United States

Department of

Agriculture

Forest Service

Southern Forest

Experiment Station

New Orleans,

Louisiana

Research Paper

SO-233

0

r4s, m

A Comparison of Tree

Volume Estimation

Models for Forest

Inventory

John F. Kelly and Roy C. Beltz
SUMMARY

Eight different linear regression models were tested for ability to predict

timber inventory for four tree species in Northeast Texas. Sample trees were

selected for the Forest Survey by the variable plot (prism) method. Each model

was tested using two weighting schemes for weighted least squares regressionprobability weights and optimal heteroscedasticity-correcting weights. In general, the probability weights performed best for inventory prediction. Spurr’s

combined formula model, incorporating DsH, fitted with probability weights

made the best predictions.

May 1987
A Comparison of Tree Volume Estimation

Models for Forest Inventory

John F. Kelly and Roy C. Beltz

INTRODUCTION

Developing forest inventory estimates often involves predicting tree volumes from only diameter at

breast height (d.b.h.) and/or merchantable height.

Prediction equations based on these two factors from

a small number of intensively measured trees can

lead to significantly different inventory estimates.

The nationwide Forest Survey is particularly susceptible to these effects because of the interval between

successive surveys and the importance of estimating

inventory trends.

The principal goal of the Forest Survey at its inception in 1928 by the McSweeney-McNary Act was to

estimate the total volume and area of the forest resource. As user demands become more intense, concerns center more on inventory components such as

species and d.b.h. classes, and estimates of change,

i.e., growth and removals. But these estimates are not

nearly as reliable as those for the total inventory

since they deal with smaller numbers of trees or relatively small changes in volume. Estimates of change

deal with differences in volumes of those surviving

trees which were measured in previous surveys and

the addition or removal of trees from the inventory.

The increasing concerns for reliable estimates of inventory components and volume changes requires

that Forest Survey volume estimation techniques be

as accurate as possible.

Techniques may differ enough between successive

surveys to cloud the change statistics. While procedures are standard, slight variations creep in as survey analysts continually try to improve their methods. Volume equations, for example, may change

slight!y !I!-%? to a chmge ifi the moclel or estimat.inn

methods. Past volumes are usually recalculated to

new standards (equations) for comparison, However,

the subtle effects of these volume equations can affect

evidence of inventory trends.

Data from the 1985 Forest Survey of east Texas

were used to investigate eight linear models and two

weighting schemes in predicting timber inventory

volumes. Specific objectives of the study were to:

(1) evaluate linear models incorporating d.b.h. and

merchantable height as explanatory components; (2)

evaluate linear models incorporating d.b.h. only; and

(3) evaluate two alternative weighting schemes for estimation of linear regression parameters: (a) probability-based weights, which emphasize compatibility

with sample selection probabilities, and (b) optimal

heteroscedasticity-correcting weights, which emphasizes conformance to a key assumption of the general

linear regression model. The basis for evaluations

were comparisons of timber inventory predictions

with estimates developed from Smalian formula tree

volumes and sample probabilities for numbers of trees

per acre.

LINEAR REGRESSIONS AND TIMBER

INVENTORY SAMPLES

The linear regression model, with parameters estimated by ordinary least squares (OLS) techniques, is

often used as a convenient means of estimating tree

and inventory volumes. This classical linear regression model involves specific assumptions, some of

which may be violated by application to timber inventory situations. Furthermore, effective use depends on

the correct specification of the models; i.e., a linear

relationship should exist between predicted variables

(such as tree volume) and independent variables

(d.b.h., height, etc.).

The general linear regression model has the form:

Yj = PO + pIX,i + @2X‘& + .*. + Ej

where Y = dependent variable;

X = indepandent vttri&lc?;

p = regression parameters;

E = variation not explained by the regression

model (error term).

The six basic assumptions regarding application of

the general linear regression model are: (1) the mean

of the error term (E) is zero; (2) the variance of the

error term is constant; i.e., the data are homoscedastic; (3) the error terms for different observations are

uncorrelated; (4) independent variables are nonJohn F. Kelly is research forester and Roy C. Beltz is project leader for FIA, Southern Forest Experiment Station, Starkville, MS.

