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IT

”as \\\\1111111111\\\\\\\lllll\\\\\\\\\\\\1\\l\\\l

293705 7 8597 iRniRY/it

Michigan Staw
University

            
 

 
   

This is to certify that the

dissertation entitled

SPATIAL PATTERNS: A STATISTICAL
FORMULATION AND ANALYSIS

presented by

Rangaswami Geetha

has been accepted towards fulfillment
of the requirements for

Ph.D. degreein Statistics

 

aLAW

____ ' .
Ma Jor professor

Date 11/5/81

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

OVERDUE FINES:
25¢ per day per item

RETURNING LIBRARY MATERIALS:

_______________
.m'y” ' Place in book retum to remove
"” 1‘ charge from circulatton records

 

 

 

 

SPATIAL PATTERNS: A STATISTICAL
FORMULATION AND ANALYSIS

By

Rangaswami Geetha

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1981

ABSTRACT

SPATIAL PATTERNS: A STATISTICAL
FORMULATION AND ANALYSIS

By

Rangaswami Geetha

In this disSertation a general study of spatial patterns is
investigated. We have given a statistical formulation to the concept
of spatial patterns, a problem which has long been overlooked by the
ecologists, computer scientists etc. Have established the character-
ization of a random spatial pattern as a realization of a Poisson point
process, through the notion of convergence of point processes. In the
sequel we have introduced the stochastic integral with respect to a
point process.

Further we have studied inferences on randomness (no spatial
interaction) through estimation of the intensity of the spatial process
and through testing hypothesis in a special subclass of spatial binary”
schemes described through near-neighbour systems. Here, we have established
the chi-square behavioUr of log-likelihood ratio under the null hypothesis
of no spatial interaction without the use of Besag's (1974) coding
method of estimation.

Finally, in our attempts to study the power of the test in this
subclass of binary schemes, we established the contiguity of the probability

measures under the specified hypotheses of interest but however realized

that the asymptotic distribution of the log-likelihood ratio under the
alternative does not have an easy tractable form. A good conjecture is

that it is a non-central chi-square distribution.

Dedicated to my mother and my late father

1'1

ACKNOWLEDGEMENT

I wish to thank Dr. v.5. Mandrekar for his guidance and encour-
agement during the preparation of this thesis. I would also like to
thank Drs. J.C. Gardiner and D.C. Gilliland for their careful reading
of the chapters and their valuable suggestions. My thanks are also
due to Dr. J. Kinney for serving on my committee, to Michigan State
University for the financial assistance and to Ms. Clara Hanna for
typing of this thesis.

Finally, I sincerely appreciate the patience of my family members

throughout my graduate study.

 

TABLE OF CONTENTS

Chapter
0 INTRODUCTION ........................................
I POINT PROCESSES AND SPATIAL PATTERNS ................
l.l Notion of a pointprocess ......................
l.2 Stochastic integral with respect to a point
. process ........................................
l.3 Laplace functional of a point process ..........
l.4 Notion of weak convergence for point
processes ......................................
l.5 Concept of spatial patterns ....................
II MARKOV RANDOM FIELDS ................................
2.1 Notion of neighbours in a set of sites .........
2.2 Markovian fields and Gibbs fields ..............
2.3 Characterization of spatial schemes ............
III INFERENCES 0N RANDOMNESS ............................
3.1 Methods of estimation of A,“ the eXpected
number of individuals per unit area ............
3.2. Spatial schemes generated by.a subclass of
Markov_random fields and testing of hypothesis
forflbinary models ........... .. .................
IV POWER UNDER A SPECIFIC ALTERNATIVE ..................
4.l Contiguity and its characterizations ...........
4.2 Power of the test in binary models .............
APPENDIX A .....................................................
APPENDIX B .....................................................
APPENDIX C .....................................................
REFERENCES .....................................................

iv

Page

MUD->43

lOl

CHAPTER 0
INTRODUCTION

The phrase spatial pattern is commonly used to describe the
distribution of individuals in space. It is one of the topics investigated
under the broader subject of pattern recognition otherwise known as
the problem of classification to the statisticians.

The problem of classification has always concerned itself
with classifying a sample of individuals into groups which are to
be distinct in some sense. These groups may either be predetermined
or determined using techniques of cluster analysis. However, in the
world of organic nature, for example, with the distribution of plants
and animals, the broad outlines of the spatial patterns are determined
by the structural features of the physical environment. Therefore,
in the study of spatial patterns it is not just enough to classify
them into groups but rather determine whether or not the patterns
exhibit randomness.

In Chapter I of this thesis we have given a statistical for-
mulation of the .above problem through the concept of point processes.
It has been repeatedly maintained in the literature that by a random
pattern is meant the pattern is a realization of a Poisson point
process. Proposition l.5.l justifies this characterization. In the

sequel, we need to introduce the notion of stochastic integral with

respect to a point process and thernythniof weak convergence for point
processes (sec. l.2 and sec. l.4).

In Chapter II we discuss the ideas that are needed to construct
valid spatial schemes through, what seems reasonable, near-neighbour
systems. Section 2.l introduces the notion of neighbours in a set
of sites from a graph theoretic viewpoint [Berge - 1962]. Characterization
of Markov random fields, Gibbs field and their equivalence for any
finite graph are all discussed in section 2.2 [Carnal - l979]. Theorem
2.2.l often referred to as the Hammersley-Clifford theorem, plays a
vital role in the construction of spatial schemes through near-neighbour
systems. We use this theorem to characterize some of the specific
spatial schemes (section 2.3).

In Chapter III, we look into some of the methods to determine
randomness of a spatial pattern. Section 3.l discusses some of the
estimators of the intensity of the spatial process and their asymptotic
properties. It also looks into the use of these estimators to study
the randomness of the spatial process. In section 3.2 we consider a
particular subclass of Markov random fields and give a method to test
for randomness in this subclass. It has been shown that the log-likeli-
hood ratio has a central chi-squared behaviour under the assumption
of complete randomnessof the spatial pattern. The proof involves
simple use of Taylor's series expansion and follow the lines of proof
of the classical theory on maximum likelihood estimation [Cramer - 1946].

In Chapter IV, we have attempted to discuss the power of the
test against a specific alternative from the subclass of auto-binary

schemes. The problem has been approached using ideas on contiguity -

a concept that describes the 'closeness' of sequences of probability
measures. Section 4.l discusses some of the characterizations of conti-
guity [Roussas - l972]. In our formulation the classical techniques

of contiguity fail. In section 4.2 using the basic principles of
contiguity we establish the contiguity of probability measures under

the null and alternative hypotheses. However, the asymptotic distribution
of the log-likelihood ratio under the alternative does not seem to have

a tractable form. A good conjecture is that the power of the test

depends on a non-central chi-square distribution.

CHAPTER I
POINT PROCESSES AND SPATIAL PATTERNS

In this chapter we introduce the concept of a spatial pattern
and the characterization of a random spatial pattern as a realization
of a Poisson point process. We have approached the problem by introducing
the notion of a point process and weak convergence of point processes.
In the sequel we have defined stochastic integration with respect to

a point process and Laplace functional of a point process.

l.l. Notion of a Point process:

 

Notations:

d be the d-dimensional

Let (D,F,P) be a probability space. Let S = R
Euclidean space and 8(5) be the family of borel subsets of S. Let
x],x2,... denote points in S. Let A denote a compact set in S
such that A contains finitely many xi's.

Let
l if x 6 A

0 otherwise

Let M+(S) = {m on B(S)l23 a countable set of points xj such that
m(A) = Z 6x.(A) = # of x-points in A, A 6 8(5) and
J

J
m(A) is finite for all compact A in 8(5)}.

Let M(S) denote thecy-algebra generated making all the mappings

4

m + m(A) A 6 8(5)

of M+(S) into 2+,” (the set of non-negative integers including

+ 00) measurable.

Definition l.l.l: The family {€(A,w)2 A 6 8(5)} describes a point

 

process if

a) €(A,w) is non-negative integer valued V A 6 8(5) and finite

for compact A

b) E(°,w) is a measure such that E(°,w) puts mass 0 or I on singleton sets
i.e. (i) For a sequence A],A2,...,Am of pairwise disjoint sets in

8(5) we have

Ill
a(.U

A.,w = A.,w
3-1 3 ) X e:( )

i=1 3
(ii) For A]:3 A2:: ... in 8(5) such that n An = ¢ we have
n

Tim g(A ,w) = O
n n

C) €(A,-) is measurable on G into 2+“n,

Definition l.l.2: A point process g on S is a measurable map from
(D,F,P) into (M+(S), M(5)) i.e., V w 6 D E(w) is an (M+(5), M(5))-

valued random variable.

Theorem l.l.l: The above two notions of a point process are equivalent.

 

Proof: We first prove the easy part: Assume that we are given an
(M+(S), M(5)) - valued random variable g defined on (n,F,P).

For every A 6 8(5) and w E 9, define

€(A,w) = €(w)(A)

Then §(°,w) satisfies all the conditions a), b) and c) of definition
l.l.l.
The converse implication of the theorem depends upon a Kolmogorov

type theorem. The proof of this part is given in Appendix A.

Remark l.l.l: Following theorem l.l.l, we shall find it convenient

 

to think of a point process sometimes as a measurable map on (Q,F,P)
and sometimes as a set function satisfying a), b) and c) of definition

l.l.l.

Definition l.l.3: The intensity of a point process 5: (Q,F,P) +

 

(M+(S), M(5)) is defined as the measure A on S such that

V A€B(S)

A(A) = EP{€(Aaw)}
i.e.
(1.1.1) A(A) = j g(A,w)P(dw)

52

Example l.l.l: Poisson point process

 

A point process {€(A,m): A 6 8(5)} is called a Poisson point

process with intensity' X > 0 if

(i) V A E 8(5), g(A,u0 (which by definition l.l.2 indicates the number
of points in A) has a Poisson distribution with parameter X(A)

(if, >\(A)-‘-'-' + ms €(A9w) = + m );

$1.. : ...

(ii) V finite collection {A1,...,Am} of disjoint sets in 8(5),
the random variables 5(A],-),...,§(Am,-) are mutually independent.
To show that the above formulation characterizes a point process

we need to show that the function

r.
- J
e X(Aj) [X(A.)]
‘I 73.

lFttB

q(A],...,Am; r],...,rm) = J

satisfies conditions A-l(i-iv) of the Appendix A, where

{Ajz j = l,...,m} is a sequence of disjoint sets and X is the

intensity of the process.

A-l(i) q(A],...,Am; r],...rm) is obviously a probability distribution

on the m-tuples of non-negative integers r],...,rm and

9(A1,A2; r],r2) = q(A2,A1; r2,r1)

A-l(ii) The functions q are consistent

i e.. Z q(A].A2; r],r2) = q(A1,r])
r2=0

co

LhS QiA 9A 3 Y‘ :r)
r2=0 1 2 1 2

 

l
o

 

 

A-l(iii) Let Al""’Am be disjoint sets such that A = A1U...UAm.

Then following remark A-l, (A-l-l) implies that

q(A’A]’A2’°"3Am; r9r19...,rm) = 0
unless r = r] + ... + rm and
q(A,A],...,Am,P1 + ... + rm, r]?°°”rm)

= Q(A1:°--9A ; r1,...,r ).

m m

A-l(iv) Let A1:: A22: ... be such that 2 An = ¢ Wthh 1mpl1es

A(An) + 0. Consequently lim q(An,0) = lim e'A(An) = l. Thus the
n n

formulation of a Poisson process characterizes a point process.

