United States 

Deoartment of Predicting Diameter Distributions hculture of Longleaf Pine Plantations: A 

Forest Sewice Comparison Between Artificia 

~etwbrks and Other Accepted 

Methodo 

Southern 

Research Station 

Research Paper Daniel J. Leduc, Thomas G. Matney, 

SRS-35 Keith L. Belli, and V. Clark Baldwin, Jr. 
The Authors 

Daniel J, Leduc, Computer Specialist, USDA Forest Service, Southern 

Research Station, Forest Management Research, Pineville, LA; Thomas Go 

Matney and Keith Lo Belli, Professors, Mississippi State University, 

School of Forest Resowces, Mississippi State, MS; and V. Clark Baldwin, 

Jr,, Research Forester, USDA Forest Service, Southern Research Station, 

Forest Inventory and Analysis, Asheville, NC, respectively. 

Cover: Longleaf pine plantation. 

Photo by Bill Lea 

September 2001 

Southern Research Station 

P.O. Box 2680 

Asheville, NC 28802 
Predicting Diameter Distributions of 

Longleaf Pine Plantations: A Comparison 

Between Artificial Neural Networks and 

Other Accepted Methodologies 

Daniel J. Leduc, Thomas G. Matney, 

Keith L. Belli, and V. Clark Baldwin, Jr. 

Abstract constructed from poor Weibull estimates will produce biased 

volume estimates. Thus, general applicability of a procedure Artificial neural networks (NN) are becoming a popular estimation 

tool. Because they require no assumptions about the form of a fitting does not mean that the procedure is always best, as growthfunction, they can free the modeler from reliance on parametric and-yield modelers are always searching for procedures to 

approximating functions that may or may not satisfactorily fit the improve d.b.h. distribution estimates. Artificial neural 

obierved data. To date there have been few applications in forestry networks (NN) may provide better estimates of d.b.h. 

science, but as better NN software and fitting algorithms become 

available, they may be used to solve a wide variety of problems- distributions that do not rely on assuming an imperfect 

particularly problems where the underlying relationship between underlying probability model. 

predicted and predictors is unknown. We benchmark tested an 

glternative to the traditional Weibull probability distribution function, 

diameter-at-breast-height moment, and direct parameter prediction 

models for approximating stand-diameter distributions. Using a 

feedforward backpropagation network, we demonstrated that NN are a 

somewhat better option. Unlike Weibull approximations, NN 

solutions cannot easily be mathematically constrained to match 

known reality constraints, but this difficulty is easy to overcome in 

practice. 

Keywords: Connectionist models, parallel distributed processing 

systems, parameter recovery, Weibull distribution. 

The thee-parameter Weibull probability distribution 

function (Weibull 195 1) can take on a wide variety of shapes, 

and has been found to be an applicable model for 

approximating tree diameter-at-breast-height (d. b. h.) 

distributions (Bailey and Dell 1973). Because of its plasticity, 

many stand-level, diameter-distribution growth-and-yield 

models in use today rely on the Weibull probability 

distribution fbnction (Bailey and Aleixo da Silva 1988, 

Matney and Sullivan 1982b, Zmoch and others 199 1). 

However, the Weibull distribution does not span the entire 

function space, and its perfomance as a d.b.h. distribution 

estimator varies widely among data sets. In some d.b.h. 

distribution modeling cases, the Weibull function tends to 

produce poor estimates in the tails of the distribution. In 

other situations, it may lock down the tails and 

overcompensate by missing badly in the middle part of the 

distribution. Because any small miss in the middle and upper 

d.b.h. ranges of a distribution can have a large impact on 

derived volume estimates, growth-and-yield models 

In general, artificial NN are approp~ate in modeling 

situations: (1) where the application is data intensive and 

dependent on multiple, interacting pameters; (2) where the 

problem area is rich in historical data or examples; (3) where 

the available data are incomplete, contain errors, and 

describe specific examples, and (4) when the function to 

determine solutions is unknown or expensive to discover 

(Bailey and Thompson 1990). All these conditions are to 

some degree met by the typical growth-and-yield database. 

