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
(Attachacopyofthispagetoeachcopyofyoursolutionpaper.)
Typeasummaryofyourresultsonthispage.Donotinclude
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
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