1
stochastic; (5) no perfect collinearity exists between

independent variables (in models containing two or

more independent variables); and (6) the number of

observations is greater than the number of independent variables.

Assumption 2 is commonly violated when using linear regressions to predict tree volume. The absence of

homoscedasticity, i.e., heteroscedasticity, occurs

when samples include a range of tree sizes, from small

trees to large ones. Since the height and form of large

diameter trees may vary considerably, due to various

stand and site factors influencing the growth, the volume will likewise vary. Small diameter trees that are

shorter and contain less total volume will usually

vary less in volume. Tree samples selected to estimate

timber inventory for a variety of forest conditions

thus may not be homoscedastic because of volume

variance attributable to factors not explicit in regression models. Figure 1 illustrates the variance of total

tree volume by diameter class of sample trees in this

study.

If heteroscedasticity is, in fact, present in a sample

of trees, there are some consequences from using OLS

techniques to estimate linear regression parameters.

Fortunately, the regression coefficients estimated

under these conditions are not biased, and their estimates tend to converge on the true parameters as the

sample size increases. The variances of the coefficient

estimates will, however, be larger than those estimated for homoscedastic samples. Furthermore, the

variances for the coefficients calculated using standard OLS methods may be biased, appearing to be

smaller than their actual value. The extreme consequence posed by this situation is the rejection of the

hypothesis that regression coefficients are equal to

zero when the hypothesis is actually true. Barring

this potential problem, however, the presence of heteroscedasticity will still allow unbiased estimates of

regression coefficients, but these estimates will not

have minimum variance.

The use of weighted least squares is a common remedy for heteroscedasticity. This technique involves

multiplying the regression equation by a factor (say,

Zi) which equalizes the variance of the error term for

the sample data. For instance, if it is determined that

the variance of the error term is proportional to the

square of an independent variable, then the appropriate weighting factor would be the inverse of the

squared variable. A technique for determining correct

weighting factors will be discussed later.

Weighting techniques may alter regression results

when sample character%tics vary according to the

weighting factors. This may occur when the inverse of

tree diameter (or some transformation such as diameter squared) is used as a weighting factor. In this case,

the influence of large diameter trees in calculating

regression parameters will be discounted, while the

influence of small diameter trees will be amplified.

Since the relationship between volume and predictive

variables (diameter, height) may vary depending on

tree diameter, the use of weights related to diameter

may skew results.

While the use of weights related to the inverse of

tree diameter may potentially skew regression results, the actual outcome will depend on the specific

relationship between the sample and weights selected. The selection of weights is particularly important for probability-related samples where the number of large trees selected is greater than that

obtained from simple random samples. Because of the

700

l

600 -

ocí 0 9 7 7

l í

l l l

í I I 1 1 I I 1 I 1 I I

5 6 7 6 9 10 II I2 I3 1 4 15 IQ 17 I6 IS 2 0 21

ONE INCH DIAMETER CLASS

Figure L-Comparison of tree volume variance. Average variance by diameter class, regression

sample.

2
emphasis on large trees in a prism sample, the use of

compatible weights is necessary to fully realize the

advantages of the sample.

In the special case where the weighting variable Zi

is the number of trees per unit area represented by a

sample tree, the product of the weight and tree volume (YiZi) would be the volume per unit area represented by that tree. These weights would then be compatible with the probability of tree selection. For

prism samples, the number of trees represented by

each sample tree is a linear function of the inverse of

the squared tree diameter.

Because of the possible consequences posed by correcting for heteroscedasticity in a prism sample, it is

not clear if the procedure is desirable. The weights for

correcting heteroscedasticity will likely be quite different from probability weights. Researchers using

probability-based homoscedastic samples have not

agreed on the desirability of using probability

weights. DuMouchel and Duncan (1983) indicated

that probability weights may not be desirable in all

situations. Holt, Smith and Winter (1980) found that

probability weighted regressions provided good results as long as selection probabilities were related to

an independent variable in the model. The current

problem, however, deals with probability-based heteroscedastic sample data and with procedures desirable for timber inventory estimation.