Theorems by Cybenko (1 989), Sun and Cheney (1 992), and 

Light (1992) show that a single output, single hidden-layer, 

feedforward network employing continuous sigmoid and 

other more general activation functions with a sufficient 

number of hidden units can approximate any continuous 

function to any desired accuracy. This makes them ideal for 

d.b.h. distributions of an unknown form. Some researchers, 

e.g. Josh 1987, point to Kolmogorov's (1 957) theorem on the 

realization of real-valued functions as strong, albeit not 

conclusive, evidence that NN models can learn to 

approximate any continuous real-valued multivariate 

function. They minimize error in the least-mean-square sense 

based only on example mapping. Hassoun (1995) gives an 

excellent discussion on the real-value function 

approximation capabilities of feedfomad neworks. 

Given the advertised great promise of neural methodology, 

we decided that a preliminary investigative benchmak 

comparison of this technique with the traditional methods of 

diameter-distribution modeling was wmanted. The model 

selected for this comparison is a simple, fully connected, 

feedfoward, backpropagation delta-learning-rule network. 

The two traditional Weibull probability function-based 

benchmark test models selected were a diameter-moment 
parameter recovery system and a direct-parameter prediction 

model. Each of these tecbaiques was then used to predict 

the dimeter distributions of three distinctly different, 

unthinned, planted longleaf pine (Pinus palustrr's Mil 1.) 

databases from stands originating under three different 

conditions. The preliminay results will help us to detemine 

the mount of additional work that will be necessary to refine 

the unthinned stand model and construct network models 

for predicting thinned stand-diameter distributions. 

We report the analysis and results of the multitude of 

evaluation criteria selected for the preliminary benchmark 

testing of the NN mode. 

Data 

The three diameter-distribution NN trainingitest data sets 

selected for study were from longleaf pine plantations on (1) 

abandoned agricultural land (old fields), (2) land that had 

received some form of site preparation, and (3) recently 

cutover sites. The three data sets were plot data assembled 

from several experiments designed to represent the growth 

response of planted longleaf pine across the Southern 

United States. The sites were considered problem free, in 

terms of having low levels of competing vegetation as well 

as good initial and subsequent survival of planted trees. 

Table 1 presents the basic descriptive statistics for each of 

the three data sets. 

The input variables selected for modeling d.b.h. distributions 

were stand age, base-age-25 site index, average height of 

dominant and codominant trees, and trees per acre. The 

output units selected for benchmark testing were trees per 

acre in each 1-inch (in.) d.b.h. class up to 20 in. In effect, by 

selecting these outputs we had the NN approximate the 

probability density function of d.b.h. at points spaced 1 in. 

apart. There are many other output formats that could be 

used, but this one fits well within the limitations of a 

backpropagation PJN. 

In regression model-building work, there is little to be gained 

by separating a data set into parts for fitting and testing 

(Nursch 1991). The only good way to test a regression model 

is to obtain independent test samples from the same 

population. However, because NN contain a large number of 

nodes, and thus many parameters, there is a good chance of 

ovedtting the training data set and capturing not just 

general patterns but smpling variation as well (Leahy 1994). 

Therefore, it is important to have both fittingitraining and 

testing data sets as insurance against overfitting. In this 

study, we randomly partitioned each of the three data sets 

into training md test data sets, designating a training data 

set approximately twice the size of the test data set. 

Model Development 

The type of W-4 we selected for evaluation is known as a 

hetero-associative, fully connected, feedfornard network. 

This means that the network is composed of an input layer, 

one or more hidden layers, and a multiple unit output layer; 

and that all of the nodes in every layer are connected to all 

of the nodes in each successive layer. It is also called a 

backpropagation network because, during training, errors are 

backpropagated from the output layer to the inner 

connections using the delta rule. While there is a large 

number of possible network models to choose from, we 

selected the backpropagation network because it is 

recognized as being best suited to function approximations 

(Hassoun l 995)- Attempting to evaluate all possible 

networks for a particular problem would be an ovenvhelming 

task. 

Figure 1 shows the general, single hidden-layer NN 

backpropagation model structure adopted for each data set. 

Between data sets, the number of nodes was varied to obtain 

the lowest mean-squared error on the outputs. Otherwise, no 

changes in the structure were necessary to obtain a 

satisfactory fit for each data set. The bias term connected to 

both the hidden layer and the output layer is a constant 

value (typically equal to one) that is analogous to the 

intercept term in a linear regression equation. 

An article by Freeman (1992) and a book by Hassoun (1995) 

include descriptive summaries and sufficient mathematics to 

explain the actual backpropagation training process. An 

abbreviated explanation of this process is shown in figure 2. 

