttothenominallevelusing9quadratsforeachpatterntocalculatetheteststatisticwhencomparingPoissonpatternswithdifferentintensities.TestcomparingPoissonandregularpatternswithsimilarintensitiesareunaffectedbythenumberofquadrats.Thesizesandpowersoftestscomparingclusteredpatternsvarybasedonthenumberofquadratsused.Thesizesoftheproposedtestscomparingclusteredpatternsareclosesttothenominallevelfortestsusinggreaternumbersofquadrats.However,thepoweroftheproposedtestusingtheWteststatisticdecreasesasthenumberofquadratsusedtocalculatetheteststatisticincreases. 110

PAGE 111

Figure3-1. Sizesoftests(at=0.01,0.05,and0.1)forhomogeneousPoissonpointpatternsatvaryingintensities(=100,250,and500).BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 111

PAGE 112

Figure3-2. Sizesoftests(at=0.01,0.05,and0.1)forMaternclusteredpointpatternsatvaryingintensities(=100,250,and500)andwithasmallradiusofinteraction(rmax=0.1forallpatterns).BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 112

PAGE 113

Figure3-3. Sizesoftests(at=0.01,0.05,and0.1)forMaternclusteredpointpatternsatvaryingintensities(=100,250,and500)andwithalargeradiusofinteraction(rmax=0.25forpatternswith=100andrmax=0.2forpatternswith=250and500).BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 113

PAGE 114

Figure3-4. Sizesoftests(at=0.01,0.05,and0.1)forsoftcoreregularpointpatternsatvaryingintensities(=100,250,and500).BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 114

PAGE 115

Figure3-5. Sizesoftests(at=0.01,0.05,and0.1)forhardcoreregularpointpatternsatvaryingintensities(=100,250,and500).BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 115

PAGE 116

Figure3-6. Sizeoftests(at=0.01,0.05,and0.1)forhomogeneousPoissonpointpatternsatdifferentintensities.Foreachtest,onepatternhasanintensityof=100.ThetoprowcomparesthispatterntoaPoissonpatternwith=250.ThebottomrowcomparesthispatterntoaPoissonpatternwith=500.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 116

PAGE 117

Figure3-7. Powersoftests(at=0.01,0.05,and0.1)forMaternclusteredpointpatternsatvaryingintensities(=100,250,and500)andwithasmallradiusofinteraction(rmax=0.1forallpatterns)comparedagainstahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 117

PAGE 118

Figure3-8. Powersoftests(at=0.01,0.05,and0.1)forMaternclusteredpointpatternsatvaryingintensities(=100,250,and500)andwithalargeradiusofinteraction(rmax=0.25forpatternswith=100andrmax=0.2forpatternswith=250and500)comparedtoahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 118

PAGE 119

Figure3-9. Powersoftests(at=0.01,0.05,and0.1)forsoftcoreregularpointpatternsatvaryingintensities(=100,250,and500)comparedtoahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 119

PAGE 120

Figure3-10. Powersoftests(at=0.01,0.05,and0.1)forhardcoreregularpointpatternsatvaryingintensities(=100,250,and500)comparedtoahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 120

PAGE 121

Figure3-11. Sizesoftests(at=0.01,0.05,and0.1)forhomogeneousPoissonpointpatternswithaverageintensitiesof250pointsanddifferentnumbersofquadratsusedtocalculatetheteststatistic.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 121

PAGE 122

Figure3-12. Sizesoftests(at=0.01,0.05,and0.1)forMaternpointpatternswithaverageintensitiesof250pointsanddifferentnumbersofquadratsusedtocalculatetheteststatistic.Theradiusofinteractionis0.1units.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 122

PAGE 123

Figure3-13. Sizesoftests(at=0.01,0.05,and0.1)forMaternpointpatternswithaverageintensitiesof250pointsanddifferentnumbersofquadratsusedtocalculatetheteststatistic.Theradiusofinteractionis0.2units.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 123

PAGE 124

Figure3-14. Sizesoftests(at=0.01,0.05,and0.1)forsoftcoreregularpointpatternswithaverageintensitiesof250pointsanddifferentnumbersofquadratsusedtocalculatetheteststatistic.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 124

PAGE 125

Figure3-15. Sizesoftests(at=0.01,0.05,and0.1)forhardcoreregularpointpatternswithaverageintensitiesof250pointsanddifferentnumbersofquadratsusedtocalculatetheteststatistic.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 125

PAGE 126

Figure3-16. Sizesoftests(at=0.01,0.05,and0.1)forhomogeneousPoissonpointpatternsatdifferentintensities.Foreachtest,onepatternhasanaverageintensityof100pointsandonepatternhasanaverageintensityof250.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 126

PAGE 127

Figure3-17. Powersoftests(at=0.01,0.05,and0.1)forMaternclusteredpointpatternswithanaverageintensityof250pointsandaradiusofinteractionrmax=0.1comparedagainstahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 127

PAGE 128

Figure3-18. Powersoftests(at=0.01,0.05,and0.1)forMaternclusteredpointpatternswithanaverageintensityof250pointsandaradiusofinteractionrmax=0.2comparedagainstahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 128

PAGE 129

Figure3-19. Powersoftests(at=0.01,0.05,and0.1)forsoftcoreregularpointpatternswithanaverageintensityof250pointscomparedagainstahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 129

PAGE 130

Figure3-20. Powersoftests(at=0.01,0.05,and0.1)forhardcoreregularpointpatternswithanaverageintensityof250pointscomparedagainstahomogeneousPoissonpatternwithsimilarintensity.BlacklinesindicateHahn's(2012)Tteststatistic.RedlinesrepresentHahn'sTteststatistic.GreenlinesrepresenttheteststatisticcalculatedusingL-function.BluelinesrepresenttheteststatisticusingthevarianceoftheestimatedK-functionfromaPoissonprocess. 130

PAGE 131

CHAPTER4APPLICATIONOFMETHODSWITHJOSEPHW.JONESECOLOGICALRESEARCHCENTERDATA 4.1IntroductionPointpatterndataarecommonineldssuchasplantecologyandforestryastheeventscanrepresentthepositionsoftheplantsoveraregion.Analysisofthepatternscanrevealimportantinformationaboutthecompetition,populationdynamics,agestructure,dispersal,andhistoryofaparticularstandorspecies( Comasetal. 2007 ; Haase 1995 ; Martinezetal. 2013 ; StoyanandPenttinen 2000 ).Evenwithouttheuseofmodeling,simplesummarystatistics,suchasRipley'sK-functionandthepaircorrelationfunctioncanprovidevaluableinformationforconservationecologistsorforforestmanagementpurposes( Haase 1995 ; StoyanandPenttinen 2000 ).TheK-functionhasbeenusedtoassessspeciesinteraction,speciesdispersal,reeffects,foreststandcomparisons,andforestmanagementdecisions,suchasplantingandthinning( Anderson 1992 ; Haase 1995 ; LynchandMoorcroft 2008 ; Pommereningetal. 2011 ; StoyanandPenttinen 2000 ).Thesesummarystatisticshavealsobeenappliedtoestimateorreconstructthestructureofforestsincomplexscenarios,accountingforthetreeage,treesize,dependenceamongmultiplespecies,andmore( PommereningandStoyan 2008 ; Wiegandetal. 2013 ).Pointprocessmodelscanbeusedtomodelcomplexecologicalsystemsincorporatingdifferentinteractionsinsituationswithhighbiodiversity.Theseinteractionscanbeestimatedandtheentiresystemmodeledundertheframeworkofdifferentpointprocessdistributions( Evansetal. 2010 ; IllianandBurslem 2007 ; Illianetal. 2008 ).Inrecentdecades,moreemphasishasbeenplacedoneldssuchasconservationecologyandsustainablesilviculturepractices( Larsen 1995 ; Parmesanetal. 2013 ).Greaternumbersofplantandanimalspeciesarebeingclassiedasendangered( AgrawalandGopal 2013 ; Dobsonetal. 1997 ).Invasivespeciesarebecominganever-increasingthreattonativespecies( Pimenteletal. 2005 ).Timbercompanies 131

PAGE 132

andpolicymakershaveincreasingpressuretoadoptmoresustainablesilviculturalpracticesandreducecarbonemissions( Angelsen 2008 ).Toaddresstheseissues,thesystemsandspeciesdynamicsmustbewellunderstood.Pointprocessmethodsallowforanindividual-basedapproachinastatisticalframework,allowingbetterassessmentofinteractionsamongindividualscomparedtoothermodelingtechniquesinwhichcountsareaggregatedoveranarea( IllianandBurslem 2007 ).Understandingtheseinteractionsamongindividualsofthesamespecies,individualsofdifferentspecies,orindividualsofaspeciesandanecologicaleventisimportantindecision-makingprocessesrelatedtoecology.InChapters2and3,newmethodsofconductinginferenceonRipley'sK-functionforspatialpointprocessesareintroduced.InChapter2,methodsofcalculatingcondenceintervalsfortheK-functionarediscussed.Chapter3discusseshypothesisteststoassesswhetherthesecond-ordermomentisthesame.Inthischapter,analysisofthreepointpatternsisconducted.WithafocusonRipley'sK-function,summarystatisticsarecomputedandinterpreted.Finally,themethodsfrompreviouschaptersareappliedanddiscussed.Theusefulnessforecologicalconservationandforestmanagementisdescribed. 4.2JosephW.JonesEcologicalResearchCenterLocatedinsouthwesternGeorgia,theJosephW.JonesEcologicalResearchCenterhasover11,000hectaresofforest,includingover6,000hectaresoflongleafpineandmixedpineforest.Thecenter'sfocusisonecologicalconservationandsustainableland-use.Oneofitsprimarymissionsistoresearchtheecologyandconservationoflongleafpinewoodlandsandtheirwildlife.Thiseffortincludesmultiplestudyplotsinwhichextensivelocation-referenceddataaremaintainedforadultandjuveniletrees.Thestudyincorporatesreplicated,experimentalsilviculturalmanipulations(includingnoharvestcontrols)in80-90-year-oldlongleaf/mixedpinestandstoexaminelong-term 132

