数学建模.docx
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数学建模.docx
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数学建模
Weconstructfourmodelstostudyleafclassification,relationshipsbetweenleafshapeandleafdistribution,correlationsbetweenleafshapeandtreeprofile,andtotalleafmassofatree.Model1dealswiththeclassificationofleaves.Wefocusprimarilyonthemostconspicuouscharacteristicofleaves,namely,shape.Wecreatesevengeometricparameterstoquantifytheshape.Thenweselectsixcommontypesofleavestoconstructadatabase.Bycalculatingthedeviationindexoftheparametersofasampleleaffromthoseoftypicalleaves,wecanclassifytheleaf.Toillustratethisclassificationprocess,weuseamapleleafasatestcase.Model2studiestherelationshipbetweenleafshapeandleafdistribution.First,wesimplifyatreeintoanidealizedmodelandthenintroducetheconceptofsolaraltitude.Byanalyzingtheoverlappingindividualshadowsthroughconsideringtherelafionshipbetweenleaflengthandinternodelengthunderdifferentsolaralfitudes,wefindthattheleafshapeanddistributionareoptimizedtomaximizesunlightexposureaccordingtothesolaraltitude.Weapplythemodeltothreetesttypesoftrees.Model3discussesthepossibleassociationbetweentreeprofileandleafshape.Basedonthesimilaritybetweentheleafveinsandbranchstructureoftrees,weproposethatleafshapeisatwo-dimensionalmimicofthetreeprofile.EmployingthemethodofModel1,wesetseveralparametersreflectingthegeneralshapeofeachtreeandcomparethemwiththoseofitsleaves.Withthehelpofstatisticaltools,wedemonstratearoughassociationbetweentreeprofileandleafshape.Model4estimatesthetotalleafmassofatreegivensizecharacterisfics.Carbondioxide(CO2)sequestrafionrateandtreeageareintroducedtoestablishthelinkbetweenleafmassandtreesize.SinceaunitmassofaleafsequestersCO2ataconstantrate,theCO2sequestrationratehasaquadraticrelafionshipwiththeageofthetree,andthesizethetreeexperienceslogisficgrowth.
Wetacklefourmainsubproblems:
•classificationofleaves,•therelationshipbetweenleafdistributionandleafshape,•therelationshipbetweenthetreeprofileandtheleafshape,and•calculationofthetotalleafmassofatree.Totacklethefirstproblem,weselectasetofparameterstoquantifythecharactersoftheleafshapeandusetheleafshapeasthemainstandardforourclassificationprocess.Forthesecondquestion,weusetheoverlappingareathatoneleaf'sshadowcastsontheleafdirectlyunderitasthelinkbetweentheleafdistributionandtheleafshape,sincetheleafshapeaffectstheoverlapping.Weassumethattheleafdistributiontriestominimizetheoverlappingarea.Asforthethirdquestion,wesetparametersforthetreeprofileandcomparethosewiththeparametersforthetree'sleafshapetojudgewhetherthereisarelationbetweentreeprofileandleafshape.Weuseagetolinkthesizeoftreeandthetotalweightofitsleaves,becausethetreesizehasanobviousrelationshipwithitsageandtheageaffectsatree'ssequestrationofcarbondioxide,whichaffectsthetotalweightofatree'sleaves.
Assumptions
•Thetreesareallindividual("opengrown")trees,suchasaretypicallyplantedalongstreets,inyards,andinparks.Ourcalculationdoesnotapplytodenselyraisedtrees,asintypicalreforestationprojectswherelargenumbersoftreesareplantedclosetogether.•Theshapeoftheleavesdoesnotreflectspecialusesforthetrees,suchastoresistextremelywindy,cold,parched,wet,ordryconditions,ortoproducefood.•Thetypeoftheleafdistribution(leaflengthandinternodedistancerelation)reflectsthetree'snaturaltendencytosunlight.•Thetreeprofilethatweconsideristhepartaboveground,includingthetrurvk,thebranches,andleaves.•Allpartsofaleafcanlieflat,andthethicknessorprotrusionofveinscanbeneglected.•Leavesaretheonlypartofthetreethatreactsinphotosynthesisandrespiration,sothatthecarbondioxidesequestrationofatreeisthesumof¿hesequestrationoftheleaves.
ACloseLookatLeaves207
•ThesequestrationofatreeoraleafisthenetamountofCO2fixedinatree,whichisthedifferencebetweentheCO2releasedinrespirationandtheCO2absorbedinphotosynthesis.
•Thetreesareinhealthy,mature,andstablecondition.Treesofthesamespecieshavesamecharacteristics.
Model1:
LeafClassification
DecisiveParameters
Toclassifytheshapeofaleaf,wesetsevenparametersandestablishadatabaseforcomparison.
RectangularityWedefinetheratiooftheareaoftheleaftotheareaofitsminimumboundingrectangleastheleaf'srectangularity(Figure1).
Figure1.Figure2.Figure3.Figure4.
ThephotographsofleavesinFigures1-4arereproduced(withoverlaysbytheauthorsofthispaper)fromKnightetal.[2010],bykindpermissionofthatpaper'sauthors.
AspectRatioTheaspectratioistheratiooftheheightoftheminimumboundingrectangletoitswidth.(Figure2).