In the course of deciding to use a single hidden-layer 

network, we tested numerous network structures to 

detemine the optimum number of hidden layers and the 

numbers of nodes in each. In a typical t\3N, there can be one 

or more hidden layers. Recent research into backpropagation 

networks shows that almost any function can be 

synthesized using a sufficiently complex network with a 

single hidden layer (Neuralware 199 1 d). We made several 

attempts to use more than a single hidden layer but 

observed no noticeable improvements in precision or 

accuracy. Thus the single hidden-layer NN was constructed 

and trained for each data set. 
Table I-Descriptive statistics for the training and testing data sets used in the comparison of neural network 

methodology to traditional Weibull models 

Old field Site prepared Gutover 

Variable Training Testing Training Testing Training Testing 

Observations (no.) 292 122 200 108 636 333 

Age (years) 

Average 14.8 14.8 15.6 15.8 18.2 18.7 

Range 8 - 30 9 - 30 8-25 10-25 9 - 48 9 - 48 

Trees per acre 

Average 715.4 71 1.7 609.9 584.3 590.7 593.3 

Range 74 -1,185 284 -1,160 123 -1,036 133 -1,089 133 -1,210 133 -1,136 

Trees per acre 

> 1 in. d.b.h. 

Average 704.7 699.7 559.2 542.8 538.6 552.2 

Range 74 -1,185 284 -1,160 123 -1,000 133 - 988 133 -1,210 133 -1,136 

Total height of 

trees with d.b.h. 

> 1 in. (ft) 

Average 32.2 32.5 20.2 20.1 34.4 35.5 

Range 15 - 62 16 - 62 7 - 38 8 - 37 8 - 80 9 - 82 

Total height of 

dominants and 

codominants (ft) 

Average 36.7 36.8 23.7 23.5 38.6 39.7 

Range 14 - 67 15- 69 9 - 41 10 - 41 8 - 86 8 - 88 

Basal area (sq. ft.) 

Average 96.4 94.9 34.2 32.8 74.5 80.1 

Range 8 -230 28 - 194 5 - 113 6 - 104 5 - 221 5 - 229 

Quadratic mean 

diameter (in.) 

Average 4.94 4.98 3.18 3.20 4.83 5.02 

Range 2.3 - 8.4 2.7 - 7.9 1.3 - 5.7 1.6 - 6.4 1.6 - 12.7 1.5 - 12.9 

Site index 

(baa. 25) (ft) 

Average 56.8 57.5 38.7 37.9 56.5 56.7 

Rage 34 - 80 36 - 76 20 - 77 20 - 77 20 - 103 26 - 104 

The number of nodes in a hidden layer is another parameter 

that can be changed in a NN (fig. I). The number used for 

each data set was determined by experimentally finding the 

optimum (based on the minimum mean-squared error 

criterion) fit of the training data set using different numbers 

of nodes. These optimums were three nodes for old-field 

sites, four nodes for cutover sites, and six nodes for 

prepared sites. 

Matching the attributes of a NN to the requirement that a 

diameter distribution must have exactly as many trees as 

there are in the stand required some additional calculations. 

First, rather than work with the actual number of trees, the 

output of this network was designed to be the percentage of 

trees in each l -in. diameter class from 1 to 20 in. Thus, in 

effect, output for any d.b.h. class is the probability of a 

given tree belonging to that class. For a network to be 
I3.b.h. 

classes 

Bias 

Bias 

Figure 1-Diagram of the basic structure used for neural networks tested in this paper. They are all similar in structure except for a variable number 

of nodes in the hidden layer. Each square block is a node. Nodes can be input like age, output like the proportion of trees in any diameter class, or 

hidden, which are just internal summation and transformation points. Each line is a connection where variable weights are applied. 

logically consistent with a probability function, all diameterclass probabilities must be nonnegative and sum to one. 

These constraints cannot be imposed on a NN of the type 

used here, but they can be obtained with minor 

postprocessing. This is done simply by (1) setting any 

output less than zero to zero, and (2) dividing each output 

by the sum of the outputs. 

While there are many computer packages that could be used 

to train a NN, most of the work for this paper was done with 

NeuralWorks Professional IIiPlus (NeuralWare 199 lc). This 

is significant because there are many assumptions and 

choices that vary among different software packages. All of 

the assumptions will not be explained here, but specific 

variations from the defaults are detailed. 

NeuralWorks allows the network developer to choose from 

among several transfer or squashing functions. We used the 

hyperbolic tangent function with a range of -1 to 1 along a 

smooth, sigmoidal curve to train all of the networks in this 

study. NeuralWare (1 99 I b) recommends the function as very 

effective in backpropagation networks. 

