addedb2t

RESEARCHARTICLEOpenAccessLinkingindividual-treeandwhole-standmodelsforforestgrowthandyieldpredictionQuangVCaoAbstractBackground:Differenttypesofgrowthandyieldmodelsprovideessentialinformationformakinginformeddecisionsonhowtomanageforests.
Whole-standmodelsoftenprovidewell-behavedoutputsatthestandlevel,butlackinformationonstandstructures.
Detailedinformationfromindividual-treemodelsandsize-classmodelstypicallysuffersfromaccumulationoferrors.
Thedisaggregationmethod,inassumingthatpredictionsfromawhole-standmodelarereliable,partitionstheseoutputstoindividualtrees.
Ontheotherhand,thecombinationmethodseekstoimprovestand-levelpredictionsfrombothwhole-standandindividual-treemodelsbycombiningthem.
Methods:Datafrom100plotsrandomlyselectedfromtheSouthwideSeedSourceStudyofloblollypine(PinustaedaL.
)wereusedtoevaluatetheunadjustedindividual-treemodelagainstthedisaggregationandcombinationmethods.
Results:Comparedtothewhole-standmodel,thecombinationmethoddidnotshowimprovementsinpredictingstandattributesinthisstudy.
Thecombinationmethodalsodidnotperformaswellasthedisaggregationmethodintree-levelpredictions.
Thedisaggregationmethodprovidedthebestpredictionsoftree-andstand-levelsurvivalandgrowth.
Conclusions:Thedisaggregationapproachprovidesalinkbetweenindividual-treemodelsandwhole-standmodels,andshouldbeconsideredasabetteralternativetotheunadjustedtreemodel.
Keywords:Disaggregation;Combinationmethod;Loblollypine;PinustaedaBackgroundInformationprovidedbygrowthandyieldmodelsisessentialforforestmanagerstomakeinformeddecisionsonhowtomanagetheirforests.
Munro(1974)classifiedgrowthandyieldmodelsintowhole-standmodelsandin-dividualtreemodels.
Hefurtherseparatedindividual-treemodelsintodistance-independentanddistance-dependentmodels.
Thewhole-standmodels(lowresolution)andindividual-treemodels(highresolution)representtwoex-tremes.
Inthemiddlearemedium-resolutionmodelssuchasdiameter-distributionmodelsandstand-tableprojectionmodels,whichprovideinformationforeachdiameterclass(Figure1).
Eachtypeofmodelhasitsownbenefitsanddraw-backs.
Whole-standmodelsoftenprovidewell-behavedoutputsatthestandlevel,buttheseoutputslackinforma-tiononstandstructures.
Detailedinformationfromindividual-treemodelsandsize-classmodels,ontheotherhand,typicallyresultsinstand-leveloutputsthatarenotasaccurateorprecisebecausetheysufferfromaccumula-tionoferrors(Garcia2001,QinandCao2006).
DanielsandBurkhart(1988)attemptedtolinkdiffer-enttypesofgrowthandyieldmodelsbydevelopingaframeworkforanintegratedsysteminwhichmodelsofdifferentresolutionsarerelatedinaunifiedmathemat-icalstructure.
Thefunctionsusedinthesemodelscanthereforebeconsideredinvariantatdifferentlevelsofdimensionality.
Zhangetal.
(1997)usedthemulti-responseparameterestimationdevelopedbyBatesandWatts(1987,1988)toconstrainanindividual-treemodelbyoptimizingforbothtreeanddiameter-classlevels.
ThisapproachwaslatermodifiedbyCao(2006)toproduceaconstrainedCorrespondence:qcao@lsu.
eduSchoolofRenewableNaturalResources,LouisianaStateUniversityAgriculturalCenter,BatonRouge,LA70803,USA2014Cao;licenseeSpringer.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense(http://creativecommons.
org/licenses/by/4.
0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycredited.
CaoForestEcosystems2014,1:18http://www.
forestecosyst.
com/content/1/1/18treemodelthatwasoptimizedforbothtreeandstandlevels.
