I'~ Northern California Forest Yield Cooperative · INTROOUCTIOH TheooJECtiveof m~111ng tree lTowthIs to pr'OOlcttheaver~ (J'OWth~1Se of an looiviroal tree gIven Its spE£ies,sizeandlevelof
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~ Northern California Forest Yield Cooperative if- Department of Forestry and Resource ManagementI University of California Berkeley Ca 94720
ResearchNote No 14 Janurary 7 1986
The use of pseudo-stochastic effects in a tree growth projection system
by
James R Koehler and Lee C Wensel
Abstract
The stoch8shcmethOOused by the Clifornic Conifer Timber (Artput Simulator
(CACTOS)fOC is present81 This methOO endmOOifyinggrowth prOOicitoos is ~plied toDBH2
total hei~t g-owth in order to capture the inherent variation in grow predicitoosA ~
~intupling praess is used which uses COO1puter-~eted rlnbn numbers tMt C8I1be
replicstEn Hencethis process is csll00psetJ(b-stoch8stic
Thehet~ticlty endskewednatureof the pr811ctlooerrors n srown end
inarptrsta1 into the pseud)-strch8sticmethtxi A cEstription of the implementetiooof this
technique into ~TOS i~ ~n AI~ en cxemple il provicrd to show the effect that
pseOOJ-stcd1aStICSh8Veon longterm IJtWthproJections
Theeuttas se Grmte Assistant~tment of StaUsticsSt8nfcrdUniversitytni
AssociateProfesDDepstmentof f(JestryendResourceM~t Universityof
CaUforniaBri818vrA
CONTENTS
Abstroct
QJ1tents i i
ListofTatles iii
Listof figures iii
INTROOUCTION I
METHODS 2
IMPLEMENTATION 8
APPLICATION 14
Literature Cited 18
Appef(jixA Break rown of sums of squares 19
List ofTables
~ Title 1181 Stlnird error (lt1)andskewness for DBH2f01 heilj1t(9) estimates
IJOWthmcxEIs 6
2 Bret ttJwnof tree weilj1ts for CRYPTOS 8
3 Meansof slow mediumandfast regionsfor heilj1t~th mtx131 10
4 BIS cbwn of tree wei~ts byspeciesforCACTOSreard ~intupli~ II
5 Growthrecordsfor 5 trees(from33) on8one-fifth ocresampleplot 14
6 CJintupled inT~le 5 TowthrEards of5 treesshown 15
7 Basalarea~owth for 30-year simulationswith andwithout pseuOOshy
stochastic effectswith varying thinning alternatives 16
list of figures
amp Title IU 1 ResiWalsfrom DBH2~owth mBE1plated00premct81DBH2~owth 3
for porwEr0S8p1ne
2 Theerror distributioo brBcbwn for PROONOSISs 8DBH2~owth mBEl
3 Error regions for CACTOS~owth models 9
INTROOUCTIOH
TheooJECtiveof m~111ngtreelTowth Is to prOOlcttheaver~ (JOWth~1Se of an
looiviroaltreegIvenIts spEpoundiessizeand level of COO1pet betweenthe fltual it ion Thedlff~
tree ITOWthandthepredictedtree ITOWthis calledtheerror in predictionsandthe VfriMCeof these
errors is C8l1ajthe unexplaine1vari8tion To improve ~th estimates 8 mcxEliertries to
minimize the unexplainOOvariation without intrcxiJcinQ biasintotheestimates
locorptYstingtree growth mcxEls into PNXTBmSCOO1puter that simulate the eff~ts of
alternatives st~ m~ent (Stege 1973 poundtprfltices is OONCOO1mon amp Monserud
1971 KruJend amp Wensel 1980 Wensel end D8uQherty 1985) CACTOSthe
CAlifcrniaCooiferTimberOutputSimulatcr (Wensel amp D8uQberty1985) usesthetree~th
projpoundctionsystemcBscribedby Wensel amp Koehler( 1985) Thepurposeof this paper is to
describethe stcd1asticseffects irxorpor8te1into CACTOSin ortB- to representthe unexplained
varlationin pratictinggrowthoftreediameterat breasthei~t (DBH)~ total hei(j1t(HT)
While the purpose of mcxElllngtree Towthis to predict the aver Towth response of trees
UNErdiversecmditians two trees that are the samesize andUNErthe sameamountof measured
canpetition w11Jnot nees56riJyTowat the samerate (BelJe 1970) This miff becausedby
errcrs in the measurements cr by fEdcrs not amilEred 1nthe mcxEls If these euroEv1atiansfrom the
8Ver8B~th are not irxorporated the simulatcr mavprcxbeSIUIJJlshN$pOOSe intothesimulatcrI
predictions of ITQWthresponses after thinnirYJSand harvest Adjitionally I the dispersion into
diameterclasseswill be retarOdif stochasticffiOOifiers rot to the ~th ~tionsn ~
(Krulend1982) Hence usinge stochasticrepresentationof the errors in predictions is
1mporttJ1t1ncreErto~ely pred1ctch8I~ 1nUmberSUIIdsl
11Notethatthisis rot ~y mean asIb1ein inventoryu~tesfor estimating effects
ResearchNoteNo14 page 1
METHODS
In CACTOS cycle ere 8S the four tree variablesthat needto be pr~jjctoofor etdI ~h
fonows
D6H diameterat breast hel~t
HT total tree height
CR live crownrlfho
TPA numberoftrees per ocrefIXIKhtree rocord
StochasticvarIationis notajjed tothecrownratio estimates Howevercrownratio is a functionof
total hell11tn st800(imlty and therefore crownrecessionw11llrxU~tly havestochasticeffects
throulj) thestochast FIXthese reesoos In1 the 15kof oota for both theic effectson heiljt I]OWth
mortelity end crown recession models stochastic effects will be romputed for only the DBHend HT
(1OWthmaEls
Since 51 independent dlna ~ cDes not exist for U5e in determining the unexplained
variations the stem analysis database that W8Sused to ~Iop the ~th ~ions will also be usa1
A OOscriptionof this ootaset is given by Wensel amp Koehler (1985) TheeilJ) of the stem
anatysts samples InclucEs the followtng elements ftrst clusters of 2 or 3 plotswereselectedand
S8OOd5to 10treeswereselectoofor felting onEBtaplot Thisallowsfor thepotentialestimationof
the voriation ~mg clusters plots ondtrees os well osdiffere11CeSbetweentime periOOs
Two~ions thatneedtobeansweredare ( I) what is thenature of the deviationsfromthe
tNfrrq (1OWthI espouseend(2) how should thesemvietionsbe irorptreted into CACTOSThe
resiliJalsfromthediameter-squared8IK1totalheilj1tJQWthm(XElsare het~tic withresptlt
to predicted IJOWthThet is the error variance is proportional to the square of predicted IJOWth
Thtshet~ttclty QI1beseen bylooktngatthereslOO8I scatterplotshownfor pordros8pineIn
Figure 1 (Wensel end [oehler 1985) Hencethe followingm(XE1is appropriate for
expressingthe~th functionsforbothDBH2800totalheilj1t
Y a = f( X a) X ( 1 + E a ) [1]IJ IJ IJ
where
Yijk = lItuelITowthoftree j onplot i f(l time periook(ejther DBH2(I total ht)
f(Xijk) = pr8ftcted~owth f(l tree j onplot1f(l t1meperjoot
ReseerchNoteNo 14 page 2
-------------------- ----------
Xij( = ~tor ofpredictorvar16bJes
E1jk-renOOmelement
Thereforethe1~ thepredicte1fTOWththeI~ theabsoluteeffEpoundtofther~ elementE
shy50
i ~ i ~-
A
A
J
A A A -A A A0 - MN A A A A A c A A _A A A A CAt AA _AoltA Ate A XOCErIgtACAM- A A A A
C I ( lIfn t OpoundIpoundgtTCte- 0 CA shy oltAMIlDCCteteA- A AA - A A A AA shy A A AA shy A A
A A
A A A
A A A A
A A
I
-0shy
A A
--
A
A A
A
A
A A
-~ A A A A
A A A
J -)0 A A
A
A A
-00 A
A
-5) C 3 5 8 2 27 30 n 3A n 42 45 5 54 ~ AJ ~ 8 9 4
PnddId dII9 inDItf2(spnd trD8I)
Figure 1 ResicUi)s from DBH2~owth mcrl11p1otte100 predictej DBH2~owth fCJ
poo03r0S8 pine
In someC8SeSthe predictive mcrl11 feXijk) canbestownto systematicallyoverCJu~
estimate6CtuellTOWth particularplotstreestimeperlltX2setc TheuocertalntyInpredIctions rCJ
m~ be raducedif a systematic trendcanbeestimatedseparately(run the IJOWthrrnxmlConsider
thefollowingpossiblecomponents Eofther51OOrnelement
fJlt= oq + Aj + ~ + iJlt + ftJlt [2J
where 0 cluster aOOplot effEpoundt fCJplot i
ItIj = treeeff~t f(Wtreej 00ploti
R83e8rd1NoteNo 14 pege 3