PAGE 133

impactsonfuels,rebehavior,andstanddevelopment( JosephW.JonesEcologicalResearchCenter 2013 ).Partofthisefforttostudylongleafpineforestsincludes18long-termmonitoredplotscreatedwithinlongleafpine/wiregrasssavannas.Theseplotsarefourhectaresinareawithvaryingdimensions.Withineachplot,thelocationsofalladulttrees(treeswithdiameteratbreastheight(dbh)10cm)wererecorded.Inadditiontothelocationofeachtree,detailsofeachtree,includingspecies,height,dbh,andbasalareawererecorded.Eachplotwasassignedoneofthreedifferentharvestingtreatments.Fourplotswereusedascontrolplots,wherenotreeswereharvested.Sixplotswereharvestedbyremovingsingletreestoproducesmallgapsintheforestcanopy.Eightplotswereharvestedbyremovinglargegroupsoftreestocreatelargecanopygaps.Nineoftheseplotshaveanunderstorycomposedprimarilyofwiregrassandtheothernineplotshaveanold-eldunderstory.Althoughlocation-specicdatahavebeencollectedat18sites,onlythreeareconsideredhere.Inthesethreeplots,thelocationsofallsaplingtrees(treeswithheight>2metersandDBH<10cm)wererecordedinadditiontothelocationsofadulttrees.Eachplotisapproximatelyfourhectaresinarea,althoughthedimensionsvary.Plot1isapproximately200metersx200meterswhilePlots2and3areapproximately267metersx150meters.Thespatialpointpatternassociatedwitheachplothaspointsrepresentingthelocationoftreeswithintheplot.Eachpointismarkedwithitsageclassication:adulttrees,juvenile(sapling)longleafpinetrees,orjuvenilehardwoodtrees.Somepointsinpatternsofjuveniletreesrepresentclustersoftrees.Inthesecases,multipletreesarelocatedatasinglelocation.Thenumberofclustersandnumberofpointsperclustervarybyplot.Theoccurrenceofclustersinthesedataisaresultofsimplifyingdatacollection.Foranalysis,theprocessesareassumedtobesimple(theprobabilityofobservingtwoormoretreesataspeciclocationis0);thus,clusters 133

PAGE 134

(multipletreesrecordedatasinglelocation)areadjustedforbysimulatingthenumberoftreeswithinagivenradiusofeachclustercenter.Theradiusforeachclusterisdeterminedbasedonthenumberoftreesintheclusterbyttingalinebetweenthelargestandsmallestclusters(determinedbythenumberoftreesintheclusters)andtheircorrespondingarea.Theareaofaclusterisdeterminedbythenearest-neighbordistancetoanotherpointinthepattern.Thatis,foranearest-neighbordistanced,theareaofaclusterisequaltod2.Incaseswheremultipleclustershavesizesequaltothesmallestorlargestcluster,theminimumnearest-neighbordistanceamongclustersofequalsizesisusedtodeterminethearea.Theareaofeachclusterobservedineachpatterncanthenbeestimatedbythepredictedvaluefromthisline,correspondingtoeachcluster'ssize.Aradiusisdeterminedfromtheestimatedareaofeachcluster.Thenumberofpointscorrespondingtoeachclusterarerandomlysimulateduniformlywithinthisradius,centeredatthelocationthatthepointwasrecorded.Severalassumptionsregardingdatacollectionandthelongleafpinesystemaremadewhenaccountingforclusters.Datacollectionisassumedtobeconsistentinthattheinterpointdistancesconstitutingindiviualpointsandclustersareequalforallclustersandallpatterns.Treeshaveanequalprobabilityoflyinganywhereinsideacircularradius.Inthesystem,aclusteredorinhibitedinteractionamongpointsinsidetheclustermightbeobserved.Aclusteredinteractionwouldresultinasmallerradiusforeachcluster,whereasaninhibitedinteractionwouldcreateasmallbufferbetweentwopointsinacluster.Thetrueareathatclustersofpointsaredistributedinislikelynotcircular,astreesmayformlinesorirregularpatches.Theareaofeachclusterisalsoassumedtohavealineartrendbasedonthenumberofpointsinacluster.Furtherinvestigationofthedatamayleadtomoreappropriatemethodsofaccountingforclustersoftrees.Theecologicalcharacteristicsandmanagementofthethreeplotsdiffer.Plots1and3haveawiregrassunderstorywhereasPlot2isanold-eldsite.Wiregrassisimportanttolongleafpinestandsasitfacilitatesthemovementofreandinuencestheground 134

PAGE 135

levelmicroclimate( Kaplan 2005 ).Becauseofthis,wiregrasshasalargeinuenceonthecompositionoftheunderstory.Siteswithhighconcentrationsofwiregrassandwhichhavebeenfrequentlyburnedhavehigherspeciesrichnessintheunderstorythansiteswithdifferentlycomposedunderstories( RodgersandProvencher 1999 ).Old-eldsitesareareasthatwereformerlyusedforagriculturalcultivation.Thecompostionofold-eldsitesdiffersinbothspeciesanddensityfromsiteswithwiregrassunderstories.Thus,removesdifferentthroughtheseforestsandburnsatdifferentintensitiesanddurations.Becauselongleafpinesarefavoredbyfrequentresthatreducecompetingvegetation,areasthathavedifferentlycomposedunderstoriesmayhavedifferentdistributionsofadultandjuvenilepines.Plots1and2havebeenmanagedusingsingletreeharvesting,inwhichadulttreesaremarkedandremovedtomaintainreproductionandgrowthofthepines.Harvestingoccurseverytwoyears,andtreesareselectedbasedoncharacteristicsofthetreeand/ortocreatesmallgapsinthecanopytopromoterengeneration.Plot3isacontrolsiteinwhichnoharvestinghasbeenconducted.Becauselongleafpinesspendmanyyearsinseedling/saplingstage,harvestinghasnotaffectedthelocationsatwhichseedsaredispersed.Harvestinghasalsonotdirectlyinuencedjuveniletreesbecausejuvenilesarenotharvested.However,changesinthespatialdistributionoftreesandcanopygapsaltertheavailableareaandforestcharacteristicswherejuveniletreesestablishthemselves,possiblychangingthedistributioninwhichsaplingsgrow.Thedensitiesoftreesineachageclassicationvaryamongplots(Table 4-1 andFigure 4-1 ).Althoughallplotsareapproximatelyequalinarea,theshapeofeachisdifferent.Thebehaviorofallmethodshasnotbeenfullyevaluatedforthecasewheresamplesubregionshavedifferentdimensions.Thus,a200x150metersubregionofeachplotisusedsothateachplothasequaldimensions.Thesubregionisrandomlyselectedfromeachplotwiththerestrictionthatitliecompletelywithintheplot.Thesethreesubregionsareusedinallsubsequentanalyses.Theareaofeachalteredplotis 135

PAGE 136

convertedto1squareunitasinLohandStein(2004).Thusplotsthatwereoriginally200metersby150metersarescaledto1.155by0.866units.TheestimatedK-functionsfromthescaledplotsareidenticaltothosefromtheoriginalplots,afterscalingthex-axisbyafactorofp 200150=p 30000andthey-axisafactorof30000.Forscaledpatterns,0.1unitsrepresentsapproximately8.65metersintheobservedpatterns. 4.3ExploratoryDataAnalysisTheobjectiveistoestimateandcomparethesecond-ordercharacteristicsoftheadultandsaplinglongleafpinetreesusingtheK-function.ForsimpleinterpretationoftheK-function,andothersummarystatistics,theprocessisassumedtobestationaryandisotropic.Althoughstatisticaltestsoftheseassumptionsexist( ChiuandLiu 2013 ; Ghorbani 2013 ; Guan 2008 ; Illianetal. 2008 ),thesetestsonlyassessparticularaspectsofstationarityandneverallofthem( Illianetal. 2008 ).Here,themeasurementofthedegreetowhichapatterndeviatesfromstationarityistakentobe S=Zr00j^K(r))]TJ /F5 11.955 Tf 13.73 2.66 Td[(^KInhom(r)jdr.(4)^K(r)isanedge-adjustedestimatorofRipley'sK-function: ^K(r)=jAj n2nXi=1Xj6=iw(xi,xj)I(jjxi)]TJ /F17 11.955 Tf 11.96 0 Td[(xjjjr),(4)wherejAjistheareaoftheplot,nistheobservednumberofpoints,andw(xi,xj)isaweightcalculatedforeachpointpairusedtoadjusttheestimatorofK(r)becausepointsoutsideoftheplotboundariesarenotobserved.^Kinhom(r)istheestimatedinhomogeneousK-function,whichdoesnotassumeaconstantintensityoverthestudyarea( Comasetal. 2009 ).TheestimatoroftheinhomogeneousK-functiondiffersfromitsstationarycounterpartonlyinthattheweightednumberofpointswithinaparticulardistanceofeachpointsisstandardizedbyanestimateoftheintensityatthatpoint.The 136