METHODS

Periodic forest surveys are conducted by the Forest

Inventory and Analysis (FIA) unit of the Southern

Forest Experiment Station, USDA-Forest Service,

as a part of a nationwide forest survey. Plot locations

are at the intersections of a three-mile grid. Information was collected for Northeast Texas according to

standard procedures (USDA Forest Service 1985).

Plots consist of 10 point-samples with point centers

systematically placed on a grid 66 feet apart. Sample

trees are selected using a 37.5 basal area factor prism.

For each tree, diameter is recorded at l-foot stump,

d.b.h., mid-saw-log height, saw-log height, mid-bole

height, and merchantable height (fig. 2). Merchantable limits are 5 inches d.b.h., and 4 inches top

diameter outside bark (d.o.b.) for poletimber. Softwood sawtimber limits are 9 inches d.b.h. and 7 inches

top d.o.b.; hardwood sawtimber limits are 11 inches

d.b.h. and 9 inches top d.o.b. Top merchantability limits may also be reached where deformities or other

defects are present. Bark thickness at d.b.h. is measured and deducted to get d.b.h. inside bark. Upper

stem bark values are derived from the ratio of

d.b.h.i.bJd.b.h.0.b. according to procedures in STX

(Grosenbaugh 1964). Cubic foot volume for each section in each tree is computed using the Smalian formula. These sectional volumes are then summed for

HEIGHT

I-+,D.B.H. (4.5 FT)

Figure 2.-Measurement points on FIA tally trees.

each tree to obtain total tree volume; these calculated

volumes are referred to as “actual” volumes throughout this paper.

Data from 676 forests plots in the Northeast Texas

survey region (fig. 3) were screened for four common

species-shortleaf pine (Pinus echinata), loblolly pine

(Pinus taeda), sweetgum (Liquidambar styraciflua),

and post oak (Quercus stellata var. stellata).

Each tree had to qualify as growing stock1 and be at

least 5 inches d.b.h. and no greater than 28 inches

d.b.h. Plots were randomly allocated to one of two

‘Growing stock is defined as trees that contain, or potentially

contain, at least 1 la-foot saw log of minimum grade specification;

rough culls, rotten culls, and dead trees do not qualify as growing

StOCk.

3
Figw s 3.-Plot locations of sample trees, Northeast Texas survey

region.

data sets. Qualifying trees in one data set were used

to develop the equations. The other set was used to

test them. Sample sizes for the two data sets are given

in table 1.

Individual tree gross volumes were used to tit 8

linear regressions using (1) probability weights for

individual tree selection, and (2) optimal weights for

correcting heteroscedasticity. Regression models

evaluated were:

1. CF = l.&-, + &D + &D2 + l&H f E

2. CF = BiD + P2D2 + &H + E

3. CF = PO + &D2H + E

4. CF = &D2H + E

5. CF = PO + &D + p2D2 + E

6. CF = PO + P1D3 + E

7. CF = PO + &D2 + E

8. CF = P1D2 + E

where

CF = gross cubic foot volume in the merchantable

bole;

D = d.b.h. outside bark;

H = merchantable height;

pi = coefficients to be estimated;

E = error term.

Equations 2, 4, and 8 are identical to 1, 3, and 7,

respectively, except for the intercept (&,) term. Equations l-4 include merchantable height and d.b.h.;

Table L---Number of trees in sample data sets, Northeast Texas

Shortleaf pine

Loblolly pine

sweetgum

Post oak

Regression sample Test sample

1,131 1,059

1,133 1,260

580 618

371 390

equations 5-8 include only d.b.h. Equation 3 is

Spurr’s combined variable formula, and equation 4 is

the constant form-factor equation (Spun: 1952). Equation 5 is a 2nd degree parabolic formula which was

used by Cunia (1964).

