CONTRIBUTIONS TO STATISTICAL TECHNIQUES FOR TWO
AND THREE DIMENSIONAL MEASUREMENT PROBLEMS
By
James Robert Lackritz
A DISSERTATION PRESENTED TC
THE UNIVERSITY
IN PARTIAL FULFILLMENT OF
DEGREE OF DOCTOR
STHE GRADUATE COUNCIL OF
SOF FLORIDA
THE REQUIREMENTS FOR THE
OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
To my family and all other nonbelievers.
ACKNOWLEDGEMENTS
There are many people to whom I owe my sincere thanks
for their help and encouragement during my graduate years
at the University of Florida.
First and foremost, I will always be indebted to
Dr. Richard Scheaffer for his inspiration and unbounded
patience in helping me in my quest for my doctoral degree.
He will always be remembered, not just for his statistical
accomplishments, but as well for his great personal qual
ities as a human being.
I also wish to express my deepest gratitude to
Dr. William Mendenhall for his helpful guidance during my
graduate years.
I would like to thank the individual members of my
committee for their help and interest in my studies.
Finally, I am grateful to Betty Rovira, for her
patience and help in typing the final draft.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . .
ABSTRACT . . . . . . . . . . .
CHAPTER
1 INTRODUCTION . . . . . . . .
1.1 History of Quantitative Microscopy
1.2 Models for Random Division of Space
1.3 Applications . . . . . . .
2 BASIC STEREOLOGICAL RELATIONSHIPS .
2.1 The Mosaic . . . . .
2.2 Basic Stereological Results .
2.3 The Spatial Covariance Function
2.4 Poisson Lines Model . . .
2.5 Meijering Cell Model . . .
2.6 Boolean Scheme . . . . .
3 LINEAL SAMPLE NG . . . . . .
Page
iii
vii
6
6
7
. . 6
. . 7
. 11
S . 16
. . 17
S 18
. . 21
3.1 Introduction . . . . . . .
3.2 Lineal Sampling .....
3.3 Model Development . . . . .
3.4 Asymptotic Distribution of the
Estimator L.//T . ....
3.5 Basic Properties of Segment Counts
3.6 Asymptotic Distribution of M.//E .
3.7 Joint Distribution of M//L . . .
3.8 Asymptotic VarlanceCovariance Matrix
of M// . . . .
3.9 Variance Approximations ...
TABLE OF CONTENTS Continued
Page
CHAPTER
3.10 The Results of Hilliard and Cahn . 46
3.11 Poisson Lines Model . . . ... 49
3.12 Meijering Cell Model .... . . . 50
3.13 Tests on the Mosaic Using Segment
Counts . . . . . .... . 51
3.14 Spatial Covariance Function Approach
to Using Lineal Sampling: Poisson
Process . . . . . . . 54
3.15 Boolean Scheme . . . .... . 55
4 AREAL SAMPLING . . . .... .. . 59
4.1 Introduction .. . . . . 59
4.2 Asymptotic Distribution of A//A .. 60
4.3 Estimation of the Variance Components. 62
4.4 Poisson Process . . . . .. 63
4.5 Meijering Cell Model .. . . . 65
4.6 Comparisons with the Results of
Hilliard and Cahn . ... . . 65
4.7 Spatial Covariance Function Approach 67
5 POINT SAMPLING .. . . ... . . . 69
5.1 Introduction .. . . ... . . 69
5.2 Random Points on the Plane . ... 69
5.3 Systematic Points on a Line .. . 70
5.4 Estimation of the Variance . . .. 73
5.5 Covariance Between Colors . . . 75
5.6 Asymptotic Distribution of the
Estimates . . . . . ... 76
5.7 Extensions of Point Sampling ... . 85
5.8 Systematic Points on Random Lines 88
5.9 Comparisons with Hilliard and Cahn 89
6 MULTISTAGE SAMPLING . . . . .. . 91
6.1 Introduction . . . . . .. 91
TABLE OF CONTENTS Continued
Page
CHAPTER
6.2 Lineal Subsampling
6.3 Point Subsampling
7 NUMERICAL RESULTS . .
7.1 Introduction . .
7.2 Results: Table I
7.3 Results: Table IT .
7.4 Results: Table III
7.5 Results: Table IV
7.6 Conclusions . .
BIBLIOGRAPHY . . . . . .
BIOGRAPHICAL SKETCH . . . .
. . 91
. . 93
. . 96
. . 96
. . 97
. . 98
. . 100
. .. 100
. . 101
. . 107
. .. 108
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
CONTRIBUTIONS TO STATISTICAL TECHNIQUES FOR TWO
AND THREE DIMENSIONAL MEASUREMENT PROBLEMS
By
James Robert Lackritz
August 1977
Chairman: Dr. Richard L. Scheaffer
Major Department: Statistics
Estimation of structural properties of two and three
dimensional figures can be achieved through areal, lineal,
and point sampling techniques. Such properties might
include real and volume fractions, surface area per unit
volume, and boundary length per unit area. Unbiased esti
mators for the individual parameters are wellknown, and
have been used for quite some time in the research associ
ated with this field. Additional properties of the indi
vidual estimators are developed, including consistency and
asymptotic normality, and practical variance approximations
are given. These properties are derived under the assump
tion of a general cell model for the construction of the
population figure. Once the general results are obtained,
special cases of more specific models are discussed.
The variance approximations for the estimators are
obtained by making use of the available sampling informa
tion, so that the usual inferential techniques of interval
estimation and hypothesis testing can be employed. Often,
the assumption of a special model yields a simplified vari
ance expression for the individual estimators.
A second approach to areal and volumefraction analy
sis makes use of the spatial covariance function of the
underlying model to generate a variance expression for the
areal or volumefraction estimator. The results from this
approach are tied in with the results obtained from the
assumption of the general cell model.
The final chapter assesses the use of numerical data
to check the variance expressions derived in previous chap
ters. Four different figures were sampled, and the numer
ical results allow for the comparisons of the individual
models.
viii
CHAPTER 1
INTRODUCTION
1.1 History of Quantitative Microscopy
The problem of estimating structural properties of
two and threedimensional figures has been studied for
over 100 years by engineers, mathematicians, and stereo
logists. DeHoff and Rhines [5] review the history of
Quantitative Microscopy in summarizing the basic results
that have been developed. The first work in this field
dates back to 1848, when Delesse proved mathematically
that the area occupied by each constituent of a cross
section of a rock is proportional to its volume in the mass
of the rock. The first work in lineal analysis was intro
duced by Rosiwal in 1898, while in 1930, Thompson looked
at measurements from point counts. Starting in 1945, more
exact statistical relationships for twodimensional sections
were discovered in independent studies, first by Saltykov,
and then by several scientists in the United States,
including Rhines. The new ideas showed the direct propor
tionality between the number of intersections of a line
with a phase on the figure and the boundary length of the
phase of a twodimensional feature or the surface area of
the phase on a threedimensional feature. Throughout the
the years, investigators have looked at the basic measure
ments that can be obtained through lineal, areal, and
point sampling techniques, with a view towards developing
estimators for the basic properties of the figure. Some
important recent results in point sampling have been out
lined by Koop [9], who studied 21 different methods of
point sampling on the plane for purposes of areal fraction
estimation. Under minimal assumptions, unbiased estimators
have been generated by the researchers for such parameters
as areal and volumefractions, feature boundary length in
two dimensions, and surface area in three dimensions. The
culmination of the individual results is a mathematical
ndimensional theorem of Miles [15], from which many of the
individual unbiased estimators can be derived. These
results will be considered in more detail in Chapter 2.
1.2 Models for Random Division of Space
The work of Miles [15], generalizing the earlier indi
vidual results into one relationship, yields unbiased esti
mators for the two and threedimensional properties with
minimal assumptions on the underlying figure or the sampling
scheme from which the measurements are taken. Many theoret
ical models for the random division of space have been
developed to try to account for the construction of the
mosaic. Several of these models will be discussed through
out the next four chapters.
One of the simplest and most theoretically investi
gated models is the Poisson Lines Model, whose properties
have been developed and discussed by many scientists,
including Miles [13] and Switzer [24]. This model allows
for the division of space into a system of convex polygons
by placing a series of random lines in a coordinate system,
where the lines serve as boundaries of the individual cells.
Miles [13] developed many important properties of the cells,
including the behavior of a line transect that is randomly
placed inside the cell configuration. Switzer [24] showed
that the alternation among phases along a randomly placed
line transect in the process is Markovian for a finite
phase Poisson Process.
The Meijering Cell Model, the result of which is
sometimes known as Voronoi polygons, is discussed in detail
by Gilbert [6], and again is further expanded on by Miles
[14]. This model arises from the random placement of
points to expand outward in all directions at a constant
rate until meeting with another cell, upon which the
boundary is defined. This model, which can be paralleled
to a germinationgrowth model, thus results in a set of
nonoverlapping convex polygons.
The third model to be used will be the Boolean Scheme,
discussed by Serra [22]. Like the Meijering Cell Model,
this model starts with a random noint process, but then
random sets are placed around.the points, allowing for set
overlap, unlike either of the previous two models. Serra
also looks at a different approach to volumefraction esti
mation, which uses the spatial covariance function of the
underlying model, if a particular model can be assumed.
The assumption of an underlying model for the mosaic
gives a probability structure to the mosaic under consid
eration. Chapters 3 through 6 serve to tie the underlying
probability structure to properties of the estimators, so
that additional properties can be developed. Among the
properties discussed will be consistency, asymptotic distri
butions, and variance approximations for the estimators.
Hilliard and Cahn [81 developed variance approximations
for arealfraction estimators under certain assumptions on
the mosaic, and their results will be shown to be a special
case of the variance approximations obtained from more
general models.
1.3 Applications
The major results discussed by DeHoff and Rhines [5]
and generalized by Miles [15] have diverse applications
into a wide range of fields, including materials science,
ecology, physiology, and biology. DeHoff and Rhines [51
discuss how a structure of spherical copper powder may be
examined under a microscope to estimate the fraction of the
powder occupied by voids between the interparticle welds.
An ecologist may be able to use an areal photograph of a
piece of land to estimate the boundary length for a partic
ular river or lake, and additional ecological examples are
mentioned in detail by Pielou [16] and Matern [12].
Weibel [25] presents a section on the utilization of
electron micrographs of liver cells for useful information
on cell sizes and orientation, especially relevant in
cancer research and drugrelated experiments. Lang and
Melhuish [10] looked at estimating lengths and diameters
of plant roots by passing a plane underground through the
root system.
