美国大学生数学建模竞赛MCM A题 树与树叶 一等奖论文.docx

美国大学生数学建模竞赛MCM A题 树与树叶 一等奖论文.docx

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美国大学生数学建模竞赛MCMA题树与树叶一等奖论文

Leaf-Mass&LeavesClassification

February14,2012

Abstract

Inthispaper,wepresentastatisticalmodeltoestimatetheleafmassandclassifyvariousleafshapesbasedonthedataobtainedthroughtreeharvestandleafremovalof21urbantrees.Wefirstexplainthediversificationofleafshapesanddescribetherelationshipbetweenleafshapesandtreeprofileinordertodefinethecorrelationbetweentreeprofileandleafshapes.Inordertoestimateleafmass,weapplytheregressionanalysiswithfourobservedindependentvariables,whichallowsustoestimatetheleaf-massofawhole-tree.Inordertoclassifyleaves,weusehierarchicalclusteringmethodaccordingtothedryleaf-massbyWard'sMethod.InthismethodweusethestandardEuclideandistanceasourprincipleofclassification.Finallyweobtainaregressionmodelwithvaryhighcoefficientofdeterminationandagoodwaytoclassifyleaves.

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2012MathematicalContestinModeling（MCM）SummarySheet

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thenameofyourschool,advisor,orteammembersonthispage.

DearEditors,

Byoperatingandanalyzingtheexperimentaldata,wehavesetupanoptimizedregressionmodeltoestimateleafmass,andhaveclassifiedleavesintoseveralspecificgroupsdependingonareasonablecriterion.Moreimportantly,wehavenoticedsomereasonableresults.

Tobeginwith,aclassicregressionmodeliscitedtouncovertheambiguousrelationshipbetweentreeformsandleafshapes.Asamatteroffact,theleafshapesarenegativelycorrelatedtothetreeforms,andtheresultenablestoexposethetruthwhynarrowerleavesonlower-crowntreesarecomparativelyprolific.

Moreover,comparedwithotherregressionmodels,anoptimalregressionmodelestimatingleafmasswasproposed.Underthespecificsignificancelevel,treemassiscloselyrelevanttosometreecharacteristicsincludingcrownheight（m）,crownradius（m）andleafmassperareaofcrownprojection（

）.Therefore,iftheseparametersaremeasured,theactualtreemasscanbeestimated.

Inaddition,hierarchicalclusteringmethodisappliedinleafclassification.Inthisway,leavescanbereasonablyclassifiedrelyingonthedifferentdrymass.

Contents

1Introduction.............................................................................................................3

1.1OutlineofOurApproach..................................................................................3

2LeavesandtheLeafShapes....................................................................................4

2.1TheDiversificationofLeafShapes..................................................................4

2.2TheRelationshipbetweenLeafShapesandTreeProfile....…………….…....5

2.3LeafMosaic......................................................................................................7

3EstimatingLeafMass..............................................................................................8

3.1AssumptionsofRegressionAnalysis...............................................................8

3.2BasicRegressionModel...................................................................................8

3.2.1IndependentandDependentVariables..............................................9

3.2.2RegressionEquation...........................................................................9

3.2.3CorrelationMatrixandMulticollinearity..........................................9

3.2.4AGlobalTestontheSetofIndependentVariables..........................10

3.2.5CheckingtheSignificanceofEachIndependentVariableandANOVATable....................................................................................11

3.2.6DiagnosticPlotsfortheRegressionModel......................................12

3.3ImprovedRegressionModel..........................................................................16

3.3.1SolutionandResultsAnalysis..........................................................17

4HierarchicalClusteringinLeaves..........................................................................19

4.1DataDescription.............................................................................................19

4.2Methodology..................................................................................................20

4.3ResultsofClustering......................................................................................21

4.4Conclusion......................................................................................................21

5References.............................................................................................................22

I.Introduction

1.1OutlineofOurApproach

Withinaplant,themassofindividualleavesvariesaccordingtoleafsize,leafdensityandthickness（WitkowskiandLamont1991）.Therearevarietiesofapproacheshavebeenexploredforrelatingleafmasstoeasilymeasureddimensionsofatree,andmuchoftheworkhasbeendoneinrelativelyunifiedstandssuchastimberplantations.ThepioneeringworkofKittredge（1944）providedanequationbetweenleafmass（W）andtrunkdiameter（D）,log（W）=b*log（D）-a.