NeuralWorks automatically scales all inputs and outputs into 

a subset of the range of the transfer function chosen. For the 

hyperbolic tangent, this range is from -0.8 to 0.8. Guan and 

Gertner (199 1) chose to use a logistic function in their 

survival-probability network model partly because it has a 

range of 0 to 1. It is desirable to have the network outputs 

already in the correct range, but NeuralWare's automatic 

scaling eliminates the need for preliminary data scaling and 

makes the choice dependent only upon the slope properties 

of the transfer function. Extensive testing showed that the 

hyperbolic tangent worked best with the NN we used. 

Standard backpropagation uses the delta error correction 

rule, which adjusts the node weights after each example is 

presented. We used the normalized, cumulative delta rule for 

all the networks trained. The cumulative delta rule 

accumulates the error for each output and does not adjust 

weights until all of the observations in an epoch are 

presented. It also avoids learning rate changes by dividing 

the sum of the errors by the square root of the number of 

observations in an epoch. An epoch may constitute the 

entire data set or any subset thereof. We used an epoch size 

of 100 observations. 

Multiplying the backpropagated error by a learning 

coefficient and adding the result to the current weight makes 

weight adjustments in standard backpropagation. Two 
Mu1;;iply input vhes 

by current 

connection weights 

Sum up weighted 

connection values for 4 

each node of the layer 

Transform sums by 

squashing fimetion 

Multiply squashed 

values by current 

connection weights 

Adjust connection 

weights based on 

calculated error 

No 

Calculate the 

Yes 

Show the values 

Figure 2-A flowchart of the training and use of a backpropagation neural network. A squashing function is 

a function such as the hyperbolic tangent, which transforms its input into a number between -1 and 1. 
common modifications to this technique were used in the 

training. The first is the addition of momentum that adds 

some portion of the previous weight adjustment to the 

cunent weight adjustment. The second, developed by Tariq 

Samad (19881, is called fast learning. This technique adds a 

multiple of the error from the next lower layer to the 

activation value before the weight update. Equations (1) 

though (3) show the change in weight for smdard 

backpropagation, backpropagation with momentum, and 

backpropagation for fast learning with momentum, 

respectively . 

AW jirsl = lcoef* ejrsl * Xi[.-11 (1) 

.Aw jilSl = Icoef* ejlSj * xifs-ll + momentum* .Aw jic,l (2) 

w jirsl = lcoef* e jlsl * (xic-, + fastlm* eirs-11) 

+momentum* A w jirsl (3) 

where 

Aw - the change in weight to be applied to the 

connection between the ith node in layer s- 1 to the jth node 

in layer s; 

ejrsl = the error at node j of layer s; 

XI[S-II == the current output of the ith node in layer s- 1 ; and 

lcoef, momentum, and fastlrn = the coefficients for learning, 

momentum, and the fast-learning adjustment, respectively 

(Neural Ware 1 99 1 a). 

A final characteristic that needs to be considered in training 

these networks is how many iterations are necessary. We did 

not set a fixed number of iterations, nor did we set a fixed 

level of error that we considered small enough. Rather, we 

observed trends in the reduction of the root-mean-squared 

error (MS) during the training process. During training, 

MS usually drops quickly in the begiming then slows and 

may even climb again, For each structure that we tested, we 

selected parmeters providing the lowest value for RMS. 

Observing the trend as well as the lowest value, helped us to 

avoid some of the spurious, apparently optimum results that 

can occur in any nonlinear iterative fitting process. When 

results appeared spurious, we reinitialized the network to see 

if similar or different optimum values would be generated. 

Weibull with Parameters Recovered from Stand 

Distribution Characteristics 

statistics such as arithmetic mean d.b.h,, quadratic mean 

d.b.h., minimum d.b.h,, and percentiles. Second, the resulting 

system of equations is used to solve for the uhown 

parameters of the dishbution (Matney and Sullivan 1982a, 

Zmoch and others 199 1). 

Matney and Farrar (1 1992) and Fmar and Matney (1 994) 

successfully used the three-parmeter Weibull distribution 

parameter recovery system selected for the benchmark test 

to simulate the unthimed stand yields of cutover, siteprepared loblolly pine (P. taeda L.) plantations and natural 

longleaf pine, respectively. The steps used to construct the 

parmeter recovery system are: 

1. Develop equations for predicting stand minimum d.b.h., 

arithmetic mean d.b.h., and quadratic mean d.b.h. from age, 

surviving trees per acre, and average height of dominants 

and codominants. The arithmetic mean d.b.h. is the firstorder moment of d.b. h., and the quadratic mean d.b.h. is 

the square root of the second-order moment of d.b.h.. 