PAGE 137

estimatedintensityateachpointiisalinearcombinationofthecoordinatesxiandyi, (i)=0+1xi+2yi+3xiyi.(4)AvalueofScloseto0indicatesapatternexhibitingstationarity.Asthisstatisticincreases,thedegreetowhichtheobservedpatternexhibitsstationaritydecreases.Table 4-2 showsthecalculatedstatisticforeachpatterninthisanalysis.AsSincreases,resultsshouldbeinterpretedwithlesscondence.ProcesseswithCSRhavestationarityandisotropy( Cressie 1991 ).BecausetheK-functionsofadulttreesareconsistentwiththetreesbeingrandomlydistributedwithinallthreeplots(Figure 4-2 ),thesepatternsareassumedtobestationary,andthevaluesofSfromtheseplotsareusedasabenchmarkforassessingthestationarityoftheotherpatterns.PatternswithagreatervalueofShavestrongerevidenceofnon-stationarity.Clustersoftreesincreasethevarianceofthisstatistic,andvaluesofSchangebasedonthelocationandproximityofclusters.Patternswithfew,denseclusterscanresultinnon-stationaritybecausemosttreesarelocatedincloseproximitytoeachother.Assumingpatternsofadulttreesarestationarywithineachoftheplots,thevaluesoftheSstatisticfortheseplots(Table 4-2 ,Column3)areusedtojudgethestationarityofpineandhardwoodjuveniles.Bothjuvenilepatternsshowindicationsofnon-stationaritytovaryingdegrees(Columns1and2ofTable 4-2 ).PatternsofhardwoodsaplingsinPlots1and2deviatethemostfromstationarity.Thisismostlikelyattributedtothehighdegreeofclusteringinthesepatterns(asseeninthenortheasternquadrantoftheplotsinFigure 4-1 ).Itisalsolikelythatthenon-stationarityofthesepatternsisafactorofthelocationsofadulttreesintheseplotsandnotthexandycoordinates.Forexample,juveniletreesmaybemorelikelytooccurnearadulttreesorincanopygaps.Exploratoryandinferentialanalysesareperformedonpatternsofadulttreesandjuvenilepines.Resultscanbeconsideredaccurateforthepatternsofadulttreesasthereisevidencethatthesepatternsarestationary.Forjuveniletrees, 137

PAGE 138

resultsstillprovidevaluableinformationalthoughitisnotclearwhethertheprocessesarestationary.Ofthesepatterns,juvenilepinetreesinPlots1and2deviatethemostfromstationarity.Analyzingadulttrees,hardwoodjuveniles,andlongleafpinejuvenilesseparately,aK-functionisestimatedforeachoftheobservedpatterns(Figure 4-2 ).TheK-functionforapatternexhibitingcompletespatialrandomnessisalsoshownineachplot.Adulttreesfromeachplotappeartoberandomlydistributed.TheestimatedK-functionsforallotherpatternssuggesttreesareclustered,withthedegreeofclusteringvaryingwiththepattern.However,severalpatternsappeartohavesimilarspatialstructures.Forexample,theK-functionsforhardwoodjuveniletreesinPlots2and3indicatestrongclusteringatdistanceslessthan20meters,whichdecreaseswithdistance.Likewise,theestimatedK-functionsofjuvenilepinesissimilaracrossallplotsdespitehavingdifferentintensities.WhereastheK-functionisameasureoftheinteractionamongindividualsofthesamespeciesorclass,thecrossK-functionprovidesinsightintotheinteractionamongindividualsofdifferentclasses.Undertheassumptionsofstationarityandisotropy,thecrossK-function,Kbc(r),betweentwoclassesofevents,bandcrepresentstheexpectationofthenumberofpointsoftypeclessthandistancerofanarbitraryobservationoftypeb,standardizedbytheintensityofc.Thatis, Kbc(r)=E[#ofpointscrfromapointoftypeb] c.(4)Kbc(r)isestimatedandinterpretedinamanneranalogoustotheestimationoftheK-function.Thatis,acrossK-functionthatisclosetothecurveKbc(r)=r2representsindependencebetweenthetypebandtypecprocesses( BaddeleyandTurner 2005 ).Atdistances10meters,thejuvenileandadulttreesappearrandomlydistributedrelativetooneanother(Figure 4-3 ).Atdistancesgreaterthan10meters,greaternumbersofjuveniletreesaremorelikelytobeobservednearadulttreesthanunder 138

PAGE 139

theassumptionofrandomlydistributedjuvenilesandadults.Thisislikelyaresultofclusteringfoundamongjuveniletrees. 4.4EstimationofCondenceIntervalsfortheK-FunctionTherealizationsofaspatialpointprocesscanvary,resultinginvariationintheestimatedK-functionandothersummarystatisticsusedtointerpretthepatterns.CondenceboundsforRipleysK-functionprovideinsightintohowcloselytheK-functionisestimated.CondenceintervalsareestimatedfortheK-functionbyrstassigningallpointstoanetwork.Ifthedistancebetweentwopointsislessthan,thepointsbelongtothesamenetwork.Theinterpointdistancebetweentwopointsbelongingtodifferentnetworksisgreaterthan.ThechoiceforisalinearcombinationofthereciprocaloftheintensityofthepatternandameasurementofhowfarthepatterndeviatesfromCSR.AsdiscussedinChapter2,anempiricalstudywasconducted,andalinearmodelusedtoestimate;thus, ^=^0+^11 +^2DA+^3DA ,whereDA=1 r0Zr00)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(L(t))]TJ /F11 11.955 Tf 11.96 0 Td[(t2dt.(4)Largerchoicesforresultinfewernetworks,witheachnetworkencompassingmorepoints.Theaveragenumberofpointswithindistancerofaparticularpointinaspecicnetwork,yi(r)fornetworki,isfoundbyaveragingthesumoftheweightsoverallpointsinadistinctnetworkforthedistancerbeingevaluated.Thatis, yi(r)=1 niniXk=1nXj6=kw(xik,xij)I(jjxik)]TJ /F17 11.955 Tf 11.96 0 Td[(xijjjr).(4)wherenisthenumberofpointsintheobservedpatternandniisthenumberofpointsinnetworki.Toestimatecondenceintervals,networksaresampledwithreplacementandwithprobabilityproportionaltosize;thatis,pi=ni=nfornetworki.ForeachbootstrapestimateofK(r),thisisrepeateduntilthesumofthepointsintheselectednetworks 139

PAGE 140

isgreaterthan=n)]TJ /F5 11.955 Tf 12.7 0 Td[(0.5n,wherenistheaveragenumberofpointsinanetwork.ThevaluesofyiforeachoftheMselectednetworksaresummedandtheK-functionisestimatedusingtheHansen-Hurwitzestimator( Lohr 2010 ): ^K(r)=jAj n1 MMXm=1y(i)m(r)(4)wherey(i)m(r)isthevalueofyi(r)fromtheithnetworkofthepopulationthatissampledonthemthdraw.Bbootstrapsamplesaredrawn,and^K(r)computedforeach.A100(1)]TJ /F11 11.955 Tf 11.96 0 Td[()%bootstrapcondenceintervalforK(r)is h2^K(r))]TJ /F5 11.955 Tf 16.1 2.66 Td[(^K(B+1)(1)]TJ /F14 7.97 Tf 6.58 0 Td[(=2)(r),2^K(r))]TJ /F5 11.955 Tf 16.1 2.66 Td[(^K(B+1)(=2)(r)i,(4)where^K(r)istheestimationofK(r)fromtheobservedpattern.Figures 4-4 4-9 showtheintervalscalculatedforadulttreesandjuvenilepinetreesobservedinPlots1oftheJosephW.JonesEcologicalResearchCenter.Asthedegreeofclusteringincreases,thecondenceboundsbecomewider.TheadulttreepatternsexhibitstrongevidenceofCSR.UnderCSR,theintervalwidthdependsontheintensityofthepatterns,butvarylittlefortheadultswithinthethreestudyplots(Figures 4-4 4-6 ,and 4-8 ).ThepatternofjuvenilepinesobservedinPlot2hasthelargeststatisticS,whichindicatesnon-stationarity.Consequently,theestimatedK-functionfallsoutsidethecondenceboundsforsmallr(Figure 4-7 ),duetotheextremeclusteringoftreesinthenortheastquadrantoftheplot(Figure 4-1 ).Patternsofhardwoodjuveniletreesalsoexhibitnon-stationarityandcondenceboundsdonotcontaintheestimatedK-functions.ThecondenceboundsprovidemoreinformationtoresearchersthananestimateoftheK-function,astheydescribethevariationthatisinherentintheunderlyingprocess.Thisvariationdiffersgreatlybasedonthetypeofprocessanditsintensity.Ifspatialinteractionataparticularradiusisimportantindecisionmaking,thevariationinthisinteractionisimportantinformationaswell.Anintervalestimateofthenumberof 140

