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\def\citename {Unger} %"Author"
\def\citeemail {unger@sfasu.edu} %Use: {\href{mailto://\citeemail} {FirstName \citename}}
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\author{
{\href{mailto:\citeemail}{Daniel~\citename}}, %Change only 1st name of 1st author
{\href{mailto:stovalljp@sfasu.edu} {Jeremy Stovall}},
{\href{mailto:boswald@sfasu.edu} {Brian Oswald}},
{\href{mailto:dkulhavy@sfasu.edu} {David Kulhavy}},
{\href{mailto:hungi@sfasu.edu} {I-Kuai Hung}}
}
\affiliation {
%---------------
\large\scshape {
} \\
%---------------
%
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%
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\small\it{\href{http://www.sfasu.edu/}{Stephen F Austin State University, Texas, USA}} \\
%$^2$\small\it{{ETH-Zürich, Brüggliäcker 37, 8050 Zürich, Switzerland.}}
}
\def\yourtitle
{{
A Test of the Mean Distance Method for Forest Regeneration Assessment
}} %need double {{for \\ e.g.: {{Title \\ Subtitle}}
\def\yourkwords
{
Key Words: sampling, plot, distance, accuracy, seedlings.
}
\def\yourabstract
{
A new distance-based estimator for forest regeneration assessment, the mean
distance method, was developed by combining ideas and techniques from the
wandering quarter method, T-square sampling and the random pairs method. The
performance of the mean distance method was compared to conventional 4.05
square meter plot sampling through simulation analysis on 405 square meter
blocks of a field surveyed clumped distribution and a computer generated
random distribution at different levels of density of 100, 50 and 25{\%}.
The mean distance method accurately estimated density on the random
populations but the mean distance method estimates were more variable than
those of 4.05 square meter plot sampling. The mean distance method
overestimated actual density and was less precise than plot sampling when
both methods were tested on the clumped populations. The optimum sample
sizes needed for the mean distance method to achieve the same precision as
4.05 square meter plot sampling at all three density levels, for both the
random and clumped spatial distributions, were at least 10 times larger than
the sample size used for 4.05 square meter plot sampling.
} %----------------------------------------------------------------------
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{\citename}\citeetal}}~(\issueyear)/\mcfnshead}}
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%\\\\ {{\bf S\l {} owa kluczowe:} Polskie slowa kluczowe.}
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\section{Introduction}
Estimating the density of regeneration, or number of seedlings per unit
area, on a given site is important to foresters for assessing existing
regeneration, determining reforestation needs, and determining if
reforestation efforts have been successful. A variety of sampling methods
have been developed for estimating regeneration density. The majority of the
methods fall into two general categories: plot sampling and distance
sampling (Payandeh and Ek, 1986). Plot sampling is the traditional approach
that involves establishing fixed size plots within an area, counting the
trees within each plot, and then converting the tree counts to a density
estimate. On the other hand, distance sampling involves measuring the
distance(s) from a sample point or tree to another tree(s) within an area
and then using this distance(s) to estimate density. Our objective was to
compare the performance of a new distance-based method, known as the mean
distance method, as an alternative to traditional plot sampling. Both
methods were evaluated through computer simulation analysis on 405 square
meter blocks (0.1 acre) of a field surveyed clumped distribution and a
computer generated random distribution at different density levels of 100,
50 and 25{\%}.
\section{Distance Sampling}
Distance sampling has attracted the attention of researchers over the past
50 years as a means of estimating density. Its main attraction is that it is
fast, easy to use, and one or more distances are always recorded at each
sample point. In contrast, plot sampling can sometimes be a very time
consuming process, boundary trees may be overlooked, and some plots may have
no tallies.
Past attempts to develop a robust distance-based density estimator have not
been very successful. In general, distance estimators are not robust and
tend to be biased when the spatial distribution of the population under
consideration does not represent a spatially random distribution (Persson,
1971). The lack of robustness is of concern because plants in natural
populations tend to be aggregated, not distributed randomly (Patil et al.,
1979). The major weakness of density estimators is that their bias is
dependent on the spatial distribution of the population (Delince, 1986).
Some distance estimators have been shown to be unbiased over a wide range of
spatial patterns if the estimators are adjusted according to the spatial
pattern, but this adjustment would necessitate additional tests to determine
a population's spatial distribution before estimating density.