2. Set the location parameter (a) of the Weibull probability 

distribution equal to one-half the estimated minimum stand 

d.b.h. 

3. Equate the first-order and second-order d.b.h. moments to 

their expected values. That is 

where 

E(dbh) = z, is the expected value of d.b.h., and the 

arithmetic mean d.b.h. (first-order moment of d.b.h.); 

Qdbh '1 = dbhq2 is the expected value of d.b.h.', and the 

square of the quadratic mean d.b.h. (second-order moment of 

d.b,h,); 

b = the scale parameter of the Weibull; 

c -. the shape pmmeter of the Weibull; and 

r =. the value of the gamma function with argument a. 

4. Solve equation (5) for the b parameter and substitute the 

result for the b parameter in equation (6). This yields a 

nonlinear equation involving only the uhown parmeter 

c and the known parameter a that is solvable for c. From 

equation (5), 

Parameter recovery systems for probability distribution 

functions have two parts. First, equations are developed to 

predict the expected values of d.b.h. moments &or order 
b= dbh, - a 

T(l+ lic) 

On substitution of equation (7) for parameter b in equation (6), 

the equation involving only parameters a and c that is in a 

convenient, nonlinear equations solution form is: 

5. Solve equation (8) for the value of c satisfying the 

condition f (c) .= 0 using an algorithm, such as the 

bisection or secant methods, for finding the roots (zeros) 

of nonlinear equations (Burden and others 198 1). 

Weibuli with Regression Predicted ParametersDirect Prediction 

To obtain regression equations to directly predict the 

parameters of the Weibull distribution, we fitted the 

cumulative disbibution function of the Weiberll to the sample 

plot empirical cumulative d.b.h, distributions using the SAS 

Gaussian NONLIN procedure (SAS Institute Inc. 1990). We 

then calculated regression equations to predict the estimated 

parameters from age, surviving trees per acre, and average 

height of dominant and codominant trees. Dell and others 

(198 1) as well as other researchers, used the direct prediction 

method to build unthinned stand-yield models. 

Methods of Model Testing and Comparison 

A NN, as used in this modeling effort, is highly nonlinear; 

and no assumptions about the residual error structure were 

made. Because of this, the NN models are compared to the 

two Weibull models using mathematical measures of 

difference that resemble standard goodness-of-fit statistics, 

but which cannot be used in actual hypothesis testing 

because the probability distributions remain ubom. 

Simply, the numbers calculated are observed to be higher or 

lower than those of competing models. In these calculations, 

the degrees of freedom used is assumed to equal the number 

of obsewations, which is equal to the number of 

distributions compared multiplied by a fixed value of 20, the 

assumed number of classes in each distribution. Each of the 

test observations was estimated with the appropriate sitespecific model using each of the three methods. Then these 

individual results were combined for an overall view of 

model quality. 

The first set of comparators calculated is based on the 

goodness-of-fit statistics that are usually presented for fitted 

models. While none of these calculations adequately 

describes model quality, as a set they provide a 

comprehensive description. Included in this set are meansquared error, fit index (Schlaegel198 l), generalized RZ 

compared to the null model (Anderson-Sprecher 1994), a 

count of the number of times the model diameter class was 

predicted correctly, the number of times the probability of a 

tree falling in the model class was closest to the actual 

probability, the number of times the predicted probability of 

a tree falling in the upper or lower quartile was closest to the 

actual probability, the number of times the calculated x2 was 

lowest, the number of times the calculated KolmogorovSmirnov statistic was lowest, the number of times the 

predicted arithmetic and quadratic means were closest to 

those of the actual distribution, and the number of times that 

the standard deviation of the predicted distribution was 

closest to that of the actual distribution. For the x%nd 

Kolmogorov-Smhov calculations, each check was done for 

the entire distribution and also for the upper and lower 

quartiles. 

It is easy to pick one's favorite test statistic and then 

comparatively rank one model above or below another based 

on it, Unfortunately, this does not tell the whole story of 

how the model performs on various segments of the data. To 

better grasp this, we calculated three indicators of 

performance along the entire diameter distribution. In each of 

the following indicators, error is calculated as the predicted 

value minus the actual value (a positive error indicates 

overprediction). The first is average error for each diameter 

class. It is an indicator of bias. The second is the average 

absolute value of errors. It provides an indicator of the 

typical misfits of the model when positive and negative 

values are not allowed to cancel. Lastly, the maximum 

absolute value of an error allows one to see how poorly the 

model can perform in extreme cases. 

