Title: Contributions to statistical techniques for two and three dimensional measurement problems
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Permanent Link: http://ufdc.ufl.edu/UF00098859/00001
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Title: Contributions to statistical techniques for two and three dimensional measurement problems
Physical Description: viii, 109 leaves : ; 28 cm.
Language: English
Creator: Lackritz, James Robert, 1950-
Copyright Date: 1977
Subject: Estimation theory   ( lcsh )
Statistics thesis Ph. D
Dissertations, Academic -- Statistics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 107-108.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by James Robert Lackritz.
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Bibliographic ID: UF00098859
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000063244
oclc - 04205116
notis - AAG8442


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James Robert Lackritz





To my family and all other non-believers.


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.


ACKNOWLEDGEMENTS . . . . . . . . .

ABSTRACT . . . . . . . . . . .


1 INTRODUCTION . . . . . . . .

1.1 History of Quantitative Microscopy
1.2 Models for Random Division of Space
1.3 Applications . . . . . . .


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 . . . . . .






. . 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 Varlance-Covariance Matrix
of M// . . . .
3.9 Variance Approximations ...



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




6.2 Lineal Subsampling
6.3 Point Subsampling


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 . . . . . .


. . 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



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 well-known, 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 volume-fraction analy-

sis makes use of the spatial covariance function of the

underlying model to generate a variance expression for the

areal or volume-fraction 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





1.1 History of Quantitative Microscopy

The problem of estimating structural properties of

two and three-dimensional 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 two-dimensional 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 two-dimensional feature or the surface area of

the phase on a three-dimensional 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 volume-fractions, feature boundary length in

two dimensions, and surface area in three dimensions. The

culmination of the individual results is a mathematical

n-dimensional 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 three-dimensional 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 germination-growth model, thus results in a set of

non-overlapping 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 volume-fraction 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 areal-fraction 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 drug-related 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.



2.1 The Mosaic

Throughout the remaining chapters, the term mosaic

will refer to a two-dimensional 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 three-dimensional figure.

(2) Boundary length of certain two-dimensional features,

which in turn, may estimate the corresponding surface

area of three-dimensional features.

(3) Possible feature orientation or homogeneity on the


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 three-dimensional phases and areas and boundary

lengths of two-dimensional 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
sample, A, and the area occupied by the j phase, denoted

by A.. The lineal transect measurements of interest are L,

the length of the transect, L the total length of the
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 n-dimensional

geometrical probability. Miles' theorem states:

To estimate a t-dimensional quantity of a

d-dimensional mosaic using an s-dimensional sampling

device yielding (s +t -d)-dimensional measurements,

for s, t, and d non-negative 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+t-d = the measure of the intersection of the
sampling device with the feature of

(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.

E(X D ) kd(st) D (2.2.1)
s d


s + 1 t +1

kd(s,t) = s- t-d 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 three-dimensional 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,
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

lower-dimensioned sample-fraction, 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 non-degenerate 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 three-dimensional

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 two-dimensional mosaic, G, of fixed over-

all size and any set B, of random size and location within

the borders of G.


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.


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]}

= 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


cov(h) = E[f(x)f(x+h)]- E[f(x)]E[f(x+h)]

= C(h) -p2

Furthermore, if ( is a two-dimensional 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).
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.


I = f(x)g(x)dx,


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,
= 0 otherwise

for which case

me S f(x)dx
g mes S
E(I2) = E{[ f(x)g(x)dx]2}

= 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 ,


G(h) = f g(x)g(x+h)dx

so that

V(I ) = E(I2) (I )

= J C(h)G(h)dh-E[ 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

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= ly-xI.
With the knowledge of the spatial covariance function
being necessary to obtain a variance expression for the
areal-fraction 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
interval (0,T). For each pair of points, fit the


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)

and variance

2 (2_2)
02 I 2 -_2) (2.4.2)

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 two-dimensional

model, Gilbert [6] showed that, with A. denoting the area

of the cell generated by y.,

E(A.) = 1.018 T- (2.5.1)


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

Bj S = {x: Bjn S ~ .
3 J

E mes (.j @S)
P(S n B) = -

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


distribution with parameter TEmes (6@S). For any point

x e G, define

i if x EB
k(x) =
0 otherwise,


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)

C2(h) = P(x,x+h e B)

= PN (S) = O]

= exp[-T{2(O-K0 -(h)].

q = C2(0) = P(x e BC) = e-TK(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),


P(x r 6 x+h E ) = q e(h)

so that

Cl(h) = P(x,x+h e I) = 1 2q +q2eTK(h)


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.