Inordertoincreasetheprecisionandaccuracyofleaf-massestimates,additionalvariablesneedtobeincludedandmultivariatestatisticalanalysesalongwithsophisticatedregressionequationsshouldbeadopted.Forinstance,inapreviousstudymadebyKershawandMaguire（1995）,alargesamplewasdrawandextensivemeasurementsweremadetoobtaintheempiricalcoefficientsforresultingequations.Branchdiameterwasconsideredtobethebestpredictorofbranchfoliagemass;infact,useofstructuralorpositionalvariableswillalwaysleadtoimprovements.Unfortunately,theattainmentofalargesamplesizeforeachplantspeciestendstobeunpracticaloutsideurbanareas.

Intermsoftheleavesclassification,clusteranalysisisfrequentlyselectedbyanumberofresearchers.Clusteranalysisissupposedtobeamethodofmultivariatedataanalysisassigningsetsofobjectswithsimilarpropertiesintoagroup.Asacommontechnique,thealgorithmisappropriateinmanyfieldslikepatternrecognition,imageanalysisandinformationretrieval.Inthispaper,wemainlyusetheHierarchicalclusteringwhichisbasedondistancesinordertogainclustersandillustratethemusingadendrogram.（Wikipedia）

II.LeavesandtheLeafShapes

2.1TheDiversificationofLeafShapes

Amongallplanttraits,leafshapeisamostcommonandvariableone.Indeed,latestresearcheshaveshownthatleafshapecanbeexplainedwithitsworkingmechanism.Itisgenerallybelievedthatplantsplayacriticalroleinglobalcarboncycling.Plantsabsorbcarbondioxideandproducingorganicmatterthroughphotosynthesismainlyusingtheirleaves.Therearethreebasicpointshere,theamountofcarbonneededtoconstructtheleaf,leaf’slifespanandthephotosynthesisrateoftheleaf（WeiMing2011）.Thesethreebasicfactorstogethershapeaprofileofleafthatwesee（WeiMing2011）.Scientistshaveconstructedmathematicalmodelincorporatedallthreefactors（carbonconsumption,leaf’slifespanandphotosynthesisrate）tosimulatethehypotheticalshapeofaleaf.Theysampledleavesofover2500plantspeciesandtheresultsseemedtobequiteconsistentwiththeirhypotheses（WeiMing2011）.

Whenitcomestoamoreexotericlevel,conclusioncanbedrawnfromthepreviousresearchesthatvariationinleafshapeisanoutcomeofadaptationandfunctionssuchasthermoregulation,waterbalance,nutrientstransport,absorptionoflightandresistancetonaturalenemiesareallrelatedtoleafshape（WeiMing2011）.Thistheoryconcernsmoreabouttheinteractionbetweenplantsandtheirenvironment.Althoughnosimpleandunifiedmechanismcanbedrawasweexpectedconsideringtheleafshapeanditshabitat.Thereisstillmuchtruthinthistheory.

●Leavesonthecrownwillreceiveadequatesunlight.Theseleavesareoftenrelativelysmalltoreducetheexposureareatoletmoresunlightgothroughtogetthelowerleaves.Incontrast,leavesbellowthecrownusuallytendtoberelativelylargertomaximizetheexposurearea.Inthisway,leavesbellowthecrowncanalsocontinuephotosynthesis（WeiMing2011）.

●Plantslivinginadryenvironmentnormallyhaveslenderandtinyleaves.Thoseleaveswithsmallareacanhelpmaintainwatergiventhatnearly90%ofthewaterabsorbedbyplantswillinevitablybelostthroughtranspirationprocessofleaves（WeiMing2011）.

●Weallhaveseensomeleavesgrowtocone;thisistopreventtheleavesfrombeingcrushedbysnowinwinter.Someleavesevolvetobeneedletothreatenthepredator.Forexample,coniferinNorthPolegrowstheirleavestobelikeneedlesothateventhehungriestanimalwon’tdaretoeat（WeiMing2011）.

Althoughenvironmentalselectionplaysanimportantroleinshapingleaves,whenexplainingmoresubtleandminordifferencesbetweendiversifiedleafshapes,weneedtocomedowntothegeneticlevel.Theleafsizeandshapeofacertainplantisnormallyconservedwithinthespecies,indicatinggeneticcontroloftheseprominentcharacteristics.Whenmutationsatmanylocioccurs,thissourceofnaturalwondersresultinalteredshapeandsize,suggestingtheinteractionof