Remark l.l.2: In example l.l.l if X(A) = c . v(A) where c is a

 

constant and v is the Lebesgue measure then the Poisson process is

known as a homogeneous Poisson process with intensity c.

l.2. Stochastic Integral with respect to a Point process:

Notations:

Let C;(5)(C;S(S)) denote the family of all non-negative continuous
functions (simple functions) on S with a compact support.

Let g be a point process with intensity X.

Definition l.2.l: Let f e C:s(5) so that

where cj's are positive constants and IA '5 are the indicator functions
J

of the disjoint sets A ,A

1,... m.

Then the stochastic integral of f with respect to a is defined

by
m
(1.2.1) I f(u)g(du,m) = X C- g(A.,w) V m 6 Q
s j=1 J 3
Note that in the above definition if A is a compact support
of f, then

5(A,w) < w v w (def. 1.1 1)
which implies by monotonicity of g(-,m) (definition l.l.l(b)) that
5(Aj,w) < m Vj = l,2,...,m; V w E 9

Consequently it follows that

f f(U)a(du,w) < w v w e n.
5

Remark 1.2.1: Let f e c;s(5)

qu f(U)€(du,w)} = 1 {I f(U)€(du,w)} we)
5 n 5

= f f(u) I 5(dU,w)P(dm) (Fubini's theorem)
5 n

= I f(u)x(du) (by 1.1.1)
5

thus, V f e CES(S) we have

(1.2.2) EP{£ f(u)g(du,w)} = g f(u)X(du).

10

Let now f e L;(A), then there exists a sequence {fn} of

simple functions in L:(X) such that
flfk - fjfl] + 0 k,j integers, k 3 3
Now

iEP{g fk(U)§(dU,w) ' é fj(U)€(dU,w)}l

5 EP é|fk(u) - fj(u)| a(du.w)

= f lfk(U) - fj(U)| A(du) (by 1.2 2)
S

= ”fk " fjii]

+ 0

Hence it follows that f fn(u)g(du,w) converges in L](P)
S

We denote this L1 - limit by

f f(U)€(du,w).
S

 

Definition 1.2.2: The stochastic integral of an f in L:(X) is
defined by
(1.2.3) I f(u)g(du,w) = L1 - lim j fn(u)g(du,w)

S n S

where fn e L;(X) are simple.

Remark 1.2.2: The above definition 1.2.2 is independent of the particular

 

sequence {fn}. For if {9”} is another sequence of simple functions

in L:(X) converging to f in the sense that

ll

f f(u)g(du,w) = Ll-lim f gn(u)g(du,w)
S n S

then the sequence {hn} where th = fn and h2n+1 = 9n 15 also
convergent to f in the same sense.
i.e.,

f f(u)g(du,w) = Ll-lim f h (u)g(du,w).

5 n S n

Consequently it follows that

L1-lim f f (u)§(du,w) = L1-lim f g (u)g(du,w) a.s.
n S n n S n

 

Proposition 1.2.1: Let f1,f2 be functions in L;(X) and let a],a2

be real numbers so that a1f1 + a2f2 is in L;(A). Also then

é [81f1(u) + 32f2(U)] €(du,w)

= a] f f1(u)E(dU,w) + a2 f f2(u)§(du,w)

S

Proof: Obvious.

Remark 1.2.3: Let f e c:(5) so that there exists a sequence {fn}

 

. +
1n CkS(S) such that
f = lim f
n n
Since C;(S)<: L;(X) we can define the stochastic integral of an

f e c135) by (1.2.3).

Consequently (1.2.2) is true for any f E CE(S)
i.e., V f E C;(S) we have

(1.2.4) Ep{g f(u)§(du,w)} = g f(u)A(du).

 

12

1.3 Laplace functional of a point process:

Definition 1.3.1: The Laplace transform of a probability measure Q

 

on the space (M+(S), M(S)) is defined by

(1.3.1) vQ(f) = f expi-f f(U)m(dU)}Q(dm) v f e C: (S).
M+(S) 5

Definition 1.3.2: The Laplace functional of a point process g is

 

defined to be the Laplace transform of its probability law P

5
Le, vrecflw
Wg(f) = f exp{-f f(u)m(du)}P (dm)
M+(S) s 5
By transformation of variables, this gives
(1 3.2) v€(f) = I 9XP{-é f(u)£(du.m)}P(db) v f e C; (s)
Q

Lemma 1.3.1: For every increasing sequence {ffi} of functions in

 

CZ(S) we have

(1.3.3) v€(l;m + fn) = lam + v€(fn).

Proof: Let f = 1im f .
-———- n n
8y (1.3.2) we have

v (tn) = f exp{-f fn(U)£(du,w)}P(dw)
n s

and

 

13

By (1.2.3)
I f(U)€(dan) = LI-lim f fn(u)g(du,w)
S ‘n _5
implies
I fn(u)s(du.b) 3599 f f(u)a(du,m)
S S
implies

exp{-f fn(u)g(du,w)} 3599 epr-[ f(u)§(du,w)}.
S S

Also,
exp{-f fn(u)g(du,w)} are bounded by 1. Thus by Lebesgue dominated
S

convergence theorem it follows that

(f )

f = '
vg( ) 11m wg n

ll

as was to be proved.

Example 1.3.1: The Laplace functional of a Poisson point process a

 

with intensity A is given by

(1.3.4) v€(f) = exp{-f [1-e-f(U)

]A(du)} v f e c;(S)
5

And conversely, a point process 5 whose Laplace functional is of the
form given by (1.3.4) V f 6 CE(S), is a Poisson point process with

intensity A.

Proof: By lemma 1.3.1 it is enough to consider functions f of the

form

14
f(u) = c1 IA (u) + ... + Cm IA (u)
l m
where c ,...,c are positive constants and I ,...,I are
1 m A1 Am
indicator functions of a set of disjoint sets A1,...,Am respectively.
Following (1.3.2) the Laplace functional of a point process a is
given by
v (f) = f epr-[ f(u)g(du,w)}P(dw)
*3 $2 5
Here g is a Poisson point process with intensity X so that P is
a probability measure determined by functions q satisfying:
r.

e'1(A11(y(Aj)) J

1 { r.1

J

 

":15

q(A],...,Am; r],...,rm) = .
J
where r1,...,rm are non-negative integers and Aj's are disjoint sets.

Consequently it follows that

m
vg(f) = f exp{- 2 c. E(A.,w)}P(dw)
n j=1 J 3
r.
- c.r. -ZX(A.) m [X(A.)] J
=Z {eJJe 111+}
r],r2,... j=l j'
-c1 -c
= exp{-Z X(A.)}exp{X(A1)e + ... + X(A )e m}
j J m
-c.
= exp{-Z X(A.)(1 - e J)}
j‘]
= exp{-£ [1 - e-f(u)1 X(du)}.

Thus, by lemma (1.3.1) the Laplace functional of a Poisson point process

a with intensity A is given by

15

f(u)

epr-f [1 - e' 1 A(du)} v f e c;(5)
5

Conversely: let Al""’Am be a finite collection of disjoint sets
in 8(S). Then the generating function of the random vector
(5(A1.-).....§(Am.-)) is given by

5(A1) 5(Am)
E(sl ... sm ) O < s. < 1, i = 1,...,m

m
E{exp[- i €(Ai,-) log éhd
1

E epr-é f(u)g(du,w)}

m
+
(for f(u) = g log §L1A(u) e ck(5))
1 1
= w€(f)
= exp{-] [1 - e-f(u)] X(du)} (by given hypothesis)
S

m
epr- i (1 - Si) A(Ai)}

which is the generating function of a product of m independent Poisson
random variables with parameters X(A]),...,X(Am) respectively.

Since generating functions determine a distribution uniquely
it follows that a point process g with intensity X whose Laplace
functional is given by (1.3.4) must be a Poisson point process with

intensity X.

1.4 Notion of weak convergence for point processes:

Definition 1.4.1: A sequence {An} of measures on (S,8(S)) is said

 

to converge weakly to a measure A on (S,8(5)) if

16

(1.4.1) I f(u)Xn(du) + f f(u)X(du)
S S

for every continuous, bounded real valued function f on 5.

Theorem 1.4.1: [Neveu - 1976, p. 282]

 

A sequence {Pn} of probability measures on (M+(S), M(S))
converges weakly to a probability measure P on the same space if and

only if

(1.4.2) yp (f) . vp(f) v f e c;(5)
11

Proposition 1.4.1: Let {gn(A,w): A 6 8(5)} be a sequence of point

processes with intensity {An} and {5(A,w): A e 8(5)} be a point

 

process with intensity X.

Then: A" + x weakly implies V f E CZ(S):

(1.4.3) (1) g f(u)gn(du,o) £1+ é f(u)g(du,-)
and
(1.4.4) (ii) Wgn(f) + W€(f).

Proof: (1) Let fec;(5).
Now

IEPII f(U)€n(du,-) - f f(u)g(du,-)}|
S S

= |£f(u)Xn(du) - é f(u)X(du)[ (by (1.2.4))

+ 0 (since by hypothesis An 4 X weakly).

17

Thus it follows that V f e c;(5) j f(u)gn(du,-) converges to
S

f f(u)§(du,-) in L](P). This establishes (1.4.3).

S

(ii) (1.4.3) implies

é f(u)tn(du.~) 3399 £ f(u)€(du,-)

implies

epr-é f(u)gn(du,-)} £599 exp{-£ f(u)g(du,-)}

and expf-f f(u)gn(du,-)} are bounded by 1. Thus by Lebesgue dominated
S 1

convergence theorem and using (1.3.2) it follows that
+
v€n(f) + vg(f) v f e ck(s)

thereby establishing (1.4.4).

Definition 1.4.2: A sequence {gn(A,m): A 6 8(5)} of point processes

 

is said to converge in distribution to a point process {€(A,m): A 6 8(5)}

if and only if

P converges weakly to P
En E

where PE and PE are respectively the probability laws of an
n

and g.
We prove in theorem 1.4.2, that the weak convergence of point
processes is guaranteed by the convergence of the corresponding finite

dimensional distributions (definition A-l of Appendix A).

18

 

Theorem 1.4.2: For {§n(A,w): A 6 8(5)} of point processes and a
point process {E(A,w): A 6 8(5)} the following three statements are

equivalent:
(i) P converges weakly to P
an a
+
.. f
(11) vgn( ) + W€(f) V f 6 Ck(5)

(iii) corresponding finite dimensional distributions converge weakly.

i.e. Prob {€n(Al’w) = r],...,gn(Am,m) = rm}

Prob {g(A],w) = r],...,g(Am,m) = rm}

fl

for A1,...,A in 8(5) and r],...,rm nonnegative integers.

m

3399:: 1°: (1) e (11) (see theorem 1.4.1)
2°: we will prove (ii) a (iii)

a) (11) = (1) (by 1° above)

and (1) = (111) [Billingsley - 1968]

thus (11) = (111).

b) (111) e (11): For

By lemma (1.3.1) it is enough to consider functions f in CES(S)

i.e. of the form

m
f(u) = 2 c. I (u)
. . A ,
J=1 J J
where cj's are positive constants and IA '5 are the indicator
.1

functions of disjoint sets A (j = 1,2,...,m).