--- -- --- ------
ik l8iOO efftpoundtf(T periOOk
6iJlt = measurementeffoct fortreej 00plot i for periooK
fJlt = unexplainederror fortree j 00plot i forperiooK
The tree effocts()maybedueto ytriatioos in the r1SPOl1Seof tree ITOWthIiJe to ~tic
endhistorical ffpoundtors oot usedby theITOWthmtXElsTheplot endcluster effocts ( od maybeIiJeto
estimetiooernrs in predicting site irdJx endcrownaxnpetitioo aswell asPnf i~~ies of site
irm to renet ~tU81 pnxiJctiYity Perioo effocts( )) maybe IiJe to ytriatioo in the ITOWth
environmentfremyeer to yeer ( amountend distributioo of reinfell tempereture) endother
enviroomentalffdors oot incllXEdin the premctioo effects( ~) maybe processThemeasurement
we toerrors1n f1eld measurementsor 1nsu~t ~ process1ngTheunexplainederror ( pound ) 1s
the erreuror left over after al1 of the other effects hErYebeen taken into fECOUnl In caseswhere plot
cluster tree periOOor measurementeffectsCtrI beestimatedmoreIDUrateestimatesof the mEm
Towth can be ootatined thus reOOcing the amount of unexplaine1 variatioo
Assum1rYJ1~ between theaxnpooentsthesumsof S(JJ8resof [ (~ioo 2) CtrIbe brokencbwnasfollows
55f = SSplot+ SStree+ SSperiOO + SSmees + SSunexpleined [3]
where thesumsof squares ere definedas follows
SSplot plot (endcluster) effocts
SStree tree effocts
SSperiooperiooeffects
SSmees meesurementernrs
SSunexp lained unexplained sumsof~
A cle51breakliJwnof the elementE into its axnpooentpsts wooldl8JIire e -(I ossecf
2silJ) for theSMtple dBte Tf18t is multiple measurements00multipletreesper plot 00 multiple
plots8t multiplepointsin time Withsuchd8tecmventionelanalysisofvsilRe tEpoundhni(JJeSart be
Rese8IChNoteNo 4 pege
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
CONTENTS
Abstroct
QJ1tents i i
ListofTatles iii
Listof figures iii
INTROOUCTION I
METHODS 2
IMPLEMENTATION 8
APPLICATION 14
Literature Cited 18
Appef(jixA Break rown of sums of squares 19
List ofTables
~ Title 1181 Stlnird error (lt1)andskewness for DBH2f01 heilj1t(9) estimates
IJOWthmcxEIs 6
2 Bret ttJwnof tree weilj1ts for CRYPTOS 8
3 Meansof slow mediumandfast regionsfor heilj1t~th mtx131 10
4 BIS cbwn of tree wei~ts byspeciesforCACTOSreard ~intupli~ II
5 Growthrecordsfor 5 trees(from33) on8one-fifth ocresampleplot 14
6 CJintupled inT~le 5 TowthrEards of5 treesshown 15
7 Basalarea~owth for 30-year simulationswith andwithout pseuOOshy
stochastic effectswith varying thinning alternatives 16
list of figures
amp Title IU 1 ResiWalsfrom DBH2~owth mBE1plated00premct81DBH2~owth 3
for porwEr0S8p1ne
2 Theerror distributioo brBcbwn for PROONOSISs 8DBH2~owth mBEl
3 Error regions for CACTOS~owth models 9
INTROOUCTIOH
TheooJECtiveof m~111ngtreelTowth Is to prOOlcttheaver~ (JOWth~1Se of an
looiviroaltreegIvenIts spEpoundiessizeand level of COO1pet betweenthe fltual it ion Thedlff~
tree ITOWthandthepredictedtree ITOWthis calledtheerror in predictionsandthe VfriMCeof these
errors is C8l1ajthe unexplaine1vari8tion To improve ~th estimates 8 mcxEliertries to
minimize the unexplainOOvariation without intrcxiJcinQ biasintotheestimates
locorptYstingtree growth mcxEls into PNXTBmSCOO1puter that simulate the eff~ts of
alternatives st~ m~ent (Stege 1973 poundtprfltices is OONCOO1mon amp Monserud
1971 KruJend amp Wensel 1980 Wensel end D8uQherty 1985) CACTOSthe
CAlifcrniaCooiferTimberOutputSimulatcr (Wensel amp D8uQberty1985) usesthetree~th
projpoundctionsystemcBscribedby Wensel amp Koehler( 1985) Thepurposeof this paper is to
describethe stcd1asticseffects irxorpor8te1into CACTOSin ortB- to representthe unexplained
varlationin pratictinggrowthoftreediameterat breasthei~t (DBH)~ total hei(j1t(HT)
While the purpose of mcxElllngtree Towthis to predict the aver Towth response of trees
UNErdiversecmditians two trees that are the samesize andUNErthe sameamountof measured
canpetition w11Jnot nees56riJyTowat the samerate (BelJe 1970) This miff becausedby
errcrs in the measurements cr by fEdcrs not amilEred 1nthe mcxEls If these euroEv1atiansfrom the
8Ver8B~th are not irxorporated the simulatcr mavprcxbeSIUIJJlshN$pOOSe intothesimulatcrI
predictions of ITQWthresponses after thinnirYJSand harvest Adjitionally I the dispersion into
diameterclasseswill be retarOdif stochasticffiOOifiers rot to the ~th ~tionsn ~
(Krulend1982) Hence usinge stochasticrepresentationof the errors in predictions is
1mporttJ1t1ncreErto~ely pred1ctch8I~ 1nUmberSUIIdsl
11Notethatthisis rot ~y mean asIb1ein inventoryu~tesfor estimating effects
ResearchNoteNo14 page 1
METHODS
In CACTOS cycle ere 8S the four tree variablesthat needto be pr~jjctoofor etdI ~h
fonows
D6H diameterat breast hel~t
HT total tree height
CR live crownrlfho
TPA numberoftrees per ocrefIXIKhtree rocord
StochasticvarIationis notajjed tothecrownratio estimates Howevercrownratio is a functionof
total hell11tn st800(imlty and therefore crownrecessionw11llrxU~tly havestochasticeffects
throulj) thestochast FIXthese reesoos In1 the 15kof oota for both theic effectson heiljt I]OWth
mortelity end crown recession models stochastic effects will be romputed for only the DBHend HT
(1OWthmaEls
Since 51 independent dlna ~ cDes not exist for U5e in determining the unexplained
variations the stem analysis database that W8Sused to ~Iop the ~th ~ions will also be usa1
A OOscriptionof this ootaset is given by Wensel amp Koehler (1985) TheeilJ) of the stem
anatysts samples InclucEs the followtng elements ftrst clusters of 2 or 3 plotswereselectedand
S8OOd5to 10treeswereselectoofor felting onEBtaplot Thisallowsfor thepotentialestimationof
the voriation ~mg clusters plots ondtrees os well osdiffere11CeSbetweentime periOOs
Two~ions thatneedtobeansweredare ( I) what is thenature of the deviationsfromthe
tNfrrq (1OWthI espouseend(2) how should thesemvietionsbe irorptreted into CACTOSThe
resiliJalsfromthediameter-squared8IK1totalheilj1tJQWthm(XElsare het~tic withresptlt
to predicted IJOWthThet is the error variance is proportional to the square of predicted IJOWth
Thtshet~ttclty QI1beseen bylooktngatthereslOO8I scatterplotshownfor pordros8pineIn
Figure 1 (Wensel end [oehler 1985) Hencethe followingm(XE1is appropriate for
expressingthe~th functionsforbothDBH2800totalheilj1t
Y a = f( X a) X ( 1 + E a ) [1]IJ IJ IJ
where
Yijk = lItuelITowthoftree j onplot i f(l time periook(ejther DBH2(I total ht)
f(Xijk) = pr8ftcted~owth f(l tree j onplot1f(l t1meperjoot
ReseerchNoteNo 14 page 2
-------------------- ----------
Xij( = ~tor ofpredictorvar16bJes
E1jk-renOOmelement
Thereforethe1~ thepredicte1fTOWththeI~ theabsoluteeffEpoundtofther~ elementE
shy50
i ~ i ~-
A
A
J
A A A -A A A0 - MN A A A A A c A A _A A A A CAt AA _AoltA Ate A XOCErIgtACAM- A A A A
C I ( lIfn t OpoundIpoundgtTCte- 0 CA shy oltAMIlDCCteteA- A AA - A A A AA shy A A AA shy A A
A A
A A A
A A A A
A A
I
-0shy
A A
--
A
A A
A
A
A A
-~ A A A A
A A A
J -)0 A A
A
A A
-00 A
A
-5) C 3 5 8 2 27 30 n 3A n 42 45 5 54 ~ AJ ~ 8 9 4
PnddId dII9 inDItf2(spnd trD8I)
Figure 1 ResicUi)s from DBH2~owth mcrl11p1otte100 predictej DBH2~owth fCJ
poo03r0S8 pine
In someC8SeSthe predictive mcrl11 feXijk) canbestownto systematicallyoverCJu~