PAGE 141

othereventsexpectedwithinagivenradiusofanobservedeventcanbeobtainedbymultiplyingthecondenceboundsoftheK-functionbytheintensity.Thisinformationcanbeusedtodesignconservationareasforendangeredanimalorplantspecies,orinlocatingindividuals,suchasaninvasivespeciesforremoval.CondenceboundsfortheK-functioncanbebenecialfromaforestmanagementperspective.Foreststhatareharvestedaimtosustainadistributionthatfacilitatesthegrowthandreproductionoftrees.HarvestsuggestionscanbemadesuchthattheK-functionfromtheobservedpatternoftreesstayswithintheestimatedcondencebounds.Harvestingcanbeconducteduntilthedesiredintensityofthepatternismet.Thus,thedistributionofthethinnedpatternwillbesimilartothatoftheoriginal,properlyallowingforgrowthandregenerationofthetreesandthesustainabilityofotherecologicalprocesses.Similarly,thesemethodscanbeusedtoreestablishprimaryforeststhathavebeenclearcutordevastatedduetosomeevent.Researchersmightbemotivatedtoreestablishaforestwithasimilardistributionastheoriginal,possiblyaccountingforfactorssuchasclimatechange( Churchilletal. 2013 ).Condenceintervalsforfunctions,suchastheK-function,canbeusedtoassureresearchersthatthespatialinteractionamongpointsiswithinerrorofsomebenchmarkdistribution.Managementsuggestions(e.g.treemarking)canbemadeinordertocorrecttheK-functionassociatedwiththecurrentstand(thestandtoberestored)andtoobtainavaluethatiswithintheintervalfoundfromthereferencesites.IftheobservedK-functionisabovethatoftheintervalproducedfromthereferencesites,thinningoftreeclumpsisprobablydesired.IftheobservedK-functionisbelowthatoftheintervalfromthereferencesites,treesaretooregularlyspacedandindividualsshouldbemarkedtoopenmorespaceforregeneration.Byincorporatingmarksforthetreespeciesormaturityofthetrees,evenmoreoftheinformationfromthestandcouldbeaccountedforinthepatterns.IfthiswasanalyzedusingmultivariateK-functions,amoreaccuratedescriptionoftheforest 141

PAGE 142

structure(includingspeciesinteractionoragestructure)couldbeobtained.Thiswouldresultinthecreationofpatternsthataremorestructurallysimilartothatofthereferencesites.OncethetreesareselectedinsuchawaythattheestimatedK-functioniswithintheintervalfortheoriginalforest,harvestingorthinningcantakeplace.Theresultscanbedeterminedatarangeofdistances,oraggregatedoveragivendistancer0. 4.5HypothesisTestingoftheK-FunctionWhetherornotthesamebiologicalprocessesaregivingrisetotheobservedspatialdistributionsindifferentplotsorindifferentageclassesisofprimaryinterest.IftheK-functiondifferssignicantlyamongpatterns,thentheconclusionwouldbethatthebiologicalprocessesaredifferent.IftheK-functionsdonotdiffer,thenthebiologicalprocessesmaybethesameorsimilar.However,becausemorethanonespatialprocesscangiverisetothesameK-function,adenitiveconclusionthatthebiologicalprocessesarethesamecannotbedrawn.Here,differencesintheinteractionamonglongleafpinetreesisassessedforpatternsoftreeswithdifferentageclassications,andforplotswithdifferentunderstorycompositionanddifferentharvestingplans.Longleafpineforestsaretypicallyrichinbiodiversity( Keddyetal. 2006 ; Peet 2006 );however,theareaandextentofpineforestsinthesoutheasthavedecreaseddramaticallyinthelastcentury( Kaplan 2005 ).Thus,effortsarebeingmadetoconserveandrehabilitatetheseforests.Understandingtheeffectsofmanagementdecisionsandecologicalevents,suchasreregimes,climatechanges,andharvestingonadultandjuvenilepinepopulations,canhelpdeterminetheoptimaltreatmentstofacilitatetheserehabilitationefforts.ThestatisticaltestsdescribedinChapter3areusedtocomparethesecond-orderstructureofthepatternsobservedintheJosephW.JonesCenterdata.Thehypothesesofinterestare 1. AretheK-functionsforpatternsofadulttrees/juvenilepinetreesequalforeachplot? 142

PAGE 143

2. AretheK-functionsforpatternsofadulttrees/juvenilepinetreesobservedonplotswithwiregrassunderstoryequaltothosefromold-eldplots? 3. AretheK-functionsforpatternsofjuvenilepinetreesobservedoncontrolplotsequaltothosefromharvestedplots?Comparingpatternsoftreeswithdifferentageclassications(Hypothesis1)providesinsightonthefuturespatialstructureoftheforest.Ifthedistributionofjuveniletreesissimilartothatofadults,afutureforestmayretainthesamecharacteristicsasthecurrentforest.Iftheinteractionamongadultsandjuveniletreesisshowntobestatisticallydifferent,theneitherthedistributionofjuvenileswillchangeovertimeasaresultofcompetition,harvesting,orecologicalprocesses,orthefuturedistributionoftheforestwillbedifferentthanthecurrentdistribution.Assessingtheinteractionamongtreesateachlifestagehelpsidentifydifferencesintheirdistributionsandleadstoinformationthatcanbeusedtomaintainhealthyforests.Toassessthersthypothesis,thedistributionsofthejuvenilepinetreesandadulttreesarecomparedineachplot.Fireisnecessaryfortheestablishmentandgrowthoflongleafpinesasitreducesothervegetationthatmayotherwiseoutcompetepinesaplings.Theunderstoryofaforestaffectstheintensityanddurationthataforestburns;thus,thedistributionsoftreesonplotswithdifferentunderstoriesmaybedifferent.Toassessdifferencesinthedistributionsresultingfromdifferentlycomposedunderstories(Hypothesis2),testsareconductedcomparingPlot1toPlot2.TheunderstoryofPlot1iscomposedofprimarilywiregrasswhilePlot2hasadifferentlycomposedunderstoryduetoitsold-eldhistory.However,bothplotsareharvestedusingsingle-treeharvestingsoharvestingwillnotbeaconfoundingfactor.Adulttreesandsaplingsarecomparedseparatelyacrossplots.Althoughharvestingdoesnotdirectlyaffectjuveniletrees(juvenilesarenotharvestedandareseededbeforeharvestingtakesplace),differentharvestingschemesmayaffectthedistributionofjuvenilesbyalteringthecompetitionamongtrees.Juveniletreescompetewithotherjuvenilesandadulttreesforresources,includingspace,water,sunlight,andnutrients.Thus,differentharvestingmethodsmayresultindifferent 143

PAGE 144

distributionsofjuveniletreesbyalteringthecharacteristicsoftheforest(availablespaceandcanopycover)andthecompetitionamongtrees.Totestthethirdhypothesis,Plots1and3arecompared.Plot1hasbeenharvestedwhilePlot3remainsunharvestedasacontrol.Bothplotshaveawiregrassunderstory.TheproposedtestusingtheL-functionandHahn'sTtestareusedtotestallthreehypothesesofinterest.ThehypothesistestdevelopedbyHahn(2012)isdesignedtotestwhethertheK-functionsfrommultiplespatialpointpatternsareequalagainstthealternativethatatleastoneoftheK-functionsisdifferentfromtheothers.Congruent,disjointsubregionsaretakenfromeachpattern.AcomparisonofthemeanK-functionsfromthesesubregions,adjustedforthevarianceobservedineachpattern,isintegratedoveragivenrangeofdistancer0.Ifthenumberofobservedpatternsisgreaterthan2,thesumofthisintegratedfunctionoverallcombinationsoftwopatternsiscalculated.Hahn'steststatisticTis T=X1i
PAGE 145

toobtainap-valueforthetest.Iftheobservedteststatisticislargecomparedtothosecalculatedfrompermutations,theresultingp-valueissmall.InChapter3,anothertestofthenullhypothesisthatK-functionsfromtwoormoreobservedpointpatternsareequalisproposed.ThistestusestheL-function,atransformationoftheK-function,andtheteststatisticL.Tocalculatetheteststatistic,marksmx(r)areassignedtoeachpoint,foreachdistancer.Themarkmx(r)equalsthesumofallweightsforpointswithindistancerofpointx.EachpatternisdividedintomisubregionsandtheK-functionisestimatedforeachsubregionusingthesemarks: ^Kij(r)=jAijj n2ijXx2Qijmx.(4)whereQijrepresentssubregionjinpatterni,jAijjrepresentstheareaofthesubregion,andnijrepresentstherespectivenumberofpoints.AnestimateoftheL-functioniscalculatedforeachsubregion, ^Lij(r)=s ^Kij(r) .(4)ThemeanL-functionisobtainedforeachindividualpatternbyaveragingtheestimatedL-functionsfromthatpattern'ssubregions.Likewise,atotalmeanL-functioniscalculatedbyaveragingtheestimatedL-functionsfromallsubregionsinallpatterns.ThesumofsquareddeviationsisfoundunderthenullhypothesisthatpatternshavethesameK-function,andunderthealternativehypothesisthattheL-functionsfordifferentpatternsaredifferent.Thatis,underthenullhypothesis SSNull(r)=gXi=1miXj=1^Lij(r))]TJ /F5 11.955 Tf 12.21 2.65 Td[(L(r)2,(4)andunderthealternativehypothesis SSAlt(r)=gXi=1miXj=1^Lij(r))]TJ /F5 11.955 Tf 12.2 2.65 Td[(Li(r)2.(4) 145