The applications are numerous, especially for the
fields associated with the physical, agricultural, and
ecological sciences. Many different useful measurements
may be obtained through the various sampling techniques,
not only for generating unbiased estimators for the param
eters, but also for obtaining variance approximations for
inferential purposes.
CHAPTER 2
BASIC STEREOLOGICAL RELATIONSHIPS
2.1 The Mosaic
Throughout the remaining chapters, the term mosaic
will refer to a twodimensional structure composed of
two or more distinctive features called phases. The
general topic under research involves the estimation of
various structural properties of the mosaic through geo
metric sampling procedures, and the development of addi
tional properties of these estimators, such as consistency,
asymptotic distribution, and variance approximations. The
details of interest on the mosaic might include:
(1) The areal fractions of the mosaic occupied by differ
ent phases, which in turn may estimate the volume
fractions if the mosaic represents a planar sectioning
of a threedimensional figure.
(2) Boundary length of certain twodimensional features,
which in turn, may estimate the corresponding surface
area of threedimensional features.
(3) Possible feature orientation or homogeneity on the
mosaic.
The information needed to generate estimates for the param
eters of the mosaic can be obtained through many different
sampling procedures, three of which will be discussed in
detail in the succeeding chapters. They are:
(1) Randomly selecting one or more areal sections from
the mosaic.
(2) Randomly locating one or more linear transects within
the boundaries of the mosaic.
(3) Randomly locating a set of points on the mosaic.
Estimation of structural properties of the mosaic through
areal, lineal, and point sampling has a rich history in
quantitative microscopy and other disciplines, as pointed
out in Chapter 1. The remaining sections of this chapter
will review some of the known results, which will be used
to build upon in the remaining chapters.
2.2 Basic Stereological Results
Throughout the years, researchers in the field of
quantitative microscopy have strived to develop unbiased
estimators for the parameters of the mosaic. The popu
lation quantities of interest include volumes and surface
areas of threedimensional phases and areas and boundary
lengths of twodimensional phases. The sampling devices
used are planes, lines, and points, so that the sample
measurements arise from the intersection between the
sampling device and the phase of the mosaic. The measure
ments from an areal section include the total area of the
th
sample, A, and the area occupied by the j phase, denoted
by A.. The lineal transect measurements of interest are L,
J
the length of the transect, L the total length of the
th
intersection of the line with the j phase, and I., the
number of intersections of the transect with the boundaries
of the jth phase. Similarly for point sampling, the number
of points, n, and the point count for phase j, N., will be
important measurements to the experimenter.
A set of unbiased estimators for the parameters of the
mosaic that were developed individually by the various
scientists over the years were generalized by Miles [151
into a single result based on an exercise in ndimensional
geometrical probability. Miles' theorem states:
To estimate a tdimensional quantity of a
ddimensional mosaic using an sdimensional sampling
device yielding (s +t d)dimensional measurements,
for s, t, and d nonnegative integers such that t< d
and s d, let
Dd = the measure of the population figure
D = the measure of the intersection of
the sampling device with the popu
lation figure
Xt = the measure of the feature of interest
on the population figure
Xs+td = the measure of the intersection of the
sampling device with the feature of
interest.
(Here, measure is defined to be volume in three
dimensions, area in two dimensions, length in one
dimension, and point counts for zero dimensions.)
Suppose the sampling device is randomly
located in the space and that D is not zero.
s
Then
E(X D ) kd(st) D (2.2.1)
s d
where
s + 1 t +1
kd(s,t) = s td i d2+ (2.2.2)
2 2
Looking at the individual parameters of interest, the
estimators obtained from this single relationship will be
the same as those individually developed, as discussed in
the history of quantitative microscopy by DeHoff and
Rhines [51.
Suppose the parameter of interest is the volume
fraction occupied by a particular phase. Let V denote the
volume of the threedimensional figure, and V. the volume
of phase j on the figure. If an areal sample is used,
then the intersection of the planar sample with phase j
will be the phase j areas on the plane. With d =t =3 and
s =2, k3(2,3)= 1, so that E(A )/E(A) =V./V. Similarly,
for a lineal transect, where d =t = and for point
sampling, where d = t = 0, one has
E(N.) E(L ) E(A.) V.
 = (2.2.3)
E(n) E(L) E A V
Also, for fixed areal fraction, A./A, when d=t =2,
J
lineal and point sampling will yield
E(N.) E(L.) A.
_____ ___ L (2.2.4)
E(n) E(L) A
It should be noted from (2.2.3) and (2.2.4) that in
estimating a volume, areal, or lineal fraction with a
lowerdimensioned samplefraction, unbiased estimators
result only when the intersection between the sampling
device and the population figure of interest is constant.
For example, if an areal sample is used, where A./A esti
mates V./V, and A is a nondegenerate random variable,
then E(A./A) #E(A.)/E(A), so that A./A will be a biased
estimator of V./V. In practice, this problem can be
eliminated by predetermination of the sample "size," and
locating the sampling scheme entirely within the borders
of the population figure.
A second parameter of interest on a threedimensional
figure is the surface area, S., for a particular phase j.
If the figure is sampled by a planar section of area A,
the intersection of the plane with the phase will be the
boundary length, B., of the phase on the plane. Therefore,
with d = 3, t = s = 2,
(2 2) = ( )r(2)
k3 (2,2) 2 2 T
3 F(T1)TT2) TT
so that
SE(B ) S.
SE(AT (2.2.5)
Similarly, if the boundary length per unit area for a
phase j is to be estimated from a lineal transect, the
sample measurements of interest are the number of inter
sections, I., the transect makes with the phase j boundary.
With d = 2, s=t=l,
k(l,l) ( l)
2, 1 r( )r(1) _
2 IT
so that
n (I. ) B.
2 E( (2.2.6)
implying that twice the number of intersections per unit
length of the transect with the phase boundary also serves
as an unbiased estimator for the surface area per unit
volume. Again, as mentioned previously, it will be essen
tial that the denominator of the estimator be fixed to
assure that the estimator is unbiased for the parameter.
For a treatment of the biased estimation case, see Davy
and Miles [4].
2.3 The Spatial Covariance Function
A large portion of Chapters 3, 4, and 5 will be
devoted to getting variance approximations for the afore
mentioned estimators of Section 2.2. The notion of a
spatial covariance function will be employed in some of
this work.
Consider a twodimensional mosaic, G, of fixed over
all size and any set B, of random size and location within
the borders of G.
Define
f(x) = 1 if a point x G falls
in set B
= 0 otherwise
so that P(x e 6) =E[f(x)]. The expectation of f() can be
found by first conditioning on B being of fixed size, and
then taking the expectation, E (*), over the sizes of B.
Thus,
E[f(x)] = E {E[f(x) I fixed size]}
E{ aJ f(x)dx
E (mes B)
mes Ca
where mes(') denotes volume in three dimensions, area in
two dimensions, length in one dimension, and point counts
in zero dimensions. For the remainder of this section, let
p = E (mes P)/mes G.
For two points, x and x+h, a distance h apart and at
an angle a from a fixed orientation, let
C(h) = E[f(x)f(x+h)] = P[x,x+h B]
= E {E[f(x)f(x+h) B fixed size]}
p
= Em f(x)f(x+h)dx}
E (mes B n 6_h
mes G
where Bh represents the set obtained by translating all
points in by a distance h in the direction opposite to a.
Note that
(i) C(O) = p
(ii) C(h) = C(h)
(iii) C(O) 2 C(h)
(iv) C(m) = p2
The spatial covariance function, cov(h), is defined
by
cov(h) = E[f(x)f(x+h)] E[f(x)]E[f(x+h)]
= C(h) p2
Furthermore, if ( is a twodimensional feature defined on
the mosaic, an important result involving the covariance
function is that the expected boundary length of 0 per unit
area is equal to ~cd cov(h)j =n C'(0).
h=0
If p is to be estimated, let the set S denote the
sampling device on the mosaic, where S might represent a
subarea, one or more line transects, or a set of points
dropped on the mosaic.
Let
I = f(x)g(x)dx,
a
14
where g(x) is some weight function defined on the sampling
scheme. In particular, g(x) could be a simple function,
defined by
g(x) = if x S,
mesS
= 0 otherwise
for which case
me S f(x)dx
g mes S
S
Now,
E(I2) = E{[ f(x)g(x)dx]2}
g[
= Er f(x)g(x)dx G f(y)g(y)dy]
= EL f f(x)f(y)g(x)g(y)dx dy]
= EE I ) f(x)f(x+h)g(x)g(x+h)dx dh]
G G E[f(x)f(x+h)]g(x)g(x+h)dx dh
S C(h) I g(x)g(x+hl)dx dh
= C(h)G(h)dh ,
where
G(h) = f g(x)g(x+h)dx
C
so that
V(I ) = E(I2) (I )
= J C(h)G(h)dhE[ I f(x)g(x)dx]E[ } f(y)g(y)dy]
= J C(h)G(h)dh f p2g(x)g(x+h)dx dh
= I cov(h)G(h)dh (2.3.1)
Thus, knowledge of the covariance function for a particular
structure can result in a variance expression for the area
occupied by the random set B. Note that when
xeS
mes S
g(x) =
0 otherwise
and the covariance function is stationary and isotropic,
then for two random points x and y located on S
V(lg) =  cov( y x )dx dy
(mes S) S S
1(es 3S)2 cov(h)b(h)dh
(mes S) O
where b(h) is the probability density function of h= lyxI.
With the knowledge of the spatial covariance function
being necessary to obtain a variance expression for the
arealfraction estimator, a model for the construction of
the mosaic would be very useful. The three models intro
duced in Chapter 1 will now be discussed in greater detail
so that they may be used in the development of additional
properties of the estimators.
2.4 Poisson Lines Model
As mentioned in Chapter 1, the Poisson Lines Model
is one of the most common and widely investigated models
for the random division of space. As defined by Switzer
[24], this model allows for the random division of space
by a set of random lines constructed as follows:
Upon location of an origin, "0," at the center of
the region, select n pairs of points {(r,i6.):
= 1,2,...} such that n has a Poisson distri
bution with parameter T, and 6. is uniform on the
1
interval (0,T). For each pair of points, fit the
line
x cos e + y sin 0. r.= 0.