estimate6CtuellTOWth particularplotstreestimeperlltX2setc TheuocertalntyInpredIctions rCJ
m~ be raducedif a systematic trendcanbeestimatedseparately(run the IJOWthrrnxmlConsider
thefollowingpossiblecomponents Eofther51OOrnelement
fJlt= oq + Aj + ~ + iJlt + ftJlt [2J
where 0 cluster aOOplot effEpoundt fCJplot i
ItIj = treeeff~t f(Wtreej 00ploti
R83e8rd1NoteNo 14 pege 3
--- -- --- ------
ik l8iOO efftpoundtf(T periOOk
6iJlt = measurementeffoct fortreej 00plot i for periooK
fJlt = unexplainederror fortree j 00plot i forperiooK
The tree effocts()maybedueto ytriatioos in the r1SPOl1Seof tree ITOWthIiJe to ~tic
endhistorical ffpoundtors oot usedby theITOWthmtXElsTheplot endcluster effocts ( od maybeIiJeto
estimetiooernrs in predicting site irdJx endcrownaxnpetitioo aswell asPnf i~~ies of site
irm to renet ~tU81 pnxiJctiYity Perioo effocts( )) maybe IiJe to ytriatioo in the ITOWth
environmentfremyeer to yeer ( amountend distributioo of reinfell tempereture) endother
enviroomentalffdors oot incllXEdin the premctioo effects( ~) maybe processThemeasurement
we toerrors1n f1eld measurementsor 1nsu~t ~ process1ngTheunexplainederror ( pound ) 1s
the erreuror left over after al1 of the other effects hErYebeen taken into fECOUnl In caseswhere plot
cluster tree periOOor measurementeffectsCtrI beestimatedmoreIDUrateestimatesof the mEm
Towth can be ootatined thus reOOcing the amount of unexplaine1 variatioo
Assum1rYJ1~ between theaxnpooentsthesumsof S(JJ8resof [ (~ioo 2) CtrIbe brokencbwnasfollows
55f = SSplot+ SStree+ SSperiOO + SSmees + SSunexpleined [3]
where thesumsof squares ere definedas follows
SSplot plot (endcluster) effocts
SStree tree effocts
SSperiooperiooeffects
SSmees meesurementernrs
SSunexp lained unexplained sumsof~
A cle51breakliJwnof the elementE into its axnpooentpsts wooldl8JIire e -(I ossecf
2silJ) for theSMtple dBte Tf18t is multiple measurements00multipletreesper plot 00 multiple
plots8t multiplepointsin time Withsuchd8tecmventionelanalysisofvsilRe tEpoundhni(JJeSart be
Rese8IChNoteNo 4 pege
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
List ofTables
~ Title 1181 Stlnird error (lt1)andskewness for DBH2f01 heilj1t(9) estimates
IJOWthmcxEIs 6
2 Bret ttJwnof tree weilj1ts for CRYPTOS 8
3 Meansof slow mediumandfast regionsfor heilj1t~th mtx131 10
4 BIS cbwn of tree wei~ts byspeciesforCACTOSreard ~intupli~ II
5 Growthrecordsfor 5 trees(from33) on8one-fifth ocresampleplot 14
6 CJintupled inT~le 5 TowthrEards of5 treesshown 15
7 Basalarea~owth for 30-year simulationswith andwithout pseuOOshy
stochastic effectswith varying thinning alternatives 16
list of figures
amp Title IU 1 ResiWalsfrom DBH2~owth mBE1plated00premct81DBH2~owth 3
for porwEr0S8p1ne
2 Theerror distributioo brBcbwn for PROONOSISs 8DBH2~owth mBEl
3 Error regions for CACTOS~owth models 9
INTROOUCTIOH
TheooJECtiveof m~111ngtreelTowth Is to prOOlcttheaver~ (JOWth~1Se of an
looiviroaltreegIvenIts spEpoundiessizeand level of COO1pet betweenthe fltual it ion Thedlff~
tree ITOWthandthepredictedtree ITOWthis calledtheerror in predictionsandthe VfriMCeof these
errors is C8l1ajthe unexplaine1vari8tion To improve ~th estimates 8 mcxEliertries to
minimize the unexplainOOvariation without intrcxiJcinQ biasintotheestimates
locorptYstingtree growth mcxEls into PNXTBmSCOO1puter that simulate the eff~ts of
alternatives st~ m~ent (Stege 1973 poundtprfltices is OONCOO1mon amp Monserud
1971 KruJend amp Wensel 1980 Wensel end D8uQherty 1985) CACTOSthe
CAlifcrniaCooiferTimberOutputSimulatcr (Wensel amp D8uQberty1985) usesthetree~th
projpoundctionsystemcBscribedby Wensel amp Koehler( 1985) Thepurposeof this paper is to
describethe stcd1asticseffects irxorpor8te1into CACTOSin ortB- to representthe unexplained
varlationin pratictinggrowthoftreediameterat breasthei~t (DBH)~ total hei(j1t(HT)
While the purpose of mcxElllngtree Towthis to predict the aver Towth response of trees
UNErdiversecmditians two trees that are the samesize andUNErthe sameamountof measured
canpetition w11Jnot nees56riJyTowat the samerate (BelJe 1970) This miff becausedby
errcrs in the measurements cr by fEdcrs not amilEred 1nthe mcxEls If these euroEv1atiansfrom the
8Ver8B~th are not irxorporated the simulatcr mavprcxbeSIUIJJlshN$pOOSe intothesimulatcrI
predictions of ITQWthresponses after thinnirYJSand harvest Adjitionally I the dispersion into
diameterclasseswill be retarOdif stochasticffiOOifiers rot to the ~th ~tionsn ~
(Krulend1982) Hence usinge stochasticrepresentationof the errors in predictions is
1mporttJ1t1ncreErto~ely pred1ctch8I~ 1nUmberSUIIdsl
11Notethatthisis rot ~y mean asIb1ein inventoryu~tesfor estimating effects
ResearchNoteNo14 page 1
METHODS
In CACTOS cycle ere 8S the four tree variablesthat needto be pr~jjctoofor etdI ~h
fonows
D6H diameterat breast hel~t
HT total tree height
CR live crownrlfho
TPA numberoftrees per ocrefIXIKhtree rocord
StochasticvarIationis notajjed tothecrownratio estimates Howevercrownratio is a functionof
total hell11tn st800(imlty and therefore crownrecessionw11llrxU~tly havestochasticeffects
throulj) thestochast FIXthese reesoos In1 the 15kof oota for both theic effectson heiljt I]OWth
mortelity end crown recession models stochastic effects will be romputed for only the DBHend HT
(1OWthmaEls
Since 51 independent dlna ~ cDes not exist for U5e in determining the unexplained
variations the stem analysis database that W8Sused to ~Iop the ~th ~ions will also be usa1
A OOscriptionof this ootaset is given by Wensel amp Koehler (1985) TheeilJ) of the stem
anatysts samples InclucEs the followtng elements ftrst clusters of 2 or 3 plotswereselectedand
S8OOd5to 10treeswereselectoofor felting onEBtaplot Thisallowsfor thepotentialestimationof
the voriation ~mg clusters plots ondtrees os well osdiffere11CeSbetweentime periOOs
Two~ions thatneedtobeansweredare ( I) what is thenature of the deviationsfromthe
tNfrrq (1OWthI espouseend(2) how should thesemvietionsbe irorptreted into CACTOSThe
resiliJalsfromthediameter-squared8IK1totalheilj1tJQWthm(XElsare het~tic withresptlt
to predicted IJOWthThet is the error variance is proportional to the square of predicted IJOWth
Thtshet~ttclty QI1beseen bylooktngatthereslOO8I scatterplotshownfor pordros8pineIn
Figure 1 (Wensel end [oehler 1985) Hencethe followingm(XE1is appropriate for
expressingthe~th functionsforbothDBH2800totalheilj1t
Y a = f( X a) X ( 1 + E a ) [1]IJ IJ IJ
where
Yijk = lItuelITowthoftree j onplot i f(l time periook(ejther DBH2(I total ht)
f(Xijk) = pr8ftcted~owth f(l tree j onplot1f(l t1meperjoot
ReseerchNoteNo 14 page 2
-------------------- ----------
Xij( = ~tor ofpredictorvar16bJes
E1jk-renOOmelement
Thereforethe1~ thepredicte1fTOWththeI~ theabsoluteeffEpoundtofther~ elementE
shy50
i ~ i ~-
A
A
J
A A A -A A A0 - MN A A A A A c A A _A A A A CAt AA _AoltA Ate A XOCErIgtACAM- A A A A
C I ( lIfn t OpoundIpoundgtTCte- 0 CA shy oltAMIlDCCteteA- A AA - A A A AA shy A A AA shy A A
A A
A A A
A A A A
A A
I
-0shy
A A
--
A
A A
A
A
A A
-~ A A A A
A A A
J -)0 A A
A
A A
-00 A
A
-5) C 3 5 8 2 27 30 n 3A n 42 45 5 54 ~ AJ ~ 8 9 4
PnddId dII9 inDItf2(spnd trD8I)
