Tree growth inference and prediction when the point of

12 Pages·2008·0.28 MB·English

JournalofTropicalEcology(2009)25:1–12.Copyright©2008CambridgeUniversityPress
doi:10.1017/S0266467408005646 PrintedintheUnitedKingdom
Tree growth inference and prediction when the point of measurement
changes: modelling around buttresses in tropical forests
C.JessicaE.Metcalf∗1,JamesS.Clark†andDeborahA.Clark‡
∗CentreforInfectiousDiseaseDynamics,PennStateUniversity,StateCollege,PA16802,USA
†NicholasSchooloftheEnvironment,DukeUniversity,Durham,NC27707,USA;DepartmentofBiology,DukeUniversity;DepartmentofStatisticalScience,Duke
University
‡DepartmentofBiology,UniversityofMissouri,St.Louis,Missouri,63121-4499USA;LaSelvaBiologicalStation,PuertoViejodeSarapiqu´ı,CostaRica
(Accepted27October2008)
Abstract: Estimationoftreegrowthisgenerallybasedonrepeateddiametermeasurements.Abuttressattheheight
ofmeasurementwillleadtooverestimatesoftreediameter.Becausebuttressesgrowupthetrunkthroughtime,it
hasbecomecommonpracticetoincreasetheheightofmeasurement,toensurethatmeasurementsremainabovethe
buttress.However,taperingofthetrunkmeansthatincreasingmeasurementheightwillbiasestimatesofdiameter
downwardbyupto10%permofheight.Thisbiascouldaffectinferenceconcerningspeciesdifferencesandclimate
effectsontreedemographyandonbiomassaccumulation.Hereweintroduceahierarchicalstatespacemethodthat
allowsformalintegrationofdataondiametertakenatdifferentheightsandcanincludeindividualvariation,temporal
effectsorothercovariates.WeillustrateourapproachusingspeciesfromBarroColoradoIsland,Panama,andLaSelva,
CostaRica.Resultsincludetrendsthatareconsistentwithsomeofthosepreviouslyreportedforclimateresponsesand
changesovertime,butdifferinrelativemagnitude.Byincludingthefulldata-setandaccountingforbiasandvariation
amongindividualsandovertime,ourapproachallowsforquantificationofclimateresponsesandtheuncertainty
associatedwithmeasurementsandtheunderlyinggrowthprocess.
Key Words: Cecropia obtusifolia, climate change, growth, Lecythis ampla, Luehea seemannii, Minquartia guianensis,
modellinguncertainty,pointofmeasurement,Simaroubaamara,tropicaltrees
INTRODUCTION et al. 2007, Condit et al. 2000, Hubbell et al. 1999).
Additionally,aswithanylong-termcensus,dataareoften
Tree growth underlies population dynamics (Clark & missing and observation errors vary due to changes in
Clark 1999), species interactions (Swetnam & Lynch protocol.
1993),carbonsequestration(Caspersenetal.2000)and Clarketal.(2007)introducedahierarchicalBayesian
speciesresponsestoclimatechange(Clark2002,2004; approach to modelling tree growth that accommodates
Clarketal.2003).Nevertheless,reliableestimatesoftree uncertainty in process, data and parameters. The
growtharelimited,asdataaretimeconsumingtocollect, modelalsoallowsinterpolationofmissingdatathrough
andtherearechallengesassociatedwithinference(Clark predictive distributions that reflect uncertainty both in
etal.2007).Annualgrowthratescanbestage-dependent themeasurementofgrowth,andthegrowthprocess,with
(Clark & Clark 1999), temporally correlated (Clark constraintsbasedonknownbiologicalinformationsuch
etal.2007),andspatiallycorrelated,primarilythrough as the fact that diameter increment growth cannot be
competition (Canham et al. 2006, Condit et al. 2000, negative(Clarketal.2007),growthgenerallydependson
Hubbell et al. 1999, 2001; Webster & Lorimer 2005). sizeandgrowthrateintheprevioustimesstep(Wyckoff
Treecensusdatashowconsiderableindividualvariation & Clark 2002), and it may vary from year to year and
in growth, variation inherent in the process of growth, accordingtoindividualeffects(Clarketal.2007).
as well as temporal and spatial variation in the Here,wemodifythisapproachtomeetthechallengesof
covariates affecting growth (Clark & Clark 1999, Clark tropicaltreedatasets,particularlythelackofconsistency
intheheightatwhichtreesaremeasuredduetogrowing
buttresses. Growth increments corresponding to years
1Correspondingauthor.E-mail:[email protected] where the height of measurement has changed may