Pollard (1971) used a maximum likelihood method to estimate forest density
from a random point using distances between points and provides examples of
using two nearest trees to a random point where the standard deviation to
the second tree is reduced by 30. He found that the number of sample points
required to estimate density with a prescribed accuracy does not depend on
the density being measured and estimates of large diameter tree densities
were reliable. Hanberry et al.~(2011) compared the Pollard (1971) and the
Morisita (1957) methods for estimation of tree density by analyzing spacing
methods and concluded that the Morisita estimator outperformed the Pollard
estimator in non-random and clustered distribution under typical forest
conditions. Although researchers have studied distance sampling in
quantifying vegetation, no study to date has provided a distance based
method of density estimation as an alternative to traditional plot sampling.
This research was undertaken to reinvigorate the desire and need for a fast
and efficient method of density estimation as an alternative to plot
sampling.
This study evaluated a new distance-based density estimator that combines
the attractive features of the wandering quarter method (Catana, 1963),
T-square sampling (Aherne and Diggle, 1978; Diggle, 1975; Diggle, 1977) and
the random pairs method (Cottam and Curtis, 1949; Cottam and Curtis, 1955).
The performance of the new distance estimator, which was evaluated through
simulation analysis, was proposed as a statistically robust, fast and easier
estimator to implement than traditional plot sampling.
\section{Methods}
The new method of density estimation, to be known as the mean distance
method, incorporates the feature of making multiple distance measurements
between trees in one general direction from the wandering quarter method
(Catana 1963). Because most natural stands tend to be aggregated,
directional measurement gives the estimator mobility and forces it out of
clumped areas into open areas, or out of open areas into clumped areas,
depending on the location of the sample point. By sampling through a
population and not remaining stationary, the estimator is better able to
determine the overall average distance between trees. This overall average
distance is used to determine the average area occupied per tree. The
inverse of the average area per tree is the density estimate which was
incorporated from the random pairs method for ease of calculation. The
feature of measuring the distance from one tree to its nearest neighbor
across a 180 degree line was incorporated from T-square sampling to simplify
locating seedlings in the field.
\begin{figure}[ht!]
\centerline{\includegraphics[width=.5\textwidth]{UngerEtAl_v2_W2T1.eps}}\vspace{-6pt}
\caption{Procedure of the mean distance method assuming $s$ equals three. The
circles represent trees, $X$ is a sample point, and $A$ is the closest tree to
sample point $X$. Measure line $A-B$ because $B$ is the nearest tree to $A$ lying beyond a
line drawn through $A$ which is perpendicular to line $X-A$; measure line $B-C$ because
$C$ is the nearest tree to $B$ lying beyond a line drawn through $B$ which is
perpendicular to line $A-B$; and measure line $C-D$ because $D$ is the nearest tree to
$C$ lying beyond a line drawn through $C$ which is perpendicular to line $B-C$.}
\label{fig1}
\end{figure}
The procedure (Fig. 1) for obtaining the necessary distances for the mean
distance method at each sample point within a population follows:
\begin{enumerate}
\item Beginning with a randomly located sample point ($X)$, locate the closest tree ($A)$.
\item Measure the distance from the closest tree ($A)$ to its nearest neighbor ($B)$ lying beyond a line drawn perpendicular to the sample point-closest tree line (line $X-A)$ which intersects the closest tree ($A)$.
\item Next, measure the distance from the last tree measured ($B)$ to its nearest neighbor ($C)$ lying beyond a line drawn perpendicular to the last measured distance line (line $A-B)$ which intersects the last tree measured to ($B)$.
\item Continue measuring distances as described in step 3, up to $s$ number of times.