J
8y (1.3.2) we have v f e c;s(5):

15 (f) = f exp{-f f(U)an(dU.w) P(dw)
n 9 S

j €n(Ajaw)}P(dw)

 

19

= r 2r exp{-§ cjrj} Prob {£n(A],w) = r]....
1, 2...
€n(Amaw) = rm}
,+ r1§r2,... exp{-§cjrj} Prob {€(A1,w) = r],...,

g(Am:w) = rm}

(by given hypothesis (111))

f eXPl‘f f(u)€(duow)}P(dw)
Q S

= 1150‘)

i.e.. v5 (f) converges to v€(f) V f e C;S(S). Thus it follows
n
from lemma 1.3.1, that

+
183nm -» 15(1) v 1‘ e ck(s)

10 and 20 together imply the theorem.

Remark 1.4.1: By theorem 1.4.2 any one of the three equivalent statements

 

imply the convergence in distribution of the sequence
{gn(A,m): A 6 8(5)} of point processes to a point process
{5(A.w): A 6 8(5)}.

1.5 Concept of spatial patterns:

Spatial pattern is most commonly used by plant ecologists to
describe the distribution of plants in a given area of study. In a
more generality, by a spatial pattern is meant the distribtuion of points
in space i.e., a spatial pattern is nothing but a realization of a point

process.

20

Study of spatial patterns is encountered widely in the areas
such as ecology, geology, medicine, forestry, image processing etc.
Some of the specific examples of spatial patterns include

. (i) distribution of stars in a galaxy;

‘(ii) distribution of a number of trees in a forest;

(iii) patterns of various rock formations on a geologic map;
(iv) texture modelling (through the description of images);
(v) epidemic spread from the map of a city.

The following proposition characterizes the behaviour of point

distributed completely randomly.

 

Proposition 1.5.1: Suppose that there are NM individuals distributed

uniformly over a region M of area |M| 'such that
NM
(l.5.l) iTMT. + a constant (say c)
then as the region of study is expanded into the plane i.e. as |M| + w
the distribution of events approaches the Poisson distribution of
events in the sense that:
If Al””’Am are any disjoint sets and 5(A1),...,5(Am)

denote respectively the number of individuals in A1,...,Am then

5(Ai): i = 1,...,m are random variables such that

 

-CV(AT) (CV(A1))ri

1 ri.

lll
lim Prob{t(Ai) = r1; r = 1,...,m} = n {e
1111» I:

v being the Lebesgue measure.

 

 

m m
Proof: Let v(A) = 1;] V<Ai)’ r = Z r1
- 1-1
Now
P b (A) 1 NM! m v(A1.) r‘1
r0 {g ' = r’i 1 = 9 am} ' I l 1 H X
1 1 r,....rm.(NM-r).- i=1 |M|
NM-r
(Ll—Ll” '; A 1
.1. 1 1
_ m MAN) ”14' 11A ”11*
- H r r' r M )
i=1 i' (NM-r)! |M|
ri N -r
1:] r1 M W .. |M|
r1
|M|+w E (V(Al)) e-Cv(A) Cr
1=1 '"1'

Y‘.
m (Cv(A1-))1 -cv(A1.)
= II -—fi——-— e
1=1 ""

T

as was to be proved.

Remark l.5.l: Limit in equation (1.5.1) is referred to as thermodynamic

 

limit.

N
Remark 1.5.2: Proposition l.5.l says that provided TNT. + a constant

 

then the finite dimensional distributions of a uniformly distributed
point process approaches the finite dimensional distributions of a
homogeneous Poisson point process. Consequently by theorem 1.4.2 it

follows that in the thermodynamic limit sense, a uniformly distributed

22

point process {€(A,w): A 6 8(5)} approaches a homogeneous Poisson
point process. Thus following proposition 1.5.1 a spatial pattern
{E(A,w): A 6 8(5)} is called random if it is a homogeneous Poisson
point process. In that case we also say that g(-,-) is a realization

of a homogeneous Poisson point process.

Remark 1.5.3: For a random pattern that is a realization of a

 

homogeneous Poisson point process with intensity 1, X is also referred
to as the expected number of individuals per unit area.
The following remark comments on the types of spatial schemes

that we shall or shall not be discussing.

Remark 1.5.4: At a formal level, we shall largely be concerned with

 

a rather arbitrary system. Later on we shall confine our attention

to specific spatial schemes. However, it is to be realized that a
spatial scheme could have developed continuously through time. For
example: incidence of spotted wilt over a rectangular array of tomato
plants. The disease is passed on by insects and after a period of time
we could expect to observe clusters of infected plants. Such spatial
schemes are referred to as spatial temporal schemes. We shall not be
dealing with such schemes in this study. Thus, we shall only be dealing
with homogeneous spatial schemes at an isolated instant of time. In
many practical situations, this is reasonable since we can only observe

the variables at a single point in time.

CHAPTER II
MARKOV RANDOM FIELDS

In Chapter I we introduced the notion of a spatial pattern
and showed why a random pattern can be considered as a realization of
a homogeneous Poisson point process. ‘In this chapter we discuss the
notion of neighbours in a set of sites from a graph theoretic viewpoint.
Further, we give a characterization of Markov random field and Gibbs
field and present a proof of the theorem which gives the equivalence of
Markov random field and Gibbs field with near-neighbour potential for
any finite graph.

Finally we have shown that the representing measure for a
Markov random field with no interaction is the Poisson measure so that
under the assumption of randomness spatial schemes have an exponential
density indexed by V. Grimmett's potential, with respect to the Poisson

measure .

2.1 Notion of neighbours in a set of sites:
Before we describe the concept of neighbours for a set of sites

we need a few definitions from the theory of graphs [Berge - 1962].

Definiton 2.1.1: A graph denoted by G = (A,F) is the pair consisting

 

of a set A and a function r mapping A into A.
Here A is known as the set of vertices (or sites) of the
graph G. The graph G is called finite if A has a finite number

of elements.
23

 

24

For 5 e A, let rs denote the image of 5 under r.

Consequently, if Ac: A then the image of A under I is the set

Definition 2.1.2: The pair (s,t) with t e F5 is called an arc of

 

the graph G.
Let u denote the set of all arcs of the graph G. We shall

use (A,P) or (A,u) interchangeably to represent a graph G.

Definition 2.1.3: Path is a sequence (u],u2,...) of arcs of a graph

 

G such that the terminal vertex of each arc coincides with the initial
vertex of the succeeding arc.
If a path v meets in turn the vertices 5],...,sk one may

also write
7 = [S],...,Sk].

The length of a path y = (u],...,uk) is the number of arcs

in the sequence (say 2(y) = k).

Definition 2.1.4: A circuit is a finite path in which the initial

 

vertex coincides with the terminal vertex, and a loop is a circuit of

length 1, consisting of the single arc (s,s).

Definition 2.1.5: In a graph G = (A,U) an edge is a set of two elements

 

s,t 6 A such that

either (s,t) E U or (t,s) 6 U.

25

Note that the concept of an edge should not be confused with

that of an arc which implies an orientation.

Example 2.1.1: Consider the following figure:

’ 0.

 

c .1.

Here U is the set of arcs (a,b), (b,a) (b,s), (s,s), (s,c), (c,s) and
(s,d).

The sequence {(a,b), (b,s), (s,c)1 represents a path of length 3.
The path {(s,c), (c,s)} is a circuit and (5,5) is a loop.

Finally, there are five edges namely (a,b), (b,s), (s,c), (s,d) and
(5,5).

With the above notations and definitions we are now ready to

describe the notion of neighbours in a set of sites.

Defintion 2.1.6: Let G = (A,r) be any undirected finite graph with

 

no loops or multiple edges. Then two sites s,t e A are called neighbours
if there is an edge between s and t.
Further, two sites s,t e A are called neighbours of rth order

if there are at most r edges between s and t.

26

Remark 2.1.1: (1) Neighbours of order 1 are also referred to as

 

nearest neighbours;
(ii) The above definition of neighbours does not imply that the

neighbours of a site are necessarily close in terms of distances.

Example 2.1.2: Consider a rectangular lattice with points labelled

 

with integer pairs (i,j). Then figure 1 represents neighbours of order

1 and figure 2 represents neighbours of order 2, of an arbitrary point

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1,1).
(lojtll (1-1,j+1) (1.1+1) (i+]:j+])
_ X X
(1-1,J) (1,3) (1+1oj) (1-1,J) (1,3) (1+1,J)
(113-1) (1-111-1) (1,1-1) (1+111-1)
Figure (1) Figure (2)

2.2 Definitions and characterizations of Markovian fields and Gibbs
b.1121:
The following exposition follows Carnal (1979).
Notations: In what follows:
Let G = (A,P) be a finite graph with (AI = n.
Let {xtz t e A} denote a family of random variables taking values
in some measure space (E,E,p) where u is a o-finite measure.
Let p 6 EA i.e. o: A + E. For Ac: A, let 8A denote the set of
neighbours of sites in A.
Let A'= A U 3A and AC denote the complement of A; o denote the empty

set.

27

Let mfi‘= TIA denote the restriction of o to A.

Further, if e e E

9L(¢) = {t e A | m(t) # e}

A cp(t) 1'1 te A

e (t) =

~T 0 otherwise.

Also when no confusion is involved we shall write at for

aIt}, the set of neighbours of site t and ot = TIt} = m(t).

Definition 2.2.l: We say that {X °

 

t e A} is a Markovian field on

Q u-a.5.
n

t‘
A if
(i) there exists an f > O on Ex...xE (n times) such that
(2.2.1) P[Xt 6 A1,...,Xt E An] = f f d e u
l n A1x...xA n

n
where Al""’An e E and A = {tl""’tn}; and
(2.2.2) (111 1(oA11AC1 = 1(oAlbaA1
for Ac: A, mé EA where
(2.2.31 1(1A1181 = onB (oAU81/1B<oB)
(2.2.4) f (w ) = f f d 8 u.

A A xE n
IACI

Remark 2.2.1: (i) Condition (2.2.2) is what is known as the near-

 

neighbour condition.

28

(ii) A Markov random field is said to be of order r (r 3 1) if

the neighbours of rth order are taken into consideration.

Definition 2.2.2: We shall call f(otloat) the local specifications

 

of {th t 6 A}.

Definition 2.2.3: We say that {X '

t' t e A} is a locally markovian

 

field if

(i) there exists an f > 0 on Ex...xE (n times) such that

PEX e A X 6 An] = f f d e u eo-a.s.

t l""’ t
l n A1x...xAn n n
and

(2.2.5) (11) f(mt|¢{t}C) = f(mtlmat).

Remark 2.2.2: It is clear that a Markovian random field is also locally

 

Markovian.

Definition 2.2.4: Let G = (A,P) be a finite graph. A set of sites

 

Kc: A is called a cligue if it either contains a single site or if
S,t E K such that S E at.

Let K denote the family of cliques of A and

A = {sz m 6 EA and K E K}.

Example 2.2.1: In the nearest neighbour scheme of example 2.1.2 (figure 1)

 

there are cliques of the form {(i,j)}, {(i-l,j), (i,j)}, {(i,j-l),

(i,j)} etc.... .

 

29

Definition 2.2.5: Every mapping V: A + R is called a potential of

 

Grimmett.