2 C.JESSICAE.METCALFETAL.
be simply omitted from analyses (Feeley et al. 2007). themusingataperingmodel,definedbyaparameterα,
Alternatively, where at every time-step diameters at andamodeldescribingobservationerror(measurement
several points of measurement are taken on potentially error).If D(o) isthediameterincmoftreeimeasuredat
i,h,t
problematic trees, so that no growth increments are heighth(o) inminyeart,wedefine
i,h,t
lost (Clark et al. 2003), suites of measurements (cid:2) (cid:3) (cid:2) (cid:4)(cid:5) (cid:4)
within an individual may be from several different Di(,oh),t ∼ N Di,1.3,texp α hi(,oh),t −1.3 ,w , (1)
heights.Bothissuesrenderproblematictheinferenceof wherethe(o)superscriptindicatesanobservation,αisthe
individualgrowthtrajectories.However,itisofparticular
tapering parameter, and variance w captures Gaussian
importancetocapturetheseobservations,asintwoofthe
observation error. The concept is illustrated in Fig-
majortropicaltreedata-setsastrongsignalofdeclining ure1.Sinceexp[α(h(o) −1.3]indicatestheproportion
growth through time has been detected (Clark 2004, i,h,t
by which diameter is decreased at increasing heights,
Clarketal.2003,Feeleyetal.2007).Ifthereiscovariance thevalueinbracketswillneedlessthan1,i.e.α willbe
betweentimeandtreesize(Feeleyetal.2007),ortimeand
less than 1. Although proportionate taper (summarized
heightofmeasurement,ortimeandindividualspresentin by the constant α, i.e. the tapering parameter) may
thepopulation,thereispotentialfortheperceiveddecline
change during the course of ontogeny, only large trees
over time to reflect biases in the estimation techniques
generally experience shifts in measurement height, so
thatneglectthemanylevelsofvariation.Herewedevelop
this assumption of a constant exponent will minimally
a method that integrates all sources of information to
affect results. Trees where several measurements of
formally model measurements taken at several heights
diameter at different heights are available within the
in the presence of uncertainty in measurements (Clark same year will contribute directly to calibrating α. For
&Bjornstad2004). Wethenshowhowthisframework
years where measurements are only available at one
canbeextendedtoallowinferencewhenlongoruneven height per individual, α will be informed indirectly by
census intervals are considered, without neglecting the
allobservations.
cumulative effects of variance in growth through time
Tocompletetheobservationmodel,wedefinepriorsfor
thatwilloccurasgrowthisinherentlystochastic. each of the parameters, α and w. A parameter’s prior
Inthefollowingmethodssectionwefirstintroducethe
is a probability density function that represents prior
two methods, then briefly outline the data-sets used for
knowledge about what the parameter’s value or range
this analysis. The resulting patterns of growth through
of values should be. For example, if previous analyses
timearethenpresented.Weconcludebydiscussingthe
indicatethatmeasurementerrorisof1mmthenwewould
insightsavailablefromourmethod,andhowourresults
centre the probability density function for w narrowly
relatetopreviousworkonthistopic.
around1,equivalenttospecifyingan‘informativeprior’.
If we have no information, we will use uninformative
priors with broad distributions. The forms of the prior
METHODS
distributions used in the full sampling procedure are
normalforαandconjugateinversegammaforw.
Datamodelling
To achieve the second step, we develop a model of
growth based on Clark et al. (2007) that exploits the
Dataconsistofrepeateddiametermeasuresonindividuals
informationonoveralltreegrowth,withameanforthe
throughtimeassociatedwithaheightofmeasurement. fullpopulationβ ,randomindividualeffectsβ (howtree
0 i
The model is developed to maximize the support drawn
i differs from the rest of the population) and fixed year
from the full dataset while emphasizing information effectsβ (therecanbesharedyear-to-yearvariationdue
t
associated with specific trees and years. Relationships
to,say,climate).Wemodelgrowthincrementsataheight
between different model components are specified as
of1.3mas,
conditional distributions, allowing estimation using a
Gibbs sampling approach. Specifically, our approach Xi,t ≡ Di,1.3,t+1−Di,1.3,t =μi,t +εi,t (2)
consists of two steps. First, we specify an observation
whereX isthediameterincrementaddedbetweenyeart
i,t
modeltoconvertdiameterobservedatanyheightintoa andt+1ataheightof1.3m.Thisisalinearequationwith
‘true’diameterat1.3m(allowingformeasurementerror
deterministic components defined by population mean
inourobservations).Second,wespecifyagrowthmodel growthrateβ ,randomindividualeffectβ ,yeareffectβ
0 i t
fordiametergrowthat1.3mtoallowdirectcomparison andprocesserrorε ,
i,t
amongindividualsandyears.
Forthefirststep,foreveryindividualforeveryyear,we μi,t =β0+βi +βt (3)
modeladiameterataheightof1.3masalatentvariable β ∼ N(0,τ2) (4)
i
denoted Di,1.3,t. To infer this value, we use information
from observations taken at different heights, and link εi,t ∼ N(0,σ2). (5)
Growthwithchangingheight 3
Useoftheinversegammapriorforvariancesisdiscussed
inClark(2007)andClarketal.(2007).Thebeginningof
thestudyisdefinedastheyearwhendiameterdatawere
firstcollectedfortheindividualataparticularheight,t .
i,h
WedefinetheendofthestudyT asthetimewhenthe
i,h
mostrecentdiameterdatawerecollectedorindividuali
dies,whichevercomesfirst.Foreveryheight,wemodel
thedataonatree-by-treebasisforallyearstbetweent
i,h
andT .
i,h
Theresultofsamplingfromthefullmodelisthatthe
latent state variable D , corresponding to diameter
i,1.3,t
measuredataheightof1.3m,isconditionallydependent
on observations that might be available for diameter at
a range of heights, D(o) , and on the unknown states
i,h,t
D and D , which will also be modelled. A
i,1.3,t+1 i,1.3,t−1
fulldiscussionofthisstructureiscontainedinClarketal.