PAGE 146

Thesumsofsquaresforthenullandalternativecasesareintegratedoveradistancer0,resultinginSSNullr0andSSAltr0.TheteststatisticLis L=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(SSNullr0)]TJ /F4 11.955 Tf 11.96 0 Td[(SSAltr0=(g)]TJ /F5 11.955 Tf 11.95 0 Td[(1) SSAltr0=(Pmi)]TJ /F4 11.955 Tf 11.95 0 Td[(g).(4)wheremiisthenumberofsubregionsinpatterni,andgisthetotalnumberofobservedpatterns.SimulationresultsinChapter3indicatethatthesizesofthesetestsarebestcontrolledbyusing9subregionsforeachpatterntocalculatetheteststatisticswhencomparingtwopatternswithdifferentintensities.Becausetheintensitiesofadultandjuveniletreesvaryacrossplots,eachtestisconductedusing9subregionsforeachpattern.Thismeanseachpatternisdividedinto9congruent,disjointsubregionstoobtainestimatesofKi(r)foreachpatternithatisobserved. 4.5.1Hypothesis1TocomparetheK-functionobservedinjuveniletreestothatobservedforadulttrees,thenullhypothesisthattheK-functionofjuvenilesisequaltothatofadulttreesistestedforeachplot.Hahn'stestisconductedbyintegratingthefunctionintheteststatistictovariousdistancesr0.Theteststatisticsareintegratedtodistancesbetween10metersand50metersatintervalsof10meterstoensurethatresultsarenotaproductofchoosinganinappropriatedistance.UsingHahn'sTtest,statisticallysignicantdifferencesarefoundwhencomparingtheinteractionamongadulttreestotheinteractionamongjuveniles(p-values<0.05).Thep-valuescalculatedinPlots1and2increaseasthedistancethattheteststatisticiscalculatedoverincreases(Table 4-3 ).Hahn'stestcomparingdistributionsinPlot3resultsinthelargestp-value(0.0482atr0=30meters).P-valuesforthisplotdecreaseatgreatervaluesofr0.TheproposedtestusingtheLteststatisticisalsocalculatedbyintegratingoverthesameintervalofdistances.DifferencesbetweentheK-functionsofadultandjuveniletreesarefoundtobesignicantormarginallysignicantatsmall 146

PAGE 147

valuesofr0(p-values<0.07forallplotsforr0=10meters).EvaluatingtheteststatistictogreaterdistancesresultsinK-functionsthatarenotsignicantlydifferent,withtheexceptionofPlot1,whichhasp-values<0.1forr030meters.Forallthreeplots,thep-valuesincreaseasr0increases.Theseresultsindicatethattheinteractionamongadulttreesissignicantlydifferentfromthatamongjuvenilepinetreesforeachofthethreeplotsatsmalldistancesr0.ConictlingresultsbetweentheTandLtestsforgreatervaluesofr0arelikelyduetotheconservativenatureoftheproposedLtest. 4.5.2Hypothesis2Totestwhethertheinteractionamongtreesineachageclassisdifferentforplotswithdifferentlycomposedunderstories,Plots1and2arecomparedbecausetheyhavethesameharvesttype.Hahn'sTteststatisticandtheproposedLteststatisticareusedtocomparetheK-functionsfromadultsandjuveniletreesintheseplots.Theteststatisticsareintegratedtodistancesbetween10metersand50metersatintervalsof10meterstoassurethatresultsarenotaproductofchoosinganinappropriatedistance.Basedonp-valuesfrombothtests,theK-functionsforthesedistributiondonotdiffersignicantly(Tables 4-4 and 4-5 ).P-valuesfrombothtestscomparingdistributionsofadultandjuveniletreesaregreaterthan0.1foralldistancesr0. 4.5.3Hypothesis3Hahn'spermutationtestandtheproposedtestusingtheL-functionareappliedtotestthedifferencesintheK-functionbetweenharvestedandcontrolplotsforpatternsofjuvenilepinetrees.Toavoidtheconfoundingeffectsofdifferentlycomposedunderstoryoftheforeststands,Plots1and3aredirectlycompared.Bothplotshaveunderstoriescomposedofwiregrass.Teststatisticsforbothtestsarecalculatedbyintegratingoverarangeofdistancesbetween10metersand50metersatintervalsof10meters.Resultsvariedforthedistributionofjuveniletrees.Hahn'stestindicatedstrongevidencethattheK-functionofjuvenilepinetreesinPlot1(singletreeharvesting)islargerthaninPlot3(noharvesting)indicatingadifferenceintheinteractionamong 147

PAGE 148

juvenilepinesbetweentheseplots(Table 4-6 ).ResultsfromtheproposedLtestvarybasedonthedistancer0.Forsmallr0,theresultsfromthistestindicatemarginallysignicantdifferencesbetweentheK-functionsfromharvestedandnon-harvestedplots(p-values=0.030and0.075forr0=10and20metersrespectively).Whenintegratedtolargerdistances,thep-valuesfromtheproposedtestindicatenosignicantdifferenceintheK-functionamongjuvenilepinetreesintheseplots. 4.6DiscussionAsexpected,basedonvisualassessmentandobservingtheK-functionsfromeachofthesepatterns,theinteractionamongadulttreesissignicantlydifferentfromthatofjuveniletrees.Regardlessofharvestingorunderstorycomposition,thedistributionsofadulttreesarenotsignicantlydifferentfromCSR.Juvenilepinetreestendtoformclusters.Basedonthecross-Kfunctionbetweenadultsandjuvenilepines,thedistributionsofadultandjuveniletreesarerandomrelativetoeachotheratsmalldistances(<10meters).Atlargerdistances,greaternumbersofjuveniletreestendtobelocatednearadulttrees.However,clusteringofjuvenilesnearadulttreesmaybearesultofthehighdegreeofclusteringamongjuveniletrees.Thecompositionofaforest'sunderstoryinuencesthespeciesthatinhabittheforestandthemannerinwhichtheforestburns( Wenketal. 2011 ).Forestswithwiregrassunderstorytendtofacilitatethemovementofrewhereasforestswithunderstoriesofothercompositionsresultinrethatburnsporadicallyandatdifferentintensities.Itissuggestedthattheunderstorycompositionaffectsthedistributionofpineforestsbyeliminatingcompetingvegetationforpinejuvenilesandenablingreproductionofpinetrees.Theunderstorymayinuencetherateatwhichpinesreproduceorgrowastheintensitywasgreaterforthewiregrassunderstory.However,theunderstorydoesnotappeartohaveanimpactonthelocationsofadultorjuveniletrees,ortheinteractionsamongindividualsofsimilarageclasses. 148

PAGE 149

Thedistributionsofjuvenilepinetreesaresignicantlydifferentforplotsharvestedwithsingle-treeharvestingandcontrolplots,especiallyatshortdistances.Basedontheseresults,clusteringofjuvenilepinetreesisgreaterinplotsthathavebeenharvestedcomparedtocontrolplots.Thecross-KfunctionforPlot3(controlplot)deviatesthemostfromrandomnessbetweentherelativelocationsoftreesfromdifferentagegroupsatdistanceslessthan20meters.Thus,thedifferencesinthedistributionofjuvenilepinetreesmaybearesultofanaltereddistributionofadulttreesfromharvesting,allowingjuvenilestoestablishthemselvesathigherdensitiesinthegaps.Thinningofadulttreesusingsingle-treeremovalmayalsoreducecompetitionforresourcesamongjuvenileandadulttreesandallowjuveniletreestogrowinamoredispersedpattern. 149

PAGE 150

Table4-1. NumberofeachtreeclassicationobservedineachplotoftheJoseph.W.JonesEcologicalResearchCenter. PlotNumberofJuv.PinesNumberofJuv.HardwoodsNumberofAdults 15481850324843488733297973321 Table4-2. EstimatedDeviationfromstationarityforeachpatternobservedateachplot.TheunitofmeasurementforeachPlotis1meterx1meter. PlotStationarityofJuv.HardwoodsStationarityofJuv.PinesStationarityofAdults 138.6821.703.23232.5621.464.01318.568.553.90 Table4-3. P-valuesfortestsofHypothesis1usingadultandjuveniletreesinallplots.DistanceRangetoCalculateTestStatistics TestPlot10meters20meters30meters40meters50meters T10.00070.00170.01140.01520.0154T20.00850.00420.00550.011730.0115T30.00570.02970.04820.02390.0072L10.00490.02420.05770.10440.1792L20.038490.10170.18470.23440.2294L30.06650.16740.33290.45360.5991 Table4-4. P-valuesfortestsofHypothesis2usingadulttreesinPlot1(wiregrassunderstory)andPlot2(old-eldplot).DistanceRangetoCalculateTestStatistics Test10meters20meters30meters40meters50meters Hahn'sT0.22520.38520.52060.54990.5938ProposedL0.34240.50160.52760.55060.5843 Table4-5. P-valuesfortestsofHypothesis2usingjuvenilepinetreesinPlot1(wiregrassunderstory)andPlot2(old-eldplot).DistanceRangetoCalculateTestStatistics Test10meters20meters30meters40meters50meters Hahn'sT0.11090.20910.32240.35710.4226ProposedL0.94600.84600.81550.71080.6098 150

PAGE 151

Table4-6. P-valuesfortestsofHypothesis3usingjuvenilepinetreesinPlot1(singletreeharvesting)andPlot3(control-noharvesting).DistanceRangetoCalculateTestStatistics Test10meters20meters30meters40meters50meters Hahn'sT0.00830.00840.01010.00950.0128ProposedL0.03040.07530.22770.40710.5561 Figure4-1. ThespatiallocationsofallobservedpatternsobservedattheJosephW.JonesEcologicalResearchCenter.Thetoprowshowsthepositionsofhardwoodjuvenile(saplings)foreachplot.Thesecondrowshowsthepositionoflongleafpinejuvenilesforeachplot.Thethridrowshowsthelocationsofadulttreesforeachplot. 151

PAGE 152

Figure4-2. TheestimatedK-functionforeachplot.Patternsofeachclassicationareanalyzedseparately.Hardwoodjuveniletreesareshowninthetoprow.Pinejuvenilesareshowninthesecondrow.Adulttreesareshowninthethirdrow.ThelightbluelineindicatestheK-functionforapatternexhibitingcompletespatialrandomness. 152

