2012美国大学生数学建模竞赛A题获奖论文(sponsored-by-Minzu-Universit

ClassifyLeafShapeandEstimateforLeafMassWebuildamathematicalmodeltoclassifyleavesandcalculatetheweightofleavesofatree.First,weconsiderthereasonsforformingvariousleafshapes.Ouransweristhatthevariousleafshapesareresponsetothevariousenvironmentalfactorssuchascarbondioxide,temperatureandwater.AfterinvestigatingtheverticaldistributionandtheleveldistributionofnaturalzonesinAlpsandChina,wemaketheconclusionthatthesizeofshapeisinnegativerelationwiththecarbondioxidelevel,whilepositivewithtemperatureandtheamountofwater.Next,inordertofigureoutwhethertheshapesmaximizeexposure,weanalyzeleafmosaicandgettheanswer‘yes’.Andbycomparingthesurfaceleavesandtheleavesbelow,weknowthatthedistributionofleavesdoinfluenceleafshapesincethesurfaceleavesaregenerallysmallerandhavemorecomplexedgesinordertoletthesunlightpenetratetotheleavesbelow,whiletheleavesbelowarelargerandhaverelativelysimplermargin.Then,toanalyzetherelationshipbetweenleafshape(LS)andtreeprofile(TP),webuildasimplelinearregressionmodel.First,weassumethatthereare5kindsofLSand8kindsofTPandtransferthemtonumbers.Thenweinvestigate120kindsoftreesrandomlyselectedfromabotanyencyclopediaandclassifytheirLSandTPbylabelingnumbers.Fromthescatter,wefindthatthetwovariablesarepositivelycorrelated.Thuswebuildasimplelinearregressionmodelandgetthelinearregressionequation1.35670.6536yx(ystandsforTP,xstandsforLS).Andsincetheregressioncoefficientis1.3567,wecancometotheconclusionthatleafshapeis,toalargeextent,similartothegeneralprofileoftree.Finallywemakeasignificantleveltestandgetthecorrelationcoefficientis0.748993,indicatingthatthepositivecorrelationbetweenLSandTPishighlysignificant.Finally,inordertocalculatetheweightoftheleaves,wedefineanormaltreewhoseoutlinecrownisrelativeaxis-symmetric,andthenweshrinktheoutlineofthecrowntoformaserialsofcrowns:theleaves'distributionandtheircharacteristicsineachspatialdefinedasasectionbetweentwoadjacentcrownsissimilar.Bysamplingfromthesection,wecanmacro-analyzetheoverallconditionofleaves;calculatetheleaves'massinthesectionandthenaddupallthesections'weighttogettheresult.Inaddition,wealsotrytomodifythismodelbyapplyingcalculustothecalculationofcomplicatedshapetreeviadividingthetreeintoseveralaxis-symmetriclittletreesassub-models.Inaword,thestrongtheoreticalbasisandsuitableassumptionmakeourmodelestimableforfurtherstudyofleaves.Combiningthismodelwithmoreaccurateinformationofthetreeshapeandleavesdensitywillmaximizeitsutility.Team#13828Page1of22AbstractWebuildamathematicalmodeltoclassifyleavesandcalculatetheweightofleaves.First,weconsiderthereasonsforleavesvaryinginshapeandsizeandfocusonthefactorsincludingcarbondioxide,temperatureandwater.Combinedwithpreviousresearches,weanalyzetheverticaldistributionandtheleveldistributionofnaturalzonesinAlpsandChinaandgettheresultthatasthetemperaturedecreasesandtheamountofwaterdeclines,theleafshapeisinclinedtobeneedle-shapedandasthedropofcarbondioxidelevel,theleaftendtobebigger.Then,weanalyzeleafmosaicandknowthatleavestendtoshapethemselvestomaximizetheexposuretosunlight.What’smore,wedrawaconclusionthatleaveslocatingatdifferentpartsofatreeoftenhavedifferentshapessincethesurfaceleavesaregenerallysmallerandhavemorecomplexedgesorlobeswhiletheleavesbelowaremuchlargerandtheleafmarginandlobesarerelativelysimpler.Nextwebuildasimplelinearregressionmodeltoanalyzetherelationbetweenleafshapeandtreeprofile,fromwhichwegetthelinearregressionequation.Besides,wemakeasignificancetestandgetthatcorrelationcoefficientis0.748993,indicatingthatthemodelisofhighsignificance.Accordingtotheregressionmodel,weknowthatthereispositivecorrelationbetweenleafshapeandtreeprofile:leafshapeis,toalargeextent,similartothegeneralprofileoftree.Finally,asfortheweightofleaves,wedividethetotaltreeintoseveralpartswhichsharethesamecharacteristics.Andweworkouttheleafmassofeachsectionandthenaddupallthesectionstoestimatethetotalleafmassofatree.Furthermore,combiningthismodelwithmoreaccurateinformationofthetreeshapeandleavesdensitywillmaximizeitsutility.Keyword:LeafShape;Leafmass;Simplelinearregression;PartitionTeam#13828Page2of22ContentsAbstract.....................................................................................1Introduction..............................................................................3Restatementandanalysisoftheproblem...............................4TheBasicAssumption...............................................................5TheTreeisUnchangeable................................................5NumbertheShape..............................................................5TheTreeisNormal...........................................................5Model........................................................................................6Question1............................................................................6Question2............................................................................8Question3............................................................................8Question4...........................................