Weighted least squares were used to estimate coefficients. Probability weights used were 1/D2; the nnmber of trees per acre represented by each tree in a

prism sample is a linear function of 1/D2. Optimal

weights for correcting heteroscedasticity were determined using a maximum likelihood function as discussed below.

The test sample was used to simulate the effect of

the equations on timber inventory estimation. Results were compared to Smalian formula volumes calculated from measurements by survey field crews.

DETERMINATION OF OPTIMUM WEXGHTS

FOR CORRECTING HETEROSCEDACTICITY

Squares of non-weighted regression residuals plotted against predicted volumes indicated that the

residuals were an approximate function of the

squared volume. Plots of squared residuals against D2

and H further indicated that variability was an approximate function of the square of each of these variables, i.e., D4 and H2. This would suggest that weights

of the form 1/D4H2 and l/D4 would be appropriate for

correcting heteroscedasticity.

Meng and Tsai (1986) reported that these weights

(1/D4H2 and 1/D4) have been traditionally considered

as appropriate by several sources, but the optimum

exponent for D, say, A, will vary with different samples.

Following an approach derived by Meng and Tsai

(1986) from Box and Cox (196412, the optimum

weights in the forms (1/DXHj2 and (1iDx)2 were determined for all species and all equations.

Optimum values for the exponent of D (h) were determined by an iterative process which estimated the

likelihood function for different A. Separate iterations

were run for each equation; weights of the form

(1/DAHj2 were used for equations l-4, and weights of

the form (1/Dx12 were used for equations 5-8. The

%menta (1971) presents a similar, though more general approach.
maximum value for the likelihood function indicated

that an optimum A had been found. Values for A were

incremented by .025 until an optimum value was

reached; i.e., likelihood functions were estimated for

values such as 2.5, 2.525, 2.55, etc.

The log likelihood function for X is

n

L, = $JSS,--X 2 ln(DJ+iln(n)-;--iln2*

i=l

where Lh = the maximized log likelihood for A;

SS, = the sum of squares for the error term

resulting from weighted regression;

n = the total number of trees in the sample;

In = the natural logarithm for the variable

indicated.

After finding the value for X providing the maximum L, a confidence interval was established for A at

a probability level of 0.95 (01=.05) as follows (Meng

and Tsai 1986):

A: Lh” - J-Q < 112 x21df, .05

where L; = the maximized log function for the optimum A;

x2iaf ,05 = the chi-squared distribution for 1 degree of freedom and a = .05.

By following these procedures for all species and

equations, the optimum values for A and confidence

intervals were produced (table 2). For this analysis

the optimum values of A were used for comparisons.

RESULTS

Regression Analysis

The F test and adjusted R2 regression statistics indicate that all eight equations are effective predictors

of volume. The F test indicated that all models were

significant at the 95 percent probability level; adjusted R2 values were at least 38.

Plots of regression residuals (on weighted scales)

indicate that the basic assumptions of zero mean for

the error term and uncorrelated error terms hold true.

Possible collinearity of D and H in equations 1 and

2 was considered. Although the relationship between

D and H approaches linearity, it is not perfectly linear, nor particularly strong. Thus, the estimated regression coefficients for equations 1 and 2 are valid

(unbiased).

Other than heteroscedasticity, no other violations

of remaining basic assumptions for linear regression

models were noted.

There were some coefficients for independent variables not statistically different from zero, all occurring in fitting equation 5 and involving the D variable. The use of probability weights produced 3 of the

4 instances where the D coefficient was statistically

insignificant at the .95 probability level. Alternately,

3 of the 4 optimally weighted regressions for equation

5 had D coefficients that were statistically significant.

Negative intercept terms occurred in most regressions for equations 5 and 7. The intercept term for

equation 1 (probability weighted) fitted to post oak

data was also negative.

Negative signs for D occurred in all regressions for

equations 1,2, and 5. In each of these equations D is

paired with D2; coefficients for the latter variable

were positive in all cases.