1 1 1
This constitutes the realization of a Poisson Process with
intensity parameter T/T. The realization of this process
leaves the area with a number of intersecting random lines,
resulting in a set of convex polygons. Properties of this
model have been discussed and developed by several mathe
maticians, including Switzer [24] and Miles [131. Miles
introduced some key results for areal fraction estimation:
(1) The points at which a line transect intersects the
random lines form a Poisson Process of density
(2) The distribution of the areas of the individual cells
has mean
1 = (2.4.1)
T
and variance
2 (2_2)
02 I 2 _2) (2.4.2)
2T
2.5 Meijering Cell Model
The construction of the Meijering Cell Model, as
discussed by Gilbert [6] and Miles [14], produces cells of
convex polygons from an underlying Poisson Point Process
of intensity T. Each point expands outward at a constant
rate until it meets with another cell, or equivalently, the
cell assigned to a point, yi, on the process contains all
of the points that are closer to y. than to any other
point y (j # i) of the process. For a twodimensional
model, Gilbert [6] showed that, with A. denoting the area
of the cell generated by y.,
2
E(A.) = 1.018 T (2.5.1)
and
V(Ai) = .228 T (2.5.2)
2.6 Boolean Scheme
As developed by Serra [22], the Boolean Scheme is a
model which allows for an overlap among cells.
Let yl1,2,... be the realization of a planar Poisson
Point Process of intensity T. Let B ,B2,... be a collection
of independently and identically distributed (i.i.d.) random
sets, with each Bi implanted at a corresponding y.. The set
B = u Bi is called the realization of the Boolean Scheme,
which will be known as phase one. The remainder of the
planar mosaic is known as phase two. Serra notes that the
Boolean Scheme "represents one of the first steps in
modeling, when one admits negligible interaction between
the particles [Bi]." This kind of model might be suggested
for use in crystalline growth and land studies.
For estimation purposes, suppose the resulting mosaic
is sampled by S, a fixed set located at random inside the
the mosaic, G, noting that S may represent a subarea of
the mosaic, one or more lineal transects, or a set of points
developed on the mosaic.
Suppose B. is located at x. Then
J
Bj S = {x: Bjn S ~ .
3 J
Therefore,
E mes (.j @S)
P(S n B) = 
SITImes G
Defining N (S) to be the.number of Bj sets that inter
sect with S, it can be shown that H (S) has a Poisson
1
I')
distribution with parameter TEmes (6@S). For any point
x e G, define
i if x EB
k(x) =
0 otherwise,
and
K(h) = I k(x)k(x+h)dx = mes (B n 6_h)
If S = x,x+h}, the set of two points differing by the
vector h, then
mes (BeS) = 2mesB mes (Bn B_h)
= 2K(O) K(h)
Therefore,
C2(h) = P(x,x+h e B)
= PN (S) = O]
= exp[T{2(OK0 (h)].
Defining
q = C2(0) = P(x e BC) = eTK(o)
2 TK(h)
C2(h) = q e
Also, with
P(x,x+h E Bc) = P(x e Bc) P(x c RC x+h E B),
then
P(x r 6 x+h E ) = q e(h)
so that
Cl(h) = P(x,x+h e I) = 1 2q +q2eTK(h)
Therefore,
cov(h) = C1(h) p2 = C2(h) q2
2 TK(h) 2
= q e q (2.6.1)
The estimation of p and q can now be achieved through the
use of the covariance function technique discussed in
Section 2.3.
With models available for the construction of the
mosaic, the following chapters will serve to not only
further develop the properties of Miles' estimators, but
also to integrate these results with the available models
for a total culmination of results available under varying
conditions and assumptions on the mosaic.
CHAPTER 3
LINEAL SAMPLING
3.1 Introduction
Consider a twodimensional mosaic composed of two or
more phases. This mosaic could arise as a cross section
of a threedimensional figure or as a subsection of a
larger planar figure. The purpose of this chapter is to
investigate properties of estimators of planar features
such as areas and boundary lengths, in the case of sampling
by line transects. An areal photograph of a section of
land may provide an environmentalist with estimates for the
fraction of the land occupied by different "phases," such
as trees or water reservoirs, along with possible additional
information such as a boundary length or the length of a
perimeter around a lake. A metallurgist may be interested
in estimating the volumefraction of the void in a piece of
metal, while a medical technologist might examine a cross
section of a liver to estimate the fraction of the wall
occupied by a certain type of cell.
The two properties of the mosaic to be studied in
detail are:
(i) The areal fraction for each of the k phases, which
may in turn estimate the corresponding volumefractions
of the phases, if the mosaic arises from a three
dimensional figure.
(ii) The boundary length for a given feature, which in
turn is related to the surface area of a corre
sponding threedimensional feature.
Estimation of these properties will be studied through
lineal, areal, and point sampling techniques, the first of
which is studied in this chapter.
3.2 Lineal Sampling
This section discusses the sampling procedure of
randomly locating one or more transects of fixed length
entirely within the borders of the mosaic, from which at
least two sets of measurements will be obtained. The first
will be the fractions of the transect passing through each
of the k phases, which will be used to estimate the corr
sponding areal fractions for each of the k phases. Another
measurement of interest will be the segment counts on the
transect for each of the kphases, which will be useful
not only in estimating the boundary length for each of the
phases but also will contribute to approximating the vari
ance of the arealfraction estimators. The segment counts
also will be used as a basis of a test for preferred
orientation for features on the mosaic.
3.3 Model Development
The first step of the estimation process is the devel
opment of some asymptotic properties of the arealfraction
estimators. This will be done through the assumption of a
model for the construction of the mosaic. The assumed
model considers the mosaic as a union of a set of nonover
lapping cells with cell sizes having finite second moments.
Each cell is independently assigned to phase j with proba
k
ability pj> 0 for j =1,2,...,k and I p. = Thus, a feature
j=l
may consist of one or more adjoining cells of the same phase.
A sampling transect of length L consists of one or more seg
ments, each segment being composed of a random number of
intersections with the original cells. Numerous phases may
appear on an individual transect, but there is no guarantee
that all k phases will show up on the transect.
Recall from Section 2.4 that both the Poisson Lines
Model and the Meijering Cell Model result in cells with
finite second moments, and hence, could result in a mosaic
as discussed above. In practice, cells may be distin
guishable in some cases, such as in observing the cross
section of the wall of the liver. However, in most cases
where such a model may be employed, the individual cells
may not be distinguishable. Also, many mosaics may not
even have an underlying cell structure, especially when a
large continuous feature such as a lake or river is studied,
but the cell model will still be assumed in order to develop
asymptotic properties of the estimators. The three specific
models for cell development from Chapter 2 will be discussed
later in more detail, but first the results will be derived
in terms of a general cell model. A few terms will now be
defined.
Let L equal the length of the sample transect randomly
located on the mosaic, and let X. denote the intercept
length of the ith cell cut by the transect. It is assumed
that the cell intercepts cut by the transect have lengths
that are independently and identically distributed with
finite mean and variance, p and o2 respectively. (This is
indeed true for the Poisson Lines Model.)
Let
1 if the ith cell cut by the
transect belongs to phase j
Oij =
0 otherwise.
and let
N = the number of intersections between
the transect of length L and the cell
boundaries.
Generally, the value for N will not be obtainable from
the data, unless the individual cells are distinguishable.
x *
Estimates for the expected value of N denoted by E(N ),
will be discussed later in this chapter.
Defining L. by
N
L. = iXi (3.3.1)
11 ': 1i~
L. will denote the total length of the intercepts of phase
3
j on the transect of length L. Note that {i X } are i.i.d.
with mean pjp and variance p.(l p.)I2 +pj2.
The cell intercepts on the transects can be viewed as
a realization of a renewal process by considering the
intersection points as the times of occurrences of different
events (phase assignment). A standard result of renewal
theory is that the average number of events per unit time
converges to the inverse of the average length of time
between events, or equivalently,
(N (3.3.2)
L 11
as L m, assuming p > 0.
In looking at the limit of E(L./L), Wald's Theorem
J
[23J can be used since
E(ij.X.i) = p.i < .
Therefore,
lim E(L./L) = pj (3.3.3)
L J
implying that I,./L is an asymptotically unbiased estimator
3
of p.. Also, by the Weak Law of Large Numbers and the fact
that
P
N /L * 
,
L /L is a consistent estimator for pj as L o.
J 3
The underlying mosaic is considered to be infinite in
size, so that a sampling transect can intersect a large
number of cells. This leads to an additional consider
ation on the mosaic. Since the cells have equal expected
th
cell size, the total areal fraction, F., for the j phase
on the mosaic, will be approximately equal to p.. Thus,
L./L can be thought of as a consistent and unbiased esti
mator of F..
J
3.b Asymptotic Distribution of the Estimators L.//E
Define the vectors L and p by
= (L12 L Lk)
P' = (Pl P2 Pk)
The next step in the estimationprocedure is to derive the
asymptotic distribution of (L), which will allow for the
usual inferential techniques of hypothesis testing and
interval estimation. Point estimation has already accounted
1
for L being a consistent and asymptotically unbiased
estimator of p. The theory behind cumulative processes
yields the theorems which give the joint asymptotic distri
bution of (Lp). To follow the definition of a cumu
lative process as stated by Smith [231, Z .ij X is defined
i=1
to be a cumulative process if:
(i) For fixed j, .X. is a sequence of i.i.d. random
variables for i= 1,2,...
*
N
(ii) .ij.X. is, with probability one, of bounded vari
i=1 l3 1
action over every finite interval on the line.
The first condition holds by assumptions already stated.
The second holds since N is finite with probability one
on any finite interval, and j..X. is nonnegative and
finite.
The asymptotic distribution of I(L p) now can be
derived as a result of Theorem 10 of Smith [23]:
n
If (ilX. i2Xi *** ikXi) are k cumulative
i=l
processes and a2 < _, then the vector J/jIL(L e)
will converge in distribution as L to a multi
variate normal random variable with zero mean
vector and variance matrix B, where bst, the stth
element of B is given by
bst = cov(is Xi PsXi. itXi PtXi). (3.4.1)
To obtain the individual terms of the asymptotic
variancecovariance matrix,
b.. = E[(4 2 p)2 X2
j = i j ) "i
= E[(i pj )2](X )
since X. and ij. are independent. Therefore
bJ = pj (1 j) (a + 2). +
__
(3.4.2)
For s t, the covariance term,
st = E is Ps ) itPt)
= E[(is s (it t)]E(X )
= ps t(a' + 2) (3.4.3)
Therefore, as L gets large, the variance of L /L can be
approximated from the above asymptotic variance. This
term, which will be denoted as vjj, can be written as
S pj p j)( + 2) p ( p )(l + )
S .v..  (3.4.4)
JJ Lp E(N )
The corresponding covariance term can now be approximated
by
U2
x Pp (i+ )
vst Ps(3.4.5)
E(N )
for s t. Now, the regular inferential procedures of
hypothesis testing can be employed on the fractions pi,.
S..,Pk once estimates can be made for p, 02, and E(N ).