(2007).Non-independenceofdataisaccommodatedby
thestochastictreatmentoftheunderlyingprocess.Here
isthejointprobabilityforasingletreeyear(conditioned
ontherestofthemodel):
(cid:2) (cid:4) (cid:2) (cid:6) (cid:4)
p Di(,oh),t,Di,1.3,t = p Di(,oh),t(cid:6)Di,1.3,t p(Di,1.3,t)
(cid:2) (cid:6) (cid:3) (cid:2) (cid:4)(cid:5) (cid:4)
= N Di(,oh),t(cid:6)Di,1.3,texp α hi(,oh),t −1.3 ,w
×N(Di,1.3,t −Di,1.3,t−1|μi,t−1,σ2)
×N(Di,1.3,t+1−Di,1.3,t|μi,t,σ2) (8)
Figure1. Illustrationofthemodelforthedataobservationprocess(Eqn.
1).Diametermeasurementsaremadeatarangeofheights.Hereone
Thedensitiesinthelastlineare,respectively,thediameter
suchisshownbytheblackcircle.Wewishtoidentifyarelationship
betweenheightanddiameterlikethatshownbythecross-hatchedarea, data model, which includes a parameter describing
definedbyataperingparameterα,sothatwecanobtainanestimateof tapering(α)toallowfordatafromseveraldifferentheights
diameteratthesameheightof1.3mforalltrees,shownherebytheblack toenterinference,andtheprocessmodelforgrowthfrom
triangleanddottedlines.Thegrowthmodel(Eqn.2)canthenbeapplied t−1totandfromttot+1.Stochasticityinourprocess
acrossthedata-settodiametervaluesatacommonheightof1.3m.
model includes process error and random effects (Eqns
Foracertainnumberoftrees,morethanonemeasurementisavailable
ataparticularcensus(Table1),sothatthetaperingrelationshipcan 2–5).
bedirectlycalibrated.Forothers,theinformationisborrowedacross We now have specified the full model, including the
theentiredataset.Resultsshownherecorrespondtoposteriorsforα formofthepriordistributionsforallparameters,andthe
forLueheaseemanniiandoneindividualmeasurementfromthatdata-
conditionalsdescribinghowthedifferentcomponentsare
set.Thecross-hatchedareaincorporatesuncertaintyinthetapering
interrelated.Thefinalstepbeforeapplyingthefullmodel
parameter,α,anduncertaintyaroundtheobserveddata−pointdueto
observationerror,w. to data is to specify values for parameters of the prior
distributions, based on knowledge of the approximate
rangesofvariabilityinmeasurements.Takeforexample
‘Process error’ εi,t is the extent to which main effects theobservationerrorparameter,w.Re-measurementin
(mean, individual and year) fail to describe diameter theLaSelvadatabaseyieldsanaveragedifferencebetween
increment.Priordistributionsareconjugatenormaland measurements of 0.03 cm, with a maximum of 0.1 cm
inversegammarespectively, (Clark&Clark2006).Wecouldsetw=0.01toreflecta
β ∼ N(b,v ), beliefthatobservationsarenormallydistributedaround
0 0
the‘true’valueofdiameterwithavarianceof0.01(which
β ∼ N(0,v ), (6)
t t moreorlesscapturesthecharacteristicsabovefromClark
σ2 ∼ IG(a ,a ) &Clark2006).However,wedonotknowforsurethatw
1 2
=0.01,andwishtoinferitsvalueusingbothinformation
For random individual effects, the variance has the fromthedataandinformationbasedonourpriorbeliefs
prior (e.g.thatitshouldbe0.01).Wethereforedefineaprior
τ2 ∼ IG(a ,a ). (7) distribution for w, taken as conjugate inverse gamma,
3 4
4 C.JESSICAE.METCALFETAL.
Table1. Dataavailableforeachofthefivespecies.DatafromLaSelva(speciesinthefirstfourrows)ismuchmoredetailed
partlyasmeasurementsweretakenyearlyratherthanevery5y.