Remark 2.2.3: The notion of potential encountered in statistical mechanics

 

describing the interaction between particles is much more general (see
[Spitzer - 1971]). Here we consider interaction only between those

particles that are near-neighbours.

Definition 2.2.6: We say that the process {X ° t e A} is a Gibbs

t‘
field on A if there exists a potential V of Grimmett such that

 

(2.2.6) PEX e A ,x e An1= f g d o p

t1 "°'° tn AIX...xA n
n
where
(2 2.7) 9(4) = c exp{ 1 V(mK)}
KEK
and Al’°"’An e E and A = {tl"°"tn}‘

 

Theorem 2.2.1: For X = {th t e A} taking values in (E,E) the following

three statements are equivalent:

(1) X is a Markovian field;
(ii) X is a locally Markovian field;
(iii) X is a Gibbs field.
Before we could give a proof of the above theorem, we need two

lemmas:

Lemma 2.2.1: If f is a density of a locally Markovian field (definition

 

2.2.3 (1)) then v t e A 5 ¢ at, c = {s,tlc, o,b', 1, n,n' 6 EA we

have e u-a.s:
n

30

f(nt.1c.ns) _ f(wt.1c.n;)
flmt.wc.ns) f(mt.wc.ngl

 

 

(2.2.8)
for an appropriate enumeration of A.

Proof: By definition:

f(mt.wc.ns) f(ntlwc.ns)
f(mt,wc.nsl f(mllwc.n;l

 

 

f(¢t|WC) .
1 “3.3.
fiitlwci : u

(by Markovian property).

f(mt.wc.ns)
, flcpéfll’cpns)

 

i.e. is independent of HS. Consequently we have

 

 

f(cptawc,ns) = f((ptawc’ns) 8 u-a S
f(mt.wc.ns) flmt.wc.ns) n

as was desired.

Lemma 2.2.2: There exists a p-null set N1: E such that V e E E\N,

 

V s,t E A, s 4 at Bc;A\{s,t} we have e u-a.s. V m 6 EA

 

 

n
f( 6(BU{S,t})C ) f( 6(BU{5,12})C eiti)
(2 2.() CPS,CPB,_C9 9 (Pt = CPS:¢PBa (P 9 S?
{s} (BU{s,t})c 1s} (Bu1s,t})C {t}
f<e¢ a $8: a? s Qt) iii-(e(p ’QB’GQ 96m )

Proof: By lemma 2.2.l, there exists a e p-null set M in Ex...xE
n
(n times) such that

 

31

f(es.wc.nt) _ f(es.wc.n£)
f(e;.1c.nt) f(¢;.wc.n;)

 

 

whenever (es.1c,nt). (es.wc.nt). (es.1c.nt). (es.wc.nt) do not belong
to M.

Consequently in order to justify equality (2.2.9) we need to show that
each of the function in the four parentheses of the equality do not

belong to M.

(Bu1s,t})c

@ .mt) e M}.

_ A
Let 9F1 - {e e E |(es.e3,e

8F , 6F ,eF are defined similarly using the other three terms appearing
2 3 4

in the equality (2.2.9).

Let m = |(B u {s,t})cl

We then have the following scheme:
Ex--- XE 1\
\—--\r'-“’

‘1\.‘£®H~1$

 

1(\\\\ 1

 

 

 

32
Since u is o-finite, we have

O=eum)=fe m%)ooanqm

n n-m m
implies
8 “(M6) = 0 u-a.S.(O)
n-m
implies
e 11(6)F ) = 0 u-a.s.(e)
n 1
Likewise we get
e “(9F.) = O u-a.s.(e); i = 2,3,4

11 '1

Therefore if e 6 E\N each of the functions appearing in
the four parentheses of equality 2.2.9, do not belong to M except

for a null set which gives the desired conclusion of the lemma.

Proof of theorem 2.2.1:

1°: (1) = (11)

 

Follows from remark 2.2.2.
2°: (11) = (111)
Given that X is locally Markovian we need to show that X is a Gibbs
field.
Let N<: E be a u-null set as in lemma 2.2.2 and let 6 e E\N be fixed.
Let u: EA + R be defined by

Q21m mm: 2 on “W 1w1m) verA

B
12
339W?)

 

33

Let now m 6 EA be fixed and A ; eL(¢). Then

(2.2.11) u(e$) = ) (.-1)'A\BI log f(e )

B
BCA R
The set of subsets of 6L(¢) is partially ordered by inclusion and

its Mobius function is given by

nwm)=CUMW1 vmase()£gA

L e

Consequently we can write (2.2.11) in the form

u(eA) = ) n(B.A)109 f(eB)
(p BQA ‘P

which yields
(2.2.12) log f(eA) = Z u(eB)
(P BCA (P

(by Mobius inversion theorem)

Claim: U(e$) = 0 unless A is a clique.

For suppose o f A e K so that A contains two sites 5 and t such

that s e at
(2.2.11) can be rewritten as:

U(e$) = ) (-1)1A\311og 1(931 +BéAI)IA\BIlog 1(92)

s,teB seB,t¢B

+ Z (-1)iA\B| 109 f(GB) + Z (-1)|A\Bllog f(eB)
BcA R BgA Q

§¢B,teB s,t¢B

 

34

(-])|A\Bl[]og f(eBU'Isst1’) _ 109 f(eBU{S}
B:A\{s,t} 1’ ‘P

_ BUIt} + B
log f(ecP ) log f(eQ)]

BU{S t} BU{S}
1 e 1 1
(-1)|A\Bl 1 _i ¢ ) _£E§L___3_.
°95 31111;} / J
B$A\{s,t} f(e‘p ) f(ecp

 

0 e u-a.s. (by lemma 2.2.2)
n

Thus (2.2.12) gives in particular for A = 6L(o)

fl?) = exp{ 2 U (6:) + U (e¢¢)} e m-a.s.
KCGLUP) ' "
K.€ K
Hence:
f(cp) = const exp{ 2 u (6:)} o u-a.s.
K.€ K

which can be written as
f(ml = const exp{ 2 V(¢K)} e p-a.s.

where

V(CPK) =

0 otherwise.

Thus, there exists a potential function V: A + R such that

 

Ill

with

f(o) = const exp{ Z V(¢K)} g u-a.s.

KEK

implies X is a Gibbs field.
3°: (111) = (1)
Only need to show that condition (2.2.2) of definition (2.2.1) is
satisfied.
i.e., need to show that V A c A, o 6 EA

9(eAleAc) = glwAlmaA) e u-a.s.
11

For simplicity we shall omit o u-a.s. Using (2.2.3) and (2.2.4) we

n
have

 

9(‘PAISPAC) = XI 9(gT§¢21A

M(TA)
1A!
)1: exp{ X V(<pK)} exp{ V .
KHAfb - 1
= KeK KEK
exp{ V(¢ )1 exp d o (e A)
Ki XE KDAfe K K “ H?
|A| KeK KEK

 

exp{ Z V(¢K)}
KnAfo
= KeK

1 exp{) who}: s 111A)
xE KllAfo ' (Al
IA] KeK

 

36

 

51m1larly we get: I g(?)d 2c ulPfic)
xE IA
g(‘1’A1°"21A) = licil 9(mld e u(e )1 d 9 “(R 1
xE x§_ AC AC 1A1 A
IAI IACI

(Here A'= A U 3A)

C I exp{ 2 Vlm )1 exp{ 1 V(¢ )1 d 9 ulm )
xE KnAfo K KnA= K (AC) AC

 

¢
=__,]ACL KEK f Kek -
c I’ {1 exp{ 2 VlmKl} exp{) V(¢K)} d 2C filmfic)}d e u(¢AT
xE xgc KOAft KnA=¢ IA | (Al
[Al IA 1 KEK K€K

exp{ V(¢K)}

 

KOAfi
= Kek ..
1 exp{ ) V(¢Kl} d 9 m(mA)
xE KOAfo |A|
|A| KeK

(since the other term does not depend upon oA)

Therefore it follows that

g(¢Ai¢Ac) = g(¢Al¢3A) 9 U‘a.$.
11

1°, 20 and 3° together imply that the three statements of the theorem

are equivalent.

Remark 2.2.4: The above theorem shows that the set of Markov (or

 

locally Markov) fields and Gibbs states with nearest neighbour potentials

are the same for any finite graph.

Remarks 2.2.5: (i) Grimmett (1973) has proved the part (ii) a (iii)

 

of theorem 2.2.1 in the case where E is countable. One needs lemma

2.2.2 to pass to the general case of carnal (1979).

37

(ii) In the case when E is discrete, Besag (1974) gives a simple
alternative proof of the part (ii) = (iii). See theorem 2.2.2 of

this section.
(iii) For E = {0,1} Preston (1974) gives a complete discussion.

For details, one is referred to his paper.

Corollary to theorem 2.2.1: If X is locally Markovian then it is

completely specified by its local specifications.

3599:; It was seen in theorem 2.2.1 that X is completely described

by the U-functions defined by (2.2.10). Therefore, it suffices to show

that the local specifications determine the U(e$) V clique K E K.
Let us consider a and o s e on A to be identical. Since

X is locally Markovian:

f(etleat) = f(etlemc)

_ f(e)
gng1dmm1

= const
g C exp U(e;) d u(et)

Likewise it can be seen that
t
c ex U 9
p ( cp)

g C exp U(e;) d m(et)

 

1c“P1219319 =

Consequently it follows that

38

t) = 10 f(etleat)
<11 9 f(etTe

 

U(e
at)

Therefore we know the form of U(e:) V clique K with a single element,
K = {t}. The proof is then completed by induction and one is referred

to Carnal (1979) for completion of this proof.

Remark 2.2.6: Carnal (1979) has also discussed the case when A is

 

a finite subset of a circle.

2.2.1 A special case of theorem 2.2.l:

 

Let there be h sites in A labelled 1,2,...,n and
{Xi: i = 1,2,...,n} be a family of random variables associated with
these n sites taking values in a discrete space E. Let
X = {Xi: i = 1,2,...,n} be a Markov field. Then the near-neighbour
condition (2.2.2) of definition (2.2.1) implies that
PCXilall other site values] = PLXilneighbours of site i].

Besag (1974) gives a representation of the probability structure
P(§) of 'Xi= (x],...,xn) through near-neighbour system in the simplest
form. Following his paper, let us assume that if P(xi) > O for each'
i then P(x) > 0 (known as the positivity condition)1

Let 51* = {X2 NY) > 0} be the sample space of all realizations
of the system. In what follows it will prove convenient to consider

the representation for the ratio POO/PUT).
Define W) =MPOO/P(ID}.

39

-—\

Lemma 2.2.3: There exists an expansion of V(X) unique on 9* given

 

by

V(Y) = X X'Fi(xi) + Z x

F (xi,x.)
lfi<jfn

1X11..1 .1
(2.2.13)

+...+ xlxz...xn F1,2,...,n(xl"'°’xn)

Proof: The F-functions are determined inductively. For example:
x.F.(x.) = V(O,...,O,x1.,0,...,0) - V(b)

l l l

with analogous difference formulae for higher order F-functions.

Theorem 2.2.2: [Besag - 1974].