3.5 Basic Properties of Segment Counts
With the establishment of the variancecovariance
approximations for the lineal estimates, the next step
is to try to estimate the modeling parameters of (3.4.4)
and (3.4.5). Note that in using L./L to estimate pj, p
and o2 are parameters of the mosaic, and will have to be
estimated to get a reasonable approximation for v.. and
JJ
vst. Additional information that can be gathered from the
sampling transect may be useful for this purpose. It is
assumed that the individual X.'s cannot be measured.
.1
Instead, the transect appears as a series of segments of
different phases. Each phase j segment is composed of a
random number of cell intercepts, that random number having
a geometric distribution with parameter p.. The segment
counts for each phase not only provide useful information
in estimating the V matrix, but also allow for boundary
length estimation of individual phases, along with a test
for orientation and homogeneity of features on the mosaic.
Define
1 if the (i 1)st cell intercepted by the
transect belongs to phase j and the ith
f.. = cell intercepted is not a phase j cell
0 otherwise
N
and let M.= 2 f for j =1,2,...,k. M. will represent
Si=2 .th
the number of individual segments of the j phase on the
transect. For a mosaic with a large number of cells, the
M.'s can be used to develop the theory behind the practical
use of the segment counts in variance approximations, bound
ary length estimates, and orientation tests. Recall from
(2.2.6) that r/2 times the number of intersections per unit
length that the transect makes with the cell boundaries
gives an unbiased estimate of the boundary length per unit
area for the phase j features, with a corresponding result
for surface area in three dimensions. In defining I. to
J
be the number of intersection points the transect makes
with the boundaries of phase j features, I. =2M. unless
J J
the test line begins or ends with a phase j region. In
any case, 2M. gives a reasonable large sample approximation
of Ij, so that any asymptotic results for I. come about by
looking at corresponding results for M..
Define
J = the phase of the nth cell cut
n by the transect
and
Fvj (x) = P(Xn
The vector M' = (M M2 *. M k) constitutes a vector of
functions defined on a Markov Renewal Process by definition
of Pyke [17], with transition matrix
(p vj = {Pj) ,
since cells are independently shaded, and sojourn times
governed by, (F vj(x)}. The joint asymptotic distribution
of L M will be derived through the use of limit theorems
developed by Pyke and Schaufele [18]. To equate the
terminology of Markov Renewal Processes with the sampling
results from the lineal transects, a few additional terms
need to be defined. Let
1 if v= j
vj 0 otherwise,
m J' = (expected number of phase v cells
crossed by the transect between two
consecutive segments of phase j)+ 6vj
= p /p ,
k
rv = 1 Pvj1 =
p.= expected return distance to phase j
on the transect
Sm(j)
V V
v
= u/p ,
and
M = expected number of cells of phase r
Swr on the line between a segment of phase
w until the next segment of phase v
S(1 6 ) + 6
p vv wr
The theorems needed to derive the asymptotic distri
bution of M are all stated in terms of the moments of f..,
so the next step is to determine these values. Without
loss of generality, the attention will be focused on phase
I, since the procedures used to look at the other phases
will be identical. For any other phase j, all subscripts
for phase one can be replaced by the pertinent phase j,
for j =2,3,...,k. To look at the moments of fil let
Sm(fl) = f ilP jm dF (X.)
f p
0
pn
0
if j =1, m 1i
otherwise ,
c (f i = f 2PjdF j(Xi)
0
if j = 1, m l
otherwise ,
k
j(f ii) =IL 3 (fm 1
'=1 i
O p
1 l pl
jil
j= 1,
k
(2)( f (2) (
j f il jm (
Using the Strong Law of
Schaufele for functions
Process,
Large Numbers given by Pyke and
defined on a Markov Renewal
0 1PO ~ lf
33
M (1) j il
I a.s. j=l j )J(fil)
1 a.s. (3.5.1)
L pl
Since
0 if j l
( fil) =
JpI J =l
and
(1)
m =1,
M1 a.s. P1(1pl)
S (3.5.2)
L p
as L'. Therefore, as L~,
M a.s. p (1 p )
j a .s.(35.3)
L p
for j = 1,2,...,k.
Through a Central Limit Theorem result of Pyke and
Schaufele [18] for functions defined on a Markov Renewal
Process, it will be shown that each M., suitably normed,
J
will have an asymptotic normal distribution, and that the
joint asymptotic distribution of the vector M, suitably
normed, will be multivariate normal with a closedform
expression for the variancecovariance matrix.
3.6 ::::'::. :c Di:str b .ion of M./jA
.1
Frsm Teorem 7.1(b) and Lemma !, 2 .:f' Pyke and 'chaufele
cis],
E N I. ,., u e. 3.6.
; i ]
1 N I
LL
.::.!i c:'',erre in distribur;io as Lo to a standard normal
ran.doT vanianile, provided the variance cf each term in
the suo (3.6.1) is finite. Recalling thac
N
M. = y f
J j2
n ii"
p.(1 p )
J U 0
..6. 1) becomes
fM. N X*
hi, U i
.: vP
L 2 ii
as '. Tui, v:orkinr ::!oh (3.6.1) ultimately yields the
:i::::. ::: : t lo of ::e er lit Cocus t nt n oni i
::,' ;i:t Nos:: .f Crrcenernitt:, cocus ttcenticr, on
 *:.. :' ,..
hi = fil XC
Lemma 4.2 of Pyke and Schaufele [18] yields the asymptotic
variance expression for (3.6.1) as
B < 2)(hil)m1)
3 (3.6.2)
+ i m(hil )r(h il))1 1M
j mli r# l J 1 imr
Following the previous definition of terms,
jm(hl) = hii(X)pjmdFjm
= jm (fi) Cp p
pm pCPm j = 1, m (3.63)
1iCI m otherwise,
( ) = [f XA ] pmdjm
= (m il) 2C1pljm il) +C m
2
where (2) = + 2+Cp Th jr=l'efore
S(h11) = (3.6.))
C p ( otherwise,
j (hil) = jm (hjl
In C1
Pl UCI
and
c(2) (h 2)(h
Sj m i n 11
(2)
C1vl
p2 ((2)
( p 2C^(p1 p) +2 C1 (2)
Now,
m() (2)
.j j Jhi1
= 1 pl 2C1(1 pl)
= 1 p 2C p(l pl)
J+ p C2 (2)
I
C2 (2)
Pl
and
2 [ jm (h (hi )m~() lM
j m/l r/1 jm r i j mr
S2 (p iCp m)(uC ) (1 6r ) +6r
r m 1 1 L l
+ I ( (PC p )(pC1) (16r) +6m
j / m '/1 r 1 1 1 r
j 1
j=1
(3.6.5)
1
(3.6.6)
1.
(3.6.7)
which reduces to
2pc (1 pl)
P 1 UC1 J (3.6.8)
Substituting into (3.6.2),
Bh 1 p 2CIl u() + (2) 2p
pi PI Pl
= C1[p + ( pl2] + C ) 2i (3.6.9)
Therefore, the asymptotic variance
S(M ,/L) = C p 2+ ( p) 2 + (3.6.10)
(j2) j *j j j
for j = 1,2,...,k. Since p (2) < ', V (M //L) < and there
fore, the asymptotic distribution of each M.//E is normal,
2 J 2f,(2) 1
2 + (1 2 7+ 21]
with mean C. and variance C.[p +(lp.)2] +C 2 .
3 J J J J 3
1
3.7 Joint Distribution of LM
With the univariate asymptotic normality of each
M //L established, the next step is to prove the joint
J
normality of M//L. The argument used to achieve the final
result is given by Smith [23].
Recalling hij = f.j X.C., write
h. = (h1 *** h
i_ ik
for i=2,3,.... Now E(h )= 0 for all i and j, and let
cov(his,hit) =ast for s /t. Note that the ast terms have
not yet been developed, but will be used in finding the
asymptotic covariance between Ms//L and Mt//L. The term,
N
T. = Ii..//L will converge in distribution to a normal
i2 i
random variable with mean zero. For any real vector
e_= (01 e2 Ok)
k k N h..
e'T = e.T. = I e 6 
j= 1 j=l ij=2
N k
S 1.h..
i=2 j=1
1 N
=L elhil + 2hi2 + + khik)
Letting
i = L(hil + 2hl + ***+ Okhik
E(Wi) = 0
and as L m,
V(W) = e V(h .)+2 I setcov(hs,ht)
L j=1 st t
= O'V .
Since 6'T converges in distribution as L to a normal
random variable with mean zero, one can see by the use of
characteristic functions that T will have a multivariate
normal distribution with zero mean vector. That is,
/L(M/L C) I N(O,V ).
3.8 Asymptotic VarianceCovariance Matrix of M//T
The covariance terms of M//L can be obtained through
Lemma 4.3 of Pyke and Schaufele [18], which yields an
expression Bst similar to the Bh introduced earlier, except
that when s / t, the expression deals with the covariances
of h. and hit functions instead of the individual variances
is it
of the h.. terms. Formally, let
Bst j(his'it)mj + j m/ 1j jmis r (hit)
J j ml r?1
+ m(h )c (h )jmi l) M (3.8.1)
jmIfl it r is 1 mr
where
jm(h is't = his hit jm dFjm
= I [fisCt +fitCs jm dFjm
+ X CsCtPjmdFjm
(2)
( CsCt j / s,t or j = m = s
s tj m = t
= (2) C C m j = s ,m s (3.8.2)
(2) CsC P CtPm j t m t
St(his hit) = jm(hishit)
(2)
(2) CsCt
= (2) CCt Cs(lPs)
S(2)Csct Ct(1 pt)
j / s,t
j = s (3.8.3)
J =t
so that
j (his' ht )mj
(2 c P P
Stpi s(2)S p t
P 1 )CsCt PCspsl p ) PCtPt(l
Now,
r r [jm (h(hi s +r(h (hit r (his)]m Mm
ij m/1 r I m it r s mr
Z y y L[{PCsm +m js( 6ms)}{uCt + ( pt)6rt
j m/l1 r m ms
+ tPm + Pm6 ( 6mrt }UCs + ( p s )6rs
(1 6)+ 6
p 1 1 mr
Pt)]. (3.8.4)
SY I [2 CsC tPm PCsP (1 t )rtCtP C p (1 s
j m71 r/1
+pm (1 Pt)6js(l 6ms)6rt + CsCtPm uCtpm(I Ps)6rs
Cs m6jt(1 6mt) +Pm( Ps )jt rs(l 6mt
' r[ (1 6 ) +mr
p P 1 rl M rme
In breaking up the expression into a series of fourteen
triple sums, and summing each up under restricted terms as
defined by the 6 functions, the whole expression simplifies
to
S 2 2
[C sPt+C p (3.85)
so that the asymptotic covariance term,
I Mt
s t 1
cov 'M = BllH t
(2) t (CS + C ) + Csp + Cp (3.8.6)
Therefore, the asymptotic variancecovariance matrix, D
of M//L is given by
2 (2)( 2 2
dst = CsEp2 +( i ( C 2+ 2 (3.8.7)
d*t = p 2+ Cp + CsC 2(2 + ) (3.8.8)
for s # t.
