U.S. Department of Agriculture

Forest Service

Research Paper SO-164

A Test of the Exponential Distribution

for Stand Structure Definition in Uneven-aged

Loblolly-Shortleaf Pine Stands

Paul A. Murphy

and

Robert M. Farrar
SUMMARY

In this study, 588 before-cut and 381 after-cut diameter (dbh) distributions of uneven-aged

loblolly-shortleaf pine stands were fitted to two different forms of the exponential probability

density function. The left truncated and doubly truncated forms of the exponential were used.

The fitted distributions were evaluated using the Kolgomorov-Smirnov one-sample test at

the 0.05 significance level. The rejection rates for the singly truncated form were 88 and 89

percent, respectively, for before-cut and after-cut stands. The rejections ran 72 and 77 a

percent for the doubly truncated function. Neither function adequately describes the structure

of the stands used in the study, but each may have utility in certain regulation applications.

1981
A Test of the Exponential Distribution

for Stand Structure Definition in Uneven-aged

Loblolly-Shortleaf Pine Stands

Paul A. Murphy

and

Robert M. Farrar

INTRODUCTION

An uneven-aged forest is composed of trees of widely

differing diameters (dbh). Selection management of

such a forest would manipulate the diameter distribution or stand structure so that periodic timber

harvests can be sustained indefinitely. It is not surprising, therefore, that many attempts have been

made to somehow describe the stand structure of

uneven-aged forests.

The French forester de Liocourt observed that the

number of trees in successive diameter classes in an

uneven-aged forest tended to decrease in a smooth

geometric progression, and that this progression was

apparently stable through time (Meyer 1952). The ratio

of the number of trees in any two adjacent diameter

classes tended to be a constant. The ratio of the

number of trees in a given diameter class to those

in the next larger class has been termed “q” in the

literature. A stand structure is balanced if this q tends

to be constant. If the q and other stand attributes tend

to remain the same over time, then the stand is stable

and exhibits little or no change as time progresses.

Meyer and Stephenson (1943) analyzed the diameter

distributions of virgin beech-birch-maple-hemlock forests in Pennsylvania using graphs of the logarithms

of numbers of trees over diameter class. A straight line

relationship indicated that the stands were balanced,

and the structures were apparently stable.

A balanced uneven-aged stand structure can be

represented by the equation

Yi = ke-axi, a >0 , (1)

where Yi is the number of trees in the ith diameter

class, Xi is the midpoint value of the diameter class

in inches, and k and a are constants. Meyer (1952)

demonstrated how the constants could be determined

graphically. The q is related to this function in the

following manner,

g = eaw. (2)

The w is the width of the diameter class. Leak

(1963) showed how equation (1) could be fitted by

least squares to determine the coefficients a and

k, and later (Leak 1964) developed a procedure to

describe stand structure for unbalanced (nonconstant

q) uneven-aged stands. In this procedure, the q’s for

4-inch diameter classes are calculated, and these values

are regressed on diameter. The q values for these

large classes exhibited a linear relationship for Leak’s

unbalanced stand data.

Leak ( 1965 1 discussed the exponential distributionthe probability form of equation (l)-in an expository

paper. It is

f(x) = re-‘x ,x>Oandr>O,

= 0, elsewhere.

(3)

There is a functional relationship q = exp( rw) between

the parameter r and q, where w is as previously

defined. He also described the left-truncated form

of the exponential p.d.f. (probability density function). The truncated form can be applied in situations

where no trees below a certain threshold diameter

are recorded. Although the functional form of the

exponential, equation (l), has been used (e.g., Schmelz

and Lindsey 1965) to describe uneven-aged stands, the

probability form apparently has not.

The Weibull p.d.f. has gained wide acceptance in

even-aged applications. It is

h(x) = (b)[(x-a)/blc-lexp(-[(x-a)/blcj,

x 2 a, b > 0, c > 0,

= 0, elsewhere.

Murphy is Principal Mensurationist, Farrar is Mensurationist, Forestry Sciences Laboratory, Monticello, Arkansas, Southern

Forest Experiment Station, Forest Service- USDA, in cooperation with the Department of Forestry and the Arkansas Agricultura1

Experiment Station, University of Arkansas at Monticello.