Comparison of Predicted Volumes for the Test

Sample

The test sample was used to compare predicted volumes with actual volumes calculated from Smalian’s

formula. Three different criteria were used-the occurrence of unreasonable values (i.e., negative or zero

tree volumes), the prediction of total inventory volumes within a 1 percent standard, and the prediction

of one-inch diameter class volumes within a 10 percent standard. Inventory volumes were estimated by

first calculating individual tree volumes, using regression coefficients, then obtaining the product of

tree volume and the number of trees per acre for the

37.5 basal area factor prism sample. Finally, values

Table 2.-Optimum d.b.h. weighting values and confia’ence intervals for linear regressions, by species and regression muo!el

Linear

regression Weight

model variable

1 (1/D”H)2

2 (l/D”H)2

3 (1/D”Hj2

4 (1lDAH)2

5 (l/DA)2

6 MY)2

7 (l/D92

8 (imv

‘At.95 probability level.

ShortIeaf pine

2.2(2.05-2.325)

1.925(1.775-2.05)

2.075c1.9752.175)

1.95t1.852.075)

2.375(2.225-2.5)

2.525C2.3752.675)

2.325(2.2-2.45)

1.65(1.525-1.775)

Optimum values and confidence intervals1 for k

Loblolly pine sweetgum

2.325C2.2252.425) 2.325(2.1?5-2.475)

1.95(1.85-2.075) 2.175(2-2.35)

1.975(1.9-2.075) 2.35(2.2-2.5)

1.75(1.65-1.85) 2.025(1.875-2.175)

2.225(2.125-2.35) 2.525(2.35-2.7)

2.55C2.4252.65) 2.8(2.625-2.95)

2.17X2.075-2.275) 2.5(2.3-2.65)

1.550.425-1.65) 2.375(2.125-2.6)

post oak

2.575(2.375-2.775)

2.5C2.3252.675)

2.375(2.175-2.575)

2.3(2.125-2.475)

2.45C2.25.2.65)

2.7(2.45-2.9)

2&X2.25-2.65)

1.875(1.675-2.1)

5
for appropriate diameter classes and species were

summed. All comparisons were made for the four species individually, and for all species (all trees) combined. One-inch diameter classes ranged from 5 to 21

inches with all trees 21-28 inches being placed in the

latter class.

Among the probability weighted regressions, negative or zero tree volumes occurred with equations 1

and 2, usually in the 5-inch diameter class. Such values also occurred in the 6- and 7-inch-diameter classes

for the pines. No negative volumes were predicted

using optimally weighted regressions. For the balance of the analysis, all negative volumes produced

from the probability weighted regressions were set to

zero.

The ability of the models to predict inventory volume was compared using the test sample. Inventory

volumes represented by individual trees were first

predicted, then mean volumes calculated both for individual species and the entire test sample. These

mean volumes were finally divided by the mean

Smalian formula inventory volume (CF x trees per

acre); the results are displayed in table 3. Instances

where the predicted volume was within 1 percent of

actual volume are noted.

Among equations including both D and H components, the probability weighted regressions generally

performed better than optimally weighted regressions

in predicting inventory volume. For the entire test

sample (all species combined), equation 3, fitted using

probability weights, predicted inventory with an

error of only .l percent; the use of optimal weights

produced an error of .6 percent. Equations 1 and 2,

fitted with probability weights, predicted inventory

for the combined sample within 1 percent error; other

regressions produced errors greater than 1 percent.

Among equations using only D as an explanatory

component, three-5,6, and 7-predicted total inventory volume within 1 percent of actual values, using

both weighting schemes. Equation 5 (optimal

weights) and equation 6 (probability weights) predicted total inventory with no error. Equations 5 and

7 (probability weights) were within .5 percent error.

Analysis of mean predicted inventory volume by

diameter class indicates distinct differences among

equations. Predicted volumes for individual trees of

the test sample, calculated using regression coefficients, were multiplied by the appropriate factor for

trees per acre to develop mean inventory volumes for

the 17 one-inch diameter classes. These mean volumes were then compared to the respective Smalian

formula inventory volumes using a Chi-square test as

described by Freese (1960). This test was formulated

to estimate ability to simultaneously predict all 17

diameter class inventory volumes within 10 percent of

actual volume. The Chi-square statistics, presented in

table 4, were calculated as follows:

17

X217a, .05 = (1.962/.102> C, [(Xi/pi) - 112

i=l

where x2iTti, ,c5 = Chi-square distribution with 17

degrees of freedom, a = .05;

xi = predicted inventory volume for

diameter class i;

pi = actual inventory volume for diameter class i.