\end{enumerate}
\begin{table}[htb]
\begin{center}\vspace{-6pt}
\caption{Seedlings per hectare of the computer generated random and field
surveyed clumped spatial distributions at 100, 50 and 25{\%} of density
level.}\vspace{3pt}
\begin{tabular}{p{1.1in}p{0.6in}l} \hline\hline
Spatial Distribution & Density Level & Actual Density \\
~ & (\%) & (seedlings/hectare) \\ \hline
Random & 100 & 70642 \\
Random & 50 & 35321 \\
Random & 25 & 17661 \\
Clumped & 100 & 70617 \\
Clumped & 50 & 35296 \\
Clumped & 25 & 17636 \\ \hline
\end{tabular}
\label{tab1}
\end{center}
\end{table}
By assuming the area occupied by a tree is represented by a hexagon and that
the average of the distances measured represents the average distance
between trees, one can apply the following formula discussed by Cottam and
Curtis (1949, 1955) to estimate density at each sample point:
\[
di{\rm =}\frac{{\rm 43560}}{{\rm 0.8661}{\left(si\right)}^{{\rm 2}}}
\]
where \textit{di} = population density estimate per acre at sample point \textit{i}, \textit{si} = average of the distances measured at sample point \textit{i } in feet.
Density can then be estimated for the entire population by using the following formula:
\[
D=\frac{\sum^n_1{di}}{n}
\]
where \textit{D} = estimated population density, \textit{n} = number of
sample points.
The estimated variance of the population (\textit{V}) is calculated by the following
formula:
\[
V=\frac{\sum^n_1{{(di-D)}^2}}{(n-1)}
\]
\begin{table*}[htb!]
\begin{center}
\caption{Average density estimate, average variance of the mean and average
coefficient of variation for the mean distance method on 100, 50 and 25{\%}
density levels of the random distribution using 25 replications of sample
size 30.}
\begin{tabular}{llllll} \hline\hline
%\begin{tabular}{|p{0.4in}|p{0.5in}|p{0.8in}|p{0.8in}|p{0.9in}|p{0.9in}|} \hline
Density & Distances & Actual & Average
Density & Average Variance & Average Coefficient \\
Level & Per Point & Density & Estimate & of the Mean & of Variation \\
(\%) & (\textit{s}) & (seedlings/hectare) & (seedlings/hectare) & ~ & (\%) \\ \hline
& 1 & 70,642 & 2,813,641 & 26,603,279,495,000 & 183.3 \\
& 2 & 70,642 & 295,807 & 6,744,809,851 & 27.8 \\
& 3 & 70,642 & 235,722 & 1,886,445,916 & 18.4 \\
& 4 & 70,642 & 207,981 & 844,705,527 & 14.0 \\
100 & 5 & 70,642 & 193,203 & 435,475,897 & 10.8 \\
& 6 & 70,642 & 187,140 & 322,285,948 & 9.6 \\
& 7 & 70,642 & 177,934 & 221,114,382 & 8.4 \\
& 8 & 70,642 & 182,906 & 205,928,119 & 7.8 \\
& 9 & 70,642 & 176,190 & 188,782,443 & 7.8 \\
~ & 10 & 70,642 & 174,352 & 147,528,450 & 7.0 \\
& 1 & 35,321 & 454,571 & 118,149,145,990 & 75.6 \\
& 2 & 35,321 & 163,961 & 2,119,511,619 & 28.1 \\
& 3 & 35,321 & 115,208 & 301,273,863 & 15.1 \\
& 4 & 35,321 & 103,411 & 146,876,446 & 11.7 \\
50 & 5 & 35,321 & 100,114 & 117,295,903 & 10.8 \\
& 6 & 35,321 & 95,796 & 82,007,273 & 9.5 \\
& 7 & 35,321 & 95,051 & 59,227,165 & 8.1 \\
& 8 & 35,321 & 94,737 & 53,315,460 & 7.7 \\
& 9 & 35,321 & 92,109 & 41,623,915 & 7.0 \\
~ & 10 & 35,321 & 93,015 & 35,810,502 & 6.4 \\
& 1 & 17,661 & 211,039 & 19,220,959,454 & 65.7 \\
& 2 & 17,661 & 78,442 & 567,246,649 & 30.4 \\
& 3 & 17,661 & 58,826 & 113,531,697 & 18.1 \\
& 4 & 17,661 & 50,704 & 38,863,435 & 12.3 \\
25 & 5 & 17,661 & 47,970 & 27,890,612 & 11.0 \\
& 6 & 17,661 & 47,555 & 23,512,661 & 10.2 \\
& 7 & 17,661 & 46,016 & 17,061,496 & 9.0 \\
& 8 & 17,661 & 46,760 & 16,726,337 & 8.7 \\
& 9 & 17,661 & 44,949 & 10,517,945 & 7.2 \\
~ & 10 & 17,661 & 45,337 & 8,560,942 & 6.5 \\ \hline
\end{tabular}
\label{tab2}
\end{center}
\end{table*}
\section{Data Analysis and Results}
The mean distance method was evaluated using computer simulation. Distances
were measured to the nearest 0.254 centimeters (0.1 inch). Simulation
allowed testing the mean distance method at different population densities
and spatial distributions, and the determination of a reasonable value of
$s$, the number of distances measured per sample point. The mean distance
method was compared to 4.05 square meter plot sampling.