3.9 Variance Approximations
In using L./L to estimate F., recall from (3.4.4)
J J
that the asymptotic variance of L./L is
p (1p ) 1+ p (1p ) 1+
L E(N*)
Under the assumption that the individual cells are not
distinguishable, a value for N would be difficult to
obtain. Since
M. a.s (1 
L U
from (3.5.1), both
k M.
(i)
j=I i
L
and
k
I M.
(ii) j= 2
k L.
1
*L
j=1
will be consistent estimators for E(N ), as would each
individual estimator,
M.
L L.
L L
for j =1,2,...,k. A comparison of the variances for each
of these estimators could be obtained using the results of
(3.8.7) and (3.8.8), but this becomes mathematically com
plex. It would appear that
k
M
k 1L.2
1
j=1
would be the most stable of the available estimators, since
it utilizes the sampling information from all k phases, and
k L. 2
also Y  should stay fairly constant, even over widely
j=1l
ranging values for each L./L.
J
It follows that a reasonable consistent estimator of
v.. would be
JJ
L L + 2
v = k (3.9.1)
Z M
1 f
s=l
It still remains to find an estimate for o2/u or at least
to get working approximations to its value.
A lower bound on the variance term would exist for a
cel] division resulting in a system of cells very close in
shape and size, so that p would be considerably greater
than o2, resulting in ao2/2 approaching zero.
One possible way to obtain estimates for j and 02 is
to use the information available from the different segments
of different phases on the sampling transect. Let N = the
number of cell intercepts composing the nth segment of phase
j. Recall from before that N has a geometric distribution
with
E(N )
Enj p
Pj
p.
V(N .) =
n (1. p )2
th
Therefore, writing Znj, the length of the n segment of
phase j, as
a .+ N
Z = n Xi.
nJ i = a .+1
nj
th
where a = the number of cells preceding the n segment
nJ
of phase j,
j = E(Znj) = E(Nn) 
3 nj nj 1p.
3
= V(Znj) V(N njp) + E(N o2)
2
 + (3.9.2)
(p)2 lpj
Rewriting the first equation as
p = (Ip )E(Znj.)
p can be estimated from each color by using the average
L.
length of a phase j segment, Y7, multiplied by (1 ).
Thus a weighted average of estimates of p over all colors
k
would yield a consistent estimate for p. Since upp =p,
j=1
one possible such estimate would be
k L. L.
jL L (3.9.3)
j=1
Rewriting (3.9.2) as
2
a2 = V(Z )( p) 
nj a 1p.
would yield an estimate of 02 by estimating p with L /L
and i by (3.9.3), and using the sample variance of the
length of the phase j segments as an estimate for V(Z j).
3.10 The Results of Hilliard and Cahn
The method just suggested in the previous section is
used by Hilliard and Cahn [8], who are responsible for many
of the applied results in volumefractions analysis. While
their results are only derived for k =2 phases, a similar
argument could extend their results for k> 2 phases. To
compare the earlier results with Hilliard and Cahn's results
for k= 2, let p=p, = p2q p, so that
1 p 2 p'
S 2 2
2 = + P
1 1 p (1 p)2
and
2 2
2 0 2 ( i P 2
So (lp)
2 p 2
Therefore, by breaking down the asymptotic variance for
k = 2 into an expression using the means and variances of
the segments,
p(lp) 1 + 02
V = 
p (1 p) 1
L
p + H
1 p p
P2l P)2 _[ + 2
p2(1 p)2 [(2 p) +pp2 p + (1 p)2
L 2 2
(1 + p)2
21 2
(M ) 2 2 + (3.10.1)
which is the HilliardCahn result. The sample estimates
gathered from the line segments would serve to estimate
~l' G2, 0, and 02 while L1/L and M estimate p and E(M1)
respectively. This approach for estimating the variance
would probably yield accurate results for a random model
with many shaded features, thus allowing for many segments
of both colors on the line segment for estimating the seg
ment parameters.
It should also be noted that the HilliardCahn result
can be obtained by looking at the segments on the line
transect as an alternating renewal process. By defining
Y = length of the it segment of phase 1
i
2 th
Y2 = length of the i segment of phase 2
the YJ's are i.i.d. with mean p ,and variance o2 for
j = 1,2. With
M Y
1 i=l
L L
a variance approximation given by Smith [23] yields
f 1 1 1 1 1
L + [ +1132 { Li 1 + j 2 21+1121
S2 2 + + 02)
L(p +U2T 1 I + 2J u
S1 2 + 1l
^1
p 2+P2 '
1+21 2
With
P2
1112
and
L
E(1) 12
the above reduces to
2I 2
E(MI)
which is (3.10.1).
For situations in which the HilliardCahn procedure
may yield inaccurate variance estimates due to a small
number of segments on the transect, it may be helpful to
study a possible model for the cell structure in order to
get estimates for p and o2. The next two sections present
possible models for consideration in such a situation.
3.11 Poisson Lines Model
The mosaic that arises from a Poisson Process, known
as the Poisson Lines Model, allows for the random division
of space by a set of random lines, as detailed in Section
2.4. Letting T be the density of the random lines on the
process, a direct result of Miles [13] is that the inter
section points of the line transect with the random lines
2T
forms a Poisson Process of density A . Therefore, the
distribution of the X.'s will be exponential with parameter
2 2 1
A. With the exponential distribution, p = 0 = 2 so that
\2'
v. = (3.11 )
J E(1J )
where p can be estimated by Lj/L and E(N ) approximated,
as mentioned before, by
k
SM
s=l
k fL 2 '
s
1 k
s=1l
which generalizes Scheaffer's [20,21] results for k=2.
Numerical results will be presented in Chapter 7, which
will allow for comparisons between this approximation and
the HilliardCahn technique from the previous section.
3.12 Meijering Cell Model
The Meijering Cell Model, as developed in Section 2.5,
also results in a set of convex polygons with identical
size distributions on the mosaic. The means and variances
of the cell sizes are tabulated by Gilbert [6] for two and
threedimensional mosaics sampled by areal sections and
lineal transects. For a twodimensional model generated
by a point process of intensity T, sampled by a lineal
transect, the intercept lengths of the line with the indi
vidual cells will have a mean u= 1.027T 2 and variance
o22 .88T1 Therefore, for this model,
S 1.18pj (I p )
V .. = ( i 
JJ E(N )
where E(N ) can be approximated as before.
3.13 Tests on the Mosaic Using Segment Counts
Consider the placing of multiple transects inside the
boundaries of the mosaic. The measurements of interest
include the individual segment counts for each phase from
each transect. The asymptotic properties of the segments
were developed in Section 3.8. This section will utilize
the results from Section 3.8 to develop a test for the
equality of mean segment counts, over all transects.
Applications for a test of this kind would arise for feature
orientation and phase homogeneity. The test for orientation
would consist of independently placing n transects of length
L at differing angles across the mosaic. If, for individual
colors, the number of segments per unit length varies signi
ficantly from transect to transect, then a possible color
orientation might exist.
A segregation test would involve the same ideas, except
that the n transects would be placed in a parallel grid
across the mosaic.
Let
M. = the number of segments of phase j
j,' on the mth transect
and let
M' = (M M *** ).
m ,m 2,m k,m
The general test, which follows Scheaffer's [191 test for
k=2, is set up as follows:
H : E(M ) = E(f2) = = E(M )
H : There is at least one j' such that
a E(Mj,) E(Mj) for j j'.
J J
Recall from Section 3.8, that if H is true, (M LC) D
o /L m 
1 F *)1
Nk(OD ) as L+ m, so that under H, (M LC)'(D ) (M LC)
has asymptotically a central X2 distribution with k degrees
of freedom. Let
n P
1 M
n m=l m
be the vector of averages for the number of segments for
each phase over the n transects. Then S(MLC) also has
/L 
an asymptotic multivariate normal distribution with zero
1 *
mean vector and variancecovariance matrix D Thus by
n
writing
M LC = (M F) + (PLC),
it can be shown, using Cochran's Theorem, that under Ho,
T n 1
T = (M M)'(D) (M F)
Lm m
m=l
has an asymptotic X2 distribution as L with (nl)k
degrees of freedom. Therefore, the test statistic will be
T 1 t(M M)'(D ) (M M)
L m m 
m=1
where D must be a consistent estimator for D Recall
that
d 2 p+2 2 2 p (2) 2
J = J 2J
and
2 2 (2) 2 2
dst = Cspt + CtP + CCt C +
for s #t, so that p. could be estimated by the total frac
tion of phase j over all n transects, and u and j(2) esti
mated from the segment information, as developed in Section
3.9. Since M./L is consistent for C., the total number of
J J
phase j segments per unit length should serve as an esti
mate for C.. With D approximated using the terms from
J
above, the test for equal means would have a rejection
region of the form
2
T X(nl)k,a
Recall from Section 3.9 and 3.10 the need for the assump
tion of random cell assignment in order to use the variance
approximation from the Poisson Lines Model. An orientation
test could be used as a prelude before such an approximation
is used.
3.14 Spatial Covariance Function Approach to Using Lineal
Sampling: Poisson Process
The approach discussed in Section 2.3 to obtain vari
ance expressions for the areal (volume)fraction estimators
depends on obtaining an estimate for the covariance function
for points on the mosaic. When estimates of the covariance
function are available, then the corresponding variance for
lineal fractions is
V = Icov(h)b(h)dh ,
0
where b(h) represents the density function for the distance
separating 2 points randomly located on the line transect.
When the sampling scheme, S, is a line transect of length L,
let Y1 and Y2 represent the distance of the two points from
the beginning of the line. The distances Y1 and Y2 are
independently distributed as uniform random variables on
(0,L), so that for 0 < h L,
P(IY1 Y21 h) = P(Y(2) Y(1) h)
l ,I(Lh)2
= L' 2! dh = i (L
0 Y2h
Therefore
2 h
(1 ) for 0 < h < L
L L
b(h) = (3.1 1)
0 otherwise
Now for any covariance function representing the under
lying mosaic, the variance of L./L can be obtained.
J
The first model to be considered is the Poisson Lines
Model, already having been defined in Section 2.4, for
which the covariance function for color j is
cov.(h) = p (1 p.)eh
where A is the underlying parameter of the process. There
fore,
2p (2 1 p ) h hei hdh
V L eL dh,
which as L gets large, will result in a comparable expres
sion to (3.11.1).