Figure 1 ResicUi)s from DBH2~owth mcrl11p1otte100 predictej DBH2~owth fCJ
poo03r0S8 pine
In someC8SeSthe predictive mcrl11 feXijk) canbestownto systematicallyoverCJu~
estimate6CtuellTOWth particularplotstreestimeperlltX2setc TheuocertalntyInpredIctions rCJ
m~ be raducedif a systematic trendcanbeestimatedseparately(run the IJOWthrrnxmlConsider
thefollowingpossiblecomponents Eofther51OOrnelement
fJlt= oq + Aj + ~ + iJlt + ftJlt [2J
where 0 cluster aOOplot effEpoundt fCJplot i
ItIj = treeeff~t f(Wtreej 00ploti
R83e8rd1NoteNo 14 pege 3
--- -- --- ------
ik l8iOO efftpoundtf(T periOOk
6iJlt = measurementeffoct fortreej 00plot i for periooK
fJlt = unexplainederror fortree j 00plot i forperiooK
The tree effocts()maybedueto ytriatioos in the r1SPOl1Seof tree ITOWthIiJe to ~tic
endhistorical ffpoundtors oot usedby theITOWthmtXElsTheplot endcluster effocts ( od maybeIiJeto
estimetiooernrs in predicting site irdJx endcrownaxnpetitioo aswell asPnf i~~ies of site
irm to renet ~tU81 pnxiJctiYity Perioo effocts( )) maybe IiJe to ytriatioo in the ITOWth
environmentfremyeer to yeer ( amountend distributioo of reinfell tempereture) endother
enviroomentalffdors oot incllXEdin the premctioo effects( ~) maybe processThemeasurement
we toerrors1n f1eld measurementsor 1nsu~t ~ process1ngTheunexplainederror ( pound ) 1s
the erreuror left over after al1 of the other effects hErYebeen taken into fECOUnl In caseswhere plot
cluster tree periOOor measurementeffectsCtrI beestimatedmoreIDUrateestimatesof the mEm
Towth can be ootatined thus reOOcing the amount of unexplaine1 variatioo
Assum1rYJ1~ between theaxnpooentsthesumsof S(JJ8resof [ (~ioo 2) CtrIbe brokencbwnasfollows
55f = SSplot+ SStree+ SSperiOO + SSmees + SSunexpleined [3]
where thesumsof squares ere definedas follows
SSplot plot (endcluster) effocts
SStree tree effocts
SSperiooperiooeffects
SSmees meesurementernrs
SSunexp lained unexplained sumsof~
A cle51breakliJwnof the elementE into its axnpooentpsts wooldl8JIire e -(I ossecf
2silJ) for theSMtple dBte Tf18t is multiple measurements00multipletreesper plot 00 multiple
plots8t multiplepointsin time Withsuchd8tecmventionelanalysisofvsilRe tEpoundhni(JJeSart be
Rese8IChNoteNo 4 pege
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
INTROOUCTIOH
TheooJECtiveof m~111ngtreelTowth Is to prOOlcttheaver~ (JOWth~1Se of an
looiviroaltreegIvenIts spEpoundiessizeand level of COO1pet betweenthe fltual it ion Thedlff~
tree ITOWthandthepredictedtree ITOWthis calledtheerror in predictionsandthe VfriMCeof these
errors is C8l1ajthe unexplaine1vari8tion To improve ~th estimates 8 mcxEliertries to
minimize the unexplainOOvariation without intrcxiJcinQ biasintotheestimates
locorptYstingtree growth mcxEls into PNXTBmSCOO1puter that simulate the eff~ts of
alternatives st~ m~ent (Stege 1973 poundtprfltices is OONCOO1mon amp Monserud
1971 KruJend amp Wensel 1980 Wensel end D8uQherty 1985) CACTOSthe
CAlifcrniaCooiferTimberOutputSimulatcr (Wensel amp D8uQberty1985) usesthetree~th
projpoundctionsystemcBscribedby Wensel amp Koehler( 1985) Thepurposeof this paper is to
describethe stcd1asticseffects irxorpor8te1into CACTOSin ortB- to representthe unexplained
varlationin pratictinggrowthoftreediameterat breasthei~t (DBH)~ total hei(j1t(HT)
While the purpose of mcxElllngtree Towthis to predict the aver Towth response of trees
UNErdiversecmditians two trees that are the samesize andUNErthe sameamountof measured
canpetition w11Jnot nees56riJyTowat the samerate (BelJe 1970) This miff becausedby
errcrs in the measurements cr by fEdcrs not amilEred 1nthe mcxEls If these euroEv1atiansfrom the
8Ver8B~th are not irxorporated the simulatcr mavprcxbeSIUIJJlshN$pOOSe intothesimulatcrI
predictions of ITQWthresponses after thinnirYJSand harvest Adjitionally I the dispersion into
diameterclasseswill be retarOdif stochasticffiOOifiers rot to the ~th ~tionsn ~
(Krulend1982) Hence usinge stochasticrepresentationof the errors in predictions is
1mporttJ1t1ncreErto~ely pred1ctch8I~ 1nUmberSUIIdsl
11Notethatthisis rot ~y mean asIb1ein inventoryu~tesfor estimating effects
ResearchNoteNo14 page 1
METHODS
In CACTOS cycle ere 8S the four tree variablesthat needto be pr~jjctoofor etdI ~h
fonows
D6H diameterat breast hel~t
HT total tree height
CR live crownrlfho
TPA numberoftrees per ocrefIXIKhtree rocord
StochasticvarIationis notajjed tothecrownratio estimates Howevercrownratio is a functionof
total hell11tn st800(imlty and therefore crownrecessionw11llrxU~tly havestochasticeffects
throulj) thestochast FIXthese reesoos In1 the 15kof oota for both theic effectson heiljt I]OWth
mortelity end crown recession models stochastic effects will be romputed for only the DBHend HT
(1OWthmaEls
Since 51 independent dlna ~ cDes not exist for U5e in determining the unexplained
variations the stem analysis database that W8Sused to ~Iop the ~th ~ions will also be usa1
A OOscriptionof this ootaset is given by Wensel amp Koehler (1985) TheeilJ) of the stem
anatysts samples InclucEs the followtng elements ftrst clusters of 2 or 3 plotswereselectedand
S8OOd5to 10treeswereselectoofor felting onEBtaplot Thisallowsfor thepotentialestimationof
the voriation ~mg clusters plots ondtrees os well osdiffere11CeSbetweentime periOOs
Two~ions thatneedtobeansweredare ( I) what is thenature of the deviationsfromthe
tNfrrq (1OWthI espouseend(2) how should thesemvietionsbe irorptreted into CACTOSThe
resiliJalsfromthediameter-squared8IK1totalheilj1tJQWthm(XElsare het~tic withresptlt
to predicted IJOWthThet is the error variance is proportional to the square of predicted IJOWth
Thtshet~ttclty QI1beseen bylooktngatthereslOO8I scatterplotshownfor pordros8pineIn
Figure 1 (Wensel end [oehler 1985) Hencethe followingm(XE1is appropriate for
expressingthe~th functionsforbothDBH2800totalheilj1t
Y a = f( X a) X ( 1 + E a ) [1]IJ IJ IJ
where
Yijk = lItuelITowthoftree j onplot i f(l time periook(ejther DBH2(I total ht)
f(Xijk) = pr8ftcted~owth f(l tree j onplot1f(l t1meperjoot
ReseerchNoteNo 14 page 2
-------------------- ----------
Xij( = ~tor ofpredictorvar16bJes
E1jk-renOOmelement
Thereforethe1~ thepredicte1fTOWththeI~ theabsoluteeffEpoundtofther~ elementE
shy50
i ~ i ~-
A
A
J
A A A -A A A0 - MN A A A A A c A A _A A A A CAt AA _AoltA Ate A XOCErIgtACAM- A A A A
C I ( lIfn t OpoundIpoundgtTCte- 0 CA shy oltAMIlDCCteteA- A AA - A A A AA shy A A AA shy A A
A A
A A A
A A A A
A A
I
-0shy
A A
--
A
A A
A
A
A A
-~ A A A A
A A A
J -)0 A A
A
A A
-00 A
A
-5) C 3 5 8 2 27 30 n 3A n 42 45 5 54 ~ AJ ~ 8 9 4
PnddId dII9 inDItf2(spnd trD8I)
Figure 1 ResicUi)s from DBH2~owth mcrl11p1otte100 predictej DBH2~owth fCJ
poo03r0S8 pine
In someC8SeSthe predictive mcrl11 feXijk) canbestownto systematicallyoverCJu~
estimate6CtuellTOWth particularplotstreestimeperlltX2setc TheuocertalntyInpredIctions rCJ
m~ be raducedif a systematic trendcanbeestimatedseparately(run the IJOWthrrnxmlConsider
thefollowingpossiblecomponents Eofther51OOrnelement
fJlt= oq + Aj + ~ + iJlt + ftJlt [2J
where 0 cluster aOOplot effEpoundt fCJplot i
ItIj = treeeff~t f(Wtreej 00ploti
R83e8rd1NoteNo 14 pege 3
