Title: Population density estimation using line transect sampling /
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Title: Population density estimation using line transect sampling /
Physical Description: viii, 92 leaves : ill. ; 28 cm.
Language: English
Creator: Ondrasik, John Anthony, 1951-
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 1979
Copyright Date: 1979
 Subjects
Subject: Plant populations -- Mathematical models   ( lcsh )
Animal populations -- Mathematical models   ( lcsh )
Statistics thesis Ph. D
Dissertations, Academic -- Statistics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 90-91.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by John A. Ondrasik.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000097635
oclc - 06591994
notis - AAL3076

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POPULATION DENSITY ESTIIMATION USING
LINE TRANSECT SAMPLING











BY

JOHN A. ONDRASIK


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1979






























To Toni

For Her Love and Support












ACKNOWLEDGMENTS


I would like to thank my adviser, Dr. P. V. Rao, for his

guidance and assistance throughout the course of this research.

His patience and thoughtful advice during the writing of this

dissertation is sincerely appreciated. I would also like to

thank Dr. Dennis D. Wackerly for the help and encouragement

that he provided during my years at the University of Florida.

Special thanks go to my family for the moral support

they provided during the pursuit of this degree. I am espe-

cially grateful to my wife, Toni, whose love and understand-

ing made it possible for me to finish this project. Her

patience and sacrifices will never be forgotten.

Finally, I want to express my thanks to Mrs. Edna Larrick

for her excellent job of typing this manuscript despite the

time constraints involved.


iii











TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS . . . . . . . . . iii

LIST OF TABLES ... . .... . . . . . . . .vi

ABSTRACT . . . . . . . . . . . . vii

CHAPTER
I INTRODUCTION . . . . . . . . . 1
1.1 Literature Revieu . . . . . . 1
1.2 Density Estimation Using Line Transects 4
1.3 Summary of Results . . .. . . . .9

II DENSITY ESTIMATION USING THE INVERSE
SAMPLING PROCEDURE . . . . . . . 13
2.1 Introduction . . . . . . . 13
2.2 A General Model Based on Right Angle
Distances and Transect Length . . .. 14
2.2.1 Assumptions . . . . . 15
2.2.2 Derivation of the Likelihood
Function . . . . . . 16
2.3 A Parametric Density Estimate . . 28
2.3.1 Maximum Likelihood Estimate for D 28
2.3.2 Unbiased Estimate for D . . .. 29
2.3.3 Variance of 6 . . . .31
2.3.4 Sample Size Determination Using u 32

2.4 Nonparametric Density Estimate . . . 34
2.4.1 The Nonparametric Model for
Estimating D . . . . . . 36
2.4.2 An Estimate for fy(O) . . .. 37
2.4.3 Approximations for the Mean and
Variance of (0) . . . 40
2.4.4 A Monte Carlo Study . . . 42
2.4.5 The Expected Value nnd Variance for
a Nonparamctric Estimate of D. . 46
2.4.6 Sample Size Determination Using DN 47








TABLE OF CONTENTS (Continued)

CHAPTER Page
III DENSITY ESTIMATION BASED ON A COMBINATION
OF INVERSE AND DIRECT SAMPLING .. . . .. 49

3.1 Introduction . . . . . . . 49
3.2 Gates Estimate . . . . . . . 50
3.2.1 The Mean and Variance of 6 .. ... 54
g
3.3 Expected Value of DCp ... ... .... 57
3.4 Variance of DCp . . . . . . 65
3.5 Maximum Likelihood Justification for DCP. 69

IV DENSITY ESTIMATION FOR CLUSTERED POPULATIONS .71

4.1 Introduction . . . . . . 71
4.2 Assumptions . . . . . . . . 73
4.3 General Form of the Likelihood Function .76
4.4 Estimation of D when p(-) and h(-)
Have Specific Forms . . . . . . 79
4.5 A Worked Example . . . . . . 86

BIBLIOGRAPHY .. . . . . . .. . .. . 90

BIOGRAPHICAL SKETCH . . . . . . . .92











LIST OF TABLES


TABLE Page
1 Number of animals, No, that must be sighted to
guarantee the estimate, D has coefficient of
variation, CV(Du) . . . . . .. . 34


2 Forms proposed for the function, g(y) . . .. 36


3 Results of Monte Carlo Study using g,(y) =e-10v 45


4 Results of Monte Carlo Study using g2(y) = -y 45


5 Results of Monte Carlo Study using g3(y) = 1-y 46


6 Number of animals, No, that must be sighted
to guarantee the estimate DN has coefficient
of variation, CV(DN) . . . . . . . 48









Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


POPULATION DENSITY ESTIMATION USING
LINE TRANSECT SAMPLING

By

John A. Ondrasik

December 1979

Chairman: Pejaver V. Rao
Major Department: Statistics

The use of line transect methods in estimating animal

and plant population densities has recently been receiving

increased attention in the literature. Many of the density

estimates which are currently available are based only on

the right angle distances from the sighted objects to a

randomly placed transect of known length. This type of sam-

pling, wherein an observer is required to travel along a line

transect of some predetermined length, will be referred to

as the direct sampling method. In contrast, one can use an

inverse sampling plan which will allow the observer to termi-

nate sampling as soon as he sights a prespecified number of

animals.

An obvious advantage of an inverse sampling plan is that

sampling is terminated as soon as the required number of

objects are sighted. A disadvantage is the possibility that

sampling may not terminate in any reasonable period of time.

Consequently, a third sampling plan, in which sampling stops

vii






as soon as either a prespecified number of objects are sighted

or a prespecified length of the transect is traversed, is of

practical interest. Such a sampling plan will be referred to

as the combined sampling method. The objective of this dis-

sertation is to develop density estimation techniques suit-

able for both inverse and combined sampling plans.

In Chapter II, both a parametric and a nonparametric

estimate for the population density are developed using the

inverse sampling approach. We will show that a primary

advantage of estimation using inverse sampling is the fact

that these estimates can be expressed as the product of two

independent random variables. This representation not only

enables us to obtain the expected value and variance of our

estimates easily, but also leads to a simple criterion for

sample size determination.

In Chapter III, we derive a parametric density estimate

that is suitable for the combined sampling method. This esti-

mate will be shown to be asymptotically unbiased. An approx-

imation to the variance of this estimate is also provided.

The density estimates developed in Chapters II and III

are based on the assumption that the sightings of animals are

independent events. In Chapter IV we relax this assumption

and develop an estimation procedure using inverse sampling

that can be applied to clustered populations--those popula-

tions composed of small groups or "clusters" of objects.


viii












CHAPTER I


INTRODUCTION



1.1 Literature Review


Our objective in this dissertation is to examine the

problem of density estimation in animal and plant populations.

The demand for new and more efficient population density esti-

mates has grown quite rapidly in the past few years. Anderson

et al. (1976, p. 1) give a good assessment of the present sit-

uation and provide some reasons for the renewed interest in

this subject in the following paragraph:

The need to accurately and precisely estimate the
size or density of biological populations has increased
dramatically in recent years. This has been due largely
to ecological problems created by the effects of man's
rapidly increasing population. Within the past decade,
we have witnessed numerous data gathering activities
related to the Environmental Impact Statement (lJEPA)
process or Biological Monitoring programs. Environmental
programs related to phosphate, uranium and coal mining
and the extraction of shale oil typically require esti-
mates of the size or density of biological populations.
The Endangered Species Act has focused attention on the
lack of techniques to estimate population size. It now
appears that hundreds of species of plants may be pro-
tected under the Act, and, therefore, we will need infor-
mation on the size of some plant populations. Estimation
of the size of biological populations was a major objec-
tive of the International Biological Program (IBP) (Smith
et al. 1975). Finally, we mention that the ability to
estimate population size or density is fundamental to
efficient wildlife and habitat management and many impor-
tant studies in basic ecological research.








The estimation of population size has always been a very

interesting and complex problem For a recent review of the

general subject area see Seber (1973). Although many of the

methods described in Seber's book are quite useful, they are

frequently very expensive and time consuming. Estimation

methods based on capture-recapture studies would fall into

this category. A further problem with many estimation meth-

ods is that they are based on models requiring very restric-

tive assumptions which severely limit their use in analyzing

and interpreting the data. For these reasons and others,

line transect sampling schemes are becoming more and more

popular. This method of sampling requires an observer to

travel along a line transect that has been randomly placed

through the area containing the population under study and

to record certain measurements whenever a member of the popu-

lation is sighted. There are several density estimation tech-

niques available using line transect data; however, the full

potential is yet to be realized.

Density estimation through line transects is typically

practical, rapid and inexpensive for a wide variety of popu-

lations. Published references to line transect studies date

back to the method used by King (See l.copold, 1933) in the

estimation of ruffed grouse populations. Since that time,

numerous papers investigating line transect models have

appeared, e.g., Webb (1942), Hayne (1949), Robinette et al.

(1954), Gates et al. (1968), Anderson and Pospahala (1970),







Sen et al. (1974), Burnham and Anderson (1976) and Crain

et al. (1978). Since it is commonly assumed by these authors

that the objects being sampled are fixed with respect to the

transect, line transect models are best suited for either

immotile populations, flushing populations (populations where

the animal being observed makes a conspicuous response upon

the approach of the observer) or slow moving populations.

Examples of such populations are:

(i) immotile birds' nests, dead deer and plants,

(ii) flushing grouse, pheasants and quail, and

(iii) slow moving desert tortoise and gila monster.

The degree to which line transect methods can be applied to

more motile populations, such as deer and hare, will depend

on the degree to which the basic assumptions are met. In any

case, one should proceed cautiously when using these models

for motile populations.

Despite the wide applicability of line transect methods,

the estimation problem has only recently begun to receive

rigorous treatment and attention from a statistical standpoint.

Gates et al. (1968) were the first to develop a density esti-

mation procedure within a standard statistical framework.

After making certain assumptions with regard to the probabil-

ity of sighting an animal located at a given right angle

distance from the transect, they rigorously derived a popu-

lation density estimate. In addition, they were the first

authors to provide an explicit form for the approximate sam-

pling variance of their density estimate.





1,

While the assumptions of Gates et al. (1968) concerning

the probability of sighting an animal did work well for the

ruffed grouse populations they were studying, it is clear

that the validity of their assumptions will be quite crucial

in establishing the validity their density estimates. If

the collected data fail to substantiate their assumptions,

large biases could occur in the estimates as seen in Robinette

et al. (1974). As a result, Sen et al. (1974) and Pollock

(1978) relaxed the assumptions of Gates et al. (1968) by

using more general forms for the sighting probability, while

Burnham and Anderson (1976) developed a nonparametric approach

as a means of providing a more robust estimation procedure.

In the following sections, we will outline the general

problem of density estimation using line transects, give

our approach to the solution of this problem and summarize

the results found in the remainder of this work.


1.2 Density Estimation Using Line Transects

The line transect method is simply a means of sampling

from some unknown population of objects that are spatially

distributed. In the context of animal or plant population

density estimation, these objects take the form of mammals,

birds, plants, nests, etc., which are distributed over a par-

ticular area of interest. From this point on, our refer-

ences will always be to animal populations with the under-

standing that the estimation methods we describe are appli-

cable to all populations which satisfy the necessary assump-

tions.








In the line transect sampling procedure, a line is ran-

domly placed across an area, A, that contains the unknown

population of interest. An observer follows the transect and

records one or more of the following three pieces of informa-

tion for each animal sighted:

(i) The radial distance, r, from the observer to the
animal.

(ii) The right angle distance, y, from the animal to the
line transect.

(iii) The sighting angle, 8, between the line transect
and the line joining the observer to the point at
which the animal is sighted.

These measurements are illustrated in Figure 1.


Figure 1. Measurements recorded using line transect sampling.
(Z is the position of an observer when an animal
is sighted at X. XP is the line from the animal
perpendicular to the transect.)