The first step of the simulation analysis involved obtaining clumped and
random seedling populations on which to conduct the density estimation.
Mapped seedling data representing a clumped distribution of 2,859 seedling
locations in a 405 square meter block within a 15-month-old clearcut in
central Pennsylvania were used. An artificial random seedling distribution
was generated in computer with 2,860 seedling locations within a 405 square
meter block, representing the same density as the field surveyed data. These
two distributions were further resampled to 50 and 25{\%} population levels
to represent 3 different populations representing 100, 50, and 25{\%} of
actual density to test the density estimator across variable population
densities (Tab. 1). A Fortran based computer program called REGEN was
written to simulate the mean distance method and the 4.05 square meter plot
sampling. The program was written to allow the following parameters to be
varied: (1) the density estimator, (2) the spatial pattern of regeneration,
(3) the seedling density, (4) the number of replications, (5) the sample
size, and (6) the number of distances measured per sample point for the mean
distance method. The program calculates the average density estimate,
average variance of the mean and average coefficient of variation over all
replications of a given sample size.
REGEN was used in initial trials of the mean distance method to determine
the optimum value of $s$, the number of distances measured per sample point.
Twenty-five replications of sample size 30, with $s$ ranging from one to ten,
were tested on three densities in the random distribution (Tab. 2). The
results from these initial trials were evaluated to determine the
proportions that the mean distance method overestimated actual density of
all three density levels of random distribution at all distances tested
(Tab. 3). The mean distance method's overestimation of actual population
density decreased as the number of distances measured per sample point
increased, but remained approximately equal for a given density regardless
of population density when the value of $s$ was greater than one. The average
variance of the mean for the mean distance method decreased as the number of
distances measured per sample point increased, but was higher at any
distance and density than the average variance of the mean for 4.05 square
meter plot sampling when tested on the three densities in the random
distribution with 500 replications of sample size 30 (Tab. 4). Average
coefficient of variation (CV) for the 4.05 square meter plot ranged from 3
-- 7{\%}, with higher average CV's at lower densities (Tab. 4). By contrast,
average CV's were 10 -- 11{\%} for the mean distance method at all densities
if 5 distances were measured, and as low as 6 -- 7{\%} when up to 10
distances were measured (Tab. 2).
\begin{table}[htb]
\begin{center}
\caption{Overestimation factors ([average density estimate -- actual
density]/actual density) for the mean distance method on three densities of
the random distribution using 25 replications of sample size 30.}\vspace{3pt}
\begin{tabular}{llll} \hline\hline
Distances & \multicolumn{3}{c}{Overestimation Factor} \\
Per Point & \multicolumn{3}{c}{ - - - Density Level - - -} \\
(\textit{s}) & 100\% & 50\% & 25\% \\\hline
1 & 38.83 & 12.87 & 11.95 \\
2 & 4.19 & 4.64 & 4.44 \\
3 & 3.34 & 3.26 & 3.33 \\
4 & 2.94 & 2.93 & 2.87 \\
5 & 2.73 & 2.83 & 2.72 \\
6 & 2.65 & 2.71 & 2.69 \\
7 & 2.52 & 2.69 & 2.61 \\
8 & 2.59 & 2.68 & 2.65 \\
9 & 2.49 & 2.61 & 2.55 \\
10 & 2.47 & 2.61 & 2.57 \\ \hline
\end{tabular}
\label{tab3}
\end{center}
\end{table}
The overestimation factors were analyzed to choose the optimum value of s.