Numberof Numberof Proportionoftrees Numberoftree-years
individual diameter Totalyearsof thatgrowand withdiametersat
trees,n observations,nD thestudy,T changeheight severalheights
Minquartiaguianensis 559 7646 18 0.17 188
Lecythisampla 400 4165 18 0.26 142
Cecropiaobtusifolia 208 1114 13 0.14 20
Simaroubaamara 415 3311 13 0.14 49
Lueheaseemannii 210 1330 23(6censuses) 0.27 128
e.g. Analysis of the model is accomplished by Gibbs
sampling (Gelfand & Smith 1990), implementation of
w ∼ IG(a ,a ), (9)
5 6 whichisdiscussedintheAppendix1.Themodeloutput
will be a sample from the posterior distributions of all
where the parameters a5 and a6 will be chosen so that parametersandoflatentdiametersforeveryindividualat
the mean of this prior distribution is around 0.01, and everytime-step.Posteriordistributionsaredensityfunc-
the variance around this mean reflects how certain we tionsdefinedbytheproductofthepriordistributionsand
arethatmeasurementerrorisclosetothisvaluevs.how conditionaldistributionsdefiningtheobservationandthe
muchthedatacandefineit.Toweightourbeliefabout processmodel.Posteriordistributionswerenotinfluenced
themeanvaluetobeahundredtimestheweightofthe by the mean values selected for β , τ2 and β because
0 t
data,wecanspecify priorswereweak.Incrementsaresampledfromtruncated
normaldistributionstoensurethattheyremainpositive.
a =100n
5 D (10)
a =0.01(a −1)
6 5
Extensionforunevensamplingintervals
wheren isthenumberofdiameterobservations(Table1
D
andTable2,seeClarketal.2007fordetailsofweighting).
Forsomedata-sets,measurementsaretakenatintervals
FortheBCIdatabase,re-measurementinformationgiven
longerthanayear,inwhichcase,thereisnoinformation
inChaveetal.(2004)suggestsweuseapriormeanfor
for growth in individual years. Furthermore, intervals
observation error variance of 1, and we use the same
maybeuneven,furthercomplicatingestimationofyearly
weightingschemeasinEqn.10.
growth increments. To model data in this case, we can
Forprocesserrorσ2,weuseapriormeanof0.5,and
explicitly model the differing time lags. We define the
wnTei=gh(cid:7)titi,tho(bTei,htw−ictei,hth).eTnhuims bisereoqfutriveaelyeenatrstoanrdeshtreiicgthintsg, mofuyletai-rysesaerpianrcarteinmgecnotnssaescu(cid:7)tiXvei,tc,eannsdus(cid:7)est,sauscthhethnautmber
possiblevaluesofσ2 tobequitecloseto0.5,sothatthe
process error will generally be large, to avoid excessive (cid:7)Xi,t ≡ Di,1.3,t+(cid:7)i,t −Di,1.3,t, (12)
smoothing,
Assuming that variation accumulates as white noise
a =2n (variance scales linearly with (cid:7)t), using the same
a1 =0.5T(a −1) (11) notation as above for μi,t (i.e. growth increment in one
2 1 time-step)andσ2(processerrorforonetime-step)thisis
Other priors are non-informative, e.g. we do not distributedas
introduce prior beliefs into their modelling. The prior
(cid:7)Xi,t ∼ N((cid:7)i,tμi,t,(cid:7)i,tσ2), (13)
distributionformeangrowthrateisnormallydistributed
withmeanb=0.8,i.e.apriormeangrowthrateof0.80 whichcanbesampleddirectly.Thisformulationcaptures
cmy−1;andv =500wherethislargevariancereflects thefactthatvariancewillincreasewithtimeelapsed.The
0
ourlackofpriorcertainty.Inferenceaboutthisparameter jointprobabilityforasingletreebecomes
(cid:2) (cid:4) (cid:2) (cid:6) (cid:4)
wpriilolrcoonnstehqeuetanptleyribnegcphaieraflmydeteetrerαmhinaesdmbeyatnheαda=ta−.T0h.1e p Di(,oh),t+(cid:7)i,t,Di,1.3,t+(cid:7)i,t = p Di(,oh),t+(cid:7)i,t(cid:6)Di,1.3,t+(cid:7)i,t
0
andvariancevα =100,whichislikewiseweak,sothat ×p(Di,1.3,t+(cid:7)i,t)
datawilldominate.Thepriorforthevarianceofrandom (cid:2) (cid:6) (cid:4)
effectsτ2 hasmeansetto1,andweightedroughly0.01 = N Di(,oh),t+(cid:7)i,t(cid:6)Di,1.3,t+(cid:7)i,texp[α(hi(,oh),t+(cid:7)i,t −1.3)],w
(cid:2)
ofthedata.Thisisaccomplishedwitha3=n/100,where ×N Di,1.3,t −Di,1.3,t−(cid:7)i,t−1|(cid:7)i,t−1μi,t−1,(cid:7)i,t−1σ2)
nisthenumberoftrees(Table1andTable2).Yeareffects
βthavepriormeanzeroandvt=1. ×N(Di,1.3,t+(cid:7)i,t −Di,1.3,t|(cid:7)i,tμi,t,(cid:7)i,tσ2).
Growthwithchangingheight 5
Table2.Priorparameters(calculatedusingthetextandTable1)andtheposteriormeans,Bayesianstandarderrors,andcredible
intervalsforthemeangrowthincrement,μ(cm)√,thetaperingparameter,α,theprocesserror,theindividualstandarddeviation
τ,andtheobservationerrorstandarddeviation w.