--- -- --- ------
ik l8iOO efftpoundtf(T periOOk
6iJlt = measurementeffoct fortreej 00plot i for periooK
fJlt = unexplainederror fortree j 00plot i forperiooK
The tree effocts()maybedueto ytriatioos in the r1SPOl1Seof tree ITOWthIiJe to ~tic
endhistorical ffpoundtors oot usedby theITOWthmtXElsTheplot endcluster effocts ( od maybeIiJeto
estimetiooernrs in predicting site irdJx endcrownaxnpetitioo aswell asPnf i~~ies of site
irm to renet ~tU81 pnxiJctiYity Perioo effocts( )) maybe IiJe to ytriatioo in the ITOWth
environmentfremyeer to yeer ( amountend distributioo of reinfell tempereture) endother
enviroomentalffdors oot incllXEdin the premctioo effects( ~) maybe processThemeasurement
we toerrors1n f1eld measurementsor 1nsu~t ~ process1ngTheunexplainederror ( pound ) 1s
the erreuror left over after al1 of the other effects hErYebeen taken into fECOUnl In caseswhere plot
cluster tree periOOor measurementeffectsCtrI beestimatedmoreIDUrateestimatesof the mEm
Towth can be ootatined thus reOOcing the amount of unexplaine1 variatioo
Assum1rYJ1~ between theaxnpooentsthesumsof S(JJ8resof [ (~ioo 2) CtrIbe brokencbwnasfollows
55f = SSplot+ SStree+ SSperiOO + SSmees + SSunexpleined [3]
where thesumsof squares ere definedas follows
SSplot plot (endcluster) effocts
SStree tree effocts
SSperiooperiooeffects
SSmees meesurementernrs
SSunexp lained unexplained sumsof~
A cle51breakliJwnof the elementE into its axnpooentpsts wooldl8JIire e -(I ossecf
2silJ) for theSMtple dBte Tf18t is multiple measurements00multipletreesper plot 00 multiple
plots8t multiplepointsin time Withsuchd8tecmventionelanalysisofvsilRe tEpoundhni(JJeSart be
Rese8IChNoteNo 4 pege
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
METHODS
In CACTOS cycle ere 8S the four tree variablesthat needto be pr~jjctoofor etdI ~h
fonows
D6H diameterat breast hel~t
HT total tree height
CR live crownrlfho
TPA numberoftrees per ocrefIXIKhtree rocord
StochasticvarIationis notajjed tothecrownratio estimates Howevercrownratio is a functionof
total hell11tn st800(imlty and therefore crownrecessionw11llrxU~tly havestochasticeffects
throulj) thestochast FIXthese reesoos In1 the 15kof oota for both theic effectson heiljt I]OWth
mortelity end crown recession models stochastic effects will be romputed for only the DBHend HT
(1OWthmaEls
Since 51 independent dlna ~ cDes not exist for U5e in determining the unexplained
variations the stem analysis database that W8Sused to ~Iop the ~th ~ions will also be usa1
A OOscriptionof this ootaset is given by Wensel amp Koehler (1985) TheeilJ) of the stem
anatysts samples InclucEs the followtng elements ftrst clusters of 2 or 3 plotswereselectedand
S8OOd5to 10treeswereselectoofor felting onEBtaplot Thisallowsfor thepotentialestimationof
the voriation ~mg clusters plots ondtrees os well osdiffere11CeSbetweentime periOOs
Two~ions thatneedtobeansweredare ( I) what is thenature of the deviationsfromthe
tNfrrq (1OWthI espouseend(2) how should thesemvietionsbe irorptreted into CACTOSThe
resiliJalsfromthediameter-squared8IK1totalheilj1tJQWthm(XElsare het~tic withresptlt
to predicted IJOWthThet is the error variance is proportional to the square of predicted IJOWth
Thtshet~ttclty QI1beseen bylooktngatthereslOO8I scatterplotshownfor pordros8pineIn
Figure 1 (Wensel end [oehler 1985) Hencethe followingm(XE1is appropriate for
expressingthe~th functionsforbothDBH2800totalheilj1t
Y a = f( X a) X ( 1 + E a ) [1]IJ IJ IJ
where
Yijk = lItuelITowthoftree j onplot i f(l time periook(ejther DBH2(I total ht)
f(Xijk) = pr8ftcted~owth f(l tree j onplot1f(l t1meperjoot
ReseerchNoteNo 14 page 2
-------------------- ----------
Xij( = ~tor ofpredictorvar16bJes
E1jk-renOOmelement
Thereforethe1~ thepredicte1fTOWththeI~ theabsoluteeffEpoundtofther~ elementE
shy50
i ~ i ~-
A
A
J
A A A -A A A0 - MN A A A A A c A A _A A A A CAt AA _AoltA Ate A XOCErIgtACAM- A A A A
C I ( lIfn t OpoundIpoundgtTCte- 0 CA shy oltAMIlDCCteteA- A AA - A A A AA shy A A AA shy A A
A A
A A A
A A A A
A A
I
-0shy
A A
--
A
A A
A
A
A A
-~ A A A A
A A A
J -)0 A A
A
A A
-00 A
A
-5) C 3 5 8 2 27 30 n 3A n 42 45 5 54 ~ AJ ~ 8 9 4
PnddId dII9 inDItf2(spnd trD8I)
Figure 1 ResicUi)s from DBH2~owth mcrl11p1otte100 predictej DBH2~owth fCJ
poo03r0S8 pine
In someC8SeSthe predictive mcrl11 feXijk) canbestownto systematicallyoverCJu~
estimate6CtuellTOWth particularplotstreestimeperlltX2setc TheuocertalntyInpredIctions rCJ
m~ be raducedif a systematic trendcanbeestimatedseparately(run the IJOWthrrnxmlConsider
thefollowingpossiblecomponents Eofther51OOrnelement
fJlt= oq + Aj + ~ + iJlt + ftJlt [2J
where 0 cluster aOOplot effEpoundt fCJplot i
ItIj = treeeff~t f(Wtreej 00ploti
R83e8rd1NoteNo 14 pege 3
--- -- --- ------
ik l8iOO efftpoundtf(T periOOk
6iJlt = measurementeffoct fortreej 00plot i for periooK
fJlt = unexplainederror fortree j 00plot i forperiooK
The tree effocts()maybedueto ytriatioos in the r1SPOl1Seof tree ITOWthIiJe to ~tic
endhistorical ffpoundtors oot usedby theITOWthmtXElsTheplot endcluster effocts ( od maybeIiJeto
estimetiooernrs in predicting site irdJx endcrownaxnpetitioo aswell asPnf i~~ies of site
irm to renet ~tU81 pnxiJctiYity Perioo effocts( )) maybe IiJe to ytriatioo in the ITOWth
environmentfremyeer to yeer ( amountend distributioo of reinfell tempereture) endother
enviroomentalffdors oot incllXEdin the premctioo effects( ~) maybe processThemeasurement
we toerrors1n f1eld measurementsor 1nsu~t ~ process1ngTheunexplainederror ( pound ) 1s
the erreuror left over after al1 of the other effects hErYebeen taken into fECOUnl In caseswhere plot
cluster tree periOOor measurementeffectsCtrI beestimatedmoreIDUrateestimatesof the mEm
Towth can be ootatined thus reOOcing the amount of unexplaine1 variatioo
Assum1rYJ1~ between theaxnpooentsthesumsof S(JJ8resof [ (~ioo 2) CtrIbe brokencbwnasfollows
55f = SSplot+ SStree+ SSperiOO + SSmees + SSunexpleined [3]
where thesumsof squares ere definedas follows
SSplot plot (endcluster) effocts
SStree tree effocts
SSperiooperiooeffects
SSmees meesurementernrs
SSunexp lained unexplained sumsof~
A cle51breakliJwnof the elementE into its axnpooentpsts wooldl8JIire e -(I ossecf
2silJ) for theSMtple dBte Tf18t is multiple measurements00multipletreesper plot 00 multiple
plots8t multiplepointsin time Withsuchd8tecmventionelanalysisofvsilRe tEpoundhni(JJeSart be
Rese8IChNoteNo 4 pege
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
-------------------- ----------
Xij( = ~tor ofpredictorvar16bJes
E1jk-renOOmelement
Thereforethe1~ thepredicte1fTOWththeI~ theabsoluteeffEpoundtofther~ elementE
shy50
i ~ i ~-
A
A
J
A A A -A A A0 - MN A A A A A c A A _A A A A CAt AA _AoltA Ate A XOCErIgtACAM- A A A A
C I ( lIfn t OpoundIpoundgtTCte- 0 CA shy oltAMIlDCCteteA- A AA - A A A AA shy A A AA shy A A
A A
A A A
A A A A
A A
I
-0shy
A A
--
A
A A
A
A
A A
-~ A A A A
A A A
J -)0 A A
A
A A
-00 A
A
-5) C 3 5 8 2 27 30 n 3A n 42 45 5 54 ~ AJ ~ 8 9 4
PnddId dII9 inDItf2(spnd trD8I)
Figure 1 ResicUi)s from DBH2~owth mcrl11p1otte100 predictej DBH2~owth fCJ
poo03r0S8 pine
In someC8SeSthe predictive mcrl11 feXijk) canbestownto systematicallyoverCJu~