3.1 Introduction

Consider a two-dimensional mosaic composed of two or

more phases. This mosaic could arise as a cross section

of a three-dimensional 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 volume-fraction 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 volume-fractions

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 three-dimensional 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 k-phases, 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 areal-fraction 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 areal-fraction

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 non-over-

lapping cells with cell sizes having finite second moments.

Each cell is independently assigned to phase j with proba-
ability pj> 0 for j =1,2,...,k and I p. = Thus, a feature
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


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.)


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

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

L. = iXi (3.3.1)
1-1 ': 1i~

L. will denote the total length of the intercepts of phase
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
[23J can be used since

E(ij.X.i) = p.i < .


lim E(L./L) = pj (3.3.3)
L- J

implying that I,./L is an asymptotically unbiased estimator
of p.. Also, by the Weak Law of Large Numbers and the fact


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
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..

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 estimation-procedure 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
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 --(L-p). To follow the definition of a cumu-

lative process as stated by Smith [231, Z .ij X is defined
to be a cumulative process if:

(i) For fixed j, .X. is a sequence of i.i.d. random

variables for i= 1,2,...

(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 non-negative and


The asymptotic distribution of I(L- p) now can be

derived as a result of Theorem 10 of Smith [23]:

If (ilX. i2Xi *** ikXi) are k cumulative
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

variance-covariance 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). +



For s t, the covariance term,

st = E is -Ps ) it-Pt)

= 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

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 variance-covariance

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

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.
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.


1 if the (i -1)st cell intercepted by the
transect belongs to phase j and the i-th
f.. = cell intercepted is not a phase j cell

0 otherwise

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
be the number of intersection points the transect makes

with the boundaries of phase j features, I. =2M. unless
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..


J = the phase of the nth cell cut
n by the transect
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 ,

rv = 1 Pvj1 =

p.= expected return distance to phase j
on the transect


= u/p ,


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


if j =1, m 1i

otherwise ,

c (f i = f 2PjdF j(Xi)

if j = 1, m l

otherwise ,

j(f ii) =IL 3 (fm 1
'=1 i

O p
1 l- pl


j= 1,

(2)( f (2) (
j f il jm (

Using the Strong Law of

Schaufele for functions


Large Numbers given by Pyke and

defined on a Markov Renewal

0 1PO ~ lf


M (1) j il
I a.s. j=l j )J(fil)
1 a.s. (3.5.1)
L pl


0 if j l
( fil) =
J-pI J =l


m =1,

M1 a.s. P1(1-pl)
S (3.5.2)
L p

as L-'. Therefore, as L-~,

M a.s. p (1 -p )
-j a- .s.(3-5.3)
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,
will have an asymptotic normal distribution, and that the

joint asymptotic distribution of the vector M, suitably

normed, will be multivariate normal with a closed-form

expression for the variance-covariance matrix.

3.6 ::::'::. :c Di:str b .ion of M./jA

Frsm T-eorem 7.1(b) and Lemma !, 2 .:f' Pyke and 'chaufele


E N I. ,., -u e. 3.6.