PAGE 153

Figure4-3. Thecross-Kfunctionforadultandjuvenilepinetreesforeachplot.Adulttreesaretheeventsfromwhichdistancesaremeasuredfromandjuvenilepinetreesareeventsinwhichdistancesaremeasuredto. 153

PAGE 154

Figure4-4. The95%condenceintervalforK(r)usingtheproposedmethodofcondenceintervalcalculationforadulttreesinPlot1. Figure4-5. The95%condenceintervalforK(r)usingtheproposedmethodofcondenceintervalcalculationforjuvenilepinetreesinPlot1. 154

PAGE 155

Figure4-6. The95%condenceintervalforK(r)usingtheproposedmethodofcondenceintervalcalculationforadulttreesinPlot2. Figure4-7. The95%condenceintervalforK(r)usingtheproposedmethodofcondenceintervalcalculationforjuvenilepinetreesinPlot2. 155

PAGE 156

Figure4-8. The95%condenceintervalforK(r)usingtheproposedmethodofcondenceintervalcalculationforadulttreesinPlot3. Figure4-9. The95%condenceintervalforK(r)usingtheproposedmethodofcondenceintervalcalculationforjuvenilepinetreesinPlot3. 156

PAGE 157

CHAPTER5FUTUREWORK 5.1EstimatingCondenceIntervalsfortheK-functionTheproposedmethodofcondenceintervalestimationforRipley'sK-functionofaspatialpointpatternisadhoc.Manyofthechoicesusedtoestimatecondenceintervalsareobtainedarbitrarilyandcanbeimprovedthroughfurtherresearch.However,currentresultsindicatethattheproposednetworkingmethodofbootstrappingaspatialpointpatterntoobtaincondenceintervalsfortheK-functionperformwellcomparedtopreviousmethods,andthevarianceofthebootstrapestimatormorecloselyapproximatesthevarianceoftheestimatorfortheK-functionbasedonsimulationenvelopes.Themethodsusedinthisstudyprovideseveralareasoffocusforfutureresearch.Inthiswork,atuningparameterischosenbasedondetailsofanobservedpattern.ismodeledasalinearfunctionofanobservedpattern'sintensityanddeparturefromCSR.Forthiswork,themodelusedtoestimateforaparticularpatternis =0+11 +2DA+3DA +e,forr0.5(5)where DA=1 r0Zr00)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(L(t))]TJ /F11 11.955 Tf 11.96 0 Td[(t2dtandL(t)=r K(t) .(5)Thismodelincorporatesdescriptiveinformationabouttheintensityandtheinteractionamongpointsinanobservedpatternandaninteractioneffectbetweenthem.Othermodelsmaybemoresuitablefor,includingtheadditionofhigher-ordertermsoradditionalcomponentsand/orinteractions.ThechoiceofDAforassessingapattern'sdeparturefromCSRisdevelopedforthepurposesofthiswork,andotherchoicestoestimatethisdeparturemaybetterdescribetheinteractionamongpointsinapointpattern. 157

PAGE 158

Thettedmodelusedtoestimateisbasedonalimitednumberofprocessesinwhichthevaluesofthatresultinapproximate95%coveragebynominal95%condenceintervalswereobtained.TodeterminethevalueofforeachprocessthatresultsinaccuratenominalcondenceintervalsforK(r),departureoftheempiricalcoveragefromthe95%condencelevelwasaggregatedoveranintervalfor0r0.5.Thisrangewaschosenarbitrarilyandotherchoicesfortheupperboundmayproducedifferentresults.Typically,theinteractionamongpointsisnotassessedtothisdistanceandasmallerupperboundmaybedesired.However,ttingthemodelbyaggregatingthedifferenceintheempiricalcoverageandthenominalleveloverasmallerrangeofdistancesmayresultindifferentestimatesforthemodelparameters.IftheappropriatevalueofdependsontheupperboundforrforwhichthecondenceintervalofK(r)isdesired,additionalresearchshouldfocusonestimatingbasedonthisupperboundofr.Necessityofamodeltoestimateforapatternisalimitationinthiswork.ThisrequirestheappropriatevaluesoftocreateaccuratenominalcondenceintervalsforK(r)formanyprocessesinordertoestimatetheparametersinthemodel.Largenumbersofsimulationsarerequiredinordertoobtaintheseparameterestimates,andadaptingthismethodtoaccountfordifferentrangesofK(r)ordifferenttypesofpointprocessesmayrequirethesesimulationstoberepeated.Thisisbothtimeconsumingandcomputationallyintensive.Othermethodsforestimatingshouldbeexplored.Othersummaryfunctionsdescribingtherstandsecondmomentsofpointprocessescouldperhapsprovideinformationfortheappropriatevalueofforanobservedpattern.couldalsobedeterminedbychoosingavaluethatpartitionspointsintoaparticularnumberofnetworksornetworksofanaveragesize.However,furtherresearchisrequiredtodeterminetheadequatenumberorsizeofnetworksnecessarytoproducecondenceintervalsforK(r)ofanobservedpointpattern. 158

PAGE 159

5.2AppropriateNumberofNetworkstoResampleThenetworkingbootstrapestimatorusesrandomselectionofnetworksuntilthenumberofpointsinthebootstrapsampleis=n)]TJ /F5 11.955 Tf 12.68 0 Td[(0.5n,wherenisthenumberofpointsintheobservedpatternandnrepresentstheaveragesizeofanetwork.ThisvalueofisusedsuchthatbootstrapestimationsofK(r)useapproximatelynpoints.Amoreappropriatebootstrapproceduremightbetokeepthenumberofnetworksconstantacrossbootstrapsamplesandallowthenumberofpointsinthebootstrapsamplestovary.Inthiscase,theexpectationofthenumberofnetworksresampledtoobtainnpointsineachsamplecouldbedeterminedbasedontheexpectationofthenumberofpointsinasinglenetworkdrawnatrandom: E[njN]=NXi=1nini n (5) =NXi=1n2i n (5) whereNisthenumberofnetworks,conditionalon,andniisthenumberofpointsinnetworki.Toobtainthenumberofnetworksinthebootstrapsamples,N,thenumberofpointsinthepatternisdividedbythisexpectation N=n PNi=1n2i n (5) =n2 PNi=1n2i. (5) Toaccountforanon-integersolutionforN,thechoicebNccanbeusedandanadditionalnetworkcanbedrawnwithprobabilitypNsuchthatbNcE[njN]+pNE[njN]=n.Here,bNcrepresentstheintegerclosesttoN,suchthatbNcN.Thischoicewouldalterthebootstrapprocedurebycreatingadditionalvarianceinthenumberof 159

PAGE 160

pointsbootstrappedandkeepingthenumberofnetworksapproximatelyconstantforeachbootstrapsample.Thisisacontrasttotheproposedmethod,whichkeepsthenumberofbootstrappedpointsapproximatelyequalandallowsthenumberofnetworksineachbootstrapsampletovary. 5.3ExtensiontoInhomogeneousSpatialPointProcessesInrecentyears,inhomogeneousPoissonpointprocesseshavebecomepopularmodelsforpointdata( Dorazio 2012 ; WartonandShephard 2010 ).TheintensityofaninhomogeneousPoissonprocessisallowedtochangeaccordingtosomefunction.Inspeciesdistributionmodeling,thelocationsofplantsoranimalnestingsitesaremodeledusingafunctionofcovariatesrecordedateachlocation.TheK-functionisusedtodescribeinteractionininhomogeneousprocesses,accountingforvaryingintensity.Inthiscase, ^KInhom(r)=1 nnXi=11 ^iXi6=jw(xi,xj)I(jjxi)]TJ /F17 11.955 Tf 11.95 0 Td[(xjjjr),(5)where^iistheestimatedintensityatlocationi.Currently,estimationofcondenceintervalsfortheK-functionofinhomogeneousprocessesisbasedonsimulationenvelopesusingMonteCarlomethods.Thistypicallyignoresinteractionamongpoints,andcondenceintervalsforK(r)maybeinaccurateasaresult.ExtensionsofthemethodsforcondenceintervalestimationforK(r)ofinhomogeneouspointprocesseswouldbeavaluableanalysistoolforresearchersusingpointprocessmodelstomodeldata.Investigationofextensionsoftheproposedmethodstoinhomogeneousprocessesmaybeginbydeterminingthenetworkparameteriindividuallyforeachpoint,basedontheestimatedintensityandameasurementofclustering.Networkscanbedeterminedbytheconnectedpointsbasedontheseestimatednetworkradii. 5.4BayesianMethodsofInferenceBognarandCowles(2004)andBognar(2006)discussaBayesianframeworkforconductinginferenceonpointprocessdistributions( Bognar 2006 ; BognarandCowles 160