estimate6CtuellTOWth particularplotstreestimeperlltX2setc TheuocertalntyInpredIctions rCJ
m~ be raducedif a systematic trendcanbeestimatedseparately(run the IJOWthrrnxmlConsider
thefollowingpossiblecomponents Eofther51OOrnelement
fJlt= oq + Aj + ~ + iJlt + ftJlt [2J
where 0 cluster aOOplot effEpoundt fCJplot i
ItIj = treeeff~t f(Wtreej 00ploti
R83e8rd1NoteNo 14 pege 3
--- -- --- ------
ik l8iOO efftpoundtf(T periOOk
6iJlt = measurementeffoct fortreej 00plot i for periooK
fJlt = unexplainederror fortree j 00plot i forperiooK
The tree effocts()maybedueto ytriatioos in the r1SPOl1Seof tree ITOWthIiJe to ~tic
endhistorical ffpoundtors oot usedby theITOWthmtXElsTheplot endcluster effocts ( od maybeIiJeto
estimetiooernrs in predicting site irdJx endcrownaxnpetitioo aswell asPnf i~~ies of site
irm to renet ~tU81 pnxiJctiYity Perioo effocts( )) maybe IiJe to ytriatioo in the ITOWth
environmentfremyeer to yeer ( amountend distributioo of reinfell tempereture) endother
enviroomentalffdors oot incllXEdin the premctioo effects( ~) maybe processThemeasurement
we toerrors1n f1eld measurementsor 1nsu~t ~ process1ngTheunexplainederror ( pound ) 1s
the erreuror left over after al1 of the other effects hErYebeen taken into fECOUnl In caseswhere plot
cluster tree periOOor measurementeffectsCtrI beestimatedmoreIDUrateestimatesof the mEm
Towth can be ootatined thus reOOcing the amount of unexplaine1 variatioo
Assum1rYJ1~ between theaxnpooentsthesumsof S(JJ8resof [ (~ioo 2) CtrIbe brokencbwnasfollows
55f = SSplot+ SStree+ SSperiOO + SSmees + SSunexpleined [3]
where thesumsof squares ere definedas follows
SSplot plot (endcluster) effocts
SStree tree effocts
SSperiooperiooeffects
SSmees meesurementernrs
SSunexp lained unexplained sumsof~
A cle51breakliJwnof the elementE into its axnpooentpsts wooldl8JIire e -(I ossecf
2silJ) for theSMtple dBte Tf18t is multiple measurements00multipletreesper plot 00 multiple
plots8t multiplepointsin time Withsuchd8tecmventionelanalysisofvsilRe tEpoundhni(JJeSart be
Rese8IChNoteNo 4 pege
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
--- -- --- ------
ik l8iOO efftpoundtf(T periOOk
6iJlt = measurementeffoct fortreej 00plot i for periooK
fJlt = unexplainederror fortree j 00plot i forperiooK
The tree effocts()maybedueto ytriatioos in the r1SPOl1Seof tree ITOWthIiJe to ~tic
endhistorical ffpoundtors oot usedby theITOWthmtXElsTheplot endcluster effocts ( od maybeIiJeto
estimetiooernrs in predicting site irdJx endcrownaxnpetitioo aswell asPnf i~~ies of site
irm to renet ~tU81 pnxiJctiYity Perioo effocts( )) maybe IiJe to ytriatioo in the ITOWth
environmentfremyeer to yeer ( amountend distributioo of reinfell tempereture) endother
enviroomentalffdors oot incllXEdin the premctioo effects( ~) maybe processThemeasurement
we toerrors1n f1eld measurementsor 1nsu~t ~ process1ngTheunexplainederror ( pound ) 1s
the erreuror left over after al1 of the other effects hErYebeen taken into fECOUnl In caseswhere plot
cluster tree periOOor measurementeffectsCtrI beestimatedmoreIDUrateestimatesof the mEm
Towth can be ootatined thus reOOcing the amount of unexplaine1 variatioo
Assum1rYJ1~ between theaxnpooentsthesumsof S(JJ8resof [ (~ioo 2) CtrIbe brokencbwnasfollows
55f = SSplot+ SStree+ SSperiOO + SSmees + SSunexpleined [3]
where thesumsof squares ere definedas follows
SSplot plot (endcluster) effocts
SStree tree effocts
SSperiooperiooeffects
SSmees meesurementernrs
SSunexp lained unexplained sumsof~
A cle51breakliJwnof the elementE into its axnpooentpsts wooldl8JIire e -(I ossecf
2silJ) for theSMtple dBte Tf18t is multiple measurements00multipletreesper plot 00 multiple
plots8t multiplepointsin time Withsuchd8tecmventionelanalysisofvsilRe tEpoundhni(JJeSart be
Rese8IChNoteNo 4 pege
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
usOOto partitioo the sums of sqJ8ressrownin ~t100 [3J ~ to estimatetheeffoctssrownin
~tion [2] Thoseeffectsthatcannotbeestimatedare lumpedwith theunexplainederror
~8te studies can be usa to estimate measurementerror aOOthereby rtJUe the
dimensionof theproblem However I errors inherentin theooteurm- stOOymeasurement werenot
Investi~ted(J1a scientific basis fertunately ~ating measurement unPJeplajned is~ errors
not very critital since the meesurementerror should be controlled and remain small Also
mecsurementerror should be inclucEdin the stochasticprocessrether that be left up to user
C8libratioo
In thestemanatysisstOOyoolya subsetof thetrees00 a given plot werefelled for stem
M8lysis with t~ other trees00theplotsbeingmeasurrofor current size~ r~ial incrementusing
~ incrementw-er Unfertunately siocethestemI1Otysisootesetoolyhastimeseriescitafor a
subsetofthetrees 00theplotsthelastfourelementsof~tioo [3J camotbeestimatooI-bwever
the plot effects were estimated separatelyand they maybe inchrl3din rxros by calibrating the
mcxElsdiroctIy(Wensel amp Daugherty 1985)
Estimatesof the plot effocts in ~tioo [2J can be sep8ratoointo cluster 600
plot-within-cluster ofthetwoIJOWth andsixspecies NyenJIAmcxEleffoctsfer ~ mcxEls anestedwasusedtoseperatethevariatiooasfollows
Elaquo h)i)j = + ~h + ~(h)i + ( (h)i)j [4J
where
E((h)i)j = residualOfjth tree00jth plotInhthcluster
JJ = overallbias (hopefullyzero)
~h = effect fer cluster h
~(h)i = effoct fcr plot j Incluster h
e((h)i)j = unexplainederror (00etotreeperioo aOOmeasurementeffocts)
The ec (h)i)j s were then processrotonndthest~ ernr nj measureof skewnessfer ~
~ies 800~h mooel(Teble I) This was 00neby using theproceOJreGLM (general linear
Ch Note No 14 ROOOOI page 5
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
m~J) of thestetisticsl prcJTem ~ (SAS 1982) Insummarytheclustern plotefffrlswere
sig1ifi(3lt in mostcaseswith the variance reOdions ranging from 5 to 45 percent Also the
unexplained with starOrdernrs between 8rerra-s were unbiase1 020 fnj 036 for hej~t IJOWth
between 034 ~ 052 for diameter ~ ~th Exceptfor the hei~t IJOWthof the pines
positivevaluesof the CDNpoundtedthird mcmentswere foondPositive~ewnessof theunexplainej
errors Is exPEpoundtedbec8lrsethe unexplainederrors are coostratnedat the lower endby -1
(correspondingtoanlpoundtual IJOWthrate of zero) anduocontrail81ontheupper end Thereforethe
error distributicm00ootfollow ecentrelnormeldistribution
TooleI StEnBdernr(u) fnj ewness() estimatesfor DBH2fnj heit IJOWthmls
Coefficients pp SP IC DF WF RF
lJBH6TowtJ
CT 46 34 49 44 49 52
e 10 03 18 13 15 23
hei[tJtTOO) CT 22 20 29 27 27 36
e -06 -07 08 05 10 07
The stewnormal distribution prOYi~ ~ ~proximahons to many near-normaL
distributions (Hodges 1978) fnj was cOOsen to represent the error distributions The
prOOabiHtytEnsityfuncUonof the sUrldBrdizedstewnormal1s ~proxlmated by (tD the orcEr of
e2)
+Jz] ~ t(z)-e+1z)
or
+Jz) ~ +00 (1 + e (z3-3zD [5)
where
+Jz)= probabilitylimity functionof the~ewed normaldistribution
+laquoz) z probabilitylimity functionof thestarOrdizednormaldistribution
+1ZJ = third lErivativeof+00 RI
e a stanOrdizedthird moment 6
SimHer1yI thecumulativedistribution function (cdf) isapproximatstby
Re5e8ICh Note No 11 pege 6