CircularityToevaluatehowroundaleafis,weconsiderthatratioiftheex-circletothein-circle.(Figure3).
FormFactorFormfactor,afamousshapedescriptionparameter,iscalculatedas
208TheUMAPJournal33.3(2012)
whereAistheareaoftheleafandPisitsperimeter.
EdgeRegularityAreaIndexAlthoughtheaspectratioandtherectangularityoftwoleavesmaybesimilar,thecontourortheexactshapeoftwoleavesmayvarygreatly.Totakethedifferentcontouroftheleafintoconsideration,wejoineveryconvexdotalongthecontouranddevelopwhatwecalltheboundingpolygonarea.Theratiobetweentheleafareaandthisboundingpolygonareaistheedgeregularityareaindex.Thecloserthisratioisto1,thelessjaggedandsmoothertheleaf'scontouris(Figure3).
EdgeRegularityPerimeterIndexSimilarly,wedevelopanotherparameter,theboundingpolygonperimeter,theperimeterofthepolygonwhenwejointheconvexdotsofaleaf.Wedefinetheratiooftheboundingpolygonperimetertotheperimeteroftheleaftobetheedgeregularityperimeterindex.Thesmallerthisratio,themorejaggedandirregularthecontouroftheleafis(Figure3).
ProportionalIndexSinceitisalsohighlycriticaltocapturethespatialdistributionofdifferentportionsofaleafalongitsverticalaxis,wedividetheminimumboundingrectangleintofourhorizontalblocksofequalheight,andthencalculatetheproportionoftheleafareainaparticularregiontothetotalleaf,whichwerefertoastheproportionalindex¡PI)forthatregion(Figure4).Hence,thePIisavectoroflengthfour.
CommonTypesofLeaves
WedevelopadatabaseofthesixmostcommonleaftypesinNorthAmerica(Figure6),usingthesevenparametersdiscussedabove.Table2givesthevaluesoftheparametersforeachleaftype,asmeasuredfromscansofphotosofleavesinKnightetal.[2010].
Comparison
Givenaspecificleaf,wecalculatethesevencharacteristicsofitandcomparethemwithourdatabasebycalculatingthesquareddeviationofeachparameterofthegivenleaffromthecorrespondingstandardparameterofeachcategory.Werealizethatsomeoftheparametersaresomehowmoreimportantthanothers.Soinanefforttomakeourmodelmoreaccurate
ACloseLookatLeaves209
Figure5.ThesixmostcommonseenleaftypesinNorthAmerica.(Thephotos,fromKnightetal.[2010],arereproducedbykindpermissionofthatpaper'sauthors.)
Table1.Parametervaluesforthesixleaftypes.
Type
RectangularityAspectRatioCircularityFormFactorERAreaIndexERPerimeterIndexPIiPI2PI3PI4
1
0.66270.86150.81400.91390.93220.87270.06490.29580.34390.2954
2
0.59020.66000.54320.62060.87800.88890.07690.35550.42430.1433
3
0.62500.18000.45640.28230.90910.93840.11790.22080.41390.2474
4
0.47720.63830.34540.24700.85000.86020.19090.38920.30470.1152
5
0.48760.47920.31230.36620.78800.82310.12990.36060.41230.0970
6
0.65760.31110.33110.49560.88950.99030.29200.41870.26770.0220
andreliable,weintroduceaweightedindexofdeviationID,with
7
ID='
whereeach/jisthesquareddeviation,exceptthat
.1'
WedeterminetheweightsviatheAnalyticalHierarchyProcess(AHP)[Saatyl982].Webuilda7x7matrixreciprocalmatrixbypaircomparison:
210TheUMAPJournal33.3(2012)
RARCFFERAIERPIPI
RARCFFERAIERPIPI
/131422\7
1/311/311/21/23
1314227
1/411/411/31/32
1/221/23114
1/221/23114
1/71/31/71/21/41/41/
ThemeaningofthenumberineachcellisexplainedinTable2.Thenumbersthemselvesarebasedonourownsubjectivedecisions.
Table2.ThemultiplicationtableofDjo.
IntensityofValue
135792,4,6,8Reciprocals
Interpretation
Requirementsiandjhaveequalvalue.Requirementihasaslightlyhighervaluethanj.Requirementihasastronglyhighervaluethanj.Requirementihasaverystronglyhighervaluethanj.Requirementihasanabsolutelyhighervaluethanj.Intermediatescalesbetweentwoadjacentjudgments.Requirementihasalowervaluethanj.
WetheninputthematrixintoaMatlabprogramthatcalculatestheweightWiofeachfactor,asgiveninTable3.
Factor
Weight
R
0.0480
Table3.AHP-derivedweights.
ARCFF
0.15830.04800.2048
ERAI
0.0855
ERPI
0.0855
PI
0.3701
WetesttheconsistencyofthepreferencesforthisinstanceoftheAHP.Forgoodconsistency[AlonsoandLamata2006,446-447]:
•TheprincipaleigenvalueA^axofthematrixshouldbeclosetothenumbernofalternatives,here7;wegetA^ax=7.05.
•TheconsistencyindexCI=(Amax~''^)/ip-~1)shouldbecloseto0;wegetCI=0.009.
•TheconsistencyratioCR=CI/RI(whereRIistheaveragevalueof
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