PAGE 161

2004 ).ThesemethodsaresimilartoDiggleetal.(2000),inwhichalikelihoodisdenedforapairwiseinteractingpointprocess,conditionalonnobservedpoints.Thelikelihoodofapointpatternwithnpointsis p(xj)=exp")]TJ /F8 7.97 Tf 12.13 14.94 Td[(n)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xi=1nXj=i+1(jjxi)]TJ /F4 11.955 Tf 11.95 0 Td[(xjjj)#Z)]TJ /F9 7.97 Tf 6.59 0 Td[(1n()(5)whereZ)]TJ /F9 7.97 Tf 6.59 0 Td[(1n()isanintractablenormalizingconstantand(jjxi)]TJ /F4 11.955 Tf 11.95 0 Td[(xjjj)isapairpotentialfunctiondescribingtheinteractionbetweentwopointsdistancejjxi)]TJ /F4 11.955 Tf 12.54 0 Td[(xjjjapart.isasetofparametersdeningtheinteractionamongpointsandisthefocusofinference.BognarandCowles(2004)useimportancesamplingwithinaMarkovChainMonteCarloalgorithmtoestimatetheratioofintractablelikelihoodfunctionsandsamplefromtheposteriordistributionwhenpriordistributionsareusedfor( BognarandCowles 2004 ).Bognar(2006)usesthepredictiveposteriordensitytosimulatepointpatternsandobtainstheposteriorexpectationfortheK-functionfromthesesimulations.ThesimulatedK-functionscanbeusedassimulationenvelopesfortheK-functionusingaBayesianframework( Bognar 2006 ).ThesesimulationenvelopesareanalogoustothosedevelopedinfrequentistframeworksdiscussedinChapter1,butallowforuncertaintyinestimatingtheparametersassociatedwiththepairwiseinteractionamongpoints.However,toourknowledge,theseposteriorsimulationenvelopeshavenotbeendirectlycomparedtofrequentistsimulationenvelopesorbootstrapmethodsforestimatingcondenceintervalsforK(r).BecausethedistributionoftheK-functionisunknown,usingBayesianmethodstodirectlyestimateK(r)forapointpatternisdifcult.However,usingaBayesianapproachtoestimatemodelparametersforortoclusterpointsintonetworksusingaBayesianclusteringapproachmayhelpproducecondenceintervalsforK(r)withempiricalcoverageclosertothenominallevel. 161

PAGE 162

5.5HypothesisTestingfortheK-functionThep-valuesassociatedwiththeproposedhypothesistestsarenotuniformlydistributed,resultinginsizesbelowthenominallevel.Furtherresearchshouldfocusonunderstandingthedistributionsofthep-valuesfromthesetests.Ifthedistributioncanbeapproximatedfromanobservedpattern,theprobabilityintegraltransformationcanbeusedtoobtainp-valueswithauniformdistribution.Thiswillresultintestswithsizesapproximatelyequaltothenominallevelandincreasedpowerunderthealternativehypothesis.Despiteconservativesizes,theproposedtestshavegreaterpowerthanHahn's(2012)testswhencomparingpatternswithintensitiesof100pointstopatternswithCSRandasimilarintensity,andwhentheteststatisticisintegratedoverdistancesr00.15unitsforpatternsobservedonaunitsquare.TheseresultsindicatethattheproposedmethodsmaybemoresuitablethanHahn'stestsforcomparingpointpatternswithintensities100.Furtherinvestigationofthesemethodsshouldbefocusedontheirperformanceonpatternswithintensitieslessthan100points.Anexpandedsimulationcandeterminetheperformanceoftheproposedtestsforprocesseswithsmallintensities.Inaddition,furtherinvestigationshouldbefocusedontheperformancesofalltestswhenthesizesofquadratsaredifferentamongthepatternsbeingcompared.TheteststatisticusingaweightfunctiontoaccountfortheheteroscedasticityofK(r)usesthevarianceofK(r)fromaPoissondistributionwithaxednumberofpointstoweighteachofthesumofsquaresofdeviationsforaparticulardistancerintheteststatistic.Thexednumberofpointsisthesumofthepointsfromtwoormoreobservedpatterns,dividedbythetotalnumberofquadratsusedtocalculatetheteststatistic.Theareausedtoapproximatethevarianceistheareaofonequadratfromapattern.Thischoiceoftheweightfunctiondoesnotequallyweighalldistanceswhenintegratingtheteststatisticovertheranger0,possiblyresultinginteststhatareconservative 162

PAGE 163

whentheteststatisticisintegratedtolargevaluesofr0.Asaresult,thesetestsarenotaspowerfulastestsinwhichthesizeisequaltothenominallevel.Otherchoicesforweightsshouldbeinvestigatedtoequallyweighalldistancesbeingevaluated.Theobservedvarianceof^K(r)calculatedfromtheobservedquadratscanbeusedforcaseswhenatleastonepointpairwithinterpointdistancelessthanrexistsforalldistancesrbeingevaluated.Ifthisconditionisnottrue,thevariancematrixisnotfullrankandtheinverseofthismatrixcannotbedetermined.Ifthisconditionistrue,theobservedvariancemayweigheachdistanceappropriately.Becausethevaluesof^K(r)arecorrelatedfordifferentvaluesofr,furtherresearchshouldalsofocusoncorrectingforthiscorrelationinthecalculationoftheteststatistics. 163

PAGE 164

REFERENCES Agrawal,A.,andGopal,K.(2013),ConceptofRareandEndangeredSpeciesandItsImpactasBiodiversity,BiomonitoringWaterandWasteWater,pp.71. Anderson,M.(1992),SpatialAnalysisofTwo-SpeciesInteractions,Oecologia,91(1),134. Angelsen,A.(2008),MovingaheadwithREDD:issues,optionsandimplicationsCIFOR(FreePDFDownload). Baddeley,A.,andMoller,J.(1989),Nearest-NeighborMarkovpointprocessesandrandomsets,InternationalStatisticalReview,57,89. Baddeley,A.,Moller,J.,andWaagepetersen,R.(2000),Non-andSemi-ParametricEstimationofInteractioninInhomogeneousPointPatterns,StatisticaNeerlandica,54(3),329. Baddeley,A.,andSilverman,B.(1984),ACautionaryExampleontheUseofSecond-OrderMethodsforAnalyzingPointPatterns,Biometrics,40,1089. Baddeley,A.,andTurner,R.(2005),Spatstat:anRpackageforanalyzingspatialpointpatterns,Journalofstatisticalsoftware,12(6),1. Barnard,G.(1963),CommentonThespectralanalysisofpointprocessesbyM.S.Bartlett,JournaloftheRoyalStatisticalSocietyB,25,294. Besag,J.(1977),CommentonModellingSpatialPatternsbyB.D.Ripley,JournaloftheRoyalStatisticalSocietyB,48,616. Besag,J.,andDiggle,P.(1977),SimpleMonteCarloTestsforSpatialPatterns,JournaloftheRoyalStatisticalSociety-AppliedStatistics,26(3),327. Bognar,M.A.(2006),OnbayesianinferencefortheKfunction,Biometricaljournal,48(2),205. Bognar,M.A.,andCowles,M.K.(2004),Bayesianinferenceforpairwiseinteractingpointprocesses,StatisticsandComputing,14(2),109. Braun,W.,andKulperger,R.(1998),ABootstrapforPointProcesses,JournalofStatisticalComputationandSimulation,60,129. Brillinger,D.(1975),StatisticalInferenceforstationarypointprocesses,StochasticProcessesandRelatedTopics,pp.55. Brillinger,D.(1978),Comparativeaspectsofthestudyofordinarytimeseriesandofpointprocesses,InDevelopmentsinStatistics,1,33. Buhlmann,P.(2002),Bootstrapsfortimeseries,StatisticalScience,17,52. 164

PAGE 165

Chiu,S.(2007),CorrectiontoKoen'scriticalvaluesintestingspatialrandomness,JournalofStatisticalComputationandSimulation,77(11),1001. Chiu,S.N.,andLiu,K.I.(2013),StationarityTestsforSpatialPointProcessesusingDiscrepancies,Biometrics,pp.1. Churchill,D.,Larson,A.,Dahlgreen,M.,Franklin,J.,andHessburg,J.(2013),Restoringforestresilience:Fromreferencespatailpatternstosilvicultralprescriptionsandmonitoring,ForestEcologyandManagement,291,442. Comas,C.,,andMateu,J.(2007),ModellingForestDynamics:APerspectivefromPointProcessMethods,BiometricalJournal,49(2),176. Comas,C.,Palahi,M.,Pukkala,T.,andMateu,J.(2009),Characterisingforestspatialstructurethroughinhomogeneoussecondordercharacteristics,StochasticEnviron-mentalResearchandRiskAssessment,23(3),387. Cressie,N.(1991),StatisticsforSpatialData,NewYork:JohnWileyandSons. Davison,A.,andHinkley,D.(1997),BootstrapMethodsandtheirApplications,Cambridge,U.K.:CambridgeUniversityPress. Diggle,P.(1977),Thedetectionofrandomheterogeneityinplantpopulations,Biomet-rics,33,390. Diggle,P.(1979),OnParameterEstimationandGoodness-of-FitTestingforSpatialPointPatterns,Biometrics,35(1),87. Diggle,P.(1986),Parametricandnon-parametricestimationforpairwiseinteractionpointprocesses,,inProceedingsofthe1stWorldCongressoftheBernoulliSociety. Diggle,P.J.,Fiksel,T.,Grabarnik,P.,Ogata,Y.,Stoyan,D.,andTanemura,M.(1994),Onparameterestimationforpairwiseinteractionpointprocesses,InternationalStatisticalReview,62,99. Diggle,P.,Lange,N.,andBenes,F.(1991),AnalysisofVarianceforReplicatedSpatialPointPatternsinClinicalNeuroanatomy,JournaloftheAmericanStatisticalAssocia-tion,86(415),618. Diggle,P.,Mateu,J.,andClough,H.(2000),AComparisonbetweenParametricandNon-ParametricApproachestotheAnalysisofReplicatedSpatialPointPatterns,AdvancesinAppliedProbability,32(2),331. Dobson,A.,Rodriguez,W.,andWilcove,D.(1997),GeographicDistributionofEndangeredSpeciesintheUnitedStates,Science,275,550. Doguwa,S.(1989),OnSecondOrderanalysisofmappedpointpatterns,BiometricalJournal,31,451. 165