The optimum value of $s$ was determined to be the smallest value where the mean
distance method overestimated the actual population densities of all three
density levels of the random distribution at approximately the same
proportions. A large number of distances would render the method ineffective
when considering the time required to complete a sample. The optimum value
of $s$ was determined to be three.
\begin{table*}[bth!]
\begin{center}
\caption{Average density estimate, average variance of the mean and average coefficient of variation for 4.05 square meter plot sampling on three densities of the random distribution using 500 replications of sample size 30.}\vspace{3pt}
\begin{tabular}{lllll} \hline\hline
Density & Actual & Average Density & Average Variance & Average Coefficient \\
Level & Density & Estimate & of
the Mean & of Variation \\
(\%) & (seedlings/hectare) & (seedlings/hectare) & ~ & (\%) \\ \hline
100 & 70,642 & 70,358 & 5,796,740 & 3.4 \\
50 & 35,321 & 35,185 & 2,970,126 & 4.9 \\
25 & 17,661 & 17,791 & 1,467,529 & 6.8 \\ \hline
\end{tabular}
\label{tab4}
\end{center}
\end{table*}
Because the mean distance method overestimated the three random distribution
densities at all values of $s$ it was inferred that the mean distance method was
underestimating the average distance between trees. The mean distance method
was measuring less than the actual distance needed, resulting in the
overestimation bias. It was determined that the average distance obtained
between trees when $s$ equals three represented only 55{\%} of the average
distance needed (Tab. 5).
\begin{table}[htb]
\begin{center}
\caption{Expansion factors (average measured distance/actual distance
needed) for the mean distance method on the three random distribution
densities using 25 replications of sample size 30 when $s$ equals three.}%\vspace{-6pt}
\begin{tabular}{lp{0.8in}p{0.7in}l} \hline\hline
Density & Average Measured & Actual Distance & Expansion \\
Level & Distance & Needed & Factor \\
(\%) & (cm) & (cm) & ~ \\ \hline
100 & 22.12 & 40.41 & 0.55 \\
50 & 31.65 & 57.15 & 0.55 \\
25 & 44.30 & 80.85 & 0.55 \\ \hline
\end{tabular}
\label{tab5}
\end{center}
\end{table}
The original program REGEN was altered by dividing the sample point average
distance obtained between trees by 0.55, thereby increasing the average
physical distance measured between trees and hence the average area occupied
per tree, to adjust for the overestimation bias and provide an accurate
average density estimate and to test for robustness via a new formula. The
adjusted formula is:
\[di=\ \frac{43560}{0.8661{\left(\frac{si}{0.55}\right)}^2}\]
where \textit{di} = population density estimate per acre at sample point $i$, \textit{si} = average of the distances measured at sample point $i$ in feet.
\begin{table*}[bt]
\begin{center}
\caption{Average density estimate, overestimation, average variance of the
mean and average coefficient of variation for the mean distance method
(using the 0.55 adjustment when $s$ equals three) on the three random
distribution densities using 500 replications of sample size 30.}\vspace{-6pt}
\begin{tabular}
%{|p{0.7in}|p{0.8in}|p{0.9in}|p{0.7in}|p{1.1in}|p{1.1in}|} \hline
{llllll} \hline\hline
Distances & Actual & Average
Density & Overestimation & Average Variance & Average Coefficient \\
Per Point & Density & Estimate & & of the Mean & of Variation \\
(\textit{s}) & (seedlings/hectare) & (seedlings/hectare) & (\%) & ~ & (\%) \\ \hline
3 & 70,642 & 71,600 & 1.014 & 185,406,741 & 19.0 \\
3 & 35,321 & 35,847 & 1.015 & 37,010,695 & 17.0 \\
3 & 17,661 & 17,715 & 1.003 & 15,394,389 & 22.1 \\ \hline
\end{tabular}
\label{tab6}
\end{center}
\end{table*}
The mean distance method was then tested on the three random distribution
densities with 500 replications of sample size 30 when $s$ equals three (Tab. 6).