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
tsfZJ ~~) - e+1zJ
tJz] ~ 4(zJ - e(z2- 1)+fZ) [6J
where
tJz] cumulativedistributim ofthestewnormaldistributim I lD1
4J(z) cumulativedistributim functionof thestand8rdiZB1normaldistributim
IMPLEMENTATION
The stochaStic techni(JJe useI in CACTOSmust allow the user to replicate a simulationrun
Thatis thedifferencebetweensimulatimruns mustberoetothefactors~ ~ ootroetothe
ranctm processusedThisenablestheforestm~ toestimatetheeffectsofvariousm~ent
regimeswittwt ~ ronfoundingstochesticeffects
FOREST(Et amp Monseruel 1974) uses a rarmn number generatortogeneratevaluesof
E(EJJJBtion [ 1J) from anormaldistributim with ISmeanof 0 ln1 vari80CeE(JJ81tothevari80Ceof
the resiliJalsdivicEdby the 8Ver~ g-owth ratefoundin their808tysis Thiswas00nefor boththe
DBH2~ hei~t IJOWth mOOelsWhile this is a direct methOOof apptyingstochastic effectsto
g-owthpNl1icttonsparallel slmulaUonswith different m8MJBffient(WUons canootberepllcatoo
Also the use of ISthe central normal distributim Bent reOect the stewed neture of the resiliJals
found in the currttlt sttDt
PROONJSIS(St to incorporate effectsfor the 1973)usesa similar methOO stochastic
D8H21 (TOWth manytree reax -dsslOO 00vfat1m whentheren theeffectsof anyoner~
00 the IJOWthrate of one tree wouldbe blenIBIwith mfIIYother tree reax-ds ~tty the
stand totals shooldbequitestableestimatesPRCXH)SIS 13S0trse reanis fer this rEIJIires
methOOto be implementedIf fewer trse reax-dsare enterej intothe simulator 8 reard-tripling
scheme Is usaI with eatI of thenewtree reaxdshaYing ofthecrioineithesamechar(Wteristics
tree except the tree wei~ts are rlWced to 1SllIId 2S1 of the origins1 tree The origins1 trees
wei~t is redm1 to the remaining 601 Thefirst newtree record represents the s10wer~owing
trees(up to the Islpercenti1e)whiJetheS8XJnd treesnewtree record represents faster~jng
1 PROONOSIS effects numJs6m notusestochastic fir Usheij)t~th
Researcht8te No 14 pege 7
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
(7shpercentileandup) Theorioinaitreeraxrdrepresentsthem1ium t~fT(YlaquoifWJ ( Islll to 7s1hpercentiles)St assumesthattheerror distribution is normalJydistributedendcalculates
the~owth mooifiers(E= zi i=slowmedfest) tobetheexpoctedvalueof ther8Jl00sof theerror
distribution representedrespectively by etdIof thethreesimiler tree rtm11s(SeeFigure 2)
PROONOSIStriplicates eIIhtree record rN8fYITOWthcycle until a limit of 1350 trees is nBhaJ
wherebytheschemesimiler to fORESTsis implemented
I AVERAGE
r- i
SLOw i~ irAq
~ 15~ 60 25
--- i i --shy -$ Zm Zf
figure 2 DBH2~th Theerror distribution breSoownfor PROONOSIS mCJEI
CRYPTOS(Krumlend amp Wensel 1980) uses e NDrd tripling schemesimiler to
PRCXNJSISthat to both the D8H2endhelt11tITOWthmCJElsI~Is BPpl1tX1 Is 8SSUmtX1
betweenthetwo~th componentsendajoint normalerror distribution is usain euroiterminingthe
approprieteweilj1tsfurther for romputetiOMIefficiencyCRYPTOSuses 1 of the 9 possibleI
reQl005ofthejointerror distribution(SeeTable 2)
Table2 Brs mwnoftreewei1ttsfor CRYPTOS
HEIGHT I DBH2 OWTH I BROWTH I Slow Ned fllt I Sum Slow I 25 I 25 MEn I 25 25 I 50 F I 25 I 25 Sum I 25 50 25 I 100
HenceCRYPTOS tree(JJ(Dupleseed1tree raordwith eEDrepl1cati00having251 of the (11011181
records wei1tlThevalues EereCOOIputed of eachregionof theerr(l distributions asthemedian
representedby that new tree raord This methOOhasbeenmined -psetJb-stochastic-sincethis
schemeincorpa-atesthe unexplainedveri8tion while notusingrarmn errors
ThesUxhastic empl~ hasthe(jsirable propertyofreplic8bUity R1 scheme byCRYPTOS
is empl~ jointly betweentheDB~ endHT~th mCKElsHowevertheff Jmptionofnormel errorsisrotvalidin thiscase
ReseerchNoteNo 14 page 8
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
rACTOSempl(1ySe stochasticmethOO but the errors ere assumErisimihr to CRYPTOS to be
from e skewrumel distribution with P8fametersgivenby Tlble I Theernr distributtoofor
eEd1lTQWthmtX21n1speciesWBSdivitid intothree reJions -- slow m~fjum ero fBStCTOWthBS
i11ustr8t~ in figure 3 Using the
AVERAGEI
w1 i SLO 1~---i FAST
j1 1 I ii I ~
Zc r Zm r Z f J 2 22
Figure 3 Error regions fCKCACTOSITQWth maEls
distributioo furction fCKthe skewnormel ts the area of thesereJions are wnputed by the
foliowirwJreletimships
A( s) = 415( z )
A(m) = t5(Z2) - t5(z)
A(f) = 1-ts( z2)
Further inoopeuiuce WBSassumed between the heitlt ero diameter IJ(JWtherror terms because 00
~te WBSavailableto investi~te alternative joint lEnsity distributions
The t record weights and CJOWth themo1ifiersare computedas follows First zi
dividi~ pointbetweenthe slowendmediumCTOWi~ Wa5 suchthotA(s) 16 or t~ found
tJztJ =01667
Hencethe m5g1neltree record weij)t for the tree recor1isrepresenti~ this reJioo is 16 of
the CKiginaltree reard wei~t with the JQWthmooifier 2s being BPI to the expected valueof the
distributioo within this r8Jion ~it WBScmirable tohaveoneof the tupled OtreerlaquooI1isto
ITQWBStNer pndictioo (CTOWthmooifier of 0) Thus z2 was foundStpoundh that the expectedv81ue
of the distrjbutioo between 21 and 22 was O Third the expectedvalue of the distributioo between
Z2end+ODfir thetrOWthmoolf1erf(J thefast~trrJ tre Zf wasaxnputmStmlis- IyI the
marginaltreertmd wei~t for this regionis the~ ofthedistributionwithinthis reJiootimes
ReseBNftNote1m 14 peg8 9
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
the(rioinai tree reard wei~t A fORTRANCOOIputerPnYT8mto COOIputethe boundsof the reJion
zl 51d z2 en1 the mMlS of the slow 51d fest regions Zs 51d zf is lIVoiloble fronr the auttD-s The
PrtXTM1 useseqJ8tion5] f(r the euroimity of the stewnormal In the lVe8Sare consistent with the
lIPProxlmaUonsQiven by equeUon(6]
Thecanputedmtrginol tree reard weitj1tswere thenused to firKt the tree reaJrd weifjlts
f(r eochof the reN tJjintupled tree ran tis Themeansof theslowmediumandfast regions are
givenin Table3 andthe relativetree weilj)tsIre givenin Table 4 Notethat for eoch()OWth
mcxElthesumofthemlrginaltreer1an1weil11ts~ uptotheoriginaltree Nnnfs well11tfDjthe
sumofthe~th mooifierstimes themlrginaltreerecordweifjltsf(r EHhregion fIijs up to O
Table 3 Divid1rq points and meEf1Sof slow mediumandfastregions for ()OWthm~1 I
Cre1f1cients pp SP IC DF WF Rf
OBH fT(JWth
-973 -969 -976 -974 -975 -9782
1211 1029 1512 1311 1386 1740 22 - 1354 - 1456 -1238 -1311 - 1281 -1167
25
O O O O O O2m
2f 1852 1594 2179 1973 2057 2365
het Towth 2 -963 -962 -972 -970 -973 -971
Z2 0863 847 1152 1075 1211 1125
-1586 - 1601 -1383 - 1427 -1354 -1398z Zm O O o o o oZf 1333 1308 1775 1663 1852 1735
A~rh Note In 14 IJ8IIe 10
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
Table4 BrB IiJwnoftree weights by speciesfer CACTOSreard (JJintuphng
PariJrt1lJ8 Pine
HEIGHT I DBHl GROWTH I GROWTH1 Slow Me f~ I Sum Slow I 1667 I 1667 Med I 1667 3465 1218 I 6350 f 85t 1 ~ I 1983 Sum I 1667 7115 1218 I 10000
5iQTPine HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Me I 1667 3104 1522 I 6293 fast 1 2040 1 2040 Sum I 1667 6811 1522 I 10000
~ ()rlr
HEIGHT I DBH2GROWTH I GROWTHI Slow Med F8St I Sum Slow I 1667 I 1667 Met I 1667 4421 0947 I 7035 Fast 1 1298 1 1298 Sum I 1667 7386 0947 I 10000
IJotQlasFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med fast I Sum Slow I 1667 I 1667 Med I 1667 4129 1107 I 6903 fast 1 1430 I 1430 Sum I 1667 7226 1107 I 10000
WhiteFir HEIGHT I DBH2GROWTH I AROWTHI Slow MBd Fast I Sum Slow I 1667 I 1667 Med I 1667 4410 1038 I 7115
f85t 1 1218 1 1218 Sum I 1667 7295 1038 I 10000
11MFir HEIGHT I DBH2GROWTH I GROWTHI Slow Med Fast I Sum Slow I 1667 I 1667 Moo I 1667 4501 0822 I 6990 fml W3 l 1343
Sum I 1667 7511 0822 I 10000
Rese8rCft ND4 pege 11 NOte
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
APPLICATION
Amix~-conifer standwith8 per Epoundrestccti~ of295 trees 265 squarefeetof basallre6
8006750 cubICfeetof volumeis usedto illustratethe processof ganeratingpstUi)-s~tic
~intuples Table 5 givesthe sizeandITOWthinfermationonthefirst 5 ofthe trees in the tree list fer thest~ Thisshowsfor examplethatthe 5-yeer predictedDBH(JOWthis 08~ ioches
whilethepredicOOhei~t ~owth is 56 feel
Table5 SizeandPredicted~owth for first 5 trees (of 33) 00a one-fifth ocreplot
(TPA = trees per tUe)
to to I ctC8I ~ CT08th no SPeCies [ampf ht Mltio TPA 6IBi 6ht
1 ampE 144 85 39 500 94 56 2 PP ~5 ~ ~ 500 65 48 3 PP 182 94 50 500 75 56 ~ IC 148 53 41 500 53 26 5 PP 221 87 32 500 65 57
The DBH and he1111t IJOWth estimatesfor the~1ntupledraord are then computedasfollows
Froo1TableI ferwhitefir (WF) thestandard nOOviatioos
cr0 =049 and (1 H = 027
8MfromTable3 for whitefir thesKewnormal OOviatioosn
DBH Hei(i1t
Zslow -1281 -135~
Zmd O O
ZrlSl 2057 1852
The values of ( I +pound) for eQUationI n thencomputedas
1+E =1+ zu
yieldi~ the slowdiameter~owth estimateof
Dslow =6D ( 1+ Eslow)
=084 [1 + (-1281)( 049)J
=084 [ 0372 ]
= 032 inches
Fo1Jowi(WJthis processthe remeini(WJveJuesof ( I +0 fer thefirst tree n given8Sfollows
IWe8rChNoteNo 14 pege 12
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
DBH Heiltlt slow 0372 0634 med O O fast 2008 IS00
Multiplyir1Jthese valuesof ( 1+pound) by the DBHarx1hei~t wowth rates predicted by rACTOS084
~ S6 feet ~tivety yields the quintuplecJ1)Wth for the first tS I rates given in T18 6
trees ThetnEr of the recoI-dswithin etEh quintuple is 8Sfollows
(1) Dmed ~I
(2) Dslowl
(3) DrSL 4
(4) 6Dmed 6Hslow
(S) 4Dmed Hrasl
Table6 PredictedDBHandhei~t after a singleS-yeer ~th cycleusingquintipulEl1
ITOWthr1lDds- f1rst5 trees(of33) ona one-fifth~ sampleplot
total a08n (1 + E) no 50 tEH ht ratio TPA DBH ht
1 tE 152 g1 O~ 221 1000 1000 W 147 g1 039 083 0372 1000 IF 160 91 O~ 052 2008 1(XX) IF 152 89 039 083 1000 0634 W 152 93 040 001 1CDJ 1D
2 PP 251 54 OSot 173 1000 1000 PP 247 54 053 083 0377 1(XX) PP 257 54 055 061 1852 1000 PP ZS 1 055 083 1(XX)0651 PP ZS 1 55 054 099 1(xx) 1293
3 PP 190 90 049 173 1CDJ 1CDJ pp 185 GO 048 083 0377 1000 PP 196 90 OSO 061 1852 1000 PP 190 88 OSO 083 1(XX)0651 pp 100 01 00 000 1000 1203
4 IC 153 5amp 040 221 1(XX)1(XX) IC 150 S6 040 083 03g3 1(XX) IC 159 5amp 040 047 2 1000 IC 153 55 040 083 1(0) 0 IC 153 57 O 065 1000 1515
5 PP 227 93 032 113 1(XX) 1(XX) PP 223 93 032 083 om 1000 PP 233 93 033 0amp1 1852 1000 PP 227 91 032 083 100001 PP 227 94 032 099 1(xx) 1293
ResearCft NOteNO 11 pege 13
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
The ~Jntuple estimates for the other trees follows in 8 similar manner using the
~propriatevaluesofthestanOOrderror (0-) andskewnormalOOviatioo(z) ffro) Tables 1 end 3
respect ively
Thetree expans100foctors for etrtI of the trees srown in Table 5 is 5 trees per ocre before
~intup1ing Theexpansionsfor the tree ND)I$ after ~intup1ing (Tlble 6) In ootain81by
multiplying the ex-iginalexpansioo by the wei~ts given fex-eatI species in Table -t Remember
that the sum of the relaquo tree wei(t1ts must be ~I to the orioinai tree wei(t1t
The impatof implementing pseuro-stasttc vlJiatioo in the CACTOSm~1 can be iIIustrat~
by 8 30-yeer slmulatioo for the sm1ple plot refernxj to above Theb8s81lJe6lOWthfor 30-year
periodis givenInTelale 7 with 00thinniryenJ with thinning from below eOOwith thinning from aboveEstimates srownffro) simulationswith andwithoutpsam-st0ch8StleeffectsIn
T~le 7 Basalarea IT(JWthfor 30 years with and without pseuOO-stochastieeffectsfor vorying thinning olternotives (All thinning ot the endof the first 5-~ cycle)
psetJ(i)- stoch8Stie effects
no ves lJ1ill
00thinni~ 1288 1309 16thin 70 ft from below 1053 1077 22 thin70 ft2 fromabove 1223 1241 15 thin 140 ft2 from below 823 837 17 thin 140 ft2 from above 1177 1197 I7
InetrtI C8SeusingpseudJ-schostatic thepnxjict~IOWthratesDiffererlCeSalsoeffectsiocs easedexistIn theD~letet1volume(JOWthrates
Re380rCh NoteNo 14 page 14
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
DISCUSSION
The tupl1ngof tree rfDXds to represent the unexplatnedvartatton tn tree I)OWth
predictionsis e useful tool for tree ~th prediction Quintupling8Susedherehasthe intuitive
Irl-Ient~ thet the TOWthrmes for oneof the tupledrecoc1isere unchengedAIS) in oontrast to
previoos tuplirj schemesthe current schemenDqIizes the skewednature of the unexplained
variationin the J(JWthpredictions Estimators from theskewoormaldistributionare usedto
represent this u~plairm varietion in formingthe~intuples
Short term predictions without m~ent intervention l3I be mne without the useof
pselXtI-st0ch8Sticvariation For sucha smallperioo therewill be little or 00differeree in the
predictions Thusthe rACTOSmoil can be US81to upmte timber inventorieswittwt loss of
~ray end withoutaffecting the inteqity of the original recocm Ifowever for longterm I
predictions8OOIorsimulations with m~ent Interventionstheuseof psetXi)-stoch8Sticsis a1vised
ReseerchNoteNo 14 page 15
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
LItereture Cited
BeJ1aI E 1970Simulatiooof IJOWthyield aoo m~ent ofaspen PhDThesis UnlvofBritishColumbiaVancouver 190 p NetLibof Card Ottawa(Diss Abstr 31 6148-B)
EkA RaooRA Mooserud 1974FORESTACOOIputer simulatingtheIJOWthaoo repndpoundtioo spociesmlXEl fCK ofmixedfCKeststfrldsUnIv Wisconsin ColleJeofIqicultural aooLife ScleocesReseerctlReport R2635 90 p
HOOs 1978J LJrNotesfCKStatistics 100B DepartmentofStatisticsUCBerkeley
KrumlendBE 1982Atree-basaj forestyield projEdioosystemfCKthenorth coestreglooof CalifCKniaPhDThesisUmvCalif Berkeley
KrumlandBEandL CWensel1980 Cryptos(I) - usersGuire C(q)erattve ra1wOOjyleldproj~t timber output slmulatCKshyinter~tiveprcq-amversion30 ResNoteNo16 CoopRedwtxx1YieldProj Deptof FCK endResMgtUCBerkeleyMimeo
SASInstituteloc 1982 SN5users GuireStatistics
S~ A R 1973 Promsis mlXElfCKstand (Evel~ment ResPaperINT -137 32p US Forest Service MOampOWUtah
Wemel L C n1 P J O8lMj1erty 1985 ~TOS UsersGui~ The~Iifornia ConiferTimber~tput Simulator ReseeIdI Note61 O Northern ~11f Forest Yield Cooperat1ve Dept of Forestry andResource M~t U C Berkeley 98 pgs
WenselLC~J RKoohler1985 Atree IJOWthprojectioo systemfor Northern California coniferous forests R8SE(jdI Note 612 Northern Calif Forest YieldCooperative UDeptof ForestryandResourceM~tCBerkeley30 pgs
ReseerchNoteNo 14 pege 16
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