In this work, we shall consider the problem of estimating

population density using only the right angle distances.

Because estimates depending only on right angle distances are

easy and economical to use, such estimates have become very

popular over the past several years.

Before any estimation procedure based on right angle

distances can be formulated, certain assumptions regarding

the population of interest must be made. A set of assump-

tions used by several workers in the area is detailed in

Section 2.2.1. One of the key assumptions in this set is

that the probability of sighting an animal located at a right

angle distance, y, from the transect can be represented by

some nonincreasing function g(y), which satisfies the equal-

ity, g(0) = 1. This function is simply a mathematical tool

for dealing with the fact that animals located closer to the

line transect will be seen more readily than animals located

further away from the transect. An alternative method of

dealing with this phenomenon is given by Anderson and

Pospahala (1970).

If g(y) is assumed to have some specific functional form

determined by some unknown parameters, then the estimate is

said to be parametric. On the other hand, if g(y) is left

unspecified except for the requirements that it is nondecreas-

ing and g(0) =1, then the estimate is said to be nonparametric.

Seber (1973) has shown that any density estimate based on

right angle distances will have the form








N
s 2LoC'

where N is a random variable representing the number of

animals seen in a line transect of length Lo and c is an

estimate for c, a parameter which depends on g(y) through

the relation


c = g(y)dy.


By noting that the density is simply the number of animals

present per unit of area, it is clear that c can be inter-

preted as one-half of the effective width of the strip actu-

ally covered by the observer as he moves along the transect.

Further examination of Ds also points out that estimating

the parameter c is the key to the estimation problem.

At this time, we would like to point out that the range

for the right angle distance, y, is allowed to go from 0 to

+-, as seen in the integral on the right hand side of the

equation for c. In practice, since we are considering only

a finite area, A, there will most certainly be a maximum

observation distance, W, perpendicular to the transect.

However, if W is large enough so that the approximation


Sg(y)dy g(y)dy (1.1)

is reasonable, then letting y range in the interval (0,+-)

will not cause any real problems. In practical terms, this

means that the probability of observing an animal located

beyond the boundary, W, should be essentially zero.







In most real life situations, W can be chosen large

enough so that the approximation given in (1.1) is valid.

Thus, in the chapters which follow, we will implicitly

assume chat relation (1.1) holds for the density estimates

that we develop.

Both parametric and nonparamecric models have been used

to derive an estimate for the parameter c, and, consequently,

for the population density. In both cases, the estimate for

c turns out to be a function of the observed right angle

distances. In the parametric case, c will simply be a func-

tion of the parameters that define the function chosen for

g(y). Examples of parametric estimates are found in Gates

et al. (1968), Sen et al. (1974) and Pollock (197S).

Estimation using the nonparametric model is more compli-

cated. Burnham and Anderson (1976) have shown that estimat-

ing 1 is equivalent to estimating fy(0), where fy(.) is the

conditional probability density function for right angle

distance given an animal is sighted. Thus, the problem of

finding a nonparametric estimate for the population density

reduces to the problem of estimating a density function at a

given point. Unfortunately, this problem has not received

much attention in the literature. Burnham and Anderson (1976)

suggest four possible estimates for fy,(0), but the sampling

variances associated with these estimates have not been

established.







Crain et al. (1978) have also considered the problem of

estimating fy(0). They derive an estimate using a Fourier

Series expansion to approximate the conditional probability

density function fy(y). Although their procedure does not

lead to a simple estimate, they do provide an approximation

to its sampling variance.

The line transect method and the corresponding population

density estimates so far described require the observer to

travel a predetermined distance, Lo, along the transect.

This methodwillbe called the direct sampling method. An

alternative to the direct method is the inverse sampling

method, wherein sampling is terminated as soon as a speci-

fied number, No, of animals are sighted. Clearly, in the

direct method, the number of animals seen is a random variable

and the total length travelled is a fixed quantity, while in

inverse sampling method, the total length travelled is the

random variable and the number of animals chat must be seen

is fixed. The main focus of this work will be to develop

density estimation techniques that are based on the inverse

sampling method. In addition, we will consider the density

estimation problem when a combination of the inverse and

direct sampling plans is used.


1.3 Summnar. of Results

In Chapter 2 we derive two estimates for the population

density, D, using an inverse sampling scheme. The set of

assumptions which justify the use of these estimates is








similar to those used by Gates et al. (1968) and several

others. The estimates have the form

D =

where No is the number of animals thac must be seen before

sampling terminates, L is a random variable representing the

length travelled on the transect and c is as previously

defined. Note the similarity of bI to Ds given in Section 1.2.

The only difference between the two estimates is that in bI

the random variables are L and E, while in s they are II and c.

However, this difference gives the inverse sampling method a

theoretical advantage over the direct sampling method. The

random variables L and E will be seen to be independent while

N and c are not. Thus, the estimate DI is the product of two

independent random variables, a fact which not only allows

us to obtain its expected value and variance easily, but also

leads to a simple criterion for sample size determination.

Both a parametric and a nonparametric estimate for the

animal population density are developed in Chapter II. In

deriving the parametric estimate, the functional form assumed

for g(y) is identical to the one used by Gates et al. (1968).

Our parametric density estimate is shown to be unbiased and

the exact variance of this estimate is also provided.

In the nonparametric case we propose an estimate for

f (0) using the method developed by Loftsgaarden and

Quesenberry (1965). We then use heuristic reasons to show







that the corresponding density estimate is asymptotically

unbiased, and derive a large sample approximation for its

variance.

The inverse sampling method does have one drawback when

there is little information available concerning the popula-

tion to be studied, namely, there exists the possibility that

an observer might have to cover a very long transect to sight

No animals. To overcome this problem, we develop a parametric

density estimate in Chapter III that is based on a combina-

tion of the inverse and the direct sampling procedures. In

the combined sampling scheme, sampling is terminated when

either a prespecified number, No, of animals are sighted or

when a prespecified length, Lo, has been travelled along the

transect. Thus, in combined sampling both the length trav-

elled and the number of animals seen will be random variables.

In deriving the density estimate based on the combined

sampling method, we again use the functional form for g(y)

proposed by Gates et al. (1968). This estimate is shown to

be asymptotically unbiased. In addition, an approximate

variance for this density estimate is provided.

The density estimates developed in Chapters II and III

are based on the assumption that the sightings of animals

will be independent events. Gates et al. (1968) showed that

this assumption failed to hold for the animal population they

were studying. In Chapter IV we relax this assumption, and

develop an estimate based on inverse sampling that can be




12

applied to clustered populations--populations in which the

animals aggregate into small groups or"cluscers." Since

the estimation procedure developed will require the use of

a high-speed computer, the last section of Chapter IV is

devoted to a worked example to illustrate the computations

that would be involved.












CHAPTER II


DENSITY ESTIMATION USING THE INVERSE
SAMPLING PROCEDURE



2.1 Introduction


In this chapter we shall propose estimates for animal

population density based on an inverse sampling procedure.

Unlike the direct sampling method considered by Gates et al.

(1968), the inverse sampling procedure specifies the number

of animals that must be sighted before the sampling can be

terminated. Thus, in the inverse case the number of animals

sighted will be a fixed rather than a random quantity.

A precise formulation of the inverse sampling method is as

follows:

1. Place a line at random across the area, A,
to be sampled.

2. Specify a fixed number, No, and sample along the
line transect until No animals are observed.

As one proceeds along the transect, certain measurements

will be made. These will be denoted by yl,y2,'... ,yj and

Z, where y. is the right angle distance from the ih animal

observed to the transect and k is the total distance trav-

elled along the transect during the observation period.

A visual depiction of these measurements is given in Figure 2.




























Figure 2. Measurements recorded using inverse sampling.



2.2 A General Model Based on Right Angle
Distances and Transect Length


The estimates for the density, D, that we will develop

are based on the right angle distances, yly2 ...' YNo and

the total distance, Z, travelled along the transect. TJo

possible approaches to the estimation of D merit consider-

ation. First, recall that density is defined as the number

of animals present per unit of area, or equivalently the

rate at which animals are distributed over some specific

area. Therefore, we can write

D =

where A is the area of interest and N is the total number

of animals present in A. In the direct sampling approach







the estimation of D is most often accomplished by first

estimating N and then dividing by A. Seber (1973) shows

that any estimate of N based on direct sampling has the

form

^* NA
2LoC

where N is a random variable denoting the number of animals

seen, L, is the length of the transect and c is an estimate

of c, a parameter which depends on the probability of sight-

ing an animal given its right angle distance from the tran-

sect. Note that in Seber's estimate, N is random and Lo is

fixed. It follows then, that Seber's estimate for D does

not depend explicitly on A and has the form

B = -N
s 2Loc

Therefore, the estimate of D is independent of the actual

size of A, a property that any reasonable estimate of D

should possess.

As an alternative, D itself can be regarded as the basic

parameter of interest and estimates for D can be derived

directly. This is the approach taken by Burnham and Anderson

(1976) and the one that we will follow in developing our

estimates.


2.2.1 Assumptions

The form of any estimate of D, the animal population

density, will depend upon the type of assumptions we can









make regarding the distribution of the animals to be censused

and the nature of the observations that will be made. The

assumptions our estimates will be based on are as follows:

Al. The animals are randomly distributed with rate or
density D over the area of interest A, i.e., the
probability of a given animal being in a particu-
lar region of Area, 6A, is 6A/A.

A2. The animals are independently distributed over A,
i.e., given two disjoint regions of area, 6A1 and
6A2'
P(n1 animals are in A1 and n2 animals are in 6A2)
= P(n1 animals are in 6A1)P(n2 animals are in 65A).

A3. The probability of sighcing an animal depends
only on its distance from the transect. In addi-
tion, there exists a function g(y) giving the
conditional probability of observing an animal
given its right angle distance, y, from the tran-
sect. In probability notation,

g(y) = P(observing an animal I y).

A4. g(0) = 1, i.e., animals on the line are seen with
probability one.

A5. Animals are fixed, i.e.. there is no confusion
over animals moving during sampling and none are
counted cwice.




2.2.2 Derivation of the Likelihood Function

We will use the maximum likelihood procedure to obtain

an estimate for D. The joint density function we are inter-

ested in is

f ,L(v .; No)

where Y = (Y ,Y2' ,. . Y ) is the vector of random variables
1 2 0







representing the right angle distances, L is the random var-

iable representing the total length travelled, and No is the

specified number of animals to be seen before sampling ter-

minates. Since the dependence of the joint density on No is

implicit throughout the rest of this chapter, it will be

dropped from our notation for convenience. Thus, from now

on we will denote the density as

fy,L ( ,'),

and all other expressions depending on No in this manner will

be handled accordingly.

The following two theorems will be very useful in the

derivation of the likelihood function.


Theorem 1: Let N(Z) denote the number of animals sighted in

the interval (0,O] along the transect. Then, N(Z) is a

Poisson process, and for some > 0,


n!

Note that the quantity O8 equals the expected number of

animals sighted per segment of length I.

Proof: In order to show that N(A) is a Poisson process,

we will show that the assumptions in Section 2.2.1 imply the

postulates necessary for a Poisson process given in Lindgren

(1968, p. 162).

First, consider two disjoint intervals, 1 and Z2' along

the transect and the corresponding areas, A(1) and A(Q2),

enclosed by lines perpendicular to the transect as shown

in Figure 3.






















Figure 3. Two disjoint areas along the transect.


Now let N1 and N2 be random variables representing the total

number of animals that occupy A(Q1) and A(r.2), respectively.