The results of the tests indicate the overestimation bias was corrected. The
mean distance method's average density estimates for all three random
population densities were within 1.5{\%} of true population density and in
close agreement with 4.05 square meter plot sampling results. The results
also show that the mean distance method's average variance of the mean when
using the 0.55 adjustment was higher than the average variance of the mean
of 4.05 square meter plot sampling for an equivalent density, sample size
and number of replications. Average CV's were relatively high for the mean
distance method, ranging from 19 -- 22 {\%} where $s$ = 3 (Tab. 6), as compared
to 3 -- 7{\%} for the 4.05 square meter plot methodology (Tab. 4).
\begin{table*}[htb]
\begin{center}
\caption{Average density estimate, overestimation, average variance of the
mean and average coefficient of variation for the mean distance method
(using the 0.55 adjustment when $s$ equals three) on the three clumped
distribution densities using 500 replications of sample size 30.}\vspace{-6pt}
\begin{tabular}{llllll} \hline\hline
Distances & Actual & Average
Density & Overestimation & Average Variance & Average Coefficient \\
Per Point & Density & Estimate & & of the Mean & of Variation \\
(\textit{s}) & (seedlings/hectare) & (seedlings/hectare) & (\%) & ~ & (\%) \\ \hline
3 & 70,617 & 654,535 & 9.269 & 131,109,814,367 & 55.3 \\
3 & 35,296 & 646,187 & 18.307 & 4,896,588,697,138 & 342.4 \\
3 & 17,636 & 44,801 & 2.540 & 396,354,779 & 44.4 \\ \hline
\end{tabular}
\label{tab7}
\end{center}
\end{table*}
\begin{table*}[htb]
\begin{center}
\caption{Average density estimate, average variance of the mean and average
coefficient of variation for 4.05 square meter plot sampling on the three
clumped distribution densities using 500 replications of sample size 30.}\vspace{3pt}
\begin{tabular}{lllll} \hline\hline
Density & Actual & Average Density & Average Variance & Average Coefficient \\
Level & Density & Estimate & of
the Mean & of Variation \\
(\%) & (seedlings/hectare) & (seedlings/hectare) & ~ & (\%) \\ \hline
100 & 70,617 & 67,910 & 113,729,757 & 15.7 \\
50 & 35,296 & 34,229 & 29,709,656 & 15.9 \\
25 & 17,636 & 17,179 & 8,314,850 & 16.8 \\ \hline
\end{tabular}
\label{tab8}
\end{center}
\end{table*}
The mean distance method was then tested on all three clumped population
densities with 500 replications of sample size 30 to evaluate its robustness
across different spatial patterns (Tab. 7). Plot sampling at 4.05 square
meters was also tested on all three clumped population densities with 500
replications of sample size 30 (Tab. 8). The average density estimates for
the mean distance method were not close to the true clumped population densities and the average variance of the mean and average coefficient of variation for the mean
distance method at all three clumped population densities when $s$ equals three
was higher than the average variance of 4.05 square meter plot sampling for
an equivalent density, sample size, and replication. Even 4.05 square meter
plot sampling performed more poorly when seedlings were clumped, with
average CV's of 16 -- 17{\%} (Tab. 8) as compared with 3 -- 7 {\%} for
randomly distributed seedlings (Tab. 2).
\begin{table*}[htb]
\begin{center}
\caption{Density estimate confidence interval (95{\%}) for 4.05 square meter
plot sampling on all three densities of the clumped and random distributions
using 500 replications of sample size 30.}\vspace{3pt}
\begin{tabular}{llllll} \hline\hline
Spatial & Actual & Average
Density & \multicolumn{3}{p{1.1in}}{95\% Confidence} \\
Distribution & Density & Estimate & \multicolumn{3}{p{1.1in}}{Interval} \\
& (seedlings/hectare) & (seedlings/hectare) & \multicolumn{3}{p{1.1in}}{(seedlings/hectare)} \\ \hline
Random & 70,642 & 70,358 & 65,541 & - & 75,174 \\
Random & 35,321 & 35,185 & 31,737 & - & 38,631 \\
Random & 17,661 & 17,791 & 15,363 & - & 20,214 \\
Clumped & 70,617 & 67,910 & 46,579 & - & 89,239 \\
Clumped & 35,296 & 34,229 & 23,327 & - & 45,129 \\
Clumped & 17,636 & 17,179 & 11,411 & - & 22,944 \\ \hline
\end{tabular}
\label{tab9}
\end{center}
\end{table*}
\begin{table*}[htb]
\begin{center}
\caption{Density estimate confidence interval (95{\%}) and optimum sample
size needed to achieve 4.05 square meter plot sampling precision for the
mean distance method (using the 0.55 adjustment when $s$ equals three) on all
three densities of the clumped and random distributions using 500