 

V(XD defined by (2.2.13) gives a valid probability structure
to the Markovian random field taking values in a discrete space E

provided the functions F 5(Xi’x "’Xs) are non-zero which

i.j..--. 3"
holds if and only if the sites i,j,...,s form a clique. Subject to

this restriction, the F-functions may be chosen arbitarily.

Remark 2.2.7: The above theorem says that given the neighbours of

 

each site, we can readily write down the most general form of V(X)
and consequently of P(X), the probability structure of the Markovian

random field taking its values in a discrete space.

2.3. Characterization of spatial schemes:

 

In practice, we shall often find that the points or sites

occur in a finite region of the euclidean space and often fall into

 

40

two categories: those which are internal to the system and those which
form their boundary (or boundaries). In the analysis of spatial patterns
one is interested in the behaviour of the system outside the region

of their occurrence. The problems at the boundary may be handled by
considering the joint distribution of the internal site variables
conditional upon fixed observed boundary values. Thus, in constructing
spatial schemes to study the behaviour of the system we need only to
specify the neighbours of a given set of sites and the associated
conditional probability structure for each of the sites. This is exactly
what is specified by Markovian random fields. Consequently, in order

to study the behaviour of a spatial pattern outside the region of
occurrence it is enough to consider schemes that represent Markov random
fields. Thus, in what follows we shall be considering only those spatial
schemes that satisfy Markovian property and we shall use the notations

of section 2.2.

Preposition 2.3.1: For a Markov random field with no interaction

 

the representing o-finite measure u is the Poisson measure.

Erggf: By remark 2.2.4 and theorem 2.2.1 the density of a Markov random

field with respect to any o-finite measure u is given by

PEX 6 A1’°°"Xt 6 An] = c f exp{é V(QK)}d e u

t
1 n Alx°'°XAn n

where c is a constant, V is the near-neighbour potential and
A1,...,An e E The term Z V(eK) is the contribution from the inter-
K

action between the particles. Consequently, the density of a Markov

41
random field with no interaction with respect to u is:

P[x e A1,...,Xt 6 An] = c f d e u

1 n Alx...xAn n

t

i.e.,

(2.3.1) M GA .,x u(A.)

1 l

11:13

6 An] = c

1”. n i

t] t

By proposition l.5.l, t.h.s. represents the Poisson law so that it
follows that the finite dimensional distributions of a spatial scheme
with no interaction has a finite dimensional Poisson law. Consequently
by theorem 1.4.2 it follows that for a Markov random field with no

interaction the representing measure u is the Poisson measure.

Remark 2.3.1: Theorem 2.2.1 says that Markov random fields have an

 

exponential density indexed by V, the potential of Grimmett with re-
spect to any o-finite measure u and proposition 2.3.1 says that the
representing measure u is the Poisson measure for a Markov random
field with no interaction. Consequently, it follows that in testing
for randomness namely V s 0 against the alternatives V > 0, the
spatial scheme will have an exponential density indexed by V with

respect to the Poisson measure.

CHAPTER III
INFERENCES ON RANDOMNESS

In the analysis of spatial patterns one of the problems of main
interest is to determine whether or not the spatial patterns exhibit
any randomness i.e. whether or not the observed pattern is a realization
of a Poisson point process. This may be done through
(i) estimation of X, the intensity of the spatial process;
and
(ii) testing hypotheses concerning the parameters namely V, Grimmett's
potential that describe the spatial interaction.

For the testing hypothesis problem (section 3.2) we consider
u of Chapter 2 to be a 0-1 variable and V(Y) of section 2.2 to be
of a particular form (N-N interaction) and compute the asymptotic
distribution of log-likelihood ratio under the null hypothesis of no
spatial interaction. This justifies the recent work of Besag (1974)
and it has been remarked that Besag's coding method of estimation is
not necessary in establishing the chi-square behaviour of the log-

likelihood.

3.1 Methods of estimation of A, the expected number of individuals
per unit area.
In the analysis of spatial patterns, estimation of A plays

an important role because it contributes to the understanding of certain

42

43

aspects of the pattern or the arrangement of individuals in space.
In this section we discuss some of the methods of estimation of A
and the asymptotic properties of these estimators. The estimators are
constructed so that they are at least asymptotically unbiased. There
are mainly two techniques described respectively as quadrat method and

distance method.

3.1.1 Quadrat Method: This method is based on field sampling and in-

 

volves choosing m disjoint quadrats each of area D from the region
of study.

Let Zi(i = 1,...,m) denote the number of individuals in the
ith quadrat. Assuming that the pattern is random by proposition l.5.l
each 21(1 = 1,2,...,m) has a Poisson distribution with parameter
XD and are independent.

Consequently the sample likelihood functions based on these

m quadrat counts is given by

 

lll
AmD ill Zi
e 62' L10) "
(3.1.1) Lq m
H Z1!

1=1
Ill
Clearly Z Zi is a complete sufficient statistic.
1=1

Using (3.1.1) the maximum likelihood estimator of A based

on quadrat counts is given by
(3.1.2) i = l-—--—

also known as the quadrat estimator of X.

44

Remarks 3.1.1: (1) It 1: clear that iq m given by (3.1.2) is an

9

 

unbiased estimator of A.
(ii) If the D's are not equal then Zi's will be independent and
distributed as a Poisson random variable with parameter

101 (i = 1,2,...,m). In this case the quadrat estimator has the form

)
—l.
11MB LMS

which again is an unbiased estimator of A.

Theorem 3.1.1: Assuming we have a random pattern (i.e., the pattern

 

is a realization of a homogeneous Poisson point process) the quadrat

estimator Xq m defined by (3.1.2) is

a) strongly consistent

i.e.,

(3.1.3) iq m 24E4-A as m 4 a.

b) asymptotically normally distributed: In fact

(3.1.4) M(iq m - 1.) 2. N(0,>./D)
Proof: a) By (3.1.2)
A g
= Z./mD
q’m 1=1 1

By proposition l.5.l {Ziz i = 1,...,m} is a sequence of independent

identically distributed (i.i.d. for short) Poisson random variables

45

with parameter AD. Thus, using the classical theory of i.i.d. random

variables it follows that

" a.s
q,m—~71 as Ill-+00.
b) Now: m
)2
4m“ -1)=/m1"=‘1 -1}

q,m mD

11 m
=-{——Z (Z -AD)}

D 411:1 ‘

Here {Zi - AD: 1 = 1,...,m} is a sequence of i.i.d. random variables
with mean zero and variance AD. Thus, by central limit theorem it

m
follows that jL— X (Z - AD) is asymptotically normally distributed

/m1=1‘
with mean zero and variance AD.

Hence

. D
Afi(Aq m - A) -+ N(0,A/D).

A

Remark 3.1.2: (3.1.4) implies that the asymptotic variance of Aq m

is 1fi%- which implies that if mD can be taken arbitrarily large then

 

A can be determined with high accuracy using the quadrat estimator.

3.1.3 Distance Method

 

Distance method involves estimation and testing of parameters
for a spatial process based on some kind of distance measurements.
Various distance measures have been studied in the literature. For

example, the distance measured may be from an arbitrarily chosen

46

(explained below) point to its nearest neighbour, second nearest neighbour
",...,rth nearest neighbour etc.; the distance measured using T-square
sampling introduced by Besag and Gleaves (1973) and many more. For

more details on T-square sampling the interested reader is referred

to the paper by Diggle, Besag and Gleaves (1976).

In this study we will confine our attention only to nearest
neighbour (N-N for short) distances which probably is one of the simplest
distance measures.

Before we can give the estimator based on N-N distance measure-
ments, we need the following proposition concerning the distribution

of N-N distances.

Proposition 3.1.1: Suppose the points come from a homogeneous Poisson

 

point process with intensity A. Let X denote the distance of an

arbitrarily chosen point from its N-N. Then the transformed variable
Y = V(X) where V(X) denotes the volume of a sphere centered at the
chosen point and with radius X, has an exponential distribution with

parameter A.

Proof: Choose an arbitrary point from a realization of a homogeneous
Poisson point process with intensity A.
For a Poisson process, the probability of capturing exactly

k individuals in a sphere of radius x is given by

e‘*V(X)£A V(X))k k = o,1,2,...

Now the event [X > x] denotes the event that no point is captured

47

in the sphere of radius x and centered at the chosen point.

Thus,

-A V(x)

PEX > x] = e x > O.

This implies

e-A V(x)

FX(X) = PIX f X] = 1 - x > 0

AV(

implies dFX(x) = A e- X) V'(x)dx x > 0. Consequently, the trans-

formed variable Y = V(X) has the distribution function given by
dGY(y) = A exp(-Ay)dy y > 0
i.e., Y has the probability density function

9(y) = A exp(-Ay) y > 0

which is an exponential density with parameter A.

Remark 3.1.3: If in proposition 3.1.1, X denotes the distance from

 

an arbitrarily chosen point to its rth (r 3 l) nearest neighbour,
then the event [X > x] desCribes that there are at most (r-l)
points captured in the sphere of radius x, centered at the chosen
point so that

r-l e-A V(x) k

P[X > x] = Z
k=0

[WM]
1::

implies

 

48

Thus, X has the p.d.f. given by

_'|'
11.) . Arexpm ¥(§))(v<x))r v (x) x . 0

Consequently, the transformed variable Y V(X) has the p.d.f. given

by

Ar e-Ay yr-l

913’) = Pu.)

which is a gamma density with parameters r and A.

Near-neighbour estimator of A based on distance measurements:

 

Sampling scheme for the distance approach involves choosing arbitrarily
a set of n sample points from the region of study. By proposition
1.5.1, the points are from a realization of a homogeneous Poisson point
process with intensity A.

Let Xj denote the distance of the jth point from its nearest
neighbour. By proposition 3.1.1, it then follows that the transformed
variable Yj = V(Xj), V(Xj) being the volume of the sphere centered
at the chosen point and with radius Xj’ has an exponential density
with parameter A. The points are so chosen that the Yj's are independent.
The sample likelihood function based on n N-N distance measurements
is:

l1

(3.1.6) Ld = An exp(-A Z Yj)
1=1

Clearly 2 Yj is a complete sufficient statistic and the maximum
1

likelihood estimator of A using (3.1.6) based on distance measurements

is given by

(3.1.7) Ad,“ = n/ E Y
He shall refer to Ad n as the N-N estimator of A.

Remark 3.1.4: In the preceding formulation for Ad," one may also
consider point-to-plant or plant-to-plant distances. However, as
remarked by Pielou (1959) a measure based on plant-to-plant distances
may not reveal any nonrandomness in the spatial distribution at all

since plants may often be present in clumps or clumps of clumps etc...

Theorem 3.1.2: Assuming we have a random pattern (i.e. pattern is a
realization of a homogeneous Poisson point process) the N-N estimator

Ad,n g1ven by (3.1.7) is:
a) strongly consistent

i.e.,

(3.1.8) Ad n é4§4~ A as n + w.

b) asymptotically unbiased.

 

In fact:
. - 81
(3.1.9) 5(1d n) — n_] , and
. 2 2
(3.1.10) Var(Ad n) = " 1 2
’ (n-l) (n-2)

Proof: By proposition 3.1.1 {sz j = 1,...,n} is a sequence of

i.i.d. exponential random variables with parameter A.