Beyond these tests of prediction quality, we subjected each 

methodology to a sensitivity analysis. This was done by 

comparing the predicted probabilities for each diameter class 

to the same prediction when the input values for age, height 

of dominant trees > 1 in. in diameter, and number of trees per 

acre > 1 in. in diameter were each individually varied by a 10- 

percent increase and a I O-percent decrease. These 

differences were observed by diameter class. The effect this 

procedure had on basal area, arithmetic mean d.b.h., and 

quadratic mean d.b.h. was also examined for each site type 

and for all sites combined. 
Results and Discussion The Weibull distribution with pameters recovered from 

stand-distribution characteristics relies on prediction 

The results of the trained NN are contained in FORTRAN equations of various stand pmameters. These prediction 

subroutines not shown in this paper because of their large equations are presented in figure 3 for each of the three data 

size involving one parameter for every comection. There are sets. 

eters for old-field sites, 120 parmeters for cutover 

sites, and 170 parameters for prepared sites. The Weibull dist~bution with regression predicted 

parameters produces actual prediction equations for the 

~~~__~_Old Field .......................... ..................... 

+ 0.0461age -- 

+ 0.0097age -- 

+ 0.008 1 age - 

~~~~Site Prepared ....................... ...................... 

0.027 1 age - 

+ 0.0007 age + --- 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Cutover ...................................... ------- 

+ 0.0 17 1 age + ------- 

+ 0.0 165 age + ---- 

+ 0.0175age +------- 

Figure 3-Equation sets for predicting the minimum, arithmetic, and quadratic mean d.b.h. in inches from age, surviving trees 

per acre (tpa), and average height in feet of dominant and codominant trees (h,) for the three longleaf pine plantation data 

sets. 
___--------------------------------------------- (')Id Field ----- - ------ --- ---- ----------------------------- 

a = 0.46, R' = 0.00, sy = 0.45 

b = 8.7143 + 0.2343age - 0.0289age(ln tpa) - 

lnc =1.7780------ 

__------------------------------------------- Site Prepared --------------------------------------------- 

a= 0.24,~~= O.OO,sy =0.53 

_---------------------------------------------- Gutover ............................................... a = 0.33, R* = 0.00, s, = 0.86 

b = 8.939 + 0.4232 age - 0.0439age(ln tpa) - 

In c = 1.3879 -- 

Figure 4-Equation sets for predicting the a, b, and c parameters of the Weibull distribution from age, surviving trees per acre (tpa), and 

average height in feet of dominant and codominant trees (hd) for the three longleaf pine plantation data sets. 

Weibull parameters. Figure 4 presents the prediction 

equation sets for each of the three data sets. 

The first comparison of note is a graphical examination of the 

performance of various diameter-distribution recovery 

methods. Figures 5 and 6 show the three predicted 

distributions as lines against a gray backdrop of actual 

example distribution. Figure 5 shows a unimodel distribution 

and simply illustrates that all of the examined methods 

produce a smooth curve that does a fair job of modeling the 

intended distribution. Figure 6 shows a multimodel 

distribution that no method fits very well, but it illustrates 

the flexibility of the NN to fit multimodel distributions. In 

each, the NN provided the best fit based on the x-est 

statistic. 

Mathematical measures of difference are shown in table 2. 

All of the models provided very good fits to the actual 

distributions, but the NN was best by almost all measures. 

Figure 7 is a plot of the average error (or bias) by diameter 

class for each of the tested methodologies. All of the models 

are least accurate in the smaller diameter classes and most 

accurate in the larger. It can also be seen that the regression 

method has the greatest deviation and the NN the smallest. 

Figure 8 shows the average absolute deviations that result 

from each of the three methods. When the positive and 

negative deviations are not allowed to cancel each other out, 

the average errors are much greater, but still the largest 

deviations occur in the smaller diameter classes where they 

have the least influence on volume and value. By this 

measure, all of the metkods are about equal. 