The critical x2 value for these calculations is 27.6;

statistics less than this value indicate the model will

predict all 17 diameter class volumes with less than

10 percent error, unless a l-in-20 chance of a random

event occurs.

Only equation 3, using either probability weights or

optimal weights, was able to predict inventory volumes by diameter class with no greater than 10 percent error for all test groups (the 4 individual species

and total sample). Equations 4 and 5 also did well

relative to other equations, meeting this test for the

total sample and at least 2 individual species; equation 4, optimally weighted, met the test for 3 individual species as well as the total sample. Equations 2,6,

and 8 did not predict volumes for any of the test

groups within this standard using either probability

weights or optimal weights. Equation 1 produced only

one instance under each weighting scheme where the

10 percent standard for diameter class volume prediction was met.

Table B.-Predicted mean inventory volumes as a proportion of actuul mean inventory volumes, by species and type of regression (test sample)

Equation

Entire sample Shortleaf Loblolly Post Entire sample Shortleaf Loblolly Post

(all trees) Dine Dine sweetfmm oak (all trees) pine pine Sweetgum oak

1 .993* .994* .979 1.019 1.003* .938 .938 .919 .982 .942

2 .991* 1.001* .971 1.011 1.003* .917 .930 .889 .946 .938

3 1.001* 1.004* .993* 1.022 .993* 1.006* 1.002* 1 .ooo* 1.030 1.012

4 .973 .994* .943 .997* .973 1.017 1.009* 1.006* 1.058 1.025

5 .997* 1.004* .975 1.023 1.028 1.000* 1.003* .981 1.025 1.028

6 1.000* .987 1.003* 1.024 .995* 1.007* .989 1.004* 1.039 1.040

7 .998* 1.005* .974 1.023 1.029 .994* 1.002* .973 1.010* 1.025

8 1.131 1.108 1.127 1.210 1.103 1.033 1.045 1.060 .947 1.017

Probability weighted regressions Optimally weighted regressions

*Predicted mean within 1 percent of actual meaus.

6
EVALUATION OF MODELS AND WEIGHTS

There are tradeoffs between the probability weights

and optimal weights used for linear regression in this

study which dealt with a prism sample for the purpose

of inventory estimation. Probability weights are used

for regression in order to realize the advantages provided by emphasizing the selection of large, highvolume trees. Optimal weights for correcting heteroscedasticity will provide estimates of regression

coefficients with smaller variances, although this did

not appear to be an important advantage in this study

because variances were typically very small. The optimal weights in this study, however, did reduce the

influence of large trees, when compared to probability

weights. Coefficients estimated using both weighting

schemes can be expected to represent true population

values. Furthermore, the probability weights reduce,

but do not eliminate, the effects of heteroscedasticity.

Correct specification of linear regression models requires linear relationships between independent and

predicted variables. Although equation 5 may appear

to violate this specification requirement (D is not linearly related to CF), the quadratic form of the equation apparently provides a powerful linear transformation for volume prediction where D is the only

explanatory component available. One reason for postulating equation 6 was the linear relationship implied between the units of the D3 and CF variables.

Results from the test sample, however, indicate that

while equation 6 predicts total inventory volume accurately, it does not perform well for predicting volumes by diameter classes.

One advantage of through-the-origin regression

models (equations 2,4, and 8) was thought to be the

avoidance of negative volume prediction. Since equations 3 and ‘I-comparable to equations 4 and 8, but

with an intercept term-produced no negative volumes, the use of the through-the-origin models for

this purpose was unnecessary. Equation l-comparable to equation 2, but with an intercept term-did

produce negative volumes, but only when fitted with

probability weights. Equation 2 also predicted negative volumes when fitted with probability weights;

these values resulted because of the size of the D coefficient, which was negative. Thus, through-the-origin

models would appear either ineffective or tmnecessary for this purpose.