1
General properties are described by Bailey and Dell

(1973). There have been two recent applications of the

Weibull to uneven-aged situations. Hyink and Moser

(1979) used the three-parameter Weibull for describing

stand structures of uneven-aged mixed hardwood stands,

and Stiffl employed the left-truncated two-parameter

Weibull for representing the diameter distributions

of mixed-species Appalachian hardwoods. The Weibull

distribution degenerates into an exponential p.d.f.

when the shape (or “c”) parameter equals one.

Although more general distributions, like the Weibull,

are available, the exponential has certain advantages.

Like the Weibull, its cumulative distribution function

has a closed form solution. But, unlike the Weibull, its

parameter can be determined from the first sample

moment and does not require iterative techniques for

parameter estimation. The exponential would obviously

be simple to use in the operational management and

regulation of uneven-aged forests. These potential

advantages justify investigating the exponential for

use in describing diameter distributions, even though

more flexible ones exist. In this study, the exponential’s

capacity to describe the diameter distributions of

uneven-aged loblolly-shortleaf pine stands managed

under the selection system was investigated.

PROCEDURE

Data

The data are from a cutting cycle study (Reynolds

1959, 1969), a methods-of-cutting study (Grano 1954),

and unpublished research at the Crossett Experimental

Forest and surrounding area in southern Arkansas.

Average site index for loblolly pine (Pinus taedu L.) is

about 90 feet at base age 50 (U.S. Forest Service

1976). The information was obtained from periodic

lOO-percent cruises of plots and compartments ranging

from 2.5 to about 40 acres in size. Records were

kept of stand inventories, harvest cuts, salvage cuts,

and thinnings by one-inch dbh classes for trees 3.6

inches and larger. Collection of most of these data

started in 1937 and continued into the late 1960’s

on a periodic basis.

Al1 species were tallied in the methods-of-cutting

study, and al1 were initially inventoried in the cutting

cycle study by nominal lo-acre subcompartments. In

1948, all hardwoods on the cutting cycle study were

controlled by injection or girdling, and thereafter

tallies were kept only for pine by 40-acre compartments. Nine of the compartments from the cutting

cycle study were inventoried in 1979 to provide

‘Stiff, C. T. 1979. Modeling the growth dynamics of natural

mixed-species Appalachian hardwood stands. Unpublished

Ph. D. Dissertation, Virginia Polytechnic Institute and State

University, Blacksburg, 206 p.

additional data; the methods-of-cutting study was

also remeasured.

Subcompartment data were used in lieu of compartment data, when available, to augment the information

base from the cutting cycle study. Stand tables before

cut and, after cut were analyzed separately for the

pines, loblolly and shortleaf (Pinus echinuta Mill.).

Hardwoods were not included. If a harvest cut occurred

within half of a growing season of the stand inventory,

an after-cut stand table was calculated by subtracting

the cut tally from the before-cut stand table. Depicting

stand structure after cut is of interest in projecting

changes in diameter distributions of harvested stands

over time. The total number of before-cut stand tables

was 588; after-cut stand tables, 381.

Analysis

Since the data were collected only for trees in the

4-inch diameter class and above, the left truncated

form of the exponential was used. It is

f(x) = re-r(x-a), x 2 a, a>O, r >O,

= 0, elsewhere.

(4)

The value of the parameter “a” is known and fixed

at 3.5 inches. The parameter r is

r = l/(v--a), (5)

In this case, the variable x is tree diameter. The

population mean of x is known, because lOO-percent

inventories were conducted on each subcompartment

or compartment. Mean stand diameter was calculated

for each diameter distribution, and r determined by

equation (5).

The left truncated exponential is bounded below by

the parameter a, but is unbounded above. Thus use of

the function implies that there is a small, but real

probability that a tree of infinitely large size can occur.