By definition, N(QI) and N(S2) are the number of animals

sighted in A(UI) and A(r2), respectively. We know from assump-

tion A2 that N1 and N2 are independent, and from assumption A3

that sighting an animal depends only on its distance from the

transect. Thus, N(C1), which depends solely on N1 and the

distances to the NI animals from the transect, is independent

of N(Q2), i.e., the number of sightings that occur in two

disjoint intervals along the transect are independent events.

Next we will show that for every >m> 0 and any h >0,

N(e)-N(m) and N(C+h)-N(m+h) are idcnticailly distributed.

First, note that the effective area sampled in seeing N(O)-N(m)

animals and N(R.+h)-N(m+h) animals is equal to A(.-m) as seen

in Figure 4.
























Figure 4. Effective area sampled in seeing N(Q)-N(m)
animals and N(Z+h)-N(m+h) animals.


Therefore, by assumptions Al, A2, and A3, and since the tran-

sect is dropped at random, it follows that

P{N(Z)-N(m) =j}=P{N(Z+h)-N(m+h) =j), j =0,1,2,...

Next we must show that for every Z > 0, and some 9 > 0,

P{N(z) = 1 = B+o(Z), as O,

where o(Z) is a function such that

lim () 0.


Again let A(Z) be the area defined by t on the transect. Noow

define B. to be the event {N(P.)=j} and E. to be the event
J J
that there are exactly j animals in area A(Z). Then it fol-

lows that
0o
P(B ) = E P(BIE )
j=l


= P(BI|Ej)P(Ej).
j=1




20


Under assumptions Al and A2, Pielou (1969, p. 81) has shown

that

-DA(C)
P(Ej) DA()
P(E ) = e- j =0,1,2... .


Also, under assumptions Al, A2 and A3, Seber (1973, Eq. (2.6))

has shown that

P(BIEl) i T

where

c = g(y)dy. (2.1)
'0

Therefore, we can write

P(B1) = 2cDe DA(Q) + 7 P(B 1Ej)P(E )
j=2
and if we show

SP(BlIE.)P(E.) = o(C)
j=2

the proof will be complete. Note that
CO M DA()
e -DA()[DA(C)1J
P(B1 E.)P(E ) Z e- .IDA(
j=2 j=2 J.
S-DA() jDA(9) -1
= DA(.) L e'
j=2
C" -DA(9.) j-1
e AIDA(e)] D
<-j DA() 7-
j=2. (j-1)!
j 2


= DA(.)I1-e-DA()I.







For any finite area A, A(Z) is 0(Z), that is

lim A() < K, for some K> 0.
', -

Therefore, as -+ 0


E P(BI E.)P(E ) = o().
j=2

and,. upon writing

0 = 2cD, (2.2)

we get, as 0,

P(B ) = e9 + o(Z).

Finally, we need to show that for every Z > 0,

E P{N() =n} =o(Z), as 0.
n>l

Note that for all n> 1, we can write


P(B ) = E P(B Ej)
j=l


= P(B IE.)P(Ej ).
j=n

Again, by using the fact that A(Z) is 0(,), it is easy to

show that

P(Bn) = o(9A), as 0,

and N(k) satisfies the four conditions necessary for a Poisson

process.



Before proceeding to the second theorem, we need to define

the following random variables. Let T. denote the random
i~







variable corresponding to the distance travelled on the

transect between sightings of the (i-1)st and ith animals,

i = 1,2,... ,N. Then the total distance travelled is given by

No
L = T..
i=l

The following theorem establishes the independence of

Y and T1,T2 ... T1 for the case No = 2, and this fact enables

us to derive the joint density function, fyL(v,,).


Theorem 2. The random variables T1, T2, Y1 and Y2 are mutually

independent.

Proof: In order to establish the independence of T 2, T2,

Y1 and Y2 we will derive the joint density

fT1 T2 1 (t t2 l '2)

and show that it can be factored into four functions, each

depending on only one of the random variables of interest.

Let 1' Y,2' t1' t2' h1, h2' g1 and g2 be non-negative

real numbers such that


t + h < I + t2'

l + g1 2


as shown in Figure 5.










TI~1


fIn


tl+t2 tl+t2+h2


Figure 5.


Areas defined by yl,y2,tl,t2',g1g2,hl, andh2.


Now let
P(h1,g81h2,g2) = P(t1 < T2 tl+hl'Y1 < Y 1 < 1+g

t2 < T2 t2 +h2'Y2 < Y2 y2 +g2).

Then we can write

P(hl,g' ,h2'g2)
fTl T Y Y (tlt2I'yl2) h lim h
2' '1 2 h. 0 hlgl2g2
gi 0
i = 1,2


provided the limit exists.


Now notice that the event whose probability we wish to
find, namely

{t < T st] +hl'I < Y1 <1 +glt2 < T2 t2 +h2 'y2
is equivalent to the intersection of the following events:
Si, the event {N(tl) = 0}
S2, the event {N(tl+hl)-N(tl) = 1} and {yl

i.e., an animal is seen in area I


Y2

Yl+gl
Yl


t1 tl+hl







S, the event (N(t +t2) N(tl+h) = 0)

S the event (N(tl+t2+h2) N(tl+t2) = 11 and

{y2.'Y2 2+g}2, i.e., an animal is seen in area II.

Now, by Theorem land Assumption A3,.the events S1, S2, S3

and S4 are independent so that we can write
P(hl,g1,h2,g2) = P(S1S2S3S4)
= P(SI)P(S2)P(S3)P(S ).

We now need to find expressions for the probabilities of

Si, S2' S3 and S Since N(9.) is a Poisson process,

P(S ) = e- ,
and

P(S3) = e-0(c-h)

However, P(S2) and P(SA) are not so easil: obtained. We will

only show how to find P(S2), since P(SL) is found in a similar

fashion. First, define S2j to be the event that there are

exactly j animals in area I. Then


P(S2) = E P(S2S2j)
j=1

= P(S2 |S2j)P(S2j).
j=l
By assumptions Al and A2, the number of animals located in

area I will be distributed as a Poisson random variable with

parameter 2Dglhl, where D is the density of the animals (see

Pielou, 1969, p. 81). Note, the factor of 2 comes in since

area I can be found on both sides of the transect. Therefore,








P(S2j) =


-2Dglhl j
e (2Dglh1)


S = 0,1,2,...


By assumption A3,
P(S21S21) = g(y),

for some yl that found in Theorem 1, as gl-0 and h-*0,
-2Dglh1 O
P(S) = 2Dglhle g(yl) .+ P(S21S2j)P(S2j)
j=2

-2Dglh+o(g
= 2Dglhle g(y{)+o(glhl).


Similarly, we can show that as -2+0 and,h2-0,

P(Se-2Dg2h2 +o(g2h2
P(S4) = 2Dg2h2e g(yn)+o(g~h2),


for some y2 for P(S1), P(S2), P(S3) and P(S4) into P(hl,gl,h2,g2)
to obtain


P(hl,g' ,h2,g2) =e


-9t1 -e(t2-h1)
e {2Dglhle


-2Dglh1
g(yl)+ C(glhl ))}


x { h-2Dg2h2 + 2h2
x {2Dg2h2e g(y)+ g2h2)}.


Consequently,

P(h,g ,h2'g2) 2 -0t1 -0t2
lil h" = 4D e e g(yl)g(y2),
h.-*0 81hlg2h2
gi0
i=1,2
which completes the proof of Theorem 2.


(2.3)








In the same manner, we can show that the independence

established in Theorem 2 will hold for any finite number of

sightings, No. In this case if T = (T, ,T2... ,TN and

Y= (Y' ,...,' ), then (2.3) becomes
No
-e~t
i=1
f T,(t,v) = 2 ODNoe 1 g(y ).
i=l
Upon using equation (2.2) in fT y(tv), we get


-e t NO
f ,Y(c,v) = e'e i=1 c No
-i=l 1

Thus, the marginal distributions for T. and Y. are

g(yi)
g (y i)

S iand c

-Ot.
f (ti) = 9e t.> 0.


Therefore, T1,T,, .. TN are independent, identically distri-

buted (iid) as Exponential random variables with parameter 8,

and

No
L = T. T.

has a Gamma distribution with parameters No and 0, i.e.,


oNo No-1 e-
S(9ro) = NO e .>0, 8>0.
L (4f(N0)


Furthermore, L is independent of Y.







The likelihood function for the estimation of 0 and c

can now be obtained by taking the product of fL(M) and

fy(z), i.e.,

NoM N N o-le-0O
L(O,c;y, ) = g(yi) F(No) (2.4)
c i=1

We will now outline how one can estimate D, the animal

population density, from the likelihood function given in

(2.4). As noted earlier, D is related to 0 and c by equation

(2.2), i.e.,

D -

Thus, the maximum likelihood estimate for D would be




where 0 and c are maximum likelihood estimates of 0 and c,

respectively, obtained from (2.4). Note that the estimate

D is the ratio of two mutually independent random variables,

one depending on L alone and the other depending on Y alone.

This property will be found to be very useful when evaluating

the moments of 6.

We have now set the framework necessary for deriving

an estimate of D. In the next section we shall obtain an

estimate for D assuming that g(y) has a particular parametric

form.








2.3 A Parametric Densicy Estimate


Any estimate for D that is derived after assuming an

explicit function for g(y) will be called a parametric esti-

mate. Gates et al. (1968), using direct sampling, derived an

estimate for D assuming

g(y) = e-

Using this same function for g(y), we will derive the corre-

sponding estimate based on inverse sampling.


2.3.1 Maximum Likelihood Estimate for D

To estimate D we need to estimate both 0 and c from the

likelihood function (2.4). In this case

g(y) = e- :>0, \>0

so that

1
c = .

Substituting for c in (2.2) yields

D = _- (2.5)

Also, by substituting for c in (2.4), the likelihood function

becomes

No
-\. 'i nr n"o-1 -Oz
L(0,A, ,(J,) = i=1e i 0 e.0, y.>0. (2.6)


The joint maximum likelihood estimates for 0 and A can

now be easily obtained. The natural logarithm of the likeli-

hood function is






No
ZnL(0,A;y,Z) = NolnA-A yi+N onO+(No-1)Zn-O-Z- Inlr(No).
i=l

Taking the partial derivatives with respect to 9 and X yields

anL(9,A;y,z) N


3ZnL(,X;y,j) _N NN



Setting these equal to 0 yields

^ No


and



i=l


Substituting these estimates for 0 and A in (2.5), the

maximum likelihood estimate for D is seen to be

^^ 2
No
D 2 No
2 E y.
i=l


2.3.2 Unbiased Estimate for D

The expected value of the estimate D,developed in


0 A



= E(6)E(A)

since 0 and A are independent. Using the fact that L has a

Gamma distribution with parameters No and 0, we obtain








= .No
E(A) = E(

^.J0e
(No-1) i

To derive an expression for E(.), first recall that

Y . ,Y are iid with the common density

f (y) = = Ae- y>0.
C

No
Therefore, E Y. is distributed as a Gamma random variable
i=l
with parameters No and \ and

E(X) = E N



1. 0
(No- 1)

Independence of U and X now yields

E(D) = E(6)E(,)



-2 2
(N o- 1)


=- D.
(No-1)

An unbiased estimate for D is, therefore, given by


u 2


No
2 Lu(2.7)
2L E Y.
i=l 1








2.3.3 Variance of D
U
No
Due to the independence of L and E Y. the variance of
i=l
D can be derived directly. We have
-1)
Var (D ) = Var (N0O
2L E Y.
(N-i=l ]


M-1) Var I
SVar N
L E Y.
i=l

(No-1) 1 2
L E Y. L E Y.
2C Y 2

i=l i=l

No
Since L and E Y. are independent it follows that
i=l 1

Va(D) (No-I)4 EL-_-\E~ 1 E_2 iyi_
Var(D) = E E E (2.8)
S Y E Y.,
i=1 / i=1

Deriving the Var(D ) now reduces to the problem of evaluating

1 1 1 1
the expected values for L' N. and Expres-
E Y. L EOY
i=l I i=l e

sions for these quantities are easily obtained by noting that
No
L and E Y. have Gamma distributions with respective parameters
i=l

No,0 and No,A. Straightforward calculations show that for

No>2,







1 0
E() = (N0- (2.9)



E Y.
E( y' (N -1)(1)
\i=1

1 0_ (2.11)



E (No- (N-2) (212)


Now using (2.5), (2.9), (2.10), (2.11), and (2.12) in (2.8)
we get
4 2 A 2 9
Var(D ) 0 4- 2 -AF
u 4 (No- )2(No- )2 (No-l)4


(to-L) 0-A j 1 1
(No-2) (1o-1)

= D2 (2 (2.13)
(No-2 )
provided No>2. Note that Var(D ) does not exist if Nos2.