Disaggregationmethodisamethodthathasbeenusedbymanyresearchersforlinkinganindividual-treemodelandawhole-standmodel(RitchieandHann1997).
Inthismethod,outputsfromtheindividual-treemodelareadjustedsuchthattheresultingstandsummarymatchespredictionfromawhole-standmodel.
Thedisaggregationmethodaboveassumesthatoutputsfromwhole-standmodelsaremorereliablethanthosefromindividual-treemodels.
Yueetal.
(2008)foundthatstand-leveloutputsfromwhole-standandindividual-treemodelscouldbecombinedtoimprovepredictions.
TheweightedaverageapproachwasextendedbyZhangetal.
(2010)toincludestand-leveloutputsfromadiameterdis-tributionmodel.
Inthispaper,thedisaggregationmethodandcom-binationmethodwereevaluatedagainsttheunadjustedindividual-treemodelbyuseofdatafromunthinnedloblollypine(PinustaedaL.
)plantations.
Reviewofmethodsforlinkingindividual-treemodelsandwhole-standmodelsStand-levelsummaryisobtainedbyaggregating(orsum-ming)tree-leveloutputsfromindividual-treemodels.
Be-causethissummaryisoftenbelievedtobenotasaccurateandpreciseasdirectpredictionfromawhole-standmodel,theindividual-treemodelcanbeadjustedsuchthattheresultingstand-leveloutputmatchesthatfromawhole-standmodel.
Inotherwords,outputfromthewhole-standmodelisdisaggregatedtotreelevelbyuseofsomedisaggregatingfunction.
RitchieandHann(1997)providedanexcellentreviewondisaggregationmethods,classifyingthedisaggregat-ingfunctionsintoadditiveandproportional.
Intheadditivegrowthmethod,thebasalareagrowthofeachtreeisequaltotheaveragetreebasalareagrowthplusanadjustmentbasedontreebasalarea(HarrisonandDaniels1988)ortreediameter(Dhote1994).
Anothercat-egoryofdisaggregationmethodsinvolvesproportionalallo-cationsthatcanbeappliedtoeithergrowthoryield.
Intheproportionalyieldmethod,predictedtreebasalareaisadjustedtomatchpredictedstandbasalarea(ClutterandAllison1974,ClutterandJones1980,PienaarandHarrison1988,NepalandSomers1992,McTagueandStansfield1994,1995).
Theproportionalgrowthmethodinvolvesadjustingpredictedtreebasalareagrowthtomatchpre-dictedstandbasalareagrowth(Campbelletal.
1979,Mooreetal.
1994),treevolumegrowthtomatchstandvolumegrowth(Dahms1983,Zhangetal.
1993),ortreediametergrowthtomatchstanddiametergrowth(Learyetal.
1979).
QinandCao(2006)evaluatedfourmethodstolinkanindividual-treemodelandawhole-standmodelbyuseofdisaggregation.
Intheproportionalyieldmethod,thepredictedtreesurvivalprobability,diameter,andtotalheightweremultipliedbyadjustmentfactors(equations1–3ofTable1).
Treediameterandheightgrowthwereadjustedintheproportionalgrowthmethod,whiletreesurvivalprobabilitywasadjustedbasedontheratioofdeadandaliveprobabilities(equations4–6ofTable1).
Theconstrainedleastsquaresmethod(Matneyetal.
1990,CaoandBaldwin1999)wasusedtoadjusttreeattributes(treesurvivalprobability,squareddiameter,ortotalheight)byminimizingthesumsofsquareddifferencesbetweenthepredictedandadjustedattributes,subjecttotheconstraintsthattheaggregationshadtomatchpredictionsfromawhole-standmodel(equations7–8ofTable1).
Finally,inthecoefficientadjustmentmethod,adjustingcoefficientswereaddedtomodifythecoefficientsoftheoriginalindividual-treemodeltoyieldstandattributesidenticaltothoseproducedbythewhole-standmodel(equations10–12ofTable1).
Thefourmethodsevaluatedproducedsimilarresults,withthecoefficientadjustmentselectedasthemethodtodisaggregatepredictedstandgrowthamongtreesinthetreelist.
