D.J. Leduc , Information Technology Specialist, & J.C.G. Goelz , Principal Forest Biometrician

United States Department of AgricultureForest Service, Southern Research Station

Diameter Distributions for Young Longleaf Pine Plantations: Initial

Conditions for a Growth and Yield Model

D.J. Leduc, Information Technology Specialist,

& J.C.G. Goelz, Principal Forest Biometrician


Why Longleaf?



Why is this a problem?


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Mature longleaf pine stands can be unimodal, ….


… but they can also be bi- or tri- modal.

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There are two known causes of this.


Suppressed longleaf trees do not die easily.


Not all trees exit the grass stage at the same time


To produce the irregular diameter distributions observed in older stands , it is essential that initial diameter distributions be irregular.

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Techniques

Weibull distribution by parameter recovery Artificial neural networks Model cohorts of trees beginning height

growth


What we have to work with

Age Basal area Site index Container stock or not Trees planted per acre Number of trees in 0-inch diameter class?


Weibull distribution

Included as baseline parametric technique

Pi=0.97 and Pj=0.17 as suggested by Zanakis (1979)

Only used for trees with dbh > 0

c

b

c

1

j

j

j

i

j

i

))p1((

X

XX

)p1()p1(

ln

lnlnln

ln


Artificial Neural Network


Artificial Neural Network

Number of grass stage trees is known Predict proportion of dbh 0 trees Do not predict proportion of dbh 0 trees

Number of grass stage trees is unknown Predict proportion of dbh 0 trees Do not predict proportion of dbh 0 trees


Predicting number of dbh 0 trees

Necessary for Weibull distribution that we used and two neural network models.

Used standard logistic model and an evolutionary algorithm


Logistic model

logit = 2.1696 + age * (-1.7565) + baa * (-0.1143) + si * 0.1705 + container * (-12.2144) + tpa * 0.0114 + age*baa * 0.00617 + age*si * 0.00920 + age*container * 0.8183 + baa*si * (-0.000960) + baa*container * 0.0383 + baa*tpa * 0.000013 + si*container * (-0.0640) + si*tpa * -0.00013 + container*tpa * 0.00184

predp =exp(logit)/(1+exp(logit)) pdc00 =predp*tsa


Evolutionary algorithm

tmp= (container*11.24+baa)/9.84 tmp2 =5.08*age+ ((tmp+tsa)/tmp)-

148.76+container pdc00=(((-19.2431+tmp2)*tmp2)/(-

46.1721*si)*baa+tmp2)*age/(-22.0538)+tmp2


Crossover (sexual recombination) X reproduction Inversion Mutation Hill climbing Migration and intermarriage

Evolutionary algorithm


Explicitly Modeling Cohorts

Seedlings exit the grass stage over several years. This is one of the main factors causing diameter

distributions to be irregular. Model diameter distribution as a mixture of

distributions for each cohort. As there are potentially several cohorts, it

seems wise to use a very simple distribution.


Epanechnikov Kernal

Ki(u) = 0.75 (1-u2)

For

(Xmin-.05)<X<(Xmax+.05) Complete distribution is:

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-1 -0.5 0 0.5 1

uk(

u)

2/05.005.02

minmax

minmax

xx

xxx

u

n

iii xKppdf

1

)(Where pi is proportion

of stand in cohort i.

United States Department of AgricultureForest Service, Southern Research StationUsing mixture of Epanechnikov-kernals in prediction Predict the proportion of trees in each cohort.

User-supplied input regarding length of time in grass stage (average length, or years for 75% to leave grass stage…).

Select “Guiding” Dmax (or Dmin). “oldest” cohort or most populous.

Develop equations to predict other Dmax and Dmin’s from guiding value, and stand variables (age,site index, etc).

Recover guiding Dmax from predicted basal area and trees/acre.


Cohort Pi Dmax Dmin

1 p1 G F(G,SI,TPA…)

2 p2 F(G,SI,TPA…) F(G,SI,TPA…)




n

iii xKppdf

1

)(

dDpdf DD 2iq


C o m b ine d Ac tua l

DBH (inc he s)

P


Preliminary Results


Predicting the number of trees in the grass stage

Criterion Logistic Function

Evolutionary Algorithm

Bias 19.2 -5.3

Largest Deviation

274.6 -171.6

Mean Absolute

Deviation

29.4 15.1

Root Mean Squared Error

61.48 34.41


Criterion

Neural Networks

WeibullPredict 0 Don’t predict 0

Known 0 Unknown 0

Known 0 Unknown 0

MSE 482 603 572 521 669

FI .84 .80 .81 .82 .77

2

lowest

16 14 8 14 23

KS lowest

11 13 11 23 19

Mean closest

18 14 12 22 9


All methods work 410 128 10

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actual

NN model 1

NN model 2

NN model 3

NN model 4

Weibull


Weibull works best 203 131 16

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actual

NN model 1

NN model 2

NN model 3

NN model 4

Weibull


Neural net works best 203 135 16

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NN model 1

NN model 2

NN model 3

NN model 4

Weibull


Conclusions

Evolutionary algorithm better than logistic function for predicting trees in grass stage.

Neural networks show promise for modeling young stand diameter distributions.

Modeling cohorts looks promising, but remains untested

The biggest problem is finding enough easily measured variables to base predictions on

Documents

D.J. Leduc , Information Technology Specialist, & J.C.G. Goelz , Principal Forest Biometrician