2.3.4 Sample Size Decermination Using D

The first problem in designing a survey using the inverse
line transact method is to determine in advance the number
of animals, No, that must be sighted before sampling terminates.
One criterion for the selection of No (see Sober, 1973) is Lhe
requirement that the design must yield an estimate of the
density, D, with a prescribed coefficient of variation,








CV =--
E(D)

where o^ and E(D) denote, respectively, the standard deviation
D

and the expected value for the estimate, D. As one can

see immediately, small values of CV are desirable since this

indicates that the estimate has a small standard deviation

relative to its expected value.

With the inverse sampling method, the value of No needed

to guarantee a preset value, C, for the coefficient of varia-

tion of D can be calculated easily. Using (2.7) and (2.13)
u
we see that, for No>2

CV(D) (2N-3)/2
CV( U) = N.
uNo-2

Then, setting C= CV(D ), it is easily shown that No is the

root of the quadratic equation

C2N2 (4C2+2)No+4C2+3 =0.

Solving for No yields the two roots

S1+(+C2)l/2
No = 2+
C

Since the variance of D exists only for No>2, the required

sample size is


N0 = 2+ 1++C2 )/2
C

For example, if C= .25, then No =35. Table 1 gives values of

No corresponding to coefficients of variation ranging from

.1 to .5.








Table I. Number of animals, No, that must be sighted to
guarantee the estimate, D has coefficient of
variation, CV(D ).
U


CV(Du) No

.50 11
.40 15
.30 25
.25 35
.20 53
.15 92
.10 203



2.4 Nonparametric Density Estimate


In this section we will consider a nonparametric estimate

for the population density, D, using inverse sampling. In

contrast to the parametric approach used in Section 2.3,

the nonparametric approach leaves the function g(y), which

represents the probability of observing an animal given its

right angle distance, unspecified.

In Section 2.2.2 we showed that an estimate for D is

given by

D -
2c

where I and c Jare tLhc stimates for 0, Lhe cxpcctcd number of

sightings per unit length of the transect, and c defined as


c = g(y)dy.
*O








If g(y) is completely specified, except for some parameters,

then the problem of estimating D reduces to the problem of

estimating 6 and the parameters in g(y). In Section 2.3 we

considered the specific case

g(y) = e-
A drawback to this approach, where we specify a functional

form for g(y), is that the function chosen must take into

account the inherent detection difficulties that are present

when a particular animal species is being sampled. If one

examines the various forms that have been suggested for g(y),

one quickly becomes aware of the problem of finding a form

that is flexible enough to accommodate the many possibilities

which exist. Some of the functions that have been proposed

for g(y) are presented in Table 2. As seen in the table, the

suggestions for g(y) represent a number of different shapes

in an effort to reflect the nature of the animal being sampled

and the type of ground cover being searched.

Because of the problems that can arise in choosing a

function for g(y), Burnham and Anderson (1976) considered

a nonparametric approach as a means of avoiding the need for

the specification of g(y). Leaving g(y) unspecified will

allow the estimation procedure to depend on the observations

that are actually miadc, not on any panrticulnr model. Thus,

a nonparametric model might provide a more robust estimation

method, that is, an estimation method that could be applied

to a much wider class of animal species.








Table 2. Forms proposed for the function, g(v).


Function


e


Author


, A>0


Gates et al. (1968)


g(y) =




g(y) =


a
1 -

0


0<: y < w


Eberhardt (1968)


'y>


1 O
0 >w


Seber (1973)


S..a- -P..
BX e-l
g(y) = e (c dx,
F(ci
y


B>0, a>0


Sen et al. (1974)


, p>O, \>0


Pollock (1978)


2.4.1 The [Jonparametric Hodel for Estimating D

Consider the estimate for D developed in Section 2.2.2,

that is


D 2a


As noted earlier, if


g(y) = e -


then


c= 1
A


and our estimate for D is


2


g(y) =


e


g(y)







Now, if g(y) is left unspecified, then an estimate for 1 may

be obtained along the same lines Burnham and Anderson (1976)

used in the case of direct sampling. By assumption A4,

f (0) = gO) 1
Y c c
1
Hence, f equals the value of the fy(') evaluated at y=0,

where fy(-) is the probability density function for the right

angle distance, Y, given an animal is seen. The problem of
1
finding a nonparametric estimate for -, therefore, reduces

to the problem of finding an estimate, y(0), for fy(0).

An estimate for D will then be given by

D = (2.14)

where 0 may be taken as the maximum likelihood estimate derived

in Section 2.3.1. That is,

S(N -1)
L

where we have replaced No by No-l to remove the bias.

2.4.2 An Estimate for f (0)
------------------ -
Burnham and Anderson (1976) suggested four possible

methods for estimating fy(0), but we are not aware of any

work which investigates the theoretical properties of any

of these estimates. Loftsgaarden and Quesenberry (1965) con-

sidered a density function estimate based on the observation

that
hat F (x+h) Fy(x-h)
fy(x) = lim 2h
h-0 h
where F y() is the cumulative distribution function. For the

purpose of estimating fy(0), their estimate takes the form








Ff(0) = {(O (' + ) (2.15)

where [ITfo + 11] is the value of ,Ton + 1 rounded off to the

nearest integer and Y is the j order statistic of the

sample ylY2',. -n

Loftsgaarden and Quesenberry (1965) showed that f (0)

as given in (2.15) is a consistent estimate, provided fy(-)

is a positive and continuous probability density function.

One nice property of y(0) is that it can be easily calcu-

lated from the data. However, evaluation of the moments of

this estimate does present some problems. In fact, the mean

and the variance may not even exist in some cases. But,

whenever ['TJo + 1] 3, i.e., whenever (Jo,,, the variance of

fy(0) is finite as shown in the following theorem.

Theorem 3. Let Y 1 Y2' .. n be a set of independent, iden-

tically distributed random variables, representing the right

angle distances, with continuous probability density function

(p.d.f.)

fy(y) = c v>O

Also, let Y( be the rh order statistic. Then

(r)r)
Efor e y i er + h

for every integer r, such that 3 r n.







Proof: The density function for Y(r) is


hr(y) = n- FrI y)[l-F(y) n-rfy)

where
y
F (y) = fy(t)dt.

Therefore,

E-) = n1 r- (y)[1-Fy(y) n-rdF,


Since g(y) represents a probability, g(y)sl and
F(y) g(t) dt < y
F(y) c C

Therefore,

nr-
E- 2 Fr(y)[l-Fy(y)]n-rdFy),
(rY c 0 y

(n-l\
Sr- r (r-2)r(n-r+l)
c r(n-1)

which completes the proof.


Simple asymptotic approximations for the mean and variance

of y(0) which work well for several densities given in

Table 2 can be developed using the first order terms in a

formal Taylor series expansion of Jy(0). The basic ideas

involved in the derivation of these approximations are pre-

sented in the following section.







2.4.3 Approximations for the Mean and Variance of f,,(0)
-1-
Let F(-) and F- (*) denote the cumulative distribution

function and its inverse for the random variable Y, the right

angle distance. Also, let

r = ['N + 1],

U = F(Y ),
r (r) '
and

+(U ) =
(r-l)F (U )
r
th
where Y(r) is the r order statistic in a random sample of

size N from F(-). Then proceeding as in Lindgren (1968,

p. 409) it is easy to see that

f,,(0) = ,(U ),

E(U) -- Pr'

and

Pr(1-Pr)
Var(Ur) N+2 (2.16)

Assuming that q(-) is continuous and differentiable once at

Pr, the first order terms in the Taylor series expansion of
F(*) at Pr yields the approximation

.(U .) t ,(p ) + (U -p ) (u) (2.17)
r du u=P (2.17)

Taking expectaLions on both sides of (2.17) yields

EI t.,(U )} J l (pr),
r -
and substicucing for r, pr and g(-) fields

E{ (O0) ) -1
S- F- 1N+1
/ N /+1







Taking the limit as N tends to infinity and noting that
F (0) is 0 and u = F(y) yields


lim E{f ,(0) lim
N- co Y I -co


] =


u=0


SdF( o) ( ).
dy y=0


(2.18)


Thus for large N, fy(0) is approximately unbiased.
An appro:-:imation for the variance of 'y(0) is found
in a similar fashion. Using (2.17) we get

Var{((Ur)) -d, upu r 2Var(Ur).
--- u=pr

Evaluating the derivative yields
d., (u) 1
du u=pr (r-l){F- (pr f

so that


Varf.+(Ur)} r -1 1 -
r (r-l)F (r Pr)


Var (U ).
r


How, using (2.16) and then making the appropriate substitu-
tions for r, pr and +*(*), we get







1 (,/ +i) (14- ,'N)
Var(f (0))- >. -

-1 2


TIN+y IY N+1 I
12


Therefore, as [1-u we have

lim ,'T(Var .',(0)} = f (O)

so that an approximation for the variance, when N is large,

is given by

f2(0)
Var(,,(0)} (2.19)


As stated earlier, the expressions obtained for the

expected value and variance of f,,(0) are only approximations.

Their adequacy for practical purposes may be evaluated by a

Monte Carlo study involving various specific forms for the

p.d.f., f,(-). In the next section we will look at the results

of just this kind of simulation study.

2.4.4 A Monte Carlo Study

A Monte Carlo study was used to examine the approximations

for E{ F,(0)) and Var(y,(O)} presented in Section 2.4.3. Three

possible shapes for Cy(-) were used in the study. Since the

shape of fy(.) depends solely on the choice of g(y), the

functions
g1(y) = e-10y, y>O

g2(y) = 1-y O






and

g3(y) = l-y2, 0 were chosen. The function gl(-) was first proposed by Gates

et al. (1968), while Eberhardt (1968) suggested both g2(-)

and g3(*). The different shapes that these three functions

represent are depicted in Figure 6.


1



82
g(y)





1 y

Figure 6. Three forms for the function g(y).


For each value of n= 25, 35, 45, 65, 80 and 100, two

thousand random samples of size n were selected from each of

the three populations defined by gi('), i= 1,2,3. These

samples were obtained by first generating observations from

a uniform distribution defined on the interval [0,1] and then

transforming these values using the appropriate density fy(*).

The UNIFORM function described in Barr et al. (1976) was used

to generate the samples from the uniform distribution. For

each set of 2000 samples, empirical estimates were calculated

for the expected value, e, the percent bias, Be, and the







standard deviation, ae, of fy(0) given in equation (2.15)

as follows:

Let fiY(0) denote the estimate from the ih sample,

i= 1,2,...,2000. Then
12000
e 1 i (0)
qe = -0 fiY(0)
i=Li

B = 00( fy(O)
e f Y(0)

and

1 2000 ( 21/2
ae i= 9 1 (f y(0)- e)-=l

All of the necessary computing was performed under release

76.6B of SAS (see Barr et al., 1976) at the Northeast

Regional Data Center located at the University of Florida.