To evaluate the impact of this phenomenon, the doubly

truncated form of the exponential p.d.f. was also fitted

to the stand table data. It is (Aroian 1965)

g(x) = reWrx ,a<xlb,O<a<b,r>O, - (6)

e-ar -e -br

= 0, elsewhere,

where x is tree diameter. The parameter a is known

and fixed at 3.5 inches, and b was set equal to the upper

limit of the largest diameter class encountered. The b

parameter varies by compartment and the date of the

lOO-percent tree tally. The parameter r is the root of

the equation

l/r - p + (ae-ar - be-br) = 0.

(e-ar -e-br)

(7)

A solution can be found by using Newton’s method for

finding roots or some alternative numerical procedure.

2
The doubly truncated exponential was fitted with a

equal to 3.5 inches, b equal to the upper limit of the

largest diameter class, and r equal to the root of

equation (7). Equation (7) can be re-written to illustrate

how the parameter r for the doubly truncated exponential differs from that of the singly truncated form

in terms of ~ 1 and the other parameters. This version is

l/r = a - (ae-ar -be-br)/(e-ar -e-br), (8)

Recall that l/r = ~ 1 - a in the singly trrmcated

exponential. In equation (S), the population mean is

reduced by a term which is a function of al1 three

parameters of the distribution.

Goodness-of-fit was evaluated by comparing the

actual cumulative distribution to the fitted cumulative

using the Kolgomorov-Smirnov one-sample test with a

0.05 significance level.

RESULTS

The summary of the tests may be seen in table 1. The

results are conclusive. Neither the singly nor doubly

truncated exponential p.d.f.‘s adequately describe the

loblolly-shortleaf pine stands used in the analysis.

The rejection rate for the singly truncated form was

88 and 89 percent respectively for before-cut and aftercut stands. For the doubly truncated distribution,

the rejection rates were 72 percent and 77 percent.

The doubly truncated version did perform somewhat

better than the singly truncated p.d.f.

Most of the rejections occurred because of lack of fit

in the lower diameter classes. Though the exponential

is a simple distribution, its simplicity is bought at

the expense of curve flexibility. Even the addition of an

upper bound, the b parameter, did not help much.

Table l.- Kolgomorou-Smirnou test for fitting the exponential

distribution to stand tables of uneuen-aged loblollyshortleaf pine

Null hypothesis rejectionsl

Stand table Number of

observations Left truncated Doubly truncated

exponential exponential

Before cut 588 519 422

After cut 381 341 291

l5-percent level.

Figure 1 illustrates a typical diameter distribution and

a fitted doubly truncated exponential p.d.f., and table 2

lists the observed stand table and the corresponding

estimates for both forms of the exponential. The

Kolgomorov-Smirnov test was significant at the 0.05

leve1 for this particular stand, an indication of inade2,000

1,600 -

OBSERVE0

? 1,200 -

>

8 \

\

\

6

s

4 6 8 10 12 14 16 18 2 0 2 2 2 4 ¿

DBH (INCHES)

Figure l.-Observed and fitted doubly-truncated exponential

diameter distribution, uneuen-aged loblolly-shortleafpine stand (39 acres).

Table 2.-Comparison of actual versus fitted diameter distributions, uneuen-aged loblolly-shortleaf pine stand

(39 acres)

Dbh

(inches) Observed Left truncatedexponential

Doubly truncated

exponential

4

5

6

7

8

9

1 0

l l

1 2

1 3

1 4

1 5

1 6

1 7

1 8

1 9

2 0

2 1

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

_----_-___----- Numberoftrees-----------------

1840 1160 1152

893 896 893

457 693 691

237 535 535

156 414 415

159 320 321

184 247 249

209 191 193

169 148 149

168 114 116

158 8 8 9 0

126 6 8 6 9

9 0 5 3 5 4

8 0 4 1 4 2

5 3 3 1 3 2

4 9 2 4 2 5

3 3 1 9 1 9

2 6 1 5 1 5

8 1 1 1 2

4 9 9

1 7 7

1 5 5

0 4 4

0 3 3

3 2 3

1 2 2

3
quate fit to the data. The smaller diameters are much

more subject to volatile change from such things as

damage from ice storms and suppression mortality

because of high stand densities. Also, these stands

were essentially regulated by volume control in the

sawtimber component and the smaller trees were not

given strict objective regulation. The preponderance of

trees in these classes together with their more erratic

distributions make these sizes especially sensitive to

goodness-of-fit tests. As seen in figure 1 and table 2,

largest deviations from the actual distribution are

in these lower classes. Furthermore, the KolgomorovSmirnov test is very rige rous in this application due