3.15 Boolean Scheme
The model for the Boolean Scheme, as developed in
detail in Section 2.6, yields a covariance function
cov(h) = q e (h)q
for a model generated from a point process of intensity T.
When S is a lineal transect of length L,
VJ} f [q2eTK(h) 1 q (1 b)dh
v L )2q
a
One possible approach to integrating the above expression
is to use a Taylor expansion, with
eTK(h) = e K(O) +hTK'(O)eTK() +o(h)
= [l +hTK'(0)] + o(h) ,
q
where K(h) is defined as in Section 2.6. Recall from
Chapter 2 that 7C'(0) equals the expected boundary length
per unit area, which in turn can be estimated by T x the
number of intersections of the transect with the cell
boundaries. With TC'(0) = Tq2TK'(0 )e K(0) = qK'(O), an
estimate for TK'(0) will be
7T M
2L 2qL "
With this approximation available, eK(h) can be approxi
mated by
i 1 (3.15.1)
q 2qL ,
so that V can be approximated by
L 1
i(L hM 2 2 h
J(q q )L (l)dh
0
L 2 h_ hM(l h dh
= ( ) L (1 L)dh
M M M
= pq = pq (3.15.2)
However, a problem with this approximation might arise if
K(h) = when h is greater than some fixed distance.
A second possibility would involve studying specific
models which could generate a mosaic from the definition
of the Boolean scheme. Unfortunately, many modeling
assumptions result in an expression for K(h) that makes
Vi mathematically complex. One of the simpler models
to work with assumes each 8. set to be a ball of radius r.
.J
2
With K(0) = r it can be shown using geometric results
mentioned by Matern [11, that
r 2 l h2
K(0) 2r sin 1 rh 1 for 0 < h 2r
\ r2
K(h) =
0 otherwise.
Therefore,
L .L L
fV  [q2eTK(h) 2 (l )dh
0
2= s2rn r h rh 1 h ]
2 h
L L)d'h,
which is mathematically difficult to integrate. An approxi
mation can be obtained through a Taylor expansion of the
term
T 2 rsin1 rh 1 12
I 2r 2
However, using the approximation from (3.15.1), V [
IL
can be approximated by
2 2 h hM h
pq 2(1 1 L 2(1 )dh
0 L
pq(Ir) pq8r2 2r2M +8Mr3
L L2 L2 31
As L gets large, the second and fourth terms should get very
L.)
small, so that V  can be approximated by
pq(lr) 2r2
L 2 (3.15.4)
L L
Note that M is a function of L, so that the second term in
the expression is really the same order of L as in the first
term, so that as L gets large, the two terms should get
close. However, it should be realized that the whole approx
imation is based on the expansion of e k(h), from which
estimates are only obtained for the first two terms. If any
additional terms of the expansion are significantly differ
ent from zero, the corresponding variance estimate may not
yield an accurate approximation.
CHAPTER 4
REAL SAMPLTNC
4.1 Introduction
In considering the parameters of the ndimensional
mosaic as defined in Section 3.1, it was mentioned that
several different sampling procedures would be considered
for obtaining information vital to the estimation of various
properties of the mosaic. Chapter 3 developed the ideas
behind the lineal sampling approach, with the asymptotic
properties of the lineal estimators being developed from a
renewal theory argument and the use of some limit theorems
on Markov Renewal Processes. This chapter will develop
estimators for the mosaic parameters when the sampling pro
cedure involves the random location of a planar section or
sections of fixed total area A entirely within the borders
of the mosaic. The fraction of total area of the planar
features of phase j gives an unbiased and consistent esti
mate of the fraction of the jth phase on the entire mosaic.
As done in Chapter 3, the next step will be the development
of the asymptotic joint distribution for the vector of areal
fraction estimators, followed by estimation of the variance
components through the utilization of other data obtainable
from the real sample.
4.2 Asymptotic Distribution of A//A
Many of the arguments used in Chapter 3 can be extended
and/or paralleled to derive the asymptotic distribution of
the vector of arealfraction estimators,assuming that the
sample is a planar section of area A randomly selected from
the mosaic. Under the assumption of the underlying cell
model for the construction of the mosaic, as developed in
Section 3.2, let
th
X. = area on the i cell on the plane,
I1
E(X.) = c
1 if the ith cell is phase j
ij 0 otherwise
N = the number of cells on the sample planar
section .
Again, assuming that the individual cells will not generally
be distinguishable, N will not be available from the data.
Letting
N
A.j i ijxi.
i=1
A. will represent the total sampled area due to phase j
J
features and A./A will be a consistent estimator for pj,
for j =1,2,...,k. Let A' = (Al A2 *** Ak). The approach
used to develop the asymptotic distribution of /A is to
VA to
think of the cells being dropped sequentially and filling
up a total area A, much as the line segments fill up the
lineal transect of length L. Assuming that the individual
cells will be small in comparison to the large sampling
area, so that of the sample will consist a large number of
cells, any border effect of the areal sample will be negli
gible. With the independent assignment of colors to the
cells, the sequential arrangement of cells will allow for
parallel arguments to the renewal theory derivations of
Chapter 3. In addition to A./A being a consistent estimator
for p it also is asymptotically unbiased, this being a
direct result of Wald's Theorem in an analagous proof to the
lineal case of Section 3.3.
Furthermore, the joint asymptotic distribution of
I(A_p) will be normal with zero mean vector and variance
covariance matrix V., so that the variance of A /A can be
approximated for large A by
p(1 1p) 1+
vj = A/ (4.2.1)
and covariance term,
v PsP (4.2.2)
st A/u
for s t. Hence the next step is to get practical estimates
for p and c2 from the information available from the sample
data.
4.3 Estimation of the Variance Components
To obtain a practical estimate of the terms of V
A./A and 1 A/A serve as consistent estimators for p. and
J J J
Sp. respectively, but the other terms are parameters of
J
the underlying scheme.
Some models may arise in which the cells are very sim
2 2
ilar in both size and shape. In this case, E(X)/ 2 is
very close to one, so a lower bound on vjj is
approximately equal to E(N ), and an estimate for E(N can
be obtained, similar to the one from Section 3.9. Defining
J 2
k A A .
A/p
Since E(N )/A * 1/p as A , A/p can be thought of as being
*
approximately estimqual to E(N ), and an estimate for E ) can
However, the feature count in two dimensions is not as
easily obtained, similar to the one from Section 3.9. Defining
since very regularly shaped of phases make counting diffi
cult. Frequently, the sample,nar simple section may have only
k f tAAS]
IM ./1 I i ;
5=1 2 J=1i"j
is a consistent estimator of E(N ).
However, the feature count in two dimensions is not as
easily obtained as the intercept count in one dimension
since very irregularly shaped phases make counting diffi
cult. Frequently, the planar sample section may have only
one continuous feature for a certain phase, such as in
viewing a large body of water in a land study situation.
Perhaps a more useful result could be obtained from a
basic stereological result introduced in Chapter 2. Recall
that E[boundary length for phase j features]/A = rC'(0),
J
where C.(h) is as defined in Section 2.3, for phase j. For
this estimate to be used, an underlying process for cell
modeling must be assumed, so that a form for C.(h) can be
obtained. The next section discusses this estimate for a
frequently used model, the Poisson Lines Model.
4.4 Poisson Process
If the underlying process constitutes the realization
of a Poisson Process of density A = /r, as stated by Miles
[131, several additional results of Miles will help to sim
plify the asymptotic variance expressions.
The individual cell areas have mean
1
and variance
2
oz 2
2nT (A )
and the expected cell boundary length is 2/A With
o2/U = ( 2 2)/2,
S p.(N ) 2A( 2( )
2E(N ) 2A(A
Recall from Section 3.14 that for a Poisson Lines Model
generated from a point process of intensity T, the covar
iance function of color j is
2T *
(j \>2A h
cov.(h) = pj(l pj)e = p ( p)e h
so that the expected boundary length per unit area of phase
j features is
E(B.)
A = 2A pj(1pj) (I.4.2)
Therefore,
B.
0 ) ( 4.4.3)
2A7ip.( p j4.
J J
gives an unbiased estimate of A.. Utilizing the sampling
information over all colors,
k
) B.
1 j=l
2AkT k A. 2
j=1 A
gives a consistent estimate for A In arguments similar
to those of Section 3.9, it would seem that this would give
a reasonable estimate of A, since the denominator expression
of I would appear to be fairly stable over varying
j=1r
arealfraction estimates.
4.5 Meijering Cell Model
For the Meijering Cell Model defined in Section 2.5,
Gilbert [61 tabled the following mean and variance for the
areas of cells arising from a point process of intensity T:
2
S= 1.018TO
and
02 = .228T .
Therefore, for this model
** p.(l p.)
v.. = 1.22
JJ E(N )
It should be noted that for both the lineal and areal cases,
this model yields less variability for the arealfraction
estimators than the Poisson Lines Model, so that it might be
preferred for mosaics with similar type features.
4.6 Comparisons with the Results of Hilliard and Cahn
A variance expression for A./A is obtained by Hilliard
and Cahn under the assumption of two rigid conditions. The
first assumption is that the number of equal sized features
of color j within the given region follows a Poisson distri
bution. Second, the areal fraction for any phase is assumed
to be very small. Their final result is
VAj 2 E(M[.) 2 + (451)
V A= o2+p] (.5.1)
A A2 } t, J
where jp and c2 denote respectively the mean and variance of
J J
a phase j feature. Recalling that a feature may be composed
of one or more individual cells, and using the arguments of
Section 3.9, it can be shown that
J 1p
and
2
2 o2 + j
j 1 j (1 P)2
j 1 p 2
or equivalently,
p = p (1 p .)
and
2 = (1p )o Pp .1,
with
x* p.(1p.) A2]
v + 3i +a
jj A/p
/ 2 P P21
= p (1pj) 1+
= u [u +o2]
E(M. )
SE(M[ 2+{(3p +2p )U2_pj2]
2= +uj 3P.+p f .p~j(4.6.2)C
(4.6.2)
which reduces to (4.6.1) when p. is very small. As a result,
the variance expression obtained by Hilliard and Cahn repre
sents a special case of an asymptotic variance from a more
generalized model. The result of (1.6.1) may be a useful
expression if there are a few individual features from which
we can obtain areal measurements within the sampling plane.
The sample mean and variance can then be used as consistent
estimators for i. and 0o respectively.