The results of the study, along with the approximate

standard deviations,
fy(0)
T -

are presented in Tables 3, 4, and 5. As can be seen from the

tables, the estimate of fy(0) has a negative bias for most

samples, generally of a magnitude less than 10% of the true value.

The ratio of OT/oe is also within 10% of one for almost

all samples considered. This is even true for the smaller

sample sizes, n<45. Also, when considering the smaller sample

sizes, the ratio was for the most part greater than one.

Based on the results of this simulation, we feel that, in

practice, the approximations obtained for the expected value

and variance of fy(0) would perform adequately.










Table 3. Results of Monte Carlo Study using gl(y) = e-


Sample a
Size e B oT o T
e e Te --
e
25 9.05 -9.5 4.47 4.55 .98
35 9.08 -9.2 4.11 4.12 1.00
45 8.87 -11.3 3.86 3.65 1.06
65 9.48 -5.2 3.52 3.65 .96
80 9.49 -5.1 3.35 3.56 .94
100 9.48 -5.2 3.16 3.13 1.01


For gl(y) the theoretical mean is 10.





Table 4. Results of Monte Carlo Study using g2(y) = l-y.


Sample B T aT
Size e e T e --
e
25 1.88 -6.0 .894 .850 1.05
35 1.88 -6.0 .822 .801 1.03
45 1.83 -8.5 .772 .674 1.15
65 1.96 -2.0 .704 .722 .98
80 1.92 -4.0 .669 .647 1.03
100 1.93 -3.5 .632 .615 1.03


For g2(y) the theoretical mean is 2.










Table 5. Results of Monte Carlo Study using g3(y)= 1-y2

o
Sample B T
B o a --
Size e e T e
e
25 1.47 -2.0 .671 .625 1.07
35 1.48 -1.3 .616 .594 1.04
45 1.44 -4.0 .579 .559 1.04
65 1.51 .7 .528 .536 .99
80 1.49 .7 .502 .506 .99
100 1.50 0.0 .474 .477 .99


For g3(y) the theoretical mean is 1.5.



2.4.5 The Expected Value and Variance for
a Nonparametric Estimate of D

Now that we have decided upon an estimate for f (0), the

problem of estimating D is straightforward. Substituting

the estimate, fy(0), defined in Section 2.4.2 into expression

(2.14) a nonparametric estimate for D is

(N.-1)
D^ = No-l) (2.20)
2L/N '1 +11)

Expressions for the expected value and variance of 5N

are easily obtained. Since L and Y,Y ...'. Yo are indepen-

dent, we can write
1 ^
E(DN) = ; E(0)E{fy,(0)},
and (see Goodman, 1960, Eq. (2))
1 ar(D)
Var(N) = 4 (6)Varf (0)}+E'lf,(O))Var(6)+Var(6)Var(@'(0)],







where
^ N -1
L '
and

fy(0) = 1
/Y([/No+l])

Then, upon substituting the appropriate expressions for the

moments of 6 and fy(0) into the above equations, we get

E(DN) D, (2.21)

and

2 (/N +1)
ar(N (No+2) (2.22)


2.4.6 Sample Size Determination Using DN

We can now determine the approximate value of No that

is needed to guarantee some preset value for the coefficient

of variation of DN, CV(DN). These values for No can then

be compared to the corresponding values for No (see Table 1)

that are needed to ensure the same coefficient of variation

with the parametric estimate, D Using (2.21) and (2.22),

we see that an approximation for the coefficient of variation

of DN is

CV(6N) 1(-1/2
S(No+2)172

and by setting C=CV(DN), one can easily show that /No is the

root of the quadratic equation

C2No /No + 2C2-1 = 0.






Solving for .''o, yields the two roots

.. t(l-4C2 (2C 1))i/
fo= 2-------------
2C
and since

(1-4C2(2C2-1))1 2 >1

whenever
? 1
C- <
2'

the required sample size for values of C- .5 is


Sj1+(1-4C2 (2C -))1/22
2C2

For example, if C= .25, then No = 284. Table 6 gives values

for No corresponding to coefficients of variation ranging

from .2 to .5.


Table 6. Number of animals, No, that must be sighted
to guarantee the estimate DN has coefficient
of variation, CV(DQI).


CV(DN) No

.50 20
.40 48
.30 142
.25 284
.20 671











CHAPTER III


DENSITY ESTIMATION BASED ON A COMBINATION
OF INVERSE AND DIRECT SAMPLING



3.1 Introduction

When sampling a population by means of line transects,

it is important to keep in mind that the transect length

that can be covered by an observer will be finite. This

poses a problem for the inverse sampling plan since there

will exist the possibility of not seeing the specified number

of animals within the entire length of the transect. There-

fore, it seems reasonable to develop a sampling scheme that

would employ a rule, which allows one to stop when either a

specified number, No, of animals are seen or a fixed distance,

Lo, has been travelled on the transect. In this chapter we

will consider a sampling plan which combines the inverse

sampling procedure discussed in Chapter II and the direct

sampling procedure of Gates et al. (1968).

More precisely, we will define the combined sampling

method as follows:

1. Place a line at random across the area, A, to be
sampled

2. Specify a fixed number of animals, No>2, and a
fixed transect length, Lo, and then continue sampling
along the transect until either N animals are seen
or a distance, Lo, has been travelled.
49







Since the above method merely incorporates the individual

stopping rules from the inverse and direct sampling methods,

it seems reasonable to use the estimate

f D if H = INo
DCP = ^u (3.1)
D if N < No,


where N is a random variable corresponding to the actual num-

ber of animals sighted using combined sampling, D is the

inverse sampling estimator given in (2.7) and D is an esti-
g
mator appropriate for the direct sampling case. In other

words, the combined sampling procedure uses the inverse sam-

pling estimate if sampling terminates after No animals are

seen and the direct sampling estimate if sampling terminates

after travelling a distance Lo. In Section 3.5 we will also

show that DCp has a maximum likelihood justification.

Before proceeding to derive the mean and variance for

DCP, we need an estimate appropriate for the direct sampling

case.


3.2 Gates Estimate

Based on the direct sampling approach and assuming

g(y) =e- y, x 0,

Gates ce al. (1968) developed the estimate

0 n = 0,1
d 2(3.2)
D dn (n -1) n > 2 (
n
2L : ,
i=1







where Lo is the fixed length of the transect, n is an observed

value for the random variable Nd, the number of animals seen

using direct sampling, and yi is an observed value for the

random variable Yi, i =1,2,...,n, the right angle distance

to the ith animal seen. In what follows, we shall show that

the variance of Dd is not finite. First, we need a result

concerning the joint density of the Yi, i =1,2,...,Nd, condi-

tional on Nd.


Theorem 3. Under the assumptions stated in Section 2.2.1,

conditional on Nd=n>0, the random variables Y1,Y 2...,YNd

are independently, identically distributed with common density

fy(y) = Ae- y y>0, A>0.

Consequently, conditional on Nd = n>0, the random variable
Nd
E Y. has a Gamma distribution with parameters n and A.
i=l

Proof: We want to show that for yi >0, i= i,2,...,Nd,

n
-A I y.
i=l
fY 1 2 (YI'' Y2 'YN INd=n) =A ne
l..YN .... YNd "

Recall that in the direct sampling procedure, the total

length travelled, Lo, is fixed, and define L to be the random

variable representing the total length travelled on the tran-

sect when the nth animal is sighted. Then the events {Nd=n}

and {Ln Lo






fY1Y Y.d(yl''2.... ( I'Ndld=n)
1 2 L1 d
n

f', 1Y 2 Y ( yl ...1 nI -- Lo.
Now by Theorem 2, Y, Y .Y L and L are mutually
12''' n n n+l
independent, and

g(yi)
f (Yi) g


Consequently,

n
fY Y .Y *(y1 y2 N LNd=n) = H f (yi
1 2 d i=1 l
Sn
= -n R g(yi).
c 1=1

n e
i=l

which completes the proof.


It is now easy to show that Var (Dd) does not exist.

N d
From Theorem 3, conditional on 1d=n>0, Z Y. has a Gamma
i=l
distribution with parameters n and X. Thus, using (2.12)

and (3.2)

0 = 0, 1
.:(iilN d=n) = T 2 2
2 n 2
Lo (n-2)







Also, since Nd is the number of sightings inatransect of

length Lo, it follows from Theorem 2 that Nd has a Poisson

distribution with parameter OL. Thus

E(D2) = E E(D2Nd)

-A2 n2 (n-l) e 1(0L)n
Sn2 (n-2) n1
4Lo
= +0 ,

showing that the variance for the estimate Dd defined in

(3.2) is infinite. In fact as long as

P(Nd= 2) > 0,

the variance of Dd cannot be finite.

The problem of infinite variance for Dd can be overcome

by replacing Dd with D where

f0 if n=0,l,2
D =" (3. 3)
g n(n1) if n>3

2L E Y
i=l

Note that the estimate, D differs from Dd only when n=2.

Since any estimate of the density based on only 2 sightings

should be effectively 0, the above modification does not

seem to be unreasonable. We will now proceed to derive

expressions for the mean and variance of D which are needed

in the sequel.







3.2.1 The Mean and Variance of D
g
A
We will first examine E(D ). Recall from Theorem 3, that

id
conditional on 1d=n, n>0, Z Y. has a Gamma distribution with
i=l
parameters n and A. Thus


Nd, d = n-
i=l

and


0 n=0,l,2
E(Dg t l=n) =nA
1, nt3


Now since Ud is distributed as a Poisson random variable with

parameter OLo, it follows that

E(Dg) = E dE(Dg INd)


C OL n
ne O(OLo)
2L n!
0 n=3

O= {l-e (I+OLL o)}
2(
Substituting the left hand side of (2.5) for in the above

yields

E(D ) = )D l-c (1+0Lo)),

and after wriLing In = OL.o, the expected number of sightings in

a transect of length Lo, we get

E(D ) = D{1-o- (l+u) (3.4)
g







Thus,D is not strictly unbiased, but the bias arises because
g
there is a positive probability of obtaining samples of size

1 or 2. However, even for moderate values of v, the bias in

D will be small since e (1+p) tends to zero exponentially
g
fast. For example, if p = 10, the relative bias is only .05%.

Next we will look at Var (Dg). Again since, conditional
Nd g
on Nd=n, n>0, E Y. is distributed as a Gamma random variable,
i=l
with parameters n and A, we know that

1 2
E 1 =n X n>2,
N 2I Nd= (n-l)(n-2)' n>2,
E Y i
i=l 1

and
2 0 if n=0,1,2
E(D INd=n) = 2 (nl) 2
4 n (n-1) if n>3


Therefore,

E(D ) = ENdE(Dg Nd)

2 0 2 n
--- n (n-l) e- (3 5)
z. (n-2) nl
n=3
4Lo

and we can write

2 0 2 n- ln
Var(D ) = n (n-2) ne D -e +)}2 (3.6)
4Lo n=3

An approximation to Var(D ) valid for large values of u

may be derived in a manner analogous to the method used by

Gates et al. (1968). After writing







2
n (n-1) 2 4
(n-- = +n+2+-

it is easy to see that for n,3

2
n2 n+2 n (n-1) 2
(n-2)n+6


Thus, lower and


and


upper bounds for E(D ) are
g


.22 e n
LB = -- Z (n 2+n+2)e -- E(Db)
ALo n=3 n

B 2 2 -uj n ^
UB = =Z n n+n+6)eE
4Lo n=3 n


Now


2 m -ui n
UB-LB = n
Lo n=3

2 -
= -7 (1-e -e


2 -u
- o-).