Posterior Bayesian Credible Credible
Priorparametervalues mean SE interval:2.5% interval:97.5%
Minquartiaguianensis
b0 b=e−0.22,vb=500 0.1722 0.000088 0.1476 0.1965
α α0=−0.01,vα=100 −0.0194 0.000001 −0.0195 −0.0192
σ a1=20124,a2=10061 0.6594 0.000015 0.6552 0.6637
τ√ a3=5.59,a4=4.59 0.2080 0.000057 0.1929 0.2245
w a5=1394600,a6=13945 0.1007 0.000000 0.1006 0.1008
Lecythisampla
b0 b=e−0.22,vb=500 0.2808 0.000151 0.2391 0.3227
α α0=−0.01,vα=100 −0.0247 0.000001 −0.0250 −0.0244
σ a1=20124,a2=10061 0.6712 0.000019 0.6660 0.6764
τ√ a3=5.59,a4=4.59 0.3131 0.000123 0.2795 0.3480
w a5=13956,a6=13.9 0.1027 0.000001 0.1026 0.1028
Cecropiaobtusifolia
b0 b=e−0.22,vb=500 0.4895 0.000257 0.4161 0.5602
α α0=−0.01,vα=100 −0.0149 0.000011 −0.0178 −0.0122
σ a1=20124,a2=10061 0.6881 0.000032 0.6794 0.6971
τ√ a3=5.59,a4=4.59 0.2992 0.000187 0.2505 0.3537
w a5=13956,a6=13.9 0.1000 0.000001 0.0998 0.1002
Simaroubaamara
b0 b=e−0.22,vb=500 0.4761 0.000189 0.4227 0.5271
α α0=−0.01,vα=100 −0.0272 0.000013 −0.0289 −0.0220
σ a1=20124,a2=10061 0.6743 0.000022 0.6682 0.6804
τ√ a3=5.59,a4=4.59 0.4230 0.000137 0.3869 0.4623
w a5=13956,a6=13.9 0.1000 0.000001 0.0999 0.1002
Lueheaseemannii
b0 b=−1,vb=500 0.3070 0.000243 0.2386 0.3743
α α0=−0.5,vα=100 −0.1190 0.000005 −0.1203 −0.1178
σ a1=2520,a2=1259.5 0.7871 0.000056 0.7718 0.8027
τ√ a3=2.1,a4=1.1 0.5529 0.000242 0.4906 0.6241
w a5=175400,a6=175399 1.0253 0.000009 1.0229 1.0277
We can model μ essentially as above; see Appendix2 with measurements of diameter taken every year for a
i,t
for further details. As above, there are truncated totalof3382individualtreesfrom10speciesfrom1983
normalpriorsonsampledincrements,toensurepositive to2006,includingMinquartiaguianensisAubl.,Lecythis
incrementsforallyears.Unlikethemodeldescribedabove, ampla Miers, Cecropia obtusifolia Bertol. and Simarouba
wemodelonlyobserveddiameters,andnotunobserved amaraAubl.whichweusetoillustratethemethodhere.
diameters for years where data is not taken. Below we Diameterwasmeasuredusingcallipersortapemeasures.
illustratethesetwoprocedureswithtwodata-sets. In some years, when on some individuals the upwards-
growth of buttresses or other irregularities was judged
to be approaching the current point of measurement
Data (POM), double measurements were made – one at the
current POM and a second at a new POM established
Forthefirstmethod,weusethelarge,long-termdataset at least 50 cm above the irregularity (Clark & Clark
gathered on tree growth and survival in an old-growth 2006).Inyearswherethepointofmeasurementchanged,
tropical wet forest in La Selva, Costa Rica, provided in diameters at both the new and old height are not
Clark&Clark(2006).Thesedatacontainmeasurements directly provided (Clark & Clark 2006). However, the
of tropical forest tree species that exhibit distinct life- value at the lower height at the later time is indirectly
historystrategies.Complexandvariablediametergrowth available through the variable DGRO, which indicates
patterns (Clark & Clark 1999) as well as strong year theincrementgrowthforeachyearfromthelastyearto
effects related to climatic conditions (Clark et al. 2003) thelowerpointofmeasurement.Weusedthistoobtain
haveallbeenreported.Thedataareunusuallydetailed, twoobservedheightsanddiametersforeveryindividual
6 C.JESSICAE.METCALFETAL.
ineveryyearwherethepointofmeasurementchanged. is considerably smaller than the non-informative prior
For the remaining years, a single diameter and height (Table 2, α posterior means); but Simarouba amara has
were available. We set the maximum yearly diameter thegreatesttaperofthefour,arounddoublethatofthe
incrementto4.5cmy−1andtheminimumto0.002cm others, in accordance with species descriptions. Across
y−1. species,Lueheaseemanniishowsthelargestlevelsoftaper;
For the second method, geared to inferring growth individualsmeasured7metersoffthegroundwouldsee
in the presence of long uneven census intervals, we theirdiameterreducedto∼54%ofthediameterat1.3m
used the diameter data available from the Center for (Table2).