Table 5 displays summary test information by indicating (1) the number of test groups where total inventory volumes were predicted within 1 percent of

actual volume; and (2) the number of test groups

where one-inch diameter class volumes were predicted (simultaneously) within 10 percent of actual

volume, as indicated by the Chi-square tests. Test

groups are the four individual species, plus the entire

combined test sample, i.e., the columns in tables 3 and

4.

In general, it appears that optimally weighted regressions provide no clear advantage over the probability weighted regressions. Although the use of optimally weighted regressions slightly improved

predictions for two equations (4 and 71, other equations provided better estimates when fitted with probability weights (1, 2, 3, and 6). Table 5 indicates no

difference for the two weighting schemes for equations 5 and 8.

Based on the evidence summarized in table 5, equation 3, fitted with probability weights, is clearly the

best performer. Equation 3 also performed well when

fitted with optimal weights.

Among models using only D as a component, equation 5, fitted with probability weights, and equation 7

fitted with optimal weights, are the best performers.

Coefficients for equations 3, 5 and 7, derived using

both weighting schemes, are presented in table 6.

In summary, results from this study indicate that

linear regressions can be an effective tool for invenTable 4.-Chi-square statistics for predicting inventory volume within 10 percent of actual volume for 17 one-inch diameter classes simultaneously at the 95 percent confidence level (a = .05)‘; by species and type of regression (test sample)

Probability weighted regressions Optimally weighted regressions

Entire sample Shortleaf Loblolly Post Entire sample Shortleaf Loblolly Post

Equation (all trees) pine pine sweetgum oak (ail trees) pine pine Sweetgum oak

1 51.5 27.1* 62.6 30.0 43.1 35.2 34.9 59.7 25.3* 31.7

2 199.7 154.5 139.7 102.9 44.4 131.7 111.7 158.8 167.1 45.4

3 11.9* 5.0* 11.7* 19.2* 22.1* 1.8* 5.8* 2.9* 16.4* 17.6*

4 24.2* 18.2* 42.5 33.7 16.0” 18.1* 18.1* 26.7* 52.5 18.1*

5 15.2* 35.4 21.3* 44.9 17.3* 6.2* 32.0 14.2* 59.7 18.3*

6 809.6 463.6 568.1 286.3 300.4 185.6 296.8 191.5 450.4 392.7

7 39.2 52.0 39.5 56.0 18.6* 27.1* 43.7 37.3 48.7 17.4*

8 1435.1 1183.8 1614.0 962.4 369.5 1012.4 1122.7 1580.8 775.9 320.9

17

1x2r~~, .eb = (l.96?.102) Jl [(Xi/pi) - 112, where xi = predicted inventory volume; h = actual inventory volume. Note: asterisks (*I indicate

statistics less than the critical value of 27.6; these predictions are within the 10 percent limit at the indicated confidence level.

7
Table 5.-Number of test groups’ meeting 1 percent total inventory and 10 percent diameter class

volume standards, by type of regression

Probability weighted regressions Optimally weighted regressions

Number within 1% Number within 10% Number within 1% Number within 10%

total inventory volume standard for total inventory volume standard for

Equation volume diameter class volume diameter class

1 3 1 0 1

2 3 0 0 0

3 4 5 3 5

4 2 3 2 4

5 2 3 2 3

6 3 0 2 0

7 2 1 3 2

8 0 0 0 0

‘Five test groups are included: 4 individual species and all trees combined.

Table 6.-Coefficients for gross cubic volume equations for probability and optimally weighted

regression moo!& 3,5, and 7

Species

Regression model,

variables’ $0

Coefficient estimates

$1 I32

Shortleaf pine GP)Po,PI(D~H) 0.2578117 0.002676204 .........

(30) 0.1971193 0.002684152 .........

(~P)Bo,PI(D)&(D~) - 2.52757 - 0.2882 0.1898756

(50) - 1.05001 - 0.619361 0.2063557

(~P)Po,BI(D~) - 3.96955 0.1771888 .........