Upon using the relationships

D = and = L
D = and I = 0Lo


we get


4D'
UB-LB =
ji


which tends to 0 as u-o

for E(Dg) is
8


(1-e -ll-ie


2- I)
u-e- ^


Thus, a reasonable approximation


L^ UB+LB
SE(D
g = 2


\
4Lo n=3


e-n
2 -e n
(n +n+4) --
n!-


1)
- (u 2+22u+4-e 11(4+6u+5ju )
U


(3.7)







From (3.7) an approximation for Var (D ) is

Var(D ) =D2{1+ 4 e ((5+ + )}-D {l-2e- (l+l+e-2 (1+p)2 }

2 2 4 -' 6 4 -2 2
=D { -+-- e (3-2u +-+ e (1+p) }

Now, as p increases, the terms involving e-P and e-2p will
2 4
tend to 0 much faster than -+-, so that for large u, we

have the approximation

Var(D ) D2 (-+ ). (3.8)

We are now in a position to derive the mean and variance

of Dp



3.3 Expected Value of DCp

Recall that in the combined sampling scheme both N, the

number of animals seen, and L, the distance travelled before

termination of sampling are random variables. Thus, the

expected value of DCp can be found directly using

E(Dp) = ENE(DCPIN).

However, before proceeding along these lines it will be help-

ful to have the following theorems.

Theorem 4. Let N be the random variable representing the

number of animals seen using the combined sampling method.

Then under the assumptions stated in Section 2.2.1,




58


f n
e n=0, ,...,N -1

P(N=n)=
om n
e 1I
S n=I]o
=Uo

where u=0Lo is the expected number of animals sighted along

a transect of length Lo.

Proof: For n
event {exactly n sightings occur in (0,Lo1). By Theorem 1,

the number of sightings in (0,Lo] is Poisson with parameter

OLo. Hence,


P(N=n) = enu n=0 ....... Io-1.

The case N=No follows since the event (N=N.J is equivalent

to the event (at least No sightings occur in (0,Lo]}.



The following three theorems establish some useful

relationships among the random variables I], LN andY ,Y2 .. YN

where 11 is as defined in Theorem 4, LI represents the total
t-h
length travelled on the transect when the th animal is

sighted and Yi, i=1,2,. ..,, represents the right angle
S th
distance to the i animal seen.







Theorem 5. Under the assumptions stated in Section 2.2.1,

the conditional p.d.f. of LN given N=n>0, is

n-1
nn
n n-i Lo
fLN (|N=n) =
fLeN ( ) -le-0 n=N

F(No) P(N=N)
where
Lo NNoNo-1e -0.
P(N=No) = e dt.
'F(No)

Proof: First we will consider the case when n
Theorem 1, seeing n
to observing n Poisson events in the interval (O,Lo]. There-

fore (see Bhat, 1972, p. 129), the joint density of

L1,L2 ....,LN conditional on the occurrence of N=n Poisson

events in (0,L]o is


L1 L2 ..LN( 1 . JN=n) = nn Os
and the marginal density of LN conditional on N=n is

n-l
fLN(I|N=n) = n! ---, 0

Next, we will consider the case where N=No. Define T.

to be the random variable corresponding to the distance

travelled on the transect between the (i-l)st and ith sight-

ing. Then the Noth observation is made at

No
L = E T..
N0 i=1




60



Now in the combined sampling approach, we will see N=Noani-

mals, if and only if che distance

N
L = T.- L.
N0 i=l I

Therefore, if O<-sLo, then

P(L tNJ=N0o)
P ( [L.= 1,
P(LIj-I N=to) = P(.=)


P(i=No)
No
P( T. .)
i=l 1
P(N=No)

Now, since the sightings are Poisson events by Theorem i,

the random variable

n0
L = E T.
o i=l

has a Gamma distribution with parameters No and 0. Thus,

0[ No N -I -0?.
f (IjN=N) = o[ e ___ I 0

Now, by Theorem 4, we have

P(N=[) = e-0L (OLo)
j=0o j

OL-z ONo No-1

0 z dzo

SOe- 0 NO )(o-1
Sr(No) de.

Substituting P(N=No) into (3.9) above completes the proof.







Theorem 6. Under the assumptions stated in Section 2.2.1,

conditional on N=n>0, the random variables Y1,Y ... N and

LN are independent.

Proof: First consider the case N=No. Let 20 and y.i0

for i=l,2,...,N. We want to show that

P(Yi yi, LNt, i = 1,2,...,NIN=No)

= P(Y iYi, i=1,2,...,No)P(LN ]|N=No).

Note that the event {N=NJ is equivalent to the event {LNo Lo},

so that we can write

P(YiYi, LN : i = 1,...,NIN=No)

= P(Y iyi, LNos-, i = 1,2,...,No I=No)

= P(Y iYi, LNosz, i = 1,2,... ,No|LN sLo)

P(Yi Yi, LNo : LNoLo, i=1,2,. .. ,N )
P(LNo : Lo)
N0


Now by Theorem 2,

Consequently,


Y1Y2,...,YNo and LNo


are independent.


P(Y iYi, LNO < LN, Lo, i=1,2,...,NO)
P(LNo L0)

P(Y P(LNo Lo)

= P(Y
= P(Y y. i=1,2,...,No)P(LNo IN=No0).
1 1- 0.







Now consider the case l=n<1o, and let 9. and v. be defined

as before. Also, define XN to be the actual length travelled

to see N animals when the combined sampling method is used,

that is,

= Lo ' L No N= N< o-0
L o, N=No

Then for n
are equivalent. Thus,

P(Yi-y L s., i = 1,2, .... ,N|N=n
= P(Y y L i = 1,2,...,nl:n=Lo)
1i n n

= P(' ., L t, i = 1,2,...,n L Lo
P(Y L -, L Lo S- n n n+l)
n nn++
Again by Theorem 2, for N=n, YL 2" n and Ln Ln+

are independent, so that

P(Y.i 1 -i n n n+l.


P(Y iYi, i=1,2 .... ,n)P(Ln- Ln<- Lo.Ln+l)
P(L nLo
= P(Y.i .Y i=l,2,...,n)P(L <. N=n


Theorem 7. Under the assumptions stated in Section 2.2.1,

conditional on [t=n>0, the random variables Y1,Y2,..' are

independently, identically distributed with common density




63


fy(y) = Ae y>0, A>O.

Consequently, conditional on N=n>0, the random variable
N
Y. has a Gamma distribution with parameters n and X.
i=l

Proof: The case N=n by noting that the random variables N and Nd are equivalent

when 0 o
is equivalent to the event {LNo

fYY2 .... YN (Y'Y2' ... 'YN IN=No)


1 f ,Y2 ..., No (Y Y21 . 'YNo ILNo Lo )
Now by Theorem 2, Y ,Y2 .... YNo and LNo are mutually inde-

pendent and


g(Yi)
fY(Yi) c

Consequently,

No
fYY2"'YNo (yl'y2' ..YN IN=N) H f (yi)
12*N. i=l i


1 No
= I g(yi),
c i=l
and substituting

-AYi
g(yi) = e


completes the proof.







We are now ready to determine the expected value of

DCp given N=n. For n = 0,1,2,
E(DCPIN=n) = E(D IN=n) = 0. (3.10)
g

Next consider the values 3~n,-No. Recall from Theorem 7

that, conditional on N=n>0, Y. has a Gamma distribution
i=l -
with parameters n and A. Then using expressions (3.1) and

(3.3) it follows that

E(DCpIN=n) = E(6 lN=n)


= E N(Nrl) il=n
2L.E Y.

n.\ 2 (3.11)
2L0
Finally for N=No it follows from Theorem 5, Theorem 6,

Theorem 7, and expressions (3.1) and (3.3) that

E(DcpI r=Nro) = E(Du J=No)

iE N-_1 ____ E '-1 [ "0
E Y L T



L 0 1 0-2 -
2i'(N=No ) o e *

Then using the transformation 9. = we get


e\0 I-, e (OL, )
E(D pN=No) =- 2P0(N=N) (3.12)
CP 2P (N nl A-12
n=N'. 1







We can now evaluate the expected value of DCp. Using

Theorem 4 and expressions (2.5), (3.10), (3.11), and (3.12),

we find that

E(DCP) = ENE(DcPIN)


n E L- P (N=n) + E nl
n3 n=No-l

= Dfl-e- (I+p)]. (3.13)
where i =0Lo.

Thus DCp is a biased estimate for the density. Note

that the bias here is equal to the bias for the modified

estimate, D in direct sampling. This is as expected since

in the combined sampling procedure, we are simply choosing

the estimate that corresponds to the reason for terminating

sampling. If we stop sampling after seeing the Noth animal,

then the inverse sampling estimate is used, and, likewise,

if sampling stops after travelling the distance Lo, then the

direct sampling estimate is used.



3.4 Variance of Dp

An expression for the variance of DCp can be found

directly using the formula

2 2
Var(DCp) = E(DCI) {E(D ,)}2. (3.14)

In the preceding section we derived E(Dcp) so that our

"2
problem reduces to evaluating E(DCp). Proceeding along

the same lines as in Section (3.3), we quickly find







2
E(DCPIN=n, n=0,1,2) =


E(D
CP N[=n, 2

(3.15)


S1 N=n, 2 *^


9 2
n' (n-L).
4Lo-(n-2)


E(DCpIrJ=No) = E
CP


(3.16)


N=N j


- E -IN=No E


LL
N2
1 '(N) -1) [
S4(No-2) 2P (N=No)

2 2''(No-l)"

2 2
4(N0-2) P(N=f0)


-, (N 2 I.) n=
4(No-2) P(N=N0) n=N


(N,- 1) 20 N -2. 3e- 9.
F (No)

0 4-2,No-3 -0.
e d.


-OL LO) n
e-L -(oLo)-
n9- .0(3.17)
<- 1


Then, using Theorem 4, expressions (2.5), (3.15), (3.L6), and

(3.17) and letting u=0Lo, it follows that

(2 2
E(IC) EN CP N)


2 o-1

4L- n=3


-, -ii n 2 22 / 2 2
I L(n- 1 ) c A I -
(n-2) nN 4 No-2
n=No-2


- n2 -1 n-2 -,u 2 2 2
,102 n u e A2 0 0-
4 (n-2) (n-2)! '-- -2
n=3 O


m -1 n
e nI
n!
n=No-2


and,


0 -l I 1
-II


S11="No







N-3
2 -3n -p 2 m -e n
= D2 (n+2) e+ (No- E e
nnN0-2 (3.18)
n=l n n1 n=No-2 n!

An expression for the variance of DCp is now evident.

Using (3.13), (3.14), and (3.18), we get

S 2 -n+2) -3 n2 -- / 2 co - n
Var(DCP)=D2 L (n 2)L N "- [ -e- l+u)]2
CP n=1 n=No-2 n

(3.19)

where p=9Lo.

Note that,

S rN_1 2 D2
lim Var(Dcp) = L -2 D
L 0-o

and

lim Var(D p) =D e -e (1+u)]
CP n n!
o-400 n=1

After some simple algebraic manipulations and using the

relationship D= one can easily show that the limit as

Lo-f and the limit as No--o are equal to the Var(D ) given

in (2.13) and the Var(D ) given in (3.6), respectively.

These limiting values are as expected, since letting Lo-0

in the combined sampling approach is equivalent to using

inverse sampling, while letting No-*o is equivalent co using

direct sampling.

We will now show that Var(DCP) can be expressed as a

function of both Var(6u) and Var(b ) given in (2.13) and

(3.6), respectively. Writing the equation in this form will

then lead directly to an approximation for Var(DCp).




68


First note that (3.19) can be rewritten as

Var(Dcp) No-3 (n+2) nn -u e 2 -1 -, n
CP (n+2) e IN-11 i e n -u 2
mn= n n +---+ 7 ---r-- ll-e (l+u)]
D n= n=No1-2
(3.20)


Adding and subtracting the terms


n=N-
n=N. -2


m n -u
IM (n+2) u e-
n=N -2 and
n=N.-2 n n!