Tropical Science plot on Barro Colorado Island, in the Errorparameterestimatesareinfluencedbyinformat-
PanamaCanal,initiatedin1981/1982andre-censused ive priors. The informative priors ensure that posterior
in 1985, 1990, 1995, 2000 and 2005. In the 1982 estimates of σ2 are cantered on 0.5. Large weights on
and1985censuses,diameterswereroundedtothelower w ensure that it dominates for individuals and years
5 mm for trees with diameters of less than 5 cm. From whereobservationsareavailable(Table2).Forallspecies,
1985, diameter was measured at a point chosen to be the posterior of individual variance is distinct from
representative of the diameter of the trunk, i.e. above the deliberately non-informative prior. In the La Selva
buttresses and roots. If buttresses later encroached on database, individual variation is highest for Simarouba
thisheight,workersincreasedtheheightofmeasurement, amara(Table2,posteriormeanforτinallLaSelvaspecies),
recordingboththeoldandnewheightsanddiameters.In a species with potentially varied life trajectories as it is
the last two or three censuses, a height well above the describedasbothlight-responsiveandshadetolerant.The
maximumrangeofbuttresseswassetasthenewheight. twoshade-tolerantspeciesLecythisamplaandMinquartia
Forthisanalysis,wefocusedonasinglespecies,Luehea guianensis show similar degrees of variation across
seemannii,atallspecieswithalargetrunkandirregular individuals (Table 2, posterior mean for τ for these two
buttresses, found abundantly in secondary growth or species).
gaps. Modelling this species is also of interest, as it has The model predicts the data well (Figure 2, 3), and
a very irregular bole surface and is likely to represent diagnostics indicate good agreement between predicted
an extreme case of trunk deformation across tropical andobserveddistributions.Inthesimplemodelpresented
tree data-sets. We set the maximum yearly diameter here, higher growth of trees with large diameters
incrementto4.5cmy−1andtheminimumto0.002cm is captured through process error, i.e. for the same
y−1, the latter being an arbitrarily low value consistent tree, imputed values of X can be larger for larger
i,t
withthepossibilitythatgrowthmaynotoccurinsome D , even though the underlying linear predictor
i,1.3,t
years. These values could be altered for application to μ is the same. For Minquartia, the mean value
i,t
species with different characteristics. Efforts were made of X inferred for time steps where the height had
i,t
to measure individuals during the first 10 mo of each changed was 0.146, vs. 0.144 where the height had
year; however, some individuals were measured very not changed, indicating no systematic bias in the
earlyin1991and1996.Thesetreesgenerallycorrespond method.
to individuals whose measurements are identified as Following Eqn 1, the height at which measurement
potentiallyerroneousduringthecourseofdata-screening, occurs can substantially affect the diameter increment
and are subsequently re-measured. Variation in timing recorded. For example, in Luehea seemannii individual 2
of measurement over the course of the year could have from Figure 3 was measured initially at 1.3 m, but in
influencedestimatesofindividualandperiodeffects.We 1990, the height of measurement was moved to 4 m.
thereforeusedthetimebetweenmeasurementsindaysto Theobserveddiameterincrementsatthismeasurement
define(cid:7)i,t,obtainingfractionalyearlyintervals(e.g.we heightwere5.4,6.4and6.9cm(in1990–1995,1995–
couldobtain(cid:7)i,t =4.55). 2000and2000–2005respectively).Usingtheposterior
forα toconvertthediameterataheightof4mintothe
diameterataheightof1.3m,wecancalculatethatthe
increments observed at 1.3 m would have been of 7.2,
RESULTS 8.6,9.2cmrespectively,i.e.considerablylargerineach
period.
In the analysis presented here, all parameters are well There is substantial shared variation represented by
identified as indicated by narrow credible intervals year effects (Figure 4); which shows similar patterns
(Table 2). The credible intervals for mean growth β across the species within the La Selva database. Year
0
of shade-tolerant species Lecythis ampla and Minquartia effects for the La Selva data-base also show a temporal
guianensis are lower than the credible intervals of the trend compatible with previous results from this area.
threeotherlight-demandingspeciesweanalysed.Forall FortheBCIdatabasethetemporaltrendisalsoobserved
LaSelvaspecies,theposteriorofthetaperingparameter (Figure 5). For both data-sets there is considerable
Growthwithchangingheight 7
Figure2.DiametertrajectoriesofthreeMinquartiaguianensisindividuals Figure3. AsinFigure1forLueheaseemannii,forthemodelincluding
over the course of the census. Solid black points are diameter varyingcensusintervals,sinceobservationsaretakenapproximately
measurements.Solidlinesarethepredictedposteriormeantrajectory only every 5 y. The degree of taper for this species is considerable.