(70) - 3.63574 0.1734781 .........

Loblolly pine C@)&J,&(D~H) 0.8668721 0.002384048 .........

(30) 0.3031817 0.002529929 .........

(~P)Po,PICWMD~) - 3.2068 - 0.0931619 0.1645083

(50) - 0.479005 - 0.681579 0.1917292

(~P)Po,MD~) - 3.68215 0.1606684 .........

(70) - 3.3994 0.1574598 .........

Sweetgum (3p)P,,P,(D2H) 0.2976087 0.002579246 .........

(30) 0.2848158 0.002605488 .........

(5~)9o,&(DMz(D~) 0.63616 - 0.908149 0.1940991

(50) 2.268433 - 1.30103 0.215419

(~P)P,,P,(D~) - 3.54821 0.1508756 .........

(70) - 2.69169 0.1375616 .........

Post oak (3p)Po,&(D2H) 0.3127841 0.002961708 .........

(30) 0.1054914 0.003102175 .........

(~P)Po,PD),P~(D~) - 1.28018 - 0.0736108 0.1071492

(50) - 0.657807 - 0.217158 0.1143979

(7~)@0~31@‘) - 1.6376 0.1039153 .........

(70) - 1.52254 0.102326 .........

IProbability weighted regression estimates are designated (p); optimally weighted regression

estimates are desigued (0).
tory estimation including component volumes by species and d.b.h. class. While some linear models perform well using either probability weights or optimal

weights for correcting heteroscedasticity, other linear

models predict volumes with substantial errors, particularly when fitted with optimal weights. Evidence

from this study indicates that data from prism samples, where the probability of selection is a function of

tree diameter, can be effectively fitted to certain linear models using probability weights. Optimal

weights do not appear to provide any distinct advantages for timber inventory estimation. The best equation in this test-using diameter and merchantable

height components combined in a single variable, and

fitted with probability weights-predicted total inventory within a fraction of a percent. This equation

also predicted inventory volumes for individual species by l-inch diameter classes within 10 percent of

actual volumes.

LITERATURE CXTED

Box, G.E.P. and Cox, D.R. 1964. An analysis of transformation. J.R. Stat. Sot. B, 26: 211-243.

Cunia, T. 1964. Weighted least squares method and

construction of volume tables. For. Sci., 10:180-

191.

DuMouchel, W.H. and Duncan, G.J. 1983. Using sample survey weights in multiple regression analysis

of stratified samples. J. of the Am. Stat. Assoc.,

78:535-543.

Freese, F. 1960. Testing accuracy. For. Sci. 6:139-

145.

Grosenbaugh, L.R. 1964. STX-FORTRAN 4 program-for estimates of tree populations from 3P

sample-tree-measurements. Pacific SW Forest and

Range Expt. Sta. U.S. For. Serv. Res. Pap. PSW-13.

49 p.

Holt, D.; Smith, T.M.F.; and Winter, P.D. 1980. Regression analysis of data from complex surveys.

J. R. Stat. Sot., 143~474-487.

Kmenta, J. 1971. Elements of econometrics. Macmillan Publ. Co., New York. 655 p.

Meng, C.H. and Tsai, W.Y. 1986. Selection of weights

for a weighted regression of tree volume. Can. J.

For. Res., 16:671-673.

Spurr, S.H. 1952. Forest Inventory. Ronald Press,

New York. 476 p.

USDA Forest Service. 1985. Forest survey inventory

work plan. Southern Forest Experiment Station,

Forest Inventory and Analysis Res. Work Unit,

Starkville, MS. Unpubl. 57 p.

9

Kelly, John F., and Beltz, Boy C. 198’7. A comparison of tree

volume estimation models for forest inventory. Bes. Pap. SO233. New Orleans, LA; U.S. Department of Agriculture,

Forest Service, Southern Forest Experiment Station. 9 p.

Tests are analyzed for eight different linear regression models to determine abilities to predict inventory for four tree

species in Northeast Texas.

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