-u n
e to the right hand side of (3.20) fields
to the right hand side of (3.20) yields
n]


Var(Dcp) o n -11
CP-_ = (n+2) e -u 2
D = I -e n +u)
D n=1


e n -
^n I ^I.-2/'


co -L n 0)
+ e z
n=No-2 n=N,-2
c -n
e uI 2N,-3 +
n=No-2 (No-2)


n -u
(n+2) u e
n nT

c n -
Z (n+2) I e
n=l n


O 2 n -u
-2 2 u e
1l-e (l+u)] 2
n=No-2 n n

Now, after multiplying both sides of the previous expression
2 0.\
by D using the relationship D= and substituting the

expressions for Var(D ) and Var(D ) given in (2.13) and (3.6),

respectively, we see that

11 n-
Var(Dp) = Var( ) +Var(D) D 2
C n=No-2 n u n=No-2n n


(3.21)


Therefore, an approximation for Var(DCP) can be simply ob-

tained by using the approximation for Var(D ) given in (3.8).
6


CO
+ Z
n=No,-2







3.5 Maximum Likelihood Justification for Dp

In Section 3.1 we stated that the estimate

D g N D =
-DU N=No

could be justified using the maximum likelihood procedure.
To show this, we first need the joint density function for

YIY2 ... YN LN and N, i.e., fY,LN N(X,S,n).

By Theorem 6, fY,LNN() N can be written as


fY,LNN(y, ,n) =fy (yiN=n)fLN(JN=n)P(N=n).

The functional form for fY,L NN() is now evident. Using

Theorems 4 and 5 and recalling from Theorem 7 that

n
-A E Y.
fy(yxN=n) = ne i=l A>O,

we obtain,

/ n
-A E Y.
ne i=l n-1 -OLo n
nZ e (OLo)
Lo n,- N=n YLN, N(,,n) =
No
N0
-A Y.
AN ei-1i N ,No-1 -A0
(No- )-I- N=No. (3.22)
As shown in Section 2.3. .themaximum likelihood estimate

for D is given by

OX
D 2








where 0 and \ are maximum likelihood estimates for 0 and ,\

respectively. Finding maximum likelihood estimates for

6 and A is now straightforward. Taking the natural logarithm

of the likelihood function and setting the partial derivatives

with respect to 6 and equal to 0 yields, for n>0,

N=n
96 =


and


Y N
Z Y.
,


I N=l, . N .


Thus, a maximum likelihood estimate for D using combined

sampling would be


9
N2
E'l


i=l 1


N=n

6 *=


2L Y.
[ 1i=

Our estimate for DCp is obtained by correcting the estimates

A and for bias and noting that 9711=n onl


y


j


exists for values of N>2.












CHAPTER IV


DENSITY ESTIMATION FOR CLUSTERED POPULATIONS



4.1 Introduction


The estimation procedures developed in Chapters II and

III are based on the assumption that the sightings of ani-

mals are independent events. These methods would be appli-

cable to animal populations that are generally made up of

solitary individuals, such as ruffed grouse, snowshoe hare

and gila monster. However, there are other types of animals

which aggregate into coveys, schools and other tight groups.

Animals behaving in this way will be said to belong to clus-

tered populations. Some examples of clustered populations

are bobwhite quail, gray partridge and porpoise. In these

cases the assumption of independent sightings is certainly

not valid, and a different procedure would have to be used.

The line transect method could be easily generalized to

provide estimates for clustered populations. As noted by

Anderson et al. (1976, p. 12), if we amend the assumptions

in Section 2.2.1 so that they refer to clusters of animals

rather than individual animals, then the results of Chap-

ters II and III are directly applicable to the estimation

of the cluster density, Dc. The estimate for Dc will be

71







based on the right angle distances to the clusters from the

random line transect. In the case where the number of ani-

mals in every sighted cluster can be determined without error,

an estimate for the population density D is given by

D = D s

where Dc is the estimate for Dc and s is the average size

of the observed clusters.

Some criticisms of the approach outlined in the preced-

ing paragraph are possible. First of all, it may not be

possible to determine the distance to a cluster as easily

(or as accurately) as the distance to an animal. How will

this distance be defined? Secondly, the simple modification

of the assumptions in Section 2.2.1, obtained by replacing

the word "animal" by the word "cluster" would imply that the

probability of sighting a cluster depends only on its right

angle distance from the line. This may not be a reasonable

assumption since the probability of sighting a larger cluster

is likely to be greater than the probability of sighting

a smaller cluster. Finally, the sighting of a cluster may

not necessarily mean that all of the animals comprising the

cluster are seen and counted by the observer. In this case,

a more reasonable assumption would be to let the probability

of sighting an animal belonging to a cluster depend on the

distance to the cluster as well as the true cluster size.

In this chapter we shall propose a density estimate for

a clustered population by assuming, among other things, that







it is possible to determine the distance to the center of the

cluster from the line transect. An estimation procedure will

then be developed using a model in which the observer's count

of the number of animals in a cluster is regarded as a random

variable with a probability distribution depending upon the

right angle distance and the size of the cluster.



4.2 Assumptions


The density estimate that we will develop is based on

the inverse sampling approach outlined in Section 2.1, with

one minor modification. In clustered populations the plan

is to continue sampling along a randomly placed cransect

until a prespecified number, Nc, of clusters (rather than

animals) are seen. As each cluster is sighted, the follow-

ing information is recorded:

1. the right angle distance, y, from the transect
to the center of the cluster

2. the observed number of animals, s, in the cluster.
(this may be less than the true size of the cluster)

3. the actual distance, Z, travelled by the observer
to sight N clusters.

The sampling procedure described above may be used to

construct an estimate of the population density under the

following set of assumptions. These assumptions closely

parallel those of Section 2.2.1 with the exception chat

they are now phrased in terms of clusters rather than indi-

vidual animals.









Bl. The clusters are randomly distributed with rate
(density) D over the area of interest, A.

B2. The clusters are independently distributed over A,
i.e., given two disjoint regions of area, 6A1 and
6A2'

P(n1 clusters are in .A1 and n2 clusters are in 6A2)

= P(nI clusters are in 6A1)P(n2 clusters are in 6A2).


B3. Clusters are fixed, i.e., there is no confusion
over clusters moving during sampling and none are
counted twice.

B4. There exists a probability mass function p(-)
defined on the set of positive integers, such that
p(r) is the probability that r is the true size
of a cluster located at a right angle distance, y,
from the transect. Note that p(r) is independent
of y. In probability notation, if R and Y denote
the random variables representing the true cluster
size and the right angle distance to the cluster,
respectively, then
P(R=rjY=y) = p(r), r = 1,2,... (4.1)

B5. The probability of observing a cluster depends
only on the size of the cluster and the distance
from the transect to the cluster.

B6. There exists a non-negative function h(-) defined
on 10,w) such that
0 5 h(.) < 1,

h(0) = 0,
and the probability of observing s animals belong-
ing to a cluster of size r 2 s located at a right
angle distance y from the transect is

(r) [h(y) ls l-h(y) r-s
That is, if Y and S denote the random variables
represent ng tLhe right angle di:sLtance to a cluster
and the observed number of animals in a cluster,
respectively, then

P(S=slR=r,Y=y) = (r)[h(y)Is[-h(y) r-s. (4.2)
\s









Closer examination of assumption B6 shows that we are

now allowing the probability of observing a cluster to depend

on both the right angle distance, y, and the true cluster

size, r. To see this, first let C be the event that a cluster

is observed. Then the probability of observing a cluster of

size r located at distance y from the transect is

r
P(CIR=r,Y=y) = E P(S=slR=r,Y=y)
s=l

= 1 P(S=0IR=r,Y=y)

= 1 [l-h(y)]r, (4.3)
which clearly depends on both y and r.

The assumption B6 also satisfies the reasonable require-

ment that for a fixed right angle distance y >0 and r1 < r2,

P(CIR=rl,Y=y) P (C R=r2,Y=y).

This follows immediately from equation (4.3). Note that

P(CIR=rl,Y=y) = 1 [l-h(y)] l

and

P(CIR=r2,Y=y) = 1 [l-h(y)]2

Now, since 0 h(y) 1, it is clear that

P(CIR=rl,Y=y) _P(CIR=r2,Y=y).

One final note with regard to assumption B6 is in order.

In the case where every cluster has size 1, i.e.,

P(R= 1) =1,









the probability of sighting a cluster located at a right

angle distance y is simply h(y). This is quickly seen by

setting r= 1 in (4.3). Thus, under these circumstances,

h(y) has the same interpretation as g(y) defined in Sec-

tion 2.2.1, that is, h(y) is the conditional probability of

sighting an animal at distance y given there is an animal

at v.


4.3 General Form of the Likelihood Function


We will use the maximum likelihood procedure to obtain

an estimate for D, the animal population density. To obtain

the likelihood function, we first need an expression for the

probability density function

fS,Y,L s'y' )

where S= (S1,S2, .... SN ) is the vector of random variables
c
representing the actual number of animals seen in the clusters,

Y= (Yi ,\Y ...2' Y ) is the vector of random variables repre-
c
senting the right angle distances from the clusters to the

transect and L is the random variable representing the total

length travelled on the transect to see H clusters. Upon

writ ing

fS ( = fS I 'Y, L(s)I'y) f [, L 1)fL (9(), (4.4)
S. 'i L S |YSI^,LilL- L

it is seen that specifying the joint probability density

function for S,Y and L is equivalent to specifying the three

functions on the right hand side of (4.4).









The density functions fyIL(ylk) and fL(k) may be derived

in a manner analogous to that used in Section 2.2. Let g (y)

denote the probability of sighting a cluster located at a

right angle distance y from the transect, that is

gc(y) =P(observe a cluster Y=y).
Since sighting a cluster located at a distance y is equiv-

alent to observing at least one animal belonging to the

cluster, we can write


gc(y) = E P(observe s animals Y=y)
s=l


= E P(S=s Y=y).
s=l

Now, for s l1,


P(S=slY=y) = E P(S=slR=r,Y=y)P(R=rlY=y).
r=s

By assumption B4, it follows that Y and R are independent

random variables. Thus, using (4.1) and (4.2) we get


P(S=sIY=y) = Z s (h(y)lsl-h(y)r-s p(r).
r=s

Therefore,


g (y) = s ( [h(y)s [l-h(y)r-s p(r). (4.5)
s=1 r=s

Now, according to assumption B6

h(0) = 1,

so that

g (0) = p(s) =1.
s=1








Therefore, the function gc(y) plays a role similar to the

role of g(y) in Section 2.2. Consequently, by regarding a

"cluster" as an "animal," the results of Section 2.2 can be

applied to clustered populations in a straightforward manner.

Let Nc () denote the random variable representing the

number of clusters seen when travelling a distance on the

transect. Then, by Theorem 1, Nc () is a Poisson process

with parameter 0*L, where 0* is the expected number of

clusters seen when travelling along a transect of length Z.

Also, from Theorem 1 we see that the respective analogs to

equations (2.2) and (2.1) are
0
D (4.6)
c 2c

where Dc is the density of clusters and
Cr
c' = gc y)dy. (4.7)


Furthermore, Theorem 2 gives us the results that L and Y are

mutually independent random variables, L is distributed as

a Gamma random variable with parameters N and 0 and the

conditional density of Y given L= R is

N
IL( fY) IT c gc(Yi). (4.8)
(c*) c i=l

Now, assumption B5 implies that the number of animals

actually observed in a cluster depends only on the right

angle distances to the animals, Y, and the size of the

cluster, R. Thus, S is independent of L, and since Y is








also independent of L it follows that

fSIY,L (sly,) = fsly(S Y)
Nc
= n P(S =s .Y.=y.). (4.9)
i=l i

We can now write an expression for the likelihood func-

tion L(e*,p(.),h(.);s,y,t). Using (4.4), (4.8) and (4.9)

and recalling that L has a Gamma distribution with param-

eters N and 0 we obtain
c
N
L(6*,p(.),h(.);s,y,t) = H P(S=sjY=y)

N N Nl -
H gc(Yi)(O*) c Z c e
x i= (4.10)
(c*) c F(N )



4.4 Estimation of D when p(*) and h(.)
Have Specific Forms

For a clustered population with a cluster density, D ,

the animal population density may be defined as

D=Dc ,

where v =E(R) is the expected cluster size. Upon using the

expression for D given in (4.6), we get
0*v
D= 0 (4.11)
2c
so that maximum likelihood estimation of D can be carried out

by using (4.11) in the likelihood function presented in (4.10).