of diameter for each individual at a height of 1.3 m; dashed lines Diametermeasurementstakenat∼7mwillbeonly∼50%ofthose
arethecorresponding95%predictiveintervals.Thetrianglesarethe takenat1.3m.Notethatforthefirstindividualshown,thesecond
predicteddiametersformeasurementstakenatthesameheightasthe observeddiameterimpliesshrinkageofmorethan30cm.Evenallowing
data (i.e. the black points). The associated vertical dashed lines are forachangeinheightofmeasurementof∼1m,thisisunlikely,andthe
the corresponding predictive intervals, including uncertainty in observeddiameteristhereforeoutsidethepredictedconfidenceintervals,
estimation of the tapering parameter, α (which is slight for these andwillbeencompassedthroughmeasurementerror(notshown).
examples). The first individual was measured at a height of 1.3m
throughout the course of the census; the second individual was
measuredataheightgreaterthan1.3mthroughoutthecourseofthe
analysed here. For example, they have been used to
census.Forthethirdindividual,measurementheightchangedoverthe
infertheeffectofglobalwarmingonforests(Clarketal.
courseofthecensus.Overall,theobservedvaluesareaccommodatedby
thepredictedmeanvaluesatthatheight,andtheirpredictiveintervals. 2003), to track directional changes in growth (Feeley
et al. 2007), to test the relationship between diversity
overlapbetweencredibleintervalsofyeareffectposterior and degree of demographic variance within and across
distributions. species(Conditetal.2006),totestfortrade-offsbetween
growth and survival (Gilbert et al. 2006), and even
to test for empirical evidence for the metabolic theory
DISCUSSION (Muller-Landauetal.2006).However,changingheights
ofmeasurementisaubiquitousfeatureofthesetropical
Inrecentyears,muchofthekeyinferenceontreegrowth tree datasets. Analyses to date where diameters from
in tropical forests has been based on the two datasets several points of measurement are not available (Clark
8 C.JESSICAE.METCALFETAL.
Figure4. Theposteriordensitiesforyeareffectsinthegrowthmodel(onesolidcurveforeachyear),withposteriormeans(dots,connectedby
solidlines)forfourspeciesfromtheLaSelvadatabase;theareaboundedbyeachcurveandtheverticaldashedlineforeachyearintegratesto
one.Minquartiaguianensis(a)andLecythisampla(b)showasimilartemporaltrendtothoseestimatedbyClarketal.(2003).Lateryears,which
correspondtohigherminimumnight-timetemperatures,havemorenegativeeffectsongrowth.However,ourresultsdifferfromClarketal.(2003)
inthemagnitudeofyeareffects,andthedetailsofpatterns.Forexample,thelastyearshowsanincreaseinyeareffectsforMinquartiaguianensis,
andforLecythisamplatheincrementfrom1995to1996islargerthanthatfrom1994to1995whereasinClarketal.(2003)theoppositepattern
isrecorded.ForLecythisampla,thetrendthroughtimeisparticularlyweakwhencredibleintervalsaretakenintoaccount,inagreementwithClark
etal.(2003),whofoundthenegativetrendtobenon-significantforthisspecies.Cecropiaobtusifolia(c)andSimaroubaamara(d)werenotconsidered
inpreviousanalyses.
et al. 2003) have either discarded such measurements ourmethod,whichprovidesastrongbasisforinferenceon
(Condit et al. 2006), or set them to a chosen value biomasschange.Furthermore,theregressionatthecore
such as the mean value for trees of the appropriate size of the model is flexible to a wide range of assumptions,
whose measurement point did not change (Feeley et al. and can be extended to include, for example, diameter,
2007). Errors are not limited to the years in which the age,orothercovariatessuchascrownclassobservations
measurementheightchanges.Treesmeasuredatgreater (Clark&Clark2006)orminimumnocturnaltemperature
heights can appear to be growing more slowly, simply (Clarketal.2003).Diversesourcesofinformationbeyond
duetothetaper.Akeywaytostrengthentheinference repeateddiametermeasurements,suchasincrementcore
obtainedfromtheseimportantdataistohaveacoherent datacanalsobebuiltontothemodeltoinforminference
approach that includes all measurements, accounts for inastraightforwardmanner(Clarketal.2007).Explicit
allsourcesofvariationincludingchangesintheheightof modelsofmeasurementerrorthatvarythroughtimeand
measurement,anddoesnotneglectpotentialsourcesof atspecificsizescouldbeimposedtoabsorbtherounding
bias.Ourapproachisfoundedonaconsistentprobability differences that occur during the course of collection of
modelforbothdataandprocess. theBCIdata.(Lessthan7%ofmeasurementsareaffected
IthasbeenshownthatforallLaSelvaspecies,growth intheBCIspeciesusedforillustrationhere,sowedidnot
varies in a complex fashion across diameter (Clark & feelitwaswarrantedinthiscase.)