to the large number of trees per observation. The

number of trees for most observations in this study

exceeded 1000 individuals. The critica1 value for the

Kolgomorov-Smirnov test at the 5-percent leve1 for

samples exceeding 35 is 1.36/vTñ (Siegel 1956), where

n is the sample size. For example with n = 100,

the critica1 value is 0.136; when n = 1000, the critica1

value declines to .043.

In conclusion, the different forms of the exponential

p.d.f. do not adequately describe the diameter distributions of uneven-aged loblolly-shortleaf pine stands

used in this test. A more flexible curve form and,

perhaps, a different density measure such as basal

area are alternatives worthy of further investigation.

However, it should be pointed out that none of the

stands in the study were regulated by consistent and

objective application of a q to specify after-cut

structures. Where such regulation is practiced, the

exponential may have real value for specifying current

residual structures and predicting future ones.

REFERENCES CITED

Aroian, L.A.

1965. Some properties of the conditional Weibull

distribution. In: nansactions of the 19th ‘I&hnical

Conference of the Ameritan Society for Quality

Control, p 361-368.

Bailey, R. L., and T. R. Dell.

1973. Quantifying diameter distributions with the

Weibull function. For. Sci. 19:97-104.

Grano, C. X.

1954. Re-establishment of shortleaf-loblolly pine

under four cutting methods. J. For. 52:132-133.

Hyink, D. M., and J. W. Moser, Jr.

1979. Application of diameter distributions for yield

projection in uneven-aged forests. In: Forest

resource inventories. p. 906-916, Vol. II, Colorado

State University, Ft. Collins.

Leak, W. B.

1963. Calculation of “q” by the least squares me$hod.

J. For. 61:227-228.

Leak, W. B.

1964. An expression of diameter distribution for unbalanced, uneven-aged forests. For. Sci. 10:39-50.

Leak, W. B.

1965. The J-shaped probability distribution. For.

Sci. 11:405-409.

Meyer, H. A.

1952. Structure, growth, and drain in balanced

uneven-aged forests. J. For. 50:85-92.

Meyer, H. A., and D. D. Stephenson.

1943. The structure and growth of virgin beechbirch-maple-hemlock forests in northern Pennsylvania. J. of Agrie. Res. 67:465-484.

Reynolds, R. R.

1959. Eighteen years of selection timber management

onthe Crossett Experimental Forest. U.S. Dep.

Agr. Tech. Bull. 1206, 68 p.

Reynolds, R. R.

1969. Twenty-nine years of selection timber management on the Crossett Experimental Forest. U.S.

Dep. Agr. For. Serv. Res. Pap. SO-40, 19 p.

South. For. Exp. Stn., New Orleans, La.

Schmelz, D. V., and A. A. Lindsey.

1965. Size-class structure of old-growth forests in

Indiana. For. Sci. ll :258-264.

Siegel, S.

1956. Nonparametric statistics for the behavioral

sciences. 312 p. McGraw-Hill, New York.

U.S. Forest Service.

1976. Volume, yield, and stand tables for secondgrowth southern pine. U.S. Dep. Agr. Misc. Publ.

50 (Rev.), 202 p.
MURPHY, PAUL A., and ROBERT M. FARRAR.

1981. A test of the exponential distribution for stand structure

definition in uneven-aged loblolly-shortleaf pine stands.

U.S. Dept. Agrie. For. Serv. Res. Pap. SO-164, 4 p. South.

For. Exp. Stn., New Orleans, La.

In this study, 588 before-cut and 381 after-cut diameter

distributions of uneven-aged loblolly-shortleaf pine stands were

fitted to two different forms of the exponential probability

density function. The left truncated and doubly truncated forms

of the exponential were used.

Additional Keywords: Selection management, diameter classes.