TheadjustedtreemodelcombinedthelowresolutionDiameter-distributionmodelIndividual-treemodelStand-tableprojectionmodelhighresolutionWhole-standmodelFigure1Relativepositionofdifferenttypesofgrowthandyieldmodelsintermsoftheresolutionsoftheoutputs.
CaoForestEcosystems2014,1:18Page2of8http://www.
forestecosyst.
com/content/1/1/18Table1ListofadjustmentfunctionsusedinrecentmethodstolinkmodelsofdifferentresolutionsCitationMethodEq.
no.
Adjustmentfunction1/QinandCao(2006)Proportionalyield1~p2;i^p2is^N2Xj^p2j0@1A2~d22i^d22is^B2=KXj^p2j^d22j0@1A3~h2i^h2is^V2a^N2bXj^p2j^d22j^h2j0@1AProportionalgrowth4~p2i^p2i^p2imp1^p2i5~d22id21is^B2=KXj~p2jd21jXj~p2j^d22jd21j0B@1CA^d22id21i6~h2ih1is^V2a^N2bXj~p2j~d22jh1jbXj~p2j~d22j^h2jh1i0B@1CA^h2ih1iConstrainedleastsquares7~p2i^p2is^N2X^p2j=n8~d22i^d22j~p2iXj~p2j^d22js^B2=KXj~p22j0@1A9~h2i^h2i~p2i~d22iXj~p2j~d22j^h2jsa^N2^V2=bXj~p22j~d42j0@1ACoefficientadjustment10~p2ip1i=1expα0α1H1α2mpd1i=Dq111~d2id1i1expβ0β1lnB1β2A1β3lnH1β4mdd1iDq1β5lnh1ihino12~h2ih1i1expγ0γ1lnB1γ2A1γ3lnH1γ4mhd1iDq1γ5h1iH1γ6lnd1ihinoCao(2006)Disaggregation13~p2i^pmp2i14~d22id21is^B2=KXj~p2jd21jXj~p2j^d22jd21j0B@1CA^d22id21iConstrainingindividual-treemodelwithdiameter-classattributes15^p2i1=1expα0α1N1α2B1α3d1i^n2;kXn1;ki1^p2i(16^d2id1iβ1A2A1β2Hβ31Bβ41dβ51^b2;kKXn1;ki1^p2i^d22i8>:Constrainingindividual-treemodelwithstandattributes17^p2i1=1expα0α1N1α2B1α3d1i^N2Xi^p2i=s(18^d2id1iβ1A2A1β2Hβ31Bβ41dβ51^B2KsXi^p2i^d22i8>>>:CaoForestEcosystems2014,1:18Page3of8http://www.
forestecosyst.
com/content/1/1/18bestfeaturesofwhole-standandindividual-treemodels.
Comparedtotheunadjustedtreemodel,theadjustedmodelperformedbetterinpredictingstandattributesintermsofstanddensity,basalarea,andvolume,especiallyforlongprojectionperiods.
Theadjustedmodelalsoprovidedcomparablepredictionsoftreediameter,height,andsurvivalprobability.
Cao(2006)evaluatedadisaggregationmethodagainsttwoapproachestoconstrainanindividual-treemodel.
Inthedisaggregationmethod,thepredictedtreesurvivalTable1Listofadjustmentfunctionsusedinrecentmethodstolinkmodelsofdifferentresolutions(Continued)Yueetal.
(2008)Combinedestimator19~B2w^B2T1w^B2S,wherewisselectedtominimizethevarianceof~B2.
Zhangetal.
(2010)Combinedestimator20~B2w1^B2Tw2^B2Sw3^B2D,wherewkisselectedtominimizeXB2~B22,andX3kwk1.