Since the random variables S and Y are independent of L,

it is easily seen from (4.10) that the maximum likelihood

estimate of O' corrected for bias is

-1
S (4.12)

However, finding estimates of v and c can be quite difficult

depending upon the nature of the functions p(.) and h(-).

Very likely, one has to resort to some iterative technique

such as the Newton-Raphson method (see Korn and Korn, 1968,

eqn. (20.2-31)) to solve the likelihood equation.

It is apparent that there exist a wide variety of func-

tions which satisfy the requirements of p(-) and h(-).

The appropriate choice in a particular problem depends on

the nature of the population under investigation. In this

work we will consider the functions

r C
p(r) = c e a > 0, r = 1,2,... (4.13)
rl(1-e )
and

h(y) = e- > 0, v > 0. (4.14)

It is easily seen that p(.) given by (4.13) represents a

truncated Poisson distribution. The expected cluster size

v is therefore given by

SC (4.15)
l-e-

The limiting case a = 0 corresponds to a population in which

the cluster size is 1 with probability 1. Thus, c =0 corre-

sponds to the model in Section 2.2.








The choice for h(-) is based on the fact that when a=0,

h(-) may be interpreted as the function g(*) defined in

Chapter II. Because

g(y) = e- y

seems to be a popular choice for g(*), we feel that

h(y) = e y

is a reasonable choice for h(-).

The likelihood function may now be regarded as a function

of 0*, a and X*, and maximum likelihood estimation of

D -
D 0 *
2c*
may be accomplished by expressing v and c as functions of

a and A We have already seen the form of v in equation

(4.15). To derive an expression for c we proceed as follows.

Recall from (4.5) that


g (y) = E P(S=sIY=y),
s=l
where

P(S=s|Y=y) = s) [h(y)]s[1-h(y)]r-s p(r).
r=s

Now using (4.13) and (4.14) in the above equation, we get
( -s
r\ 4 yst -A 'k f r -a
s e -e
P(S=sJY=y)= E s
r=s rl(1-e-)
(ae- ~ys -*y r-s
S(ae y)s e- [ra (1-e y)]
-e ) (r-s)1

-* -A y
(ee y)s -e"
s(1ee-s (4.16)
sl(l-e -)








Then, substituting for P(S=s|Y=y), we get

-oe y -A v s
g(y) = e e
(I-e ) s=l1

l-e
ea (4.17)
(l-e )

Therefore, using (4.7) and (4.17), we get


c g c(y)dy


--eA
--z) dy (4.18)
(L-e a)

To evaluate c note that
0 \ *y
I -- e )dy "li-e y
(-e )dy = lim { e dy) (4.19)
,3 X -*+? *-3o0

By letting

t = e

in the integral in the right hand side of (4.19), we can show

"" A N V 1 -aC t
ae dy 1 e dt
ee dy = t

e

1 (-1) aj(l-e ) j
V +--- g
A j=l j -j

and upon substituting inco (4.19), we get

o e 1 0 ( 1 ) a ( L e )
(l-e- )dy = lim L- c
yo j=1 j j!









Since the sum above is absolutely convergent, we can take

the limit inside the sum and obtain

o -a*e 1 J-
(l-e )dy = ), i)
0o j=l j

Then, using (4.18), it follows that

c = a) (4.20)
S(l-e-0)

where

a(a) = E ) (A.21)
j=l J

We can now write the likelihood function in terms of

6*, \* and a. Using equations (4.10), (4.16), (4.20) and

(4.21), we get

N N N *
c c c -A Yi
E s. y.s. -a e
i=l i=l i=l
a e e
L('*,X*,a;s,, ) = ;eN e
N c
(le-") c H s.!
(l-e ) i=l

N N -ae Y N N -1
( c c -ae )(0i c c9.
0. ) T (1-e )e
i=l
N
[a(a)] c F(Nc)
(4.22)

Using the likelihood function given in (4.22), we can

now obtain an estimate for the population density D.

Recall from (4.11) that we can write
*
0 V
D -
2c*








After substituting for c and v using equations (4.15),

(4.20) and (4.21), the expression for D becomes

D 0 ) k
2 2a-(i)

Thus, an estimate for D would be



2a(a)

where 0 X and a are maximum likelihood estimates for

0 A and a, respectively. As noted earlier in this section,

S and Y are independent of L so that O0 can be estimated

using equation (4.13). However, this still leaves us the

problem of estimating A* and a. Instead of estimating A*

and a separately, we can reparamecerize the likelihood

equation in (4.22) by letting

0

and

a = a.

Then, our estimate for D becomes


D = (4.23)

The advantage of this reparameterization is that it makes

use of the fact that L is independent of both S and Y. Thus,

the estimate for D given in (4.23) is now the product of two

independent esLimates, 0 which depends on L alone and D which

depends on S and Y. As a result the variance of can now

be found easily. Using the formula (see Goodman, 1960) for








the variance of the product of two independent estimates,

we get

Var(D ) =E (6)Var(p)+E (p)Var(6) +Var(6 )Var(p)


(4.24)
Since L is distributed as a Gamma random variable with
an* (^ *
parameters Nc and 0 exact expressions for E(6 ) and Var( )

can be obtained using (2.4) and (2.11), i.e.,

E(0 ) = ,
and
*2
Var(e ) -=N -- (4.25)
c
Expressions for the variance and expected value of p can be-

come quite complicated. An iterative scheme would be needed

to find the solutions for p and a that would maximize the

reparameterized version of the likelihood function given

in (4.22). There are computer programs available that can

provide maximum likelihood estimates for p and a along with

numerical approximations for the variance covariance matrix

of the estimates. In the next section we will demonstrate

the use of one such program with a set of hypothetical data.








4.5 A Worked Example

In this section we will present a worked example to

demonstrate the use of a computer program to find the estimate

D* and its approximate variance. Because we are not aware of

any real data that have been collected according to the sam-

pling plan described in Section 4.2, we shall use an artifi-

cial set of data in the example.

Suppose that sampling was continued until N = 25 clusters

were sighted, and that a transect length of = 25 miles was

needed to sight the 25 clusters. Suppose further that the

observed right angle distances and the cluster sizes were as

follows, where the first number in the pair is the right

angle distance, y, measured in yards and the second number

in the pair is the corresponding cluster size, s:

(1,1), (3,2), (7,1), (10,1), (2,3)

(5,5), (4,1), (7,2), (15,1), (22,1)

(6,1), (3,6), (2,1), (12,1), (28,3)

(9,2), (18,1), (36,7), (17,6), (5,1)

(4,1), (3,1), (8,2), (3,4), (13,1).

As noted in Seccion 4.4, an estimate for 0 is

N -1
c-
c = .96,

and an estimate for the variance of ,U is


Var(0 ) = .001.
c








In order to estimate p, the reparameterized version of

the likelihood function given in (4.22) will have to be max-

imized. The Fortran subroutine ZXMIN, found in IMSL (1979)

may be used for this purpose. This program uses a quasi-

Newton iterative procedure to find the minimum of a function.

Thus, we first need to take the negative of the likelihood

equation before we can use this subroutine to our advantage.

On output, this subroutine not only provides the values

at which the function is minimized, but also provides numer-

ical estimates for the second partial derivatives of the

function evaluated at the minimization point. Thus, when

used with the negative of the likelihood function this pro-

gram will provide the maximum likelihood estimates, p and i,

as well as the matrix of negative second partial derivatives

of the likelihood, L(*), evaluated at p and a. We will denote

this matrix by






2 2
/ 2IL(*) 2 ZnL()





2
-1

a2nL() aInL ( ) /a=a
\ a 9p p2 /

For our data, the use of the subroutine ZXMIN with

initial values ai = 2.24 and pI = .16 yielded

a = 2.844,

p = .0907,


and





88



S 7.687 -161.229

-161.229 5098.985

The initial value used for a was the mean of the observed

cluster sizes, i.e.,

25
r S.
i=l -
L 25- = s.
25

Since our model does not assume all animals belonging to a

cluster are seen, s would underestimate the expected cluster

size, i.e.,

s < E(R) .
1-e

Thus, s seems to be a good starting value for a.

In choosing an initial value for P, first recall that


0 = a(O- ,

where a(a) is given in equation (4.21). Since our initial

value for a is s, all we need is a starting value for A.

If every animal in the cluster was seen with probability 1,

the density of clusters would be estimated by the method

described in Chapter II. In this case, the maximum likelihood

estimate for .\ would be 1/7 where

25

y' =-








Thus, as the initial value for p we used

S
1 -
ya(s)

The estimate for the density can now be calculated.

Using (4.23) and substituting the values we obtained for

6 and p, we get
A*
D = 76.7 animals/square mile.

Now if we can obtain a large sample approximation for

the variance of p, then we can use (4.24) as an approxima-

tion for the variance of D Now, under the usual regularity

conditions, V will be a large sample approximation to the

inverse of the variance-covariance matrix of a and p. Further-

more, the approximate variance of D* can be obtained from

equation (4.24) after substituting the element in the matrix

corresponding to the approximate variance of p along with

the other appropriate quantities. Straightforward calcula-

tions show that

SVar(D*) 26.2 animals/square mile.

The use of this Fortran subroutine required a minimal

amount of programming to enter the appropriate likelihood

function. It was run using the computer facilities of the

Northeast Regional Data Center located in Gainesvillc, Florida.

Less than two seconds of CPU time was needed for the estimates

to converge to values that agreed to four significant digits

on two successive interations.












BIBLIOGRAPHY


Anderson, D. R., Laake, J. L., Crain, B. R., and Burnham, K. P.
(1976), Guidelines for Line Transect Sampling of Biological
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Anderson, D. R., and Pospahala, R. S. (1970), "Correction of
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Bhat, U. N. (1972), Elements of Applied Stochastic Processes,
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Crain, B. R., Burnham, K. P., Anderson, D. R., and Laake, J. L.
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Goodman, L. A. (1960), "On the Exact Variance of Products,"
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Korn, G. A., and Korn, T. M. (1968), Mathematical Handbook
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Leopold, A. (1933), Game Management, New York: Charles
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BIOGRAPHICAL SKETCH


John Anthony Ondrasik was born on August 17, 1951, in

New Brunswick, New Jersey. Shortly thereafter his parents

moved to Palmerton, Pennsylvania, where he grew up and

attended high school. After graduation in June, 1969, he

entered Bucknell University in Lewisburg, Pennsylvania, and

received the degree of Bachelor of Science with a major in

mathematics in June, 1973.

It was during his studies at Bucknell that he became

interested in statistics through the influence of the late

Professor Paul Benson. In September, 1973, he matriculated

in the graduate school at the University of Florida and

received the degree Master of Statistics in 1975.

-. lile pursuing his graduate studies, he worked for the

Department of Statistics as an assistant in their biosta-

tistics consulting unit. In November, 1978, he accepted the

position of biostatistician with Boehringer Ingelheim, Ltd.

John Ondrasik is married to the former Anntoinette M.

Lucia. Currently they reside in Danbury, Connecticut.




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