; i -]
1 N I

-.::.!i c:'',err-e in distribur;io as L-o to a standard normal

ran.doT vanian-ile, provided the variance cf each term in

the suo (3.6.1) is finite. Recalling thac

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

where (2) = + 2+Cp Th jr=l'efore

S(h11) = (3.6.))
C p ( otherwise,

j (hil) = jm (hjl

In C1


c(2) (h 2)(h
Sj m i n 11

p2 ((2)
( -p 2C^(-p1 p) +2 C1 (2)


m() (2)
.j j Jhi1

= 1 pl 2C1(1 pl)

= 1 p 2C p(l pl)

J+ p C2 (2)

C2 (2)



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) (1-6r) +6m
j / m '/1 r 1 1 1 r

j 1





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 +(l-p.)2] +C 2 .
3 J J J J 3

3.7 Joint Distribution of LM

With the univariate asymptotic normality of each

M //L established, the next step is to prove the joint
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,
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)


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 Variance-Covariance 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


jm(h is't = his hit jm dFjm

= -I [fisCt +fitCs jm dFjm

+ X CsCtPjmdFjm

( 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 C-tPm j t m t

St(his hit) = jm(hishit)

(2) CsCt

= (2) CCt- Cs(l-Ps)

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


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 )rt-CtP -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


S 2 2
[C sPt+C p (3.8-5)

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 variance-covariance 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)
that the asymptotic variance of L./L is

p (1-p ) 1+ p (1-p ) 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 -

from (3.5.1), both

k M.
j=I i


I M.
(ii) j= 2
k L.

will be consistent estimators for E(N ), as would each

individual estimator,

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 1L.2

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
ranging values for each L./L.
It follows that a reasonable consistent estimator of

v.. would be

L L + 2
v = k (3.9.1)

1- f

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


E(N )
Enj p

V(N .) =
n (1.- p )2

Therefore, writing Znj, the length of the n segment of

phase j, as

a .+ N
Z = n Xi.
nJ i = a .+1

where a = the number of cells preceding the n segment
of phase j,

j = E(Znj) = E(Nn) -
3 nj nj 1-p.

= V(Znj) V(N njp) + E(N o2)

-- + (3.9.2)
(-p)2 l-pj

Rewriting the first equation as

p = (I-p )E(Znj.)

p can be estimated from each color by using the average
length of a phase j segment, Y7, multiplied by (1- ).

Thus a weighted average of estimates of p over all colors
would yield a consistent estimate for p. Since upp =p,
one possible such estimate would be

k L. L.
jL L (3.9.3)

Rewriting (3.9.2) as

a2 = V(Z )( p) -
nj a 1-p.

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 volume-fractions 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

2 2
2 0 2 ( i P 2
So (l-p)-
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(l-p) 1 + 02
V = ------

p (1 -p) 1


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 Hilliard-Cahn 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 Hilliard-Cahn 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
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

1 i=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

p 2+P2 '
1+21 2




E(1) 12

the above reduces to

2I 2

which is (3.10.1).

For situations in which the Hilliard-Cahn 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
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

v. = (3.11 )
J E(1J )

where p can be estimated by Lj/L and E(N ) approximated,

as mentioned before, by

k fL 2 '
1- k

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 Hilliard-Cahn 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

three-dimensional mosaics sampled by areal sections and

lineal transects. For a two-dimensional 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 .88T-1 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.


M. = the number of segments of phase j
j,' on the m-th 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(M-LC) also has
/L -
an asymptotic multivariate normal distribution with zero
1 *
mean vector and variance-covariance matrix D Thus by

M -LC = (M -F) + (P-LC),

it can be shown, using Cochran's Theorem, that under Ho,

T n -1
T = (M -M)'(D-) (M -F)
L-m --m-

has an asymptotic X2 distribution as L with (n-l)k

degrees of freedom. Therefore, the test statistic will be

-T 1- t(M -M)'(D ) (M M)
L -m -m -

where D must be a consistent estimator for D Recall


d 2 p+2 2 2 p (2) 2
J = J -2J


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
phase j segments per unit length should serve as an esti-

mate for C.. With D approximated using the terms from
above, the test for equal means would have a rejection

region of the form

T X(n-l)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 ,

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(L-h)2

= L' 2! dh = i (L
0 Y2-h


2 h
-(1 -) for 0 < h < 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.
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.)e-h

where A is the underlying parameter of the process. There-


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

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) ,

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

L 2 h_ hM(l h dh
= ( ) L (1 L)dh

= 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

V-i mathematically complex. One of the simpler models

to work with assumes each 8. set to be a ball of radius r.
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.