50
Therefore by strong law of large numbers we have:

Y.
.= J
-—J————; —-——+a°s‘ E(Y]) =JA' as n +00

t~1a

and consequently it follows that

* a.s.
” . -1
b) Let now Y = Z Y. so that Ad = nY Since each
3°:] J 9“

Y.(j = 1,...,n) is exponential with parameter A it follows that Y
is a Gamma random variable with parameters n and A.
t n n-le-Ay

. - A y
1.e. P[Y < t] - f dy

 

Consequently it fellows that

F 1(t) = Ptv“ < t]
y- , _

 

II
—J
I
0% (‘f'
".1
A
3
V
CL
‘<

For r > 0, it can easily be seen that

1(1‘1) = r 1 tT'111—F _](t)]dt
o 1
implies: 5(1’1) = A/(n-l)
and 5(1'2) = AZ/(n-l)(n-2)

Hence it follows that

E(A = nA/(n-l) and

d,n)

nZAZ

(411201-21

 

Var(Ad n) =

51

clearly (3.1.9) implies the asymptotic unbiasedness of Ad n'

Remarks 3.1.5: (1) (3.1.9) says that the N-N estimator Ad n is

slightly biased. However if the spatial distribution is not random
(i.e. if it is not a realization of a homogeneous Poisson point process)

this estimator may give serious bias.

(ii) Using the classical theory on i.i.d. random variables it follows

21.

m inel- - {-1 9-» N(0.1/A2)

that

Consequently it follows that

 

mid - 1) 2.. N(O,A2)
,n
(iii) Using the quadrat estimator Aq m and the N-N estimator Ad n
let us define an index:
m
A Z] Zi/mD
a = “q,m = 1-
Ad n n
’ n/ 2 Y.
1=1 3
Under randomness assumption, it was noted earlier that both Aq m
and Ad n converge to A so that a + 1. Thus if one calculates

a from the observed data and finds that it differs significantly from
1 then it can be assumed that the spatial distribution is not random.
Also in an aggregated population we would expect higher values to

a and in a regularly dispersed population we would expect low values
to a. Pielou (1959) has given approximate confidence intervals for

a. For more details one is referred to Pielou's paper.

52

(iv) Another way to test for randomness: In the formulation for

’ J
tribution with parameters n and A under the randomness assumption.

- n
N-N estimator Ad n it was noted that Y = Z Yj has a Gamma dis-
=1

Consequently under randomness v = 2A Z Yj has a chi-square distribution
with 2n degrees of freedom. This y may be used to test for random-

ness rather than a.

3.1.3 Estimator of A based on both Quadrat counts and N-N distance

measurements:

 

By considering two independent realizations of the Poisson
point process with intensity A, one can have two independent sets of
data namely m quadrat counts and n N-N distance measurements. Thus
it seems reasonable to use these two independent sets of data to look
at an estimator of A and the sample likelihood function based on m

quadrat counts and n distance measurements is therefore:

m
n+ Z Z. Dizi
(3.1.11) L = A i=1 1 exp{-A(mD + Z Y.)}————r
i

where Zi (i = 1,...,m) are the quadrat counts and Yj = V(Xj) re-
presenting the volume of the sphere with radius Xj - the distance from
the jth point to its N-N, are the distance measurements.

In (3.1.11) the r.h.s. is a product of two exponential families
of distributions and hence is itself an exponential family of distributions.
Further, it can be noted from (3.1.11) that there is no single sufficient
statistic for A but (2 Z1,E Yj) is a sufficient statistic pair.

Using (3.1.11) the maximum likelihood estimator of A based on both

quadrat counts and distance measurements is:

53

n + Z Z

x = 1
(3.1.12) ‘ Am,n ‘JY—mDi- YJ'

Theorem 3.1.3: Assuming we have a random pattern (in the sense that it is
a realization of a-homogeneous Poisson point process) and n/mD + w as

n,m + .. the estimator im n defined by (3.1.12) is:

9

a) strongly consistent;
b) asymptotically normally distributed.

In fact:

(3.1.13) 3 - J5 (A - A) Z N(O,A2/2)

111,11

Proof: a) the hypothesis that éE- + A as n,m + m is redundant for this

 

part.
n + Z Z
* _ i
By (3.1.12): Am,” ' mD + y.
j J
1‘
n l__ Z
= mD’+ mD i=1 1
n
n 1
1 + —-- Z Y
mD n j=1 3

By proposition l.5.l {Z1: 1 = 1,2,...,m} is a sequence of i.i.d.

Poisson random variables with parameter AD so that by SLLN:

a.s.
Z. ——-+ A as m + w

(3.1.14)
1 1

"ME

.1.
m0 i

Similary by proposition 3.1.1 {sz j = 1,...,n} is a sequence of
i.i.d. exponential random variables with parameter A so that again

by SLLN:

 

54

(3115) 1"
'° FEYa.S.l

Thus (3.1.14) and (3.1.15) imply that as n,m + m

X a.s. A
111,11

. n + Z Z.
- = 1
b) V/Il- (Amm A) JET-{mi - A}

Rearranging terms this can be rewritten as:

 

x 111 n
(in -A=—’1—{—L 2-111-1— .-.1.
m’" ) "‘D/rT1g1(‘ ) 4?in 1)}
.2331 li](Z-AD)--°—Z(Y-l—)}
mD n ['51:] 1 ,fn— j A
L11
1 + m0 n j Yj

Now (i) {21 - AD: i = 1,...,m} is a sequence of i.i.d. random variables

wiht mean zero and variance AD;

(ii) {Yj - %—: j = 1,...,n} is a sequence of i.i.d. random variables
with mean zero and variance eg—.
A

Thus, by multivariate central limit theorem it follows that

1 1 1
——- Z. - D , -—- (Y. --—

using the independence of Zi's, Yj's and %%-+ A as n,m + m we have

. 1) 2
Alum,” - A) + N(O,A /2)

 

55

Remark 3.1.7: (3.1.13) implies that the asymptotic variance of

 

Am n is A2/2n which is less than the asymptotic variance A/mD of
Aq m and also than the asymptotic variance AZ/n of the unbiased
version of the N-N estimator. In practice however either Aq m or
Am," 15 used and of course the ch01ce between Aq,m and Am,n W111

affect the performance of the test.

In the following section we consider a particular subclass of
spatial Markov random fields and become more specific with some of the
spatial schemes generated by this subclass. In the later part of the
section we look at a test for randomness for spatial binary schemes

in this subclass.

3.2 Spatial schemes generated byia subclass of Markov random fields

 

and testing of hypothesis for binary models:

 

3.2.1 One dimensional problem:

 

Let there be n sites labelled 1,2,...,n and a set of neighbours
for each site. Let Xi: i = 1,2,...,n denote the site variables.
Then following section 2.2.1 a class of valid probability structure

associated with these site variables is given by

(3.2.1) poo = P(o‘) exp V(TO

where

56

With Fi,j,...,s(xi’xj’°'

i,j,...,s form a clique. Subject to this restriction, the F-functions

.,xs) non-zero if and only if the sites

may be chosen arbitrarily.

We shall use the function pi(°) to denote the conditional
probability distribution (or density function) of Xi given all other
site values. However, by the Markovian property pi(-) will be a
function of xi and of the values at sites neighbouring site i.

Within this framework we consider a particular subclass of Markov random

fields for which V(?) is well defined and has the form:

(3.2.2) V(X) = 2 xi Fi(xi) + .2. Bi,j xixj
1 1,3
where 8i j = 0 unless sites i and j are neighbours of each other.

i.e., in particular the only non-zero parameters are those associated
with the cliques consisting of single sites and of pairs of sites.

Spatial Markov random fields whose probability structure is given
by (3.2.1) with V(75 given by (3.2.2) are known as Auto-models and
{Bi,j} are called the parameters of the models that describe spatial
interaction between the sites.

We shall specifically be dealing with the subclass of auto-
models for which Bi,j = B V i and j so that 8 describes the spatial

interaction between near-neighbour sites. In such cases, the auto-

models are said to be homogeneous.

 

Remark 3.2.1: In view of (3.2.2), the homogeneous auto-models have

 

conditional probability structure satisfying:

Pi(x1-;...) _
(3.2.3) pi(0;"') - exp{xiEF1(xi) + 5 g xjjl

 

57

where Z xj will denote the sum of the values at sites neighbouring
site i. The models can further be classified into auto-normal, auto-
logistic, auto-binomial according as pi(-) taking normal, logistic

or binomial form.

3.2.2 One-dimensional auto-logistic model for binary data:

In this special case the site variables Xi take 0-1 values.
For any finite system of binary variables, the only situation in which
the non-zero F-functions can contribute to V(Y) (given by 3.2.2)
are those upon which each of the arguments is unity. We may therefore
replace the non-zero functions by arbitrary parameters. However, since
8 is the one that describes the spatial interaction, without any loss
of generality we may replace Fi(xi) by a constant namely a. Thus,
the spatial binary scheme has a probability structure given by (3.2.1)
for which V(Y) has the form:

.1
V(X) = a 2 xi + 8 Z xix.
j lfifjfn 3

Consequently it follows from (3.2.3) that:

exp{xita + B Z x.3}
J J

 

(3.2.4) pi(xi;...) =
l + expCa + B Z x.]
.j J

where 2 xj = x1_] + Xi+l is the sum of the values at sites neighbouring
j

site i.
The model specified by (3.2.4) is the classical logistic model
and thus in this case the spatial model is known as the auto-logistic

model.

 

58

Remark 3.2.2: Auto-logistic models are quite useful in practice for

 

example in an ecological context the variables may correspond to an
array of plants each of which is either infected (1) or healthy (0),

or to the presence (1) or absence (0) of a plant at a site. Moreover,

the models having once been established are easy to interpret.

Remark 3.2.3: A homogeneous first-order (or nearest neighbour) scheme

 

for zero-one variables for a rectangular lattice with sites labelled

by integer pairs (i,j) is given by [Besag - 1974]:

PEXiJIXi-l,i’ Xi+i.j’ Xi.j-1’ Xi.j+13

*
expEx.. t. .]
(3.2.5) - U W

‘ ~27
l + exp[t1,j]

 

where t:,j = a + 81(Xi-l,j + xi+1,j) + 82 (xi,j-l + Xi,j+l)‘
The parameters 81 and 82 are the ones that control the
clustering (or spatial interaction) in the lattice. 8] controls the
clustering in the E-w direction and 82 controls the N-S clustering.
The lst-order binary scheme described by (3.2.5) is said to be"
isotropic if a] = 82 = 8 (say). Thus, a homogeneous isotropic lst-

order auto-logistic model is described by

exp{[a + B t. .l x }

__ ,._ 1,3 M“
(3.2.6) Ptxi,j|N N] 1 + expta + 8 ti j]

’

 

where t. sum of the N-N values

1,j

- . . + . . . . . .
Xl-1,J xl+1,J + Xl,J-1 + X1,J+1

59

Remark 3.2.4: The number of parameters required in a binary scheme

 

depends upon the order of the near-neighbours considered.

For Example: A second-order model involves cliques of size three and

 

four so that the expression for the conditional probability structure

is given by
I I I I = ex [X T]
PExIt,t , u,u , v,v , w,w ] T_E_EYEFTJ
where:

T = a + 81(t + t') + 82(U + U') + v1(V + V') + v2(w + W')
+ 51(tu + u'w + w't') + g2(tv + v'u' + ut')
+ 53(tw + w'u + u‘t') + 54(tu' + uv + v't')
+ m(tuv + t'u'v' + tu'w' + t'uw')

The above scheme will be auto-logistic only if all the g and n

parameters are zero.