replications of sample size 30.}\vspace{3pt}
\begin{tabular}{lllllll} \hline\hline
Spatial & Actual & Average
Density & \multicolumn{3}{p{1.2in}}{95\% Confidence} & Optimum \\
Distribution & Density & Estimate & \multicolumn{3}{p{1.2in}}{Interval} & Sample \\
& (seedlings/hectare) & (seedlings/hectare) & \multicolumn{3}{p{1.2in}}{(seedlings/hectare)} & Size \\ \hline
Random & 70,642 & 71,600 & 44,366 & - & 98,832 & 960 \\
Random & 35,321 & 35,847 & 23,680 & - & 48,014 & 374 \\
Random & 17,661 & 17,715 & 9,868 & - & 25,560 & 315 \\
Clumped & 70,617 & 654,535 & 0* & - & 1,378,717 & 34,584 \\
Clumped & 35,296 & 646,187 & 0
\# & - & 5,071,834 & 4,944,441 \\
Clumped & 17,636 & 44,801 & 4,982 & - & 84,617 & 1,430 \\\hline
\multicolumn{2}{l}{${}^{*}$Actual Value: -69,647} & & & & & \\
\multicolumn{2}{l}{${}^{\#}$Actual
Value: -3,779,461} & & & & & \\
\end{tabular}
\label{tab10}
\end{center}\vspace{.2in}
\end{table*}
Ninety-five percent confidence intervals of density estimate for both 4.05
square meter plot sampling and the mean distance method, plus the optimum
sample size needed for the mean distance method to achieve 4.05 square meter
plot sampling precision, were determined for all three density levels of the random and clumped spatial distributions (Tabs. 9-10). All confidence
intervals of the density estimate for 4.05 square meter plot sampling were
smaller than the confidence intervals of density estimate for the mean
distance method for an equivalent density level and spatial pattern when $s$ equals
three. The sample sizes needed for the mean distance method to achieve 4.05
square meter plot sampling precision at any density level and spatial
pattern were at least ten times larger than the sample size of 30 used by a
4.05 square meter plot.
\section{Discussion}\addtolength{\textheight}{-.2truein}
When calculating density at each sample point, the mean distance method's
average density estimates were in close agreement with the random
distributions, as were the density estimates for 4.05 square meter plot
sampling, but the mean distance method was more variable than plot sampling.
The mean distance method overestimated actual density and was less precise
than 4.05 square meter plot sampling when both methods were tested on the
clumped distributions. Since the optimum sample sizes needed for the mean
distance method to achieve the same precision as 4.05 square meter plot
sampling at all three density levels of the random and clumped spatial
distributions were at least 10 times larger than the sample size used by
4.05 square meter plot sampling, it would not be practical to implement, and
would prohibit the use of the mean distance method as proposed.
Although the mean distance method looked promising in theory, its lack of
success was due to the inclusion of very small distances. Including very
small distances, as compared to larger distances, increased and spread out
the density estimates considerably which resulted in the mean distance
method being less precise than 4.05 square meter plot sampling. Although the
mean distance method proved to be accurate at calculating density estimates,
the high variability and overestimation bias would preclude its practical
application towards quantifying forest regeneration.
Since the overestimation bias can be adjusted by incorporating an
overestimation bias constant the mean method shows promise at calculating
density in situation such as pole size stand or larger when the average
distance between trees would be greater. This would reduce the
overestimation bias and decrease the variability of the mean distance method
and make it a more likely alternative to traditional milacre plot sampling.
The mean distance method was shown not to be a valid replacement for
traditional milacre plot sampling for quantifying forest regeneration due to
the close spacing of seedlings on a forest floor. However, the mean distance
method did perform fairly well within a random distribution but not within a
clustered or clumped population. The mean distance method may be a better
choice in rangeland-shrub or dry forest communities where regeneration is
less clustered or within large diameter tree conditions where the average
distance between trees is greater.
\section*{Acknowledgements}
The authors wish to express their gratitude to the peer-review process and in particular to the reviewers of this paper for their insightful comments and suggestions.
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