Cao(2010)121~p2i^pm2iTreesurvival222~p2i^p2i^p2imp1^p2i323~p2i1=1expmpα0α3d1i424~p2i1=1expα0α1N1α2B1mpd1i525~p2i^p2is^N2Xj^p2jsN1Xj^p2j0@1A1^p2iCao(2010)126^d2id1imddβ51Treediametergrowth227^d2id1iβ1A2A1β2Hβ31Bβ41dmd1328~d22id21is^B2=KXj~p2jd21jXj~p2j^d22jd21j0B@1CA^d22id21i1/Notation:A1=standageatthebeginningofthegrowthperiod.
A2=standageattheendofthegrowthperiod.
H1=dominantheightatageA1.
N1=numberoftreesperhaatageA1.
^N2=predictednumberoftreesperhaatageA2.
B1=standbasalareaatageA1.
^B2=predictedstandbasalareaatageA2.
^B2D=predictedstandbasalareaatageA2fromadiameterdistributionmodel.
^B2S=predictedstandbasalareaatageA2fromawhole-standmodel.
^B2T=predictedstandbasalareaatageA2fromanindividual-treemodel.
~B2=combinedestimatorforstandbasalareaatageA2.
^V2=predictedvolumeperhaatageA2,Dq1=quadraticmeandiameteratageA1.
aandb=parametersoftheindividualtreevolumeequation,viabd2ihi.
vi,di,andhi=treevolume,dbh,andtotalheightoftreei,respectively.
s=plotsizeinha.
K=π/40000=constanttoconvertdiameterincmtoareainm2.
n=numberoftreesintheplot.
d1iord1j=dbhoftreeiorjatageA1.
^d2ior^d2j=predicteddbhoftreeiorjattheendofthegrowthperiod.
~d2i=adjusteddbhoftreeiattheendofthegrowthperiod.
h1i=totalheightoftreeiatageA1.
^h2ior^h2j=predictedtotalheightoftreeiorjattheendofthegrowthperiod.
~h2i=adjustedtotalheightoftreeiattheendofthegrowthperiod.
p1i=survivalprobabilityoftreeiatageA1.
^p2ior^p2j=predictedsurvivalprobabilityoftreeiorjattheendofthegrowthperiod.
~p2ior~p2j=adjustedsurvivalprobabilityoftreeiorjattheendofthegrowthperiod.
α0…α3=parametersofthetreesurvivalequation.
β0…β5=parametersofthetreediametergrowthequation.
γ0…γ6=parametersofthetreeheightgrowthequation.
n1,k=numberoftreesofthekthdiameterclassatageA1.
^n2;k=predictednumberoftreesofthekthdiameterclassatageA2.
^b2;k=predictedbasalareaofthekthdiameterclassatageA2,andmp,md,andmh=adjustmentcoefficientstobeiterativelysolvedtoensurethattheresultingnumberoftreesperha,standbasalarea,andstandvolume,respectively,matchthoseproducedbythewhole-standmodel.
CaoForestEcosystems2014,1:18Page4of8http://www.
forestecosyst.
com/content/1/1/18probabilitywasadjustedwithasimplepowerfunction,inwhichthepowerwasiterativelysolvedsuchthattheadjustedsurvivalprobabilitysummeduptothepre-dictedstanddensity(equation13ofTable1).
Thepro-portionalgrowthformulawasusedinadjustingdiametergrowth(equation14ofTable1).
Theindividual-treemodelwasconstrainedbydiameter-classattributes(equations15–16ofTable1)byuseofthemulti-responseparameterestimationmethod(Zhangetal.
1997,BatesandWatts1987,1988).
Alsoincludedintheevaluationwasasimilarapproachtoconstraintheindividual-treemodelbystandattributes(equations17–18ofTable1).
Cao(2006)foundthatwhilethetwoconstrainedmodelsperformedslightlybetterthantheunconstrainedtreemodelinpredictingtreeandstandattributes,thedisaggregationmethodprovidedthebestpredictionsoftree-andstand-levelsurvivalandgrowth.
Cao(2010)listeddifferentdisaggregationmethodsforpredictingtreesurvivalanddiametergrowth.
Thesein-cludefivedisaggregationmethodsforadjustingtreesur-vivalprobability(equations21–25ofTable1)andthreemethodsfordiametergrowthadjustment(equations26–28ofTable1).