L .L L
fV - [q2eTK(h) -2 (l )dh

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


T 2 rsin-1 -rh 1 12
I- 2r 2

However, using the approximation from (3.15.1), V [-
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
small, so that V -- can be approximated by

pq(lr) 2r2
L 2 (3.15.4)

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.



4.1 Introduction

In considering the parameters of the n-dimensional

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 areal-fraction 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

X. = area on the i cell on the plane,

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.


A.j i ijxi.

A. will represent the total sampled area due to phase 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 1-p) 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


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
S-p. respectively, but the other terms are parameters of
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 .

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),
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


and variance

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

-A- = 2A pj(1-pj) (I.4.2)


0 ) ( 4.4.3)
2A7ip.( -p j4.

gives an unbiased estimate of A.. Utilizing the sampling

information over all colors,

) 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
areal-fraction 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:

S= 1.018T-O


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 areal-fraction

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
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 1-p


2 o2 + j
j 1 j (1 -P)2
j 1 p 2

or equivalently,

p = p (1 p .)


2 = -(1-p )o -Pp .1,


x* p.(1-p.) A2]
v +- 3i +a
jj A/p

/ 2 P P21

= p (1-pj) 1+

= u [u +o2]

E(M. )
SE(M[ 2+{(3p +2p )U2_pj2]
2= -+uj 3P.+p f .-p~j(4.6.2)C


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.

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

areal-fraction 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


-2A h
cov (h) = pj(l -p.)e-

In studying the areal-sampling 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

2C h
bS(h) = +o (h2) (4.7.1)
S A 2

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 (1-p )e-2 h gets

very small. Also, as A gets large, the second term of
bS(h) becomes very small, Bo that V can be approximated


2pp I( p ) lie-21* dh

p (1 p )

2( ) 2A

yielding the same result as (4.4.1).



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 two-phase planar mosaic. Several of

the simpler techniques will be discussed and extended for

the general k-phased 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-
tively, such that Y A. =A. As noted earlier in Chapter 3,
the mosaic itself may represent a planar section of a two

or three-dimensional population figure, for which each A./A
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
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


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
lineal transect with phase proportions ,..., respec-
1, L
tively, the results will be similar, except that L./L will
be used instead of F..

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 one-dimensional mosaic, in which the

vector L is constant. The other situation arises when the

transect is randomly located in a higher-dimensioned 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, each-interval 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].


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 .


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.]

= F. ( -Fj )(sA) +p. .

cov(u.j i+ .) = F (1 F.) (sA)
13J + ,3 3 3


so that,

Var = Fl-F- + 2 I F.(1-F.)(n-m)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.



E(ij 1+s,j

5.4 Estimation of the Variance


j 1 if usj = 1 and utj =O
0 otherwise.


n-s .
s= y J ,i+s

can be used to study the transition out of phase j for

points sA apart.

E(M') = (n-s)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 (n-s)F.(1 -F (5.4.2)

[N. ]
Therefore, V can be estimated by

N. N. N.) N.
U 2 1 1 n M
n n _+ n n -(n-m) 1 -
n n2 mN T N
(n-m) 1tr

N. N. n-1 .
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
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

= l F s[1 T(mA)]Ft(n n)
n m=l

S[n-1l n-1
S Fst (n -m) ) (n -m)7(mA)
n m=1 m=l

= n-Fs F 2 FsFt (n- m)T(mA) (5.5.1)
n1 m=l


Ns NI -FsFt n-1
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) =
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,


S ) 1 I -ihE ( dh .
rs 2 rs


Then, the vector

n 1

n F
n 2


will be asymptotically normal as n- with vari-

ance-covariance 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 m-t. 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

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)


I s(m.,m.) = E(u u Um ).
rs a n m.r m s


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.