3.2.3 Test of randomness in case of one-dimensional auto-logistic model:

 

In the class of Markov random fields given by (3.2.4) the sub-
class of Markov random fields with no spatial interaction is characterized
by B = 0. Consequently testing for randomness amounts to testing
8 = 0 against 8 f 0, indicating a spatial interaction.

In terms of notations we are interested in testing

H : 8 = 0 (i.e. no spatial interaction)

H : s f 0 (i.e. there is a spatial interaction)

60

Under HO (3.2.4) gives
' ozX.i

 

P [x.(N-NJ = e
01_ 1+eoz

which is independent of the N-N values. Thus, under H0: {Xiz i > 1}
is a sequence of i.i.d. binary random variables. Consequently the joint

distribution of (X],...,Xn) under H0 is

(X ,... - ——————————

implies
n (1

Further under Ha (3.2.4) gives

expEaXi + 3x1.(xM + xi+])]
l + expEa + 8(Xi-l + Xi+l)]

 

PatxilN-N] =

Thus, the joint distribution of (X "Xn) under Ha is

1,”

 

f (x ,...,x ) = 3 eaxl + Bxl(xl" + x1+1)
a 1 ” 1=1 l + e“ + 8(Xi-l + Xi+l)
implies
9" fa(xl’ ’Xn) = a 2 xi + B Z Xi(Xi-l + Xi+l)
_ En». [l + ea + BUM + XM)]
1

Let Ln(e) denote the log-likelihood function so that we have:

 

 

 

61

n
(3.2.8) L (0) = a Z x. - ngh (l + e“)
n i=1 1
n
Ln(8) = 1;){axl + BX1(X1 1 + Xi+])
o: + 8(X. + X. )}
-2/n[1+ e 1-1 1+1J
1.e.,
n
(3.2.9) Ln(s) = 2 21(8) (say)
i=1
where
a+ 8(X. +X. )
I - 1‘1 1+1
(3.2.9) 2i(s) — aXi + 8X1.(X1._1 + Xi+1)1h{l + e }.

The likelihood equation is:

a n a n
0 = SE-Ln(8) = j 55-21(8) = ( 11(8)
n . n 82 HI *
= ; {zi(0) + Bzi (0) + 3r-21 (8 1}
* .
where [a l < Isl. (By Taylor's series expansion of £%(B) about
8 = 0)
or
_ l._£L. _ l n . B n n
0 ‘ n as Ln(B) ' 6'1 (1 (0) + 5' 1;] £1 (0)
2 n *
+ %fi_ 2 1 HI(B )

(where (5*) < (Bl)

62

 

 

 

 

2
_ .fi.
(3.2.10) 0 - B0 + B1 3 + 82 2 .
where
g 1 E (0)
=._ 1'
O n i=1 1
(1
_ 1 n e (Xi-l T Xi+1)
_ _-i§l [X1(X1._1 + Xi+l) - 1 + ea 1,
B1 = %' E £1"(0)
i=1
2a
(3 2 11) - 1 n -(Xi'1 + X1+1) e
. . ’3': 1: (12 :19
1=1 (l + e )
n 'k *
32 = 1.21 21mm 1 <181< 1811
1:
* +s*(x +X )
a . .
1 E ( 1_, + xi+])3e“+5 (Xi-1 + Xi+l)[e "‘ ‘+1 -11
= —- {
n .3 *
L 1 1 [1+ ea+8 (Xi-1 + Xi+1)]3

 

Let én be the maximum likelihood estimate of 8. Then using the classical

theory for i.i.d. random variables
8 -4—4 0 under HO'

So that V e > O, V w E 9 and sufficiently large n there exists

a constant M < m such that
Pot/Elénl > M] g 6

This implies that it is enough to consider the behaviour of the terms

B B and B on the set [(énl f M/Vfij. (*).

0’ 1 2

63
Using 6" (3.2.10) gives
(3.2.12) 0 = B + B

* * A
(where B in the definition of B is 3 ls l < Ian] - being justified

2
for each m E a)

Claim 1 under H0 B0 converges to zero in probability, as n + m.

3599:; Under HO: {Xiz i 3 l} is a sequence of i.i.d. binary random
variables. Thus by theorem B-l of Appendix B the sequence

{X]: i g l} is stationary and ergodic, and these random variables
clearly have finite second moments. From theorem B-3 of Appendix B,

it therefore follows that under H0:

 

a
B mzmxxw e (EX +EX1)

 

0 1 2 1+90: 1
ea

= 2 EX] EX2 - 2 EX1
l+ea

(by independence of the Xi‘s)

 

 

01
= 2(EX )2 - 2 e EX (a- X.'s are id. dist.)
1 1+ a 1 l
e
ea
= Zero, since EX1 = under HO.
l+ea

Claim 2: Under H0 B1 §4§4 - k2 where k2 3 0

Proof: From (3.2.11)

64

2 a
(Xi-l + Xi+l) e
l (1+ea)2

 

again by using theorem B-3 of Appendix B we have under HO

 

a.s1 ea 2 2
31 , - (1+ea)2 [EX1 + EX1 + 2E(X]X2)J
a
_ e 2
_ - (1+ea)2 [EX1 + EX1 + 2(EX1) J

(i.i.d. property of the Xi's and the fact X? = X1 for Xi binary)

a
- e

--——-7? 2 EX (1 + EX )

(Heel) 1 1

2EX](l + EX])ea
> 0

(1+e°‘12 '

 

-k2 where k =

Claim 3: Under H0 as n + m, B is asymptotically bounded.

2

Proof: From (3.2.11) we have

*

 

 

 

*
3 a+8 (X. + x. ) a+8 (X. + x. )
B = l. E {(X1._1 + Xi+l) e 1-1 1+1 [e 1-1 1+1
2 ” i=1 a+8;(X + x ) 3
[1 + e 1-1 i+1 1
* A
(where B is such that |B*I < IBnl)
* *
3 a+8 (X. + X. ) a+8 (X. + X- 1
(X. + X. ) 1~l 1+1 1-l 1+1
f qu 1-1 1+1 2 [e -1]
15‘5” [1 + e°‘+8 (Xi-1 + X1+1)13
+2* +2*
< 23 ea 8 [ea 8 ~11
' (1+e°‘13
(since xi 5 are binary m1n(Xi_] + Xi+l) = 0 and max(Xi_] + Xi+l) = 2)

 

which is bounded as n + w.

Consequently it follows that as n + m under H0 82 is asymptotically

bounded. (3.2.12)can be rewritten as

 

 

,. .4730
6—8” = g
- B - B _r_1_
1 2 2
1.e.
2
. Jn‘BO/k
(3.2.13) Men = ,
B1 3
.. .. B .2...
If 2 2k2

using claims 2 and 3 and the fact that en E 0 under ”0’ it follows
that the denominator of (3.2.13) converges to l in probability as
n + m.

Thus the asymptotic distribution of /n én depends upon the asymptotic
distribution of #5 BO'

Asymptotic Distribution of /fi 80 under H0:

Using (3.2.11) we have

 

__l_ n e0‘(X- + X~ 1
n 1" 1+ ea
1.e.
/’ 1 E (
n B = ——- 5 say)
0 J5 1=1 1

(I
e (X. + X. )
1+1) - 1"] 1+] :1>1.
l + ea ‘

'where ti = X.(X + X

'l

 

i-l

Under H0: {Xiz i 3 l} is a sequence of iai.de binary random variables.

Therefore we have

E 5i = 0 V i

and that ti, gi+3,... are independent so that {€i: i 3 l} is a

2-dependent stationary process with mean zero. Consequently, by theorem

'1
3-2 of Appendix B it follows that Jfi'ao =-l- Z a, has a limiting
JR i=1

normal distribution with mean zero and variance

2e a
2 _ 2 = e (3 + e 1
°t ‘ E a1 + 25 g152 + 25 51‘53 (1 + ea)4

 

Hence it follows from (3.2.13) that /H én has a limiting normal

distribution with mean zero and variance oE/k4.

Asymptotic distribution of the log-likelihood ratio An(én,0) =

Ln(sn) - Ln(0), under H0:

 

Expanding Ln(0) by Taylor's deries expansion about én(v w 6 9) we 1

get

* A
where l8 | < [an]

Hence: An(5n,0) = Ln(en) - Ln(0)
_ALI" énz 11*
- 8n n (8”) ' —2—-Ln (B )

 

67

i.e., . 2 . 2
A 0 _ A I “ B" 11 * 11 8n 11
(3.2.14) An(sn, ) - BnLn (an) - 2 [Ln (3 ) - Ln (0)] - L (0)

 

 

Since Ln'(én) = o and under H0 én éeée-o it follows that the lst

term on the r.h.s. of (3.2.14) converges to zero in probability under Ho’

 

 

~ 2
8” ll
3rd term - - 2 Ln (0)
= (“-71 “T ( - fiLn"(0))
og/k 2k
1411?; 1)
From the preceding pages ———fl?—- + N(0,l) and by claim 2
o /k
E
- %—Ln"(0) ELEA k2 under H0. Consequently the 3rd term on the r.h.s.
of (3.2.14) converges in distribution to (N(0,l))2 ogz/Zkz.
A 2
_ 8n 11 II *
2nd term - 2 [Ln (0) - Ln (8 )3

"(8*13

2 l u
((fiEn) Efi'an (0) - Ln

—m 2 6n
(Vth) '5—

By following an argument analogous to that of Borwankar et at [1971]
it follows that IE”) +-0 as n + m (For a proof see Appendix B).
Hence under H0 the 2nd term also converges to zero in probability.
Thus: under H0, An(én,0), the log-likelihood ratio,follows a central

chi-squared distribution, and the associated degrees of freedom is 1.

3.2.4 Test of randomness for a homogeneous lst-order (N—N) isotropic

 

auto-logistic model:

 

Following remark 3.2.3, a homogeneous lst-order isotropic auto-

logistic model is given by

68

 

 

+ . . . .
eta B t133x1,3
PLXile-Nl - o+sti.
1+e 3
where t. . = x. . + x. . + x. . + x. .
1,j 1-l,j 1+l,j i,j-l 1,j+l.

The hypothesis of interest are:

H o

0. B = 0 (no spatial interaction)

Ha: 8 f 0 (there is a spatial interaction)

The procedure to get the asymptotic distribution of the log=likelihood

 

ratio An(8n,0) = Ln(3n) - Ln(0), (3” being the maximum likelihood
estimator of B and Ln(B), the log-likelihood function) under H0 is
analogous to the one given in section 3.2.3 for the one-dimensional

case. So we only sketch the main lines of proof.

a+8t1j
= . . + . . - 3
Ln(s) izj {ox},J 8t1,in;j mil + e 11
= t. -(8)
131' "3
0+8 ti j
Likelihood equation is:
- ii. _ EL
0 - as Ln(8) _Z. 38 21 J(B)
1,3
or
1 8 1 3
o = ——-L (B) = ——- Z ——-2 (8)
32'33 n n2 1,3 as 1.3

If B” denotes the maximum likelihood estimator of s we have then

69

 

 

l ' A
0 = — Z 2 .(8)
n2 i,j i,j n
- 1 | + A ll énz III *
- “—2 iijfliJm) an 21.,J.(0)+ —2—- 2mm )1
* x
(where (8 I < |8n| being justified V w 6 o.)
1.e.
A 2
. 8n
(3.2.15) 0 - BO + 81 8n + 82 -2—
where
(' BO =-l§ Z 2; .(0)
n i,j ’3
a t.
1 e 1.3
=-— Z (t. .X. . - a }
n2 i,j 1,3 1,3 1 + e
(3.2.16) 1
B1 = --lf .2. (t? . ea/(l + ea)2}
n i,j ’
a+8*t a 8*t
t3 e 1’JEe 1’3 ~11
B = l— X{ 1"] }
2 2 . . *
(1+ e ’ )

. = , . + . . . . . . * ‘
where tl,J X1_1,J x1+l,J + x1,j-l + X1,J+1 and IB 1 < leni-

As dealt with in section 3.2.3 we have under H0:

a.s.
B0 -——+-O as n + w

and B is asymptotically bounded as n + m.