Hisresultsshowedthatthedifferentmethodsproducedsimilarresults.
Cao(2010)alsofoundthatuseofobservedratherthanpredictedstandattri-butesfordisaggregationledtoimprovedpredictionsfortreesurvivalanddiametergrowth,i.
e.
thequalityofthetree-levelpredictionsindisaggregationdependedonthereliabilityofthestandpredictions.
Yueetal.
(2008)usedthemethodintroducedbyBatesandGranger(1969)andNewboldandGranger(1974)tocombinestand-leveloutputsfromwhole-standandindividual-treemodels.
Thecombinedestimatorisaweightedaverageofoutputsfrombothmodels(equation19ofTable1).
Theoptimumweightswereselectedtominimizethethevarianceofthecombinedestimator.
Zhangetal.
(2010)extendedthisapproachtoalsoin-cludestand-leveloutputsfromadiameterdistributionmodel(equation20ofTable1).
Theleast-squaresesti-mateoftheweightswascomputedaccordingtoTang(1992,1994).
MethodsStand-andtree-levelgrowthmodelsdevelopedbyCao(2006)wereusedinthisstudy.
Thewhole-standmodelconsistedofequationsforpredictingstanddensityintermsofnumberoftreesandbasalareaperhectareasfollows:^N2;iN1;i=1exp16:3197–42:4204RS1;i–0:7466H1;i–0:0269N1;i=A150:2622=A1;1^B2;iB1;i=1exp–3:3259–0:7800B1;i=A141:0393=A12where:N1,i=numberoftreesperhainplotiatageA1,^N2;i=predictednumberoftreesperhainplotiatageA2,H1,i=averagedominantandcodominantheight(m)ofplotiatageA1,RS1,i=(10,000/N1,i)0.
5/H1,i=relativespacingofplotiatageA1,B1,i=standbasalarea(m2/ha)ofplotiatageA1,and^B2;i=predictedstandbasalarea(m2/ha)ofplotiatageA2.
Theindividual-treemodelincludedequationsforpre-dictingtreesurvivalprobabilityanddiametergrowthasfollows:^pij1exp1:35860:0010N1;i0:1042B1;i0:2902d1;ij13^d2;ijd1;ij0:7168A2A12:0192H1;i1:0111B1;i0:3166d1;ij1:51174where:^pij=predictedprobabilitythattreejinplotiisaliveatageA2,giventhatitwasaliveatageA1,d1,ij=diameters(cm)oftreejinplotiatageA1,and^d2;ij=predicteddiameters(cm)oftreejinplotiatageA2.
DataEquations(1,2,3and4)abovewerederivedfrom100plotsfromloblollypine(PinustaedaL.
)plantationsintheSouthwideSeedSourceStudy,whichinclude15seedsourcesplantedat13locationsacross10southernstates(WellsandWakeley1966).
Datausedinthisstudywerefromanother100plots,alsorandomlyselectedfromtheSouthwideSeedSourceStudy.
Each0.
0164haplotconsistedof49trees,plantedata1.
8m*1.
8mspacing.
Treediametersandsurvivalwererecordedatages10,15,20,and25years,resultinginatotalof300growthperiods(Table2).
MethodsevaluatedInadditiontotheindividual-treemodel(equations3and4),thedisaggregationandcombinationmethodswereevaluatedinthisstudy.
DisaggregationmethodThetreesurvivalprobability(^pij)predictedfromequation(3)wasadjustedbyuseofCao's(2010)methodasfollows:CaoForestEcosystems2014,1:18Page5of8http://www.
forestecosyst.
com/content/1/1/18~pij^pija5whereαistheadjustmentcoefficientusedtomatchthesumofadjustedtreesurvivalprobabilities(~pij)topredic-tionsfromthestandsurvivalmodel(equation1).
Fromequation(4),theprojectedtreediameter(^d2;ij)wasadjusted(Cao2010)sothattheresultingstandbasalareamatchesthepredictionfromthewhole-standmodel(equation2):~d22;ijd21;ijβ^d22;ijd21;ij6where:βsBb2;i=KX~pijd21;ijX~pij^d22;ijd21;ijhi,and:K=π/40000.