J J
4.7 Spatial Covariance Function Approach
Recall from Section 2.2 that if S represents the
sampling scheme on the mosaic, then the variance of the
arealfraction estimate is
Scov(h)b (h)
where bS(h) is the density function for h, the distance
between 2 points randomly located on S. Even for a recog
nizable covariance function, the variance term may be
extremely complicated mathematically. The one modelling
scheme that has been studied in the results of Chapter 3
and 4 has been the Poisson Lines Model, in which for color
j,
2A h
cov (h) = pj(l p.)e
In studying the arealsampling approach to volume fraction
estimation, let S represent a'convex set of total area A on
the plane. A geometrical result due to Borel and given by
Mat6rn [111 is that
2
2C h
bS(h) = +o (h2) (4.7.1)
S A 2
A
as h 0, where
Cs = total perimeter of boundary
length of S.
Note that this approximation is only applicable when h is
small, but as h gets large, covj(h) =p (1p )e2 h gets
very small. Also, as A gets large, the second term of
A.
bS(h) becomes very small, Bo that V can be approximated
by
2pp I( p ) lie21* dh
p (1 p )
2( ) 2A
yielding the same result as (4.4.1).
CHAPTER 5
POINT SAMPLING
5.1 Introduction
The final method of estimation to be discussed involves
the use of point sampling on the mosaic. By placing points
within the boundaries of the mosaic, the fraction of points
falling upon a certain phase can be used as an estimator of
the phase proportion. Koop [9] has discussed 21 methods of
point sampling for the twophase planar mosaic. Several of
the simpler techniques will be discussed and extended for
the general kphased mosaic. Also, the asymptotic distri
bution for the phase proportion estimators will be developed
under restricted conditions.
5.2 Random Points on the Plane
For a planar mosaic of area A, let A1,A2,...,Ak denote
the total area on the mosaic for phases 1,2,...,k respec
k
tively, such that Y A. =A. As noted earlier in Chapter 3,
j=1
the mosaic itself may represent a planar section of a two
or threedimensional population figure, for which each A./A
J
would serve as an estimator for the population proportion
of phase j. For a point located at random within the
boundaries of the planar mosaic, the probability that the
point falls on a phase j feature is A./A = F.. Random
.1 3
sampling involves the independent location of a fixed
number, n, of points at random. Let N. equal the number
J
of the n points belonging to phase j features for j = 1,2,.
..,k. The distribution of N' = (N N2 *** N ) is multi
nomial with probabilities F1,...,F k This is, of course,
a standard result of random point sampling. Individually,
Nj/n is an unbiased estimator of F with
3 J
VK  F~l (5.2.1)
n n
and
N N M FsFt
cov ,  (5.2.2)
for s t. As n gets large, the distribution of N//T
approaches normality. For random points sampled from a
L L
lineal transect with phase proportions ,..., respec
1, L
tively, the results will be similar, except that L./L will
J
be used instead of F..
J
5.3 Systematic Points on a Line
A second method of point sampling that is widely used
in many different kinds of experiments involves systematic
sampling of points on one or more lines passing through the
mosaic. This method has some practical advantages in its
use over the random sampling technique. It generally will
offer the experimenter a more convenient design or layout
of points on the mosaic, and estimators obtained from
systematic point counts also are unbiased, with variances
inversely proportional to n.
For an individual line, denote the lineal fraction of
phase j by L./L, as defined in Chapter 3. It should be
mentioned that two possible situations may exist to allow
for systematic sampling. The first will occur if the line
by itself constitutes a onedimensional mosaic, in which the
vector L is constant. The other situation arises when the
transect is randomly located in a higherdimensioned mosaic,
as discussed in Chapter 3. This will be the case under
consideration in this section, so that for each transect,
the values for {L.j will be random, with L./L repre
senting only an estimator of Fj. The process of locating a
fixed number, n, of points on the line will start by dividing
the line into n intervals, eachinterval of width A = L/n.
The first point is then placed at random in the first inter
val, with each succeeding point a distance A from its near
est neighbors, as defined by Cox [3].
Define
1 if the ith point on the line
belongs to a phase j segment
uij =
0 otherwise.
Due to the random location of the line and the first point,
E(u j) =F. and V(u. )= F.(1 F.). Letting
i 1 u
n in
n n
E :1 = F., so that N./n would appear to be a reasonable
n ) [ N.
estimator of F., since it is unbiased. To look at V ,
it is first necessary to develop the
cov(uj ui+s,j) = E(uij,ui+s,j) F .
Define
n(h) = P(2 points a distance h apart belong
to the same cell).
= P(uj = 1 i+s,j =1)
= P(uij =1)P(U, .= 1 uij = 1)
= F.[T(sA) + {1 n(sA)}F.]
2
= F. ( Fj )(sA) +p. .
J J J
cov(u.j i+ .) = F (1 F.) (sA)
13J + ,3 3 3
(5.3.1)
so that,
Var = FlF + 2 I F.(1F.)(nm)n(mA).(5.3.2)
a n n m=l
To make use of the usual inferential techniques, an esti
mate for n(mA) must be found. The approach is intuitively
parallel to the derivation of Section 3.9.
Now,
Therefore,
E(ij 1+s,j
5.4 Estimation of the Variance
Define
j 1 if usj = 1 and utj =O
st
0 otherwise.
Then,
ns .
s= y J ,i+s
can be used to study the transition out of phase j for
points sA apart.
E(M') = (ns)E( i)
s I,i+s
= (n s)P(usj = 1)P(utj = 0 usj =1)
= (n s)F [1 r(uA)](1 F )
= (n s)F (1 F.)[ T(sA)] (5.4.1)
3 J
so that
E(M )
f(sA) = 1 (ns)F.(1 F (5.4.2)
J J
[N. ]
Therefore, V can be estimated by
N. N. N.) N.
U 2 1 1 n M
n n _+ n n (nm) 1 
n n2 mN T N
(nm) 1tr
N. N. n1 .
N J ] 1 1MJ (5.4.3)
n n 2 s
n s=1
Note that as n gets large, A becomes smaller, so that the
points get closer and closer on the line, allowing for more
transitions in and out of a particular phase. However, in
looking at (5.4.3), each variance approximation uses an
estimate of n(sA) in terms of the individual phase j.
Rather than use different estimates in each individual
variance expression, it would seem far more reasonable to
utilize all of the information in some form of weighted
expression that could be used for all variance and covar
iance terms, independent of the particular phase under
analysis. Following the same line of reasoning employed in
the lineal case,
(n s)T(sA)F.(l F.) = (n s)F (1 F.) E( )
for j = 1,2,...,k, so that
k k k
(n s)7T(s A) K F = (n s) 1 E(M)
j=1 j=1 j=1
resulting in
((sA) = 1  (5.4.4)
k ( Nt2
(n s) 1 I
t=11 n
as a stable estimate for 7n(;A). Averaging over all values
of s, V i can now be estimated by
n n n 2 k 2 (5.4.5)
n n2 n nI s1 k MNt 2
1F
t= n
5.5 Covarjance Between Colors
If comparisons are to be made between colors, the covar
lance term needs to be established. For s #t,
Ns N 1 n n
E s  = 1 iE u I s I l uit ]
nn =l1 2=1
1 n
i E u. u. + u u.
2 1= ls uit is mt
n i=l i m
The first term of the above expression is zero, since a
point cannot be in two different colors. Thus,
E j P(u. s1 = 1)
n imi inUmt
n1
= l F s[1 T(mA)]Ft(n n)
n m=l
S[n1l n1
S Fst (n m) ) (n m)7(mA)
n m=1 m=l
n1
= nFs F 2 FsFt (n m)T(mA) (5.5.1)
n1 m=l
Therefore,
Ns NI FsFt n1
ncov n = 2FFt (n m)T(mA) (5.5.2)
n m=l
with estimation of n(mA) from (5.4.4).
5.6 Asymptotic Distribution of the Estimates
It will be shown that the distribution of _N will be
asymptotically normal under the assumption of the develop
ment of the mosaic from a Poisson Lines Model. Unlike the
other area (volume)fraction estimators that have been
obtained through lineal, area, and random point measure
ments the systematic point count estimators cannot be writ
ten as an average of independent indicator functions. As
discussed in Sections 5.3 and 5.4, the systematic points on
the line are dependent with covariance function cov.(h) =
J
F.(1 F. )(h) for points h units apart. Therefore, the
3 j
asymptotic normality of the vector of estimates must be
achieved through a Central Limit Theorem for dependent ran
dom variables.
The theorem to be used is a multivariate result of
Hannan [7], stated as follows:
Let X1,X2,...,Xn be k x vectors, defined by
Xil uil F
S i2 Ui2 F
Xik uik k F
Sk 2
such that
(i) Each vector of observations satisfies the strong
mixing property.
(ii) irsw (mlm2) = E(Xr X ms X )
for mI and m2 integers such that 1 m ml m2 n n
is finite when summed over all possible values
for mI and m2.
(iii) The covariance spectrum, [f(e)] is continuous at
S= 0 and trace [f(E)] is uniformly bounded,
where
S ) 1 I ihE ( dh .
rs 2 rs
(5.6.1)
Then, the vector
N
F F
n 1
N
n F
n 2
Nkk
will be asymptotically normal as n with vari
ancecovariance matrix 2r[f(0)1.
In the general case, the first condition (i) will not neces
sarily be true and can be extremely tedious to prove from a
mathematical approach. However, under the assumption of a
Poisson Lines Model used for the development of the cell
structure of the mosaic (Section 2.4), the number of inter
sections the transect makes with the cell boundaries has a
Poisson distribution. With this additional property, the
combination of two theorems of Switzer [24] and Billingsley
[2] yield the strong mixing property. Switzer proved that
for the ordered set of collinear points on a planar Poisson
Process, fu i=1 will have the Markov property for j =1,
2,...,k, and thus the set of X. vectors will satisfy the
Markov property. The general strong mixing property will
be satisfied if for two events, G defined as a function of
the information from the first i points, and B defined as a
function of the information from the points after the
(i+m 1)t point, then
IP(GB) P(G)P(6)[ < a(m) + 0,
as mt. Billingsley [2] showed that a stationary Markov
Process satisfies the strong mixing property, so that con
dition (i) has been satisfied for the Poisson Lines Model.
For this model, the covariance function is
cov.(h) = F.( F.)e h
J J J
which will now be used in showing that conditions (ii) and
(iii) both hold.
To show the existence of the second condition, consider
Irsw (m1m2) = E( ur ums nu2w (5.6.2)
(5.6.3)
I s(m.,m.) = E(u u Um ).
rs a n m.r m s
Then
Vrsw(ml'm2) = Irsw (mim2) Frsw(ml,m2)
FIrw(,m2) wIrs ('m) +2FrFsF (5.6.2)
Note that r,s,w, represent different phases on the mosaic
so that several cases must be studied.