Clark1999).Here,wedidnotexplicitlymodelaneffectof For large mapped plots where thousands of trees are
diameterongrowth,althoughprocesserrorinthemodel measured at each census, e.g. Condit et al. (2006), it is
shouldcapturesuchpatterns,e.g.ε willincreasewith almostinevitablethatnotalltreescanbemeasuredinall
i,t
tree diameter because the deviation between the mean years.Itisalsolikelythatcensusintervalscanbestaggered
predictedfromthelinearpredictor,μ andtheobserved overyears,asittakesmorethanasingleyeartomeasure
i,t
data will be larger for large trees. However, we have all individuals and verify unlikely measurements (this
estimatesofalllatentdiameterstates,auniqueaspectof occursforexampleintheBCIdatasetin1995).Thiswill
Growthwithchangingheight 9
poor. However, the details of patterns differ between
our analyses and that of Clark et al. (2003) (Figure 4).
Additionally,ourresultsalsoallowexactquantification
of the yearly growth effects. In the best year (1983),
theposteriormedianindicatesthatMinquartiaguianensis
grows∼2mmmorethanaverage,andintheworstyear
(1997),∼2mmlessthanaverage.Bycontrast,Lecythis
ampla,adeciduous,andthereforeperhapsmoreresilient
species, shows little yearly trend, and the magnitude of
yearly deviations for this species is smaller. Our results
also indicate a sharp decline with growth rate in later
yearsforCecropiaobtusifolia(notanalysedbyClarketal.
2003), likely to also be related to increased nocturnal
temperatures. The magnitude of this effect is around
twicethatobservedinMinquartiaguianensis,suggesting
thatthispioneerspeciesexploitsgoodyearsthroughfast
growth, but suffers proportionally more in bad years.
Figure5.PeriodeffectsforLueheaseemannii;thefirstperiodstartsbetween
By contrast, Simarouba amara shows very little yearly
1981and1983;seedescriptionofdatafordetails.Thisresultissimilar
tothatobtainedinFeeleyetal.(2007),withparticularlypoorgrowth trend.Althoughlight-responsiveandwithahighaverage
duringthe1990–1995period,credibleintervalsbeforeandafteroverlap growth rate, this species is described as shade-tolerant,
substantially.Thisresultmaybepartlyanartefactlinkedtohaving so its life-history may lead to more averaging of the
thesameBCIcensussupervisorsfrom1995on,probablyleadingtoa
environment, similar to Minquartia guianensis. Species
moreconstant(ifnotrigorous)methodologysincethen(R.Pe´rezpers.
differences in magnitude of yearly effects coupled with
comm.).
data on their relative abundances will provide another
keytounderstandingthepositivefeedbackoftreegrowth
meanthatgrowthintervalsfordifferentindividualswill declines to on-going atmospheric CO accumulation
2
beofdifferentlengths.Growthisastochasticprocess,and (Clarketal.2003).
varianceingrowthwillaccumulatethroughtime,sothat For the BCI data-set, in agreement with Feeley
diameter measurements separated by longer intervals et al. (2007), year effects isolated by our analysis show
will be more unalike than they would be if growth a negative trend, through a sharp decline in growth
weredeterministic.Consequently,estimatinggrowthrate between 1990 and 1995, followed by slow growth
by simply taking the average growth increment and thereafter;themagnitudeofyearlyvariationissimilarto
dividing it by the time interval could be misleading thatfoundforCecropiaobtusifolia.However,usingsimple
(Clark & Clark 1999). If census intervals are not too ANCOVA approaches, this distinct feature of continued
long,ourapproachallowsstraightforwardinferenceon slowgrowthafterthe1990–1995periodisonlyclearly
missing diameters (Clark et al. 2007). Where census capturedifdiameterincrementsassociatedwithchange
intervals are long, the second method we introduce intheheightofmeasurementareincluded(Figure6).The
allows inference on observed diameters based on a incrementmeasurementsmadeunavailablebyachange
processmodelthatincludesmeasurementerror,process inheightatmeasurementmaybekeytorevealingyears
error, and period effects based around yearly growth of particularly poor growth, particularly where census
increments. The yearly time step facilitates comparison intervalsarelong.Ourmethodprovidesawayofexplicitly
acrossdata-setswithdifferentcensusperiods;andexplicit modelling these increments and long census intervals.
modelling of increasing variance with increasing time Incorporatingalldataintoasingleframeworkincreases
increments prevents this aspect from biasing resulting our power to detect whether patterns are global, and
inference. whetherthemagnitudeofgrowthvariesbetweenyears
In both models, all components are estimated in a orperiods.Giventhekeyrolethesetropicaltreedata-sets
single framework, so credible intervals on year and havehadinarangeofaspectsofecologicalinference,the
period parameters represent the integrated knowledge benefitsofraisinganalysistothenextlevelislikelytobe
obtained from all other components of the model. In high.
terms of inference about broad-scale climatic patterns, To conclude, if buttresses or other features of trees
our analysis supports the overall results presented in lead to a requirement that the height of measurement
Clarketal.(2003).YearlyincrementsforLecythisampla be changed during the course of the study, either
and Minquartia guianensis obtained here are similar to across individuals or within individuals through time,
those presented in Figure 1 in Clark et al. (2003). In estimation of growth effects will be biased if changing
bothanalyses,growthinyears1994–1997isparticularly height is not taken into account. Our results indicate
10 C.JESSICAE.METCALFETAL.
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