PAGE 166

Dorazio,R.(2012),PredictingtheGeographicDistributionofaSpeciesfromPresence-OnlyDataSubjecttoDetectionErrors,Biometrics,68,1303. Efron,B.,andTibshirani,R.(1993),AnIntroductiontotheBootstrap,NewYork,NY:ChapmanandHall. Evans,G.,Illian,J.B.,andKing,R.(2010),SpatialPointProcessesforModellingPlantCommunitiesinthePresenceofInteractionUncertainty,,. Ghorbani,M.(2013),Testingtheweakstationarityofaspatio-temporalpointprocess,StochasticEnvironmentalResearchandRiskAssessment,27(2),517. Guan,Y.(2008),AKPSStestforstationarityforspatialpointprocesses,Biometrics,64(3),800. Haase,P.(1995),SpatialPatternAnalysisinEcologyBasedonRipley'sK-Function:IntroductionandMethodsofEdgeCorrection,JournalofVegetationScience,6(4),575. Hahn,U.(2012),AStudentizedPermutationTestfortheComparisonofSpatialPointPatterns,JournaloftheAmericanStatisticalAssociation,107(498),754764. Hall,P.(1985),ResamplingaCoveragePattern,StochasticProcesses,20,231. Hall,P.,andWilson,S.(1991),TwoGuidelinesforBootstrapHypothesisTesting,Biometrics,47,757. Ho,L.,andChiu,S.(2006),TestingCompleteSpatialRandomnessbyDiggle'sTestwithoutanArbitraryUpperLimit,JournalofStatisticalComputationandSimulation,76(7),585. Ho,L.,andChiu,S.(2009),UsingWeightFunctionsinSpatialPointPatternAnalysiswithApplicationtoPlantEcologyData,CommunicationsinStatistics-SimulationandComputation,38,269. Hope,A.(1968),AsimpliedMonteCarlosignicancetestprocedure,JournaloftheRoyalStatisticalSocietyB,30,582. Illian,J.,andBurslem,D.(2007),ContributionsOfSpatialPointProcessModellingToBiodiversityTheory,JournaldelaSocieteFrancaisedeStatistique,148(1),9. Illian,J.,Moller,J.,andWaagpetersen,R.(2009),Hierarchicalspatialpointprocessanalysisforaplantcommunitywithhighbiodiversity,EnvironmentalandEcologicalStatistics,16(3),389. Illian,J.,Penttinen,A.,Stoyan,H.,andStoyan,D.(2008),StatisticalAnalysisandModellingofSpatialPointPatterns,WestSussex,England:JohnWileyandSons. Jenson,L.(1993),AsymptoticNormalityofEstimatesinStatialPointProcesses,ScandinavianJournalofStatistics,20,97. 166

PAGE 167

Jenson,L.,andMoller,J.(1991),PseudolikelihoodforExponentialFamilyModelsofSpatialPointProcesses,AnnalsofAppliedProbability,1(3),445. JosephW.JonesEcologicalResearchCenter(2013).URL:http://www.jonesctr.org/research/projects/llp management/llp management main.html Kaplan,J.A.(2005),Therelationofunderstorygrassesinlongleafpineecosystemstoreandgeography,PhDthesis,UniversityofNorthCarolina. Keddy,P.,Smith,L.,Campbell,D.,Clark,M.,andMontz,G.(2006),PatternsofherbaceousplantdiversityinsoutheasternLouisianapinesavannas,AppliedVegeta-tionScience,9(1),17. Koen,C.(1991),ApproximatecondenceboundsforRipley'sstatisticforrandompointsinasquare,BiometricalJournal,33,173. Kunsch,H.(1989),TheJackknifeandBootstrapforGeneralStationaryObservations,TheAnnalsofStatistics,17,1217. Lahiri,S.(1993),OntheMovingBlockBootstrapUnderLongRangeDependence,Statistics&ProbabilityLetters,18,405. Larsen,J.(1995),Ecologicalstabilityofforestsandsustainablesilviculture,ForestEcologyandManagement,73,85. Liu,R.Y.,andSingh,K.(1992),Movingblocksjackknifeandbootstrapcaptureweakdependence,Exploringthelimitsofbootstrap,225,248. Loh,J.,andStein,M.(2004),BootstrappingaSpatialPointPattern,StatisticaSinica,14,69. Lohr,S.L.(2010),Sampling:designandanalysisThomsonBrooks/Cole. Lynch,H.,andMoorcroft,P.(2008),ASpatiotemporalRipley'sK-functiontoanalyzeinteractionbetweensprucebudwormandreinBritishColumbia,Canada,CanadianJournalofForestResearch,38(12),3112. Martinez,I.,Taboada,F.,Wiegand,T.,andObeso,J.(2013),Spatialpatternsofseedling-adultassociationsinatemperateforestcommunity,ForestEcologyandManagement,296,74. Moller,J.,andWaagpetersen,R.(2003),StatisticalInferenceandSimulationforSpatialPointProcesses.,BocaRaton:ChapmanandHall/CRC. Nguyen,X.,andZessin,H.(1979),IntegralanddifferentialcharacterizationsoftheGibbsprocess,MathematischeNachrichten,88,105. 167

PAGE 168

Ohser,J.(1983),OnEstimatorsforthereducedseondmomentmeasureofpointprocesses.,MathematischeOperationsforschungandStatistik,SeriesStatistics,14,63. Parmesan,C.,Burrows,M.,Duarte,C.,Poloczanka,E.,Richardson,A.,Schoeman,D.,andSinger,M.(2013),Beyondclimatechangeattributioninconservationandecologicalresearch,EcologyLetters,16,58. Peet,R.K.(2006),Ecologicalclassicationoflongleafpinewoodlands,inTheLongleafPineEcosystemSpringer,pp.51. Pimentel,D.,Zuniga,R.,andMorrison,D.(2005),Updateontheenvironmentalandeconomiccostsassociatedwithalien-invasivespeciesintheUnitedStates,Ecologicaleconomics,52(3),273. Politis,D.N.,andRomano,J.P.(1992),Acircularblock-resamplingprocedureforstationarydata,Exploringthelimitsofbootstrap,pp.263. Politis,D.,andRomano,J.(1994),LargeSampleCondenceRegionsBasedonSubsamplesunderMinimalAssumptions,TheAnnalsofStatistics,22,2031. Pommerening,A.,LeMay,V.,andStoyan,D.(2011),Model-basedanalysisoftheinuenceofecologicalprocessesonforestpointpatternformation-Acasestudy,EcologicalModeling,222,666. Pommerening,A.,andStoyan,D.(2008),Reconstructingspatialtreepointpatternsfromnearestneighborsummarystatisticsmeasuredinsmallsubwindows,CanadianJournalofForestResearch,38,1110. Ripley,B.(1976),TheSecond-OrderAnalysisofStationaryPointProcesses,JournalofAppliedProbability,13. Ripley,B.(1979),Testsofrandomnessforspatialpointpatterns,JournaloftheRoyalSocietyofStatisticsB,41,369. Ripley,B.(1988),StatisticalInferenceforSpatialProcesses,Cambridge,UK:CambridgeUniversityPress. Ripley,J.(1981),SpatialStatistics,NewYork:JohnWileyandSons. Rodgers,H.L.,andProvencher,L.(1999),AnalysisoflongleafpinesandhillvegetationinnorthwestFlorida,Castanea,pp.138. Schabenberger,O.,andGotway,C.(2005),StatisticalMethodsforSpatialDataAnaly-sis,1stedn,BocaRaton,FL:Chapman&Hall/CRC. Stoyan,D.,Kendeall,W.,andMecke,J.(1995),StochasticGeometryanditsApplica-tions.2ndEdition,NewYork:JohnWileyandSons. 168

PAGE 169

Stoyan,D.,andPenttinen,A.(2000),RecentApplicationsofPointProcessMethodsinForestryStatistics,StatisticalScience,15(1),61. Stoyan,D.,andStoyan,H.(1994),Fractals,RandomShapes,andPointFields,NewYork,NY:JohnWiley. Warton,D.I.,andShephard,L.(2010),PoissonPointProcessModelsSolvethePseudo-AbsenceProblemforPresence-OnlyDatainEcology,AnnulsofAppliedStatistics,4,1383. Wenk,E.,Wang,G.,andWalker,J.(2011),Within-standvariationinunderstoreyvegetationaffectsrebehaviourinlongleafpinexericsandhills,InternationalJournalofWildlandFire,20(7),866. Wiegand,T.,He,F.,andHubbell,S.(2013),Asystematiccomparisonofsummarycharacteristicsforquantifyingpointpatternsinecology,Ecography,36,092103. 169

PAGE 170

BIOGRAPHICALSKETCH MichaelHymanwasbornin1983inRaleigh,NorthCarolina.HereceivedaBachelorofScienceinmathematicsfromtheUniversityofNorthCarolina,ChapelHill.MichaelreceivedhisMasterofStatisticsfromtheUniverstiyofFloridainAugust,2009andcontinuedthepursuitofaPh.D.ininterdisiplinaryecologywithaconcentrationinstatisticsundertheguidanceofDr.LindaJ.YoungandDr.ChristinaStaudhammer.Uponcompletionofhisdegree,MichaelbecameastatisticianfortheNationalAgriculturalStatisticsServiceswiththeUSDAinFairfax,VA. 170