Both figures 7 and 8 show average performance of the 

presented methodologies. Figure 9 shows the worst-case 

performance of them. Once again, the lowest accuracy is in 

the smallest diameter classes and the methods are all about 

the same. The maximum deviations are strongly negative and 
--t Neural network 

H I 6 I I Parameter recovery 

- I Regression 

0 2 4 6 8 10 12 14 16 18 20 

Diameter at breast height (inches) 

Figure 5-Three methods of predicting a diameter distribution compared to an actual unimodel distribution fkom a cutover 

site at age 2 1, with a height of dominant trees equal to 48 feet and 790 trees per acre. 
= = = 8 Neural network 

Parameter recovery 

- Regression 

Diameter at breast height (inches) 

Figure &Three methods of predicting a diameter dislribution compared to a complex multimodel distribution from a cutover 

site at age 43, with a height of dominant trees equal to 8 1 feet and 334 trees per acre. 
Table 2Corne mathematical. measures of difference for artificial neural networks and two methods 

of fitting Weibull distributions 

Comparator 

Neural Parameter Regression 

network recovery Weibull W&bu11 

M ean-squared error 

Fit index 

Generalized R2 

Model class same as actual 

Percentage of trees in model class closest to actual 

Percentage of trees in lower quartile closest to actual 

Percentage of trees in upper quartile closest to actual 

Lowest x2 

Lowest x' in lower quartile 

Lowest x2 in upper quartile 

Lowest Kolmogorov-Smirnov statistic 

Lowest Kolmogorov-Smirnov statistic in lower quartile 

Lowest Kolmogorov-Smirnov statistic in upper quartile 

Arithmetic mean d.b.h. closest to actual 

Quadratic mean d.b.h. closest to actual 

Standard deviation of d.b.h. closest to actual 

disturbingly large. The strong negative bias shows that one 

large error encountered is the presence of trees in a diameter 

class when none is predicted. One cause of this is that, in 

general, smaller diameter trees grow or die out over time, but 

there are several cases where a small diameter tree will 

stagnate and neither grow nor die. The ability to predict this 

phenomenon would greatly enhance our ability to model the 

diameter distributions of longleaf pine stands. 

Our results show that NN can perform at least as well as 

traditional methods and often better. They may be reducible 

to nonlinear models (Sarle 1994), however unwieldy, but their 

strength is that the model form does not have to be specified 

in advance. This is a great advantage because, in spite of 

many efforts in process modeling, we still do not understand 

the processes of growth that would allow us to create 

models not tied to empirical data. Even the Weibull function 

has no biological meaning. It is simply a mathematically 

handy fknction with the ability to assume a variety of 

appropriate shapes (Weibull 195 I). 

There are three principal weaknesses in NN as used in this 

paper. The first is that although a model does not have to be 

specified, the number of hidden nodes and layers and the 

transfer function to be used still must be determined. A 

correct choice of options can make a great difference in a 

modeling effort's success. Also, there are many rules of 

thumb for selecting variables, but it really comes down to a 

matter of trial and error. 

The second limitation is that one cannot count on some of 

the desired constraints that can easily be imposed by 

mathematical model selection. Constrained least squares 

d.b.h. moment-recovery algorithms such as those presented 

in Matney and others (1 990) and in Matney and Belli (1995) 

provide for the calculation of logically constrained residual 

and future d.b.h. distributions. On the other hand, natural 

constraints built into the data sets may allow for 

development of NN models that are almost logically 

constrained without imposing formal constraints. For many 

data sets, the strength of the data should generate NN 

solutions that are naturally constrained. In general, an 

unconstrained solution to a problem will have lower meansquare error than a constrained solution to the same 

problem. In linear space, a constrained model with one or 

more parameter constraints imposed always has a higher 

mean-square error than the same model with no parameter 

constraints. Hence, unnecessary constraints should not be 

imposed on a solution to a problem. However, NN are 

nonlinear and the theory for linear systems is not directly 

applicable. It is possible to have a constrained nonlinear 

estimator with lower mean-square error than the same model 

without parameter constraints. 
Diameter at breast height (inches) 

Figure 7-Average error in probability estimates (predicted to actual) in each diameter class for each of three 

methods of estimating diameter distributions. 
Diameter at breast height (inches) 

Figure 8-Average absolute value of the error in probability estimates in each diameter class for each of three 

methods of estimating diameter distributions. 
Diameter at breast height (inches) 

Figure 9-Maximum absolute value of error in probability estimates (predicted to actual) in each diameter class for 

each of three methods of estimating diameter distributions. Absolute values were used to determine magnitude, but 

signs were kept to indicate direction. 
If constraints are necessary, they can be difficult to apply to 

the output of a NN model when using commercial software. 

