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

~ 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