CASE I: Consider r= s= w. Without loss of generality, call
this phase 1. Then
11 (l,mlm2) = P(ul = uml = 1 ,u m2 = ),
which can occur in any one of three ways:
A: All three points belong to the same phase 1 cell.
B: The first two points belong to the same phase 1
cell, while the third belongs to a different
phase 1 cell.
C: The second and third points fall in the same
phase 1 cell, but the first point is on a dif
ferent phase 1 cell.
D: All three points are on different phase 1 cells.
Therefore,
Illl(l,ml,m2) = P(A) +P(B)+ F(C)+ P(D)
A(m2l)A
P(A) = Fle
P(B) = Ile e ,
P(C) = F21 e (mll)A] (m2ml)A
P(C) = FI1 e e 1e 2
P(D) = FL 1 e Arenl] LJ
Therefore,
Illl(m ,m2) = e
F 2 2
(F1 F1 F
A(ml1)A 2 X(m2l)A
(F1 F) +e
3 (mmA )1> 2 3 3
+ F ) +e (F1 F3) +F (5.6.5)
+ 1 1 1
I (ml,m2) = Fle
= F1(
A(m2ml)A 12 e
F 1je
A(m2m )A
1 Fl)e +
A (m2m )A
2
F (5.6.6)
When these expressions are substituted into the expression
for ~1(mlm2 )
(mA (m2mI )
Plll(ml,m2) = Gl(Fl)e
where G1(F1) is a third degree polynomial function of F.
To prove that this moment is finite over all values of mI
and m2, observe that
SmAA(m 1) AA(m21)
Z CI(pG )e = mlG(Fl)e <
m2=1 ml=1 m2=1
by result of the ratio test.
where
CASE II: Let r=s7w. To simplify notation, let r=l,
w= 2,
112(mlm2) = P(u11 =1, mll = 1 Um2,2 =),
which occurs when
A: The first two points belong to the same cell of
phase 1, and the third in a phase 2 cell.
B: All three points fall in different cells, the
first two being phase 1 cells and the third
being a phase 2 cell.
P(A) = Fle (m1) e A(m2 F ,2 (5.6.7)
P(B) = FIl e (l F1 i 1 e 2 F2. (5.6.8)
Therefore, j112 (m,m2) again reduces to the form of
\(m2l)A
G2(F1,F2)e
which when summed over all values for mi and m2 is finite.
CASE III: Let s=wX r, and let r = and s=2. Then
1122 ( ,2) = P(ull = ,u =1,2 1, 2u,2 =1),
which occurs when:
A: The second and third points share a common phase
2 cell.
B: All three points appear in different cells.
i 1(mll)lA An(m2m1 )A
P(A) = FI_ e ] 1(m Fe F (m
P(B) = A(m) (m2m, 2 .
P(B) = Fl[l e~ m ] jFpL 217 e
Therefore,
2 A(ml1)a
I22(m,m2) = F1F2e
+ FF2 ( F2)e
F1F2( F2)e
S(m2ml )
so that
122nm2) = 3(FF)e(m 1)A
V122(mlm2) = G3(FI,F2)e
S m2
CASE IV: Let r / s X w. To simplify notation, let r= 1,
s = 2, w = 3,
1123(ml'm2) = P(u11= 1 u 2 = 1 u 3 = 1)
= FF2F3[1 e (m l m2 m1)A]
(5.6.11)
Therefore,
X(m21)A
U123(ml,m2) = G4(F1,F2,F3)e
(5.6.9)
(5. 6.10)
A(m21)A
which is finite when summed over mI and m2, and the condi
tion (ii) has been satisfied.
The third condition needed to satisfy Hannan's theorem
states that the matrix of spectral density functions,
[f(E)], is continuous at e= 0 and that the trace [f(E)] is
uniformly bounded over all values of e. As previously
defined by (5.6.1),
() (F iF)
( ) r r2 f e e AAIhldh
rr 27
F (1
Fr(lFr) At
(FA )2 + .2 (5.6.12)
T (AA)2 +2
For r s,
Crs(h) = F(1e h)F Fr
=FF eFAAh
rs
so that
FF
f (c) r s A (5.6.14)
rs 7 (AA) + e
Noting that each frs () is continuous at e =0, and
tr[f(E)] = iF(l F),
(AA)2 + e2 r=1
which is bounded by tr[f(0)1. Therefore, all conditions
have been satisfied, and
N
1 F
has an asymptotic normal distribuiton with zero mean vector
and asymptotic variancecovariance matrix, V(n) where
(n) 2F. (lF.)
and
rs At
for r Xs. To estimate the terms of V(n) recall from
(5.4.2) that
MN
(n )F (1 F )
is an unbiased estimator of
1 (sA) = e
for a Poisson Lines Model. Using a Taylor expansion on
I M'
AsA
e yields
j=1 s
(n s) F
as an estimator of isA. Thus,
k
nl Y M3
1
1 Y j=1 s
ni1 k k = (5.6.15)
(n s) 1 Y F sA
[ J=1 *J
is a reasonable estimator of A. With N./n estimating F.,
(N.J J J
Var can now be estimated. Also, recalling from the
lineal estimation procedure that
L.j 2p.(1 p,)
V =
L XAL
for a lineal Poisson Process of intensity A, the result from
systematic point counts yields a similar expression with L
being approximately equal to nA, and F. estimating p..
5.7 Extensions of Point Sampling
In developing the theory behind point sampling on the
line, it was noted that the line represents a onedimen
sional cut from a higherdimensioned mosaic. Koop's 1976
results come from 21 different variations of point sampling in
two dimensions. Two methods will be discussed, since their
results can be tied in with the results already obtained in
this chapter.
The first sampling procedure of random points on random
lines locates b lines randomly and independently inside the
mosaic, such that a points are placed at random locations on
each line, accounting for n= ab points on the plane. Define
th th
if the i point on the m line
belongs to a phase j feature
otherwise,
Pm = the proportion
S jth phase.
p. = the proportion
jth phase.
of the line occupied by the
of the plane occupied by the
Then
P(u. = 1 Iline m) = p .,
im mj
E(u ) = E E(u Jline m) = p .
Im m im
Thus,
a b
X. I m u
n i mn
n n
gives an unbiased estimate of pj, with
N. a b
V [i b m var(uj ) + m cov(uJmu j
n Li=1 m=1 mit
+ I cov(ui mUj ) + Y cov(u )]
s mt im st
u
im
0* 
and let
Due to the random and independent location of the lines on
the plane,
cov(u ,uJ ) =
im I st
for all m #t, so that only the first and third terms of the
variance expression need to be considered. Now
V( ) = pj(l p ),
and
cov(uJ u)
Im sm
= E cov(u ,u line m) + cov E(u m line m)
m irm' sm im sm
m
= cov(prjp m) = Var(p) =  2
L
where
S= v..
L. J
UI
L
as previously derived in Section 3.4.
Thus,
VN P (1 Pj ab(b 1) 2
n n 2 L. '
n 3
p.(1 P ) 1
+ (5.71)
n L.
L
which yields the same result as Koop [9]. Note that the
presence of o2 in the variance expression reflects the two
SJi
L
stage process of first sampling lines from the mosaic, then
subsampling points from each transect. Note also that one
needs a value for 2L even though point sampling is employed.
_1
L
5.8 Systematic Points on Random Lines
The other sampling procedure to be discussed again
begins with the independent location of b random lines
inside the mosaic. On each line is placed a systematic
array of a points, again resulting in n =ab points on the
plane. Using the same notation as in Section 5.7,
a b
S_ i= m=l1
n n
gives an unbiased estimate of pj, with
V 1 var(u m) + I X cov(u j u
V(UI) = V E(u line m) +E V(u line m)
= Vm(mj) + Em mj( Pmj
= p (1 p ),
and
cov(u u~j )
Uim sm
= Ecov(u mu line m) +cov E(uj ,u line m)
m sm mim sm im m
m
= E p' (1 Pmj)(iisI A) +co(p mjp mj)
m
= w( i s A)[p (1 p ) 2 ] + (5.8.1)
L L
Thus, the resulting expression for VK!j will look like
(5.3.2) with additional terms of .2 to account for the
L
twostage sampling procedure.
5.9 Comparisons with HilliardCahn
Hilliard and Cahn's work with onedimensional random
point counts gives an expression that is the same as
(5.7.1). Their results from systematic point counts are
derived under very rigid restrictions, one being that a
phase j feature will not occupy more than 1 point on the
line. To equate this with the conditions described earlier
in Section 5.3 would necessitate
(i) cells being extremely small or equivalently n being
small and A being large in relation to L.
(ii) F. being very small.
Under all of these restrictions, the HilliardCahn variance
expression becomes
90
N. F,
V 1  '
which would yield a very special case of the result from
(5.3.2).
CHAPTER 6
MULTISTAGE SAMPLING
6.1 Introduction
As mentioned earlier, it is possible that the lineal
and point sampling procedures discussed in Chapters 3, 4,
and 5 may have resulted from a subsampling of a higher
dimensional sectioning of the mosaic. When this occurs,
multiple variance terms must be accounted for at each stage.
The use of a lineal transect may yield estimates for struc
tural features of a twodimensional mosaic, but that planar
mosaic may, in fact, be a crosssection of a threedimen
sional figure. The volumefraction of the void in a piece
of copper might only be studied from looking at a cross
section of the specimen under a microscope. If lineal or
point sampling is used, the sample fraction due to the void
will in essence be estimating the areal fraction of the void
on the planar sample. Thus, any results that would be
obtained from a point count or lineal measurement may have
to account for one or more additional stages of sampling.
6.2 Lineal Subsampling
The first case to be considered involves passing a
lineal transect through a planar section of area A, which
has been sampled from a higherordered mosaic. With
notation as defined in Chapters 3 and 4, let L. denote the
total length of phase j segments on the line. Finding the
expectation of L./L will now be a 2step process, first
conditioning on the planar section, C, and then taking the
expectation over the entire mosaic. As proven by Miles [15]
in (2.2.4),
L A '
and the expectation of A./A over the entire mosaic will be
3
V./V = p..
J J
To obtain the variance of L./L as both L and A go to
infinity, the twostage process uses the asymptotic variance
approximations from Chapters 3 and 4. One has that
S = V(E i G + E AV G]
[ LJ L 2 i G L
fA. A. A.]
J V + LE 102 2) J ]
G A G A A
where p and 02 denote respectively the mean and variance of
a cell intercept. Therefore,
^L. 2 o rA.'l
A
A A_