CombinationmethodThecombinedestimatorofstandsurvivalwastheweightedaverageofstand-levelpredictionsfromthewhole-standmodel(equation1)andtheindividual-treemodel(equation3).
TheweightswerecomputedaccordingtoamethoddescribedbyTang(1992,1994)andappliedbyZhangetal.
(2010).
Asimilarprocedurewasappliedtocomputethecombinedestimatorforstandbasalarea.
Predictionsfromtheindividual-treemodelwerethenadjustedfromthecombinedestimatorsforstandsurvivalandbasalarea,usingthedisaggregationmethoddescribedearlier.
EvaluationcriteriaTheperformanceoftheunadjusted,disaggregation,andcombinationmethodswasevaluatedatbothstandandtreelevels,basedonthefollowingstatistics.
Meandifference:MDXyi^yi=n7Meanabsolutedifference:MADXyi^yijj=n8FIXyi^yi2=yiy29Log-likelihood:2lnL2XpilnpiX1piln1pihi10where:yiand^yi=observedandpredictedvaluesattheendofthegrowthperiodofstandvariables(standsurvivalandbasalarea)ortreevariables(treediameterandsurvivalprobability),y=averageofyi,n=numberofobservations,andpi=predictedsurvivalprobabilityoftreei.
ResultsanddiscussionTable3showsthatthewhole-standmodel(disaggrega-tionmethod)producedthebestMDandMADvaluesforstanddensitywhilethecombinationmethodyieldedthebestFIvalue.
Forstandbasalarea,allofthebestevaluationstatisticscamefromthewhole-standmodel(Table3).
Attreelevel,thedisaggregationmethodreturnedthebestevaluationstatisticsforbothtreesur-vivalprobabilityandtreediameter(Table3).
DisaggregationmethodFromTable3,itisclearthatthewhole-standmodelwasmoreaccurate(lowerMD)andprecise(lowerMADandhigherFI)inpredictingstanddensityandbasalareathantheindividual-treemodel.
Thedifferencesweresubstantial.
Comparedtotheindividual-treemodel,thewhole-standmodeldecreasedMDby88and97%,de-creasedMADby15and46%,andincreasedFIby8and28%forstanddensityandstandbasalarea,respectively.
Predictedstandattributesfromthetree-levelmodelwerenotasreliablebecausetheywereobtainedthroughsummationofindividual-treepredictions,resultinginaccumulationoferror.
QinandCao(2006)showedthatatree-levelmodel,afterbeingadjustedfromobservedstandattributesthroughdisaggregation,outperformedtheunadjustedtreemodel.
Theyinferredthattheperformanceofdisag-gregationmodelsdependedlargelyonhowclosethestandpredictionsweretotheobservedvalues.
Thewhole-standmodelseemedagoodcandidateinthiscase,yieldingFIvaluesof0.
825and0.
862inpredictingstanddensityandbasalarea,respectively.
Thetree-levelstatisticssupportthishypothesis:thedisaggregationmodelreducedMDby26and19%,andMADby14and11%fortreesurvivalprobabilityandtreediameter,re-spectively,ascomparedtotheunadjustedtreemodel.
ItTable2Means(andstandarddeviations)ofstandandtreeattributes,byageAttributeStandage(years)10152025Dominantheight(m)9.
1(1.
3)13.
4(1.
6)16.
9(1.
9)19.
9(2.
2)Numberoftrees/ha1696(627)1448(548)1143(350)1013(334)Basalarea(m2/ha)19.
2(5.
6)28.
8(5.
9)33.
2(8.
1)37.
4(9.
4)Treediameter(cm)11.
6(3.
1)15.
4(4.
1)18.
7(4.
6)21.
0(5.
2)Fitindex:CaoForestEcosystems2014,1:18Page6of8http://www.
forestecosyst.
com/content/1/1/18alsodecreased–2ln(L)fortreesurvivalby11%andin-creaseFIfortreediameterby1%.