Users of grourth-and- yield models require that the residual 

stand after-thinning and before-thinning diameter 

distributions be logical in relation to each other. Likewise, 

they insist that projected diameter distributions be logically 

related to the initial diameter distribution. For thinned stands 

this means, for example, that the estimated diameter 

distribution should not have more trees in a diameter class 

after thiming than before thinning, and that the afterthinning stand should have the desired basal area or trees 

per acre. For growth projections, users expect the future 

distribution to shift to the right and have a nondecreasing 

mean diameter. The NN methods already are strictly 

empirical; and achieving these kinds of constraints is easy if 

one is willing to use additional programming in computer 

models when necessary, or to combine neural methodologies 

with other, more traditional statistical approaches. 

The third limitation is that the network that results from a 

training program appears to be a black box. You can see the 

inputs and the outputs, but the process of moving from one 

to the other is not obvious. In this case, it is implemented as 

a C or FORTRAN program, but an examination of this 

program just reveals so many interacting equations that clear 

relationships are hard to determine. Fortunately, this latter 

difficulty can be easily addressed through the use of 

sensitivity analysis (Klimasauskas 199 1 ). 

We conducted a sensitivity analysis on the input parmeters 

of age, height of dominant trees, and number of trees per 

acre to see what effect changes in these parameters would 

have upon stand basal area, arithmetic mean d.b.h., and 

quadratic mean d.b.h. These results for all site types 

combined are shorn in table 3. This table shows that all of 

the methods are very similar in their sensitivity to input, with 

only the age influence on basal area showing a change that 

exceeded the change in input. Also it shows that trees-peracre has a moderate influence, and the height of dominant 

trees has almost none. These results are rather similar across 

all three site types (not expanded here) with old-field sites 

being the least sensitive and prepared sites the most. 

Conclusion 

Artificial NN are an excellent alternative to the traditional 

method of predicting unthinned stand-diameter distributions 

with estimated Weibull probability functions. The superiority 

of the NN arises because the Weibull is not the correct 

model for the data in this study. On other data sets, when the 

Weibull distribution fits better, either the NN or Weibull 

Table 3-Percentage deviations in some key outputs caused by 10-percent changes in the input 

variables for each of the three examined models 

Result changed 

Age Height of dominant trees Trees per acre 

Increase Decrease Increase Decrease Increase Decrease 

Basal area 

Arithmetic mean d.b.h. 

Quadratic mean d.b.h. 

Basal area 

Arithmetic mean d.b.h. 

Quadratic mean d.b.h. 

Basal area 

Arithmetic mean d.b.h. 

Quadratic mean d.b.h. 

Artificial neural network 

Parameter recovery Weibull 

Regression Wei bull 
probability distribution approaches will perfbrm well. The 

clear advantage of the NN over the parametric function 

modeling techniques is that in almost all cases a solution 

will minhize the RMS error. If the Weibull dishbution or 

other ssumed probability function do not fit the data, the 

modeler is left with the very difficult task of piecing together 

and/or finding a new parametric model form for the problem. 

The next evolutionary step for applying NN in growth-andyield modeling is to find network models for estimating 

diameter distributions after thinning and for estimating 

future diameter distributions using the current distribution. 

Feedforward networks, as used in this paper, are just one of 

many kinds of NN; and investigation of other forms may lead 

to even better results. The NN theory is a rapidly expanding 

arena of study that will find application in many fields. The 

field of forestry is an area ripe with potential for NN 

exploration. 

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Artificial neural networks (NN) are becoming a popular estimation tool. Because they require no 

assumptions about the form of a fitting function, they can free the modeler from reliance on 

parametric approximating functions that may or may not satisfactorily fit the observed data. To 

date there have been few applications in forestry science, but as better MN software and fitting 

algorithms become available, they may be used to solve a wide variety of problems-particularly 

problems where the underlying relationship between predicted and predictors is unknown. We 

benchmark tested an alternative to the traditional Weibull probability distribution function, 

diameter-at-breast-height moment, and direct parameter prediction models for approximating 

stand-diameter distributions. Using a feedforward backpropagation network, we demonstrated that 

NN are a somewhat better option. Unlike Weibull approximations, NN solutions cannot easily be 

mathematically constrained to match known reality constraints, but this difficulty is easy to 

overcome in practice. 

Keywords: Connectionist models, parallel distributed processing systems, parameter recovery, 

Weibull distribution. 
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