Illl(l,ml,m2) = P(A) +P(B)+ F(C)+ P(D)

P(A) = Fle

P(B) = Ile -e ,

P(C) = F21 -e (ml-l)A] -(m2-ml)A
P(C) = FI1 e -e 1e 2

P(D) = FL 1 e Are-nl] LJ


Illl(m ,m2) = e

F 2 2-
(F1 -F1 -F

-A(ml-1)A 2 -X(m2-l)A
(F1 -F) +e

3 -(m-mA )1> 2 3 3
+ F ) +e (F1 -F3) +F (5.6.5)
+ 1 1 1

I (ml,m2) = Fle

= F1(

-A(m2-ml)A 12 e
F 1j-e

-A(m2-m )A
1 -Fl)e +

-A (m2-m )A

F (5.6.6)

When these expressions are substituted into the expression

for ~1(mlm2 )

(m-A (m2-mI )
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
Sm-AA(m -1) -AA(m2-1)
Z CI(pG )e = mlG(Fl)e <
m2=1 ml=1 m2=1

by result of the ratio test.


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


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(ml-l)lA -An(m2-m1 )A
P(A) = FI_ e ] 1(m Fe F (m

P(B) = -A(m-) (m2-m, 2 .
P(B) = Fl[l -e~ m ] jFpL- 217 e


2 -A(ml-1)a
I22(m,m2) = -F1F2e

+ FF2 ( -F2)e

-F1F2( -F2)e

S(m2-ml )

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]



U123(ml,m2) = G4(F1,F2,F3)e


(5. 6.10)


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(l-Fr) At
(FA )2 + .2 (5.6.12)
T (AA)2 +2

For r s,

Crs(h) = F(1-e- h)F Fr

=-FF e-FAAh

so that

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

-1 F

has an asymptotic normal distribuiton with zero mean vector

and asymptotic variance-covariance matrix, V(n) where

(n) 2F. (l-F.)


rs At

for r Xs. To estimate the terms of V(n) recall from

(5.4.2) that

(n )F (1 F )

is an unbiased estimator of

1 (sA) = -e

for a Poisson Lines Model. Using a Taylor expansion on

I M'
e yields

j=1 s

(n s) F

as an estimator of isA. Thus,

n-l Y M3
1 Y j=1 s
n-i1 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 =

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 one-dimen-

sional cut from a higher-dimensioned 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


Pm = the proportion
S j-th phase.

p. = the proportion
j-th phase.

of the line occupied by the

of the plane occupied by the


P(u. = 1 Iline m) = p .,
im mj

E(u ) = E E(u Jline m) = p .
Im m im


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

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 ),


cov(uJ u)
Im sm

= E cov(u ,u line m) + cov E(u m line m)
m irm' sm im sm

= cov(prjp m) = Var(p) = -- 2



S= v..
L. J

as previously derived in Section 3.4.


VN P (1 -Pj ab(b 1) 2
n n 2 L. '
n 3

p.(1 P ) 1
--+ (5.71)
n L.

which yields the same result as Koop [9]. Note that the

presence of o2 in the variance expression reflects the two-
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.

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 ),


cov(u u~j )
Uim sm

= Ecov(u mu line m) +cov E(uj ,u line m)
m sm mim sm im m

= E p' (1 -Pmj)(ii-sI A) +co(p mjp mj)

= w( i s A)[p (1 -p ) 2 ] + (5.8.1)


Thus, the resulting expression for VK!j will look like

(5.3.2) with additional terms of .2 to account for the
two-stage sampling procedure.

5.9 Comparisons with Hilliard-Cahn

Hilliard and Cahn's work with one-dimensional 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 Hilliard-Cahn variance

expression becomes


N. F,
V 1 -- '

which would yield a very special case of the result from




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 two-dimensional mosaic, but that planar

mosaic may, in fact, be a cross-section of a three-dimen-

sional figure. The volume-fraction 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 higher-ordered 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 2-step 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
V./V = p..
To obtain the variance of L./L as both L and A go to

infinity, the two-stage 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 ]

where p and 02 denote respectively the mean and variance of

a cell intercept. Therefore,

^L. 2 o rA.'l

A A_

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