CombinationmethodInthisstudy,combiningstandpredictionsfromthewhole-standandindividual-treemodelsresultedinpre-dictionsofstanddensityandbasalareathatwerebetterthanthosefromtheindividual-treemodel,butnotasgoodasthosefromthewhole-standmodel.
Amongsixevaluationstatisticsconsidered,thecombinationmethodonlyedgedthewhole-standmodelinfitindex(0.
830versus0.
825),whilecameinsecondfortheremainingstatistics.
Thiswascontrarytopastreportsofsuperiorperformancebythecombinationmethod(Yueetal.
2008,Zhangetal.
2010).
InastudybyZhangetal.
(2010),similarfitindexvalues,rangingfrom0.
9466to0.
9494,wereobtainedforpredictedstandbasalareafromthreedifferenttypesofmodelsforthevalidationdataset.
Inthisstudy,aconsiderabledifferenceinfitindexofstandbasalareapredictionbetweentheindividual-treemodel(0.
676)andthewhole-standmodel(0.
862)mightresultinmediocreperformanceofthecombinationmethod(FI=0.
699forstandbasalarea).
Thetreesurvivalmodelthatwasdisaggregatedfromthecombinedestimatorgavesimilarevaluationstatisticsasdidtheunadjustedtreesurvivalequation(Table3).
Ontheotherhand,thetreediametermodelfromthecombin-ationmethodperformedworsethantheunadjustedtreediametergrowthequation(Table3).
Tree-levelpredictionsweredisaggregatedfromthewhole-standmodelforthedisaggregationmethodandfromthecombinedestimatorforthecombinationmethod.
Basedonthedatafromthisstudy,thedisaggregationmethodwasbetterforpredictingbothtreesurvivalanddiameterintermsofallevaluationstatistics.
ConclusionsThedisaggregationmethodinvolvesadjustingoutputsfromtheindividual-treemodeltomatchpredictionsfromthewhole-standmodel.
Itwasshowninpreviousfindingsandalsointhisstudythatthismethodprovidedbetterpredictionsoftreesurvivalanddiametergrowth.
Comparedtothewhole-standmodel,thecombinationmethoddidnotshowimprovementsinpredictingstandattributesinthisstudy.
Thecombinationmethodalsodidnotperformaswellasthedisaggregationmethodintree-levelpredictions.
CompetinginterestTheauthordeclaresthathehasnocompetinginterests.
AcknowledgementFundingforthisprojectwasprovidedinpartbytheMcIntire-Stennisfunds.
Received:25July2014Accepted:4September2014ReferencesBatesDM,WattsDG(1987)AgeneralizedGauss-Newtonprocedureformulti-responseparameterestimation.
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ForEcolManage69:219–232Table3Stand-levelandtree-levelevaluationstatisticsforthreemethodsStatistic1/UnadjustedtreemodelDisaggregationmethodCombinationmethodStandlevelStanddensity(trees/ha)MD28.
13.
52/4.
1MAD176.
1148.
8149.
4FI0.
7650.
8250.
830Standbasalarea(m2/ha)MD2.
050.
061.
89MAD3.
992.
173.
87FI0.
6760.
8620.
699TreelevelTreesurvivalprobabilityMD0.
0190.
0140.
019MAD0.
2390.
2060.
2392ln(L)516746154976Treediameter(cm)MD0.
160.
130.
27MAD0.
940.
841.
03FI0.
9390.
9520.
9271/Notations:MD=Xyi^yi=n;MAD=Xyi^yijj=n;FI=Xyi^yi2=yiy2;2ln(L)=2[∑piln(pi)+∑(1pi)ln(1pi)],whereyiand^yi=observedandpredictedvaluesattheendofthegrowthperiodofstandvariables(standsurvivalandbasalarea)ortreevariables(treediameterandsurvivalprobability);y=averageofyi;n=numberofobservations,andpi=predictedsurvivalprobabilityoftreei.
2/Foreachevaluationstatistic,thebolditalicnumberdenotesthebestamongthreemethods.
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