2

7O

Rewriting (3.2.15) we get:

g=_§_0_,__
n 8n
"31"2‘32
implies
. nBO/k§
ns=————.
n B1 5
_ __nB

As before, the denominator converges to launder H0 so that the asymptotic

distribution of nén depends upon that of n3

0.
Further:
1 Z eatlj
nB = —- (t. .x. . - ——-—J——- }
0 n i,j 1.31.3 1+eo:
.1. a
n 1.3 1.3
where
on
e t. .
gi,j _ _ 19J

— t. .x. .
1931M] 1+ea

= . . . . + . . . .
[xl-1,JX1,J x1+1,Jxl.J

+ Xi.i-I"i.i + Xi.i+1xi.iJ

O.

 

1 + ea [Xi-1.3 + Xi+1.j + Xi,j-1 + Xian]

is under H0 a 4~dependent stationary process with mean zero. An

71

extension of theorem B-3 of appendix implies that n80 has a limiting

. . . . . 2
normal distribut1on w1th mean zero and some variance 0g and consequently

nén is asymptotically normal with.mean zero and variance oé/k?.

Finally following exact similar lines of proof as in §ection
3.2.3, it follows that An(8n,0) = Ln(8n) - Ln(0) has under H0 a
central chi-square distribution and the associated degrees of freedom

is 3.

Remark 3.2.5: [Besag ~1974] considers the above problem and states

 

the chi-square behaviour without giving proper justification. Also

it was seen that the coding method of estimation as suggested by Besag
is not necessary in establishing the chi-square behaviour. For further
details on coding method of estimation one is referred to Besag's

paper.

CHAPTER IV
POWER UNDER A SPECIFIC ALTERNATIVE

In Chapter III, we looked at some of the estimators of A,
the intensity of the spatial process, and the asymptotic properties
of these estimators. It was also seen that in a particular subclass

of auto-binary schemes the test statistic for testing randomness (no

 

spatial interaction) has a chi-square behaviour under the null hypothesis
of complete randomness. However, the value of a test statistic is
increased if one can discuss the power of the test to detect departure
from'randomness.

In this chapter, we try to look at the distribution of the test
statistic under a specific alternative. We shall confine our attention
to the subclass of auto-binary schemes where the parameter describing
the spatial interaction has a specified form under the alternative.

The problem has been approached using ideas on contiguity (discussed

in Section 4.1). It has been shown that in our particular formulation
the classical techniques of contiguity fail. Even though the measures
are shown to be contiguous using the basic principles of contiguity, the
distribution of the test statistic under the alternative does not seem
to have an easy tractable form. A conceivable conjecture is that it

depends upon a non-central chi—square distribution (Section 4.2).

72

73

4.1 Contiguity and its characterizations

Our exposition follows Roussas [1972].

The concept of contiguity was first introduced by Professor Le Cam
as a measure of 'nearness' of sequences of probability measures. It
plays an important role in the study of asymptotic theory in deriving

asymptotic properties of the tests under much less assumptions.

Definition 4.1.1: Let {(X,An)} be a sequence of measurable spaces

 

and {Pn} and {0”} be two sequences of probability measures defined
on (X,An). The sequence {0"} is said to be contiguous with respect

to {Pn} if and only 1f V An 6 An

(4.1.1.) [Pn(An) + 0 implies Qn(An) + 0]

In such a case, we also say that the densities qn are contiguous
to the densities pn where pn and qn are respectively the densities

of Pn and Qn with respect to some dominating o-finite measure.

Remark 4.1.1: Contiguity implies that any sequence of random variables

 

converging to zero in Pn-probability converges to zero in Qn-probability.

Definition 4.1.2: The sequence Pn of probability measures is said

 

to be relatively compact if for every subsequence {n'} of In} there

exists a further subsequence {n"} of {n'} such that Pn converges

weakly to a probability measure P (definition 1.4.1).

Alternative characterizations of contiguity;

 

Let pn and qn be the densities respectively of Pn and Qn

with respect to some dominating o-finite measure.

74

Define the log-likelihood ratio An as follows:

109(qnlpn) on {pnqn > 0}
(4.1.2) A =

arbitrary otherwise.

For each determination of An defined by (4.1.2) let

r-n
ll

LEAnIPn]
(4.1.3) ‘

L _ LEAnIQn]

Therorem 4.1.1: (Roussas [1972] PP 11-14)

 

The following three statements are equivalent:
(i) {Pn} and {0"} are contiguous;
(ii) {Ln} and {Ln'} are relatively compact (for each determination
of An);
(iii) {Ln} is relatively compact and if F is the limiting distribution

function and X ~ F then

f ex dF(x) = 1.

 

Remarks 4.1.2: (i) Ll-norm convergence implies contiguity
1-e-a "PH " an'l '+ 0
implies {Pn} and {0“} are contiguous where “P“ - Qn“l is defined

by
HPn - QnH] = 2 sxp IPn(A) - Qn(A)l.

(For a proof see Roussas [1972], p. 9)

75

(ii) converse of (i) is not true

i.e., contiguity does not imply Ll-norm convergence.
'.1)

where “n + u and “n + u' and u,u'(p # u'g arezboth finite. It
u - u'
. _ . _ n n
can ea51ly be seen that An — (on H”) X + -——§—————-

Example: Let (X,An) = (R,B). Take Pn a N(un,l) and Qn = N(un

 

 

so that
‘(“h - ”11)2 . 2
Ln - L(An|Pn) - NE 2 , (un - un) J
. (u; - un12 ' 2
Ln - L(Alen) - NE 2 . (“n - un) 1-

clearly {Ln} and {La} are relatively compact and consequently by

theorem 4.1.1 {Pn} and {0”} are contiguous. However “Pn - an1 h 0.

(iii) Contiguity does not imply mutual absolute continuity:

Example: Let (X,An) = (R,B)
Take Pn = U(- %31), the uniform measure on (- %3 l)

and On = U(O,1 + %) the uniform measure on (0, l + %)

dPn(x)

 

 

Further pn(x) = do = 527- , - %—< x < 1
d0 (X)
and qn(x) = d2 = 521' D < x < l + %-.

v being the Lebesgue measure.
= .__2__ 00 ‘ '
Then “Pn - Qnufi n+1 + D as n + so that by (11) above {Pn}

and {0“} are contiguous.

76

However {Pn} and {0”} are clearly not mutually absolutely continuous.

(iv) Mutually absolutely continuous need not imply contiguity:
Example: Let (X,An) = (2,8)

Take Pn e N(un,1) Qn 5 N(ua,1) where u" + -w, and pa + + w Then
{Pn} and {0”} are mutually absolutely continuous but not contiguous
for taking An = (on - l, “n + l) we see that 'Pn(An) = .68 k 0 but
ohm") -> o.

(v) The above (i) - (iv) imply that contiguity is weaker than the
Ll-norm convergence and is distinct from the mutual absolute continuity

notion.

4.1.1 Some results following from contiguity:

 

In this section we shall discuss some important consequences of

the notion of contiguity and their use in statistical applications.

Lemma 4.1.1: [Roussas - 1972, p. 15]

 

Any one of the three equivalent statements of theorem 4.1.1

implies that Pn(Bn) + l and Qn(Bn) + l where 8n = {pnqn > 0}.

Remark 4.1.3: Lemma 4.1.1 says that contiguous measures Pn and Qn

 

eventually rest on Bn i.e. eventually are mutually absolutely continuous.
Consequently if {Pn} and {0“} are contiguous, one may assume without
loss of generality that Pn and Qn are mutually absolutely continuous
for all sufficiently large n. Under this assumption the log-likelihood

ratio

An = log(dQn/dPn)

77

is well-defined a.s. (Pn), (Q ).

n
Define
r
* -
Ln - L[(An,Tn)|PnJ
(4.1.4) (
l* _
L” — L[(An.Tn)|Qn1

 

where {Tn} is a sequence of k-dimensional random vectors such that

Tn lS An-measurable.

Theorem 4.1.2: [Roussas - 1972, p. 34]

 

*
Suppose {Pn} and {0"} are contiguous and Ln and L6*

*
are defined by (4.1.4). Further assume that Ln e L*, a probability

measure.
1*

Then Ln =1L'* where
dL'*
-—:r- = exn(x)
dL

 

Corollary 4.1.1: [Roussas - 1972, p. 35]
If L(An|Pn) a N(u,02) then u = -e o .

Corollary 4.1.2: If L(An|Pn) = N(-% 02,02) then

 

L(Anlon) =~ Nov. 023:2).

4.1.2 Interpretation of contiguity in simple versus simple hypotheses

testing:

Consider a sequence {pn,qn} of simple hypothesis pn against

 

78

simple alternatives qn defined on measurable spaces (Xn’An) re-
spectively.

According to Neyman-Pearson lemma, for any event An in An’
there exists a function on and an integer kn: 0 < kn < m such

 

that

f

1 .

1 1f qn > knpn

(4.1.5) on = A dog a: 1) 1f on = knpn

L0 1f qn < knpn
and that

Pn(An) = f on dPn
and

Qn(An) f f ¢n dQn
Thus contiguity (definition 4.1.1) will follow if we can show that
[j on dPn + 0] implies [f on dQn + 01

for critical functions of the type (4.1.5).

Remark 4.1.4: Using the equivalence of the statements (i) and (iii)

 

of theorem 4.1.1, it can easily be observed that if A", the log-likeli-
hood ratio, is asymptotically normal (-% 02,02) under Pn’ then the
densities qn and pn are contiguous (For a proof see Hajek & Sidek

1967, pp 203-205).

79

Suppose we have an experiment {(Xn,An, ®): (3 S Rk} and are

interested in testing

(4.1.6)

where on 4 0 and {hn} is a bounded sequence in Rk such that

hn + h e Rk.
The following proposition says that under certain conditions the
probability measures P and P are contiguous.
n,90 n,8n

Prop051t1on 4.1.1: Cons1der 60 and 6n = 60 + on hn w1th on + D

k).

 

and {hn} bounded and hn + h. (hn,h E R
Suppose there exists an An-measurable function Tn(e) and a positive

definite covariance matrix P such that

e

a) An(en,eo) - h Tn(eo) + k h reoh + 0 (Pn’eo)
(4.1.7)

b) LETn(eO)|Pn,e J + N(0, To )

0 0
then Pn.e0 and Pn’en are cont1guous where
dee"
An(8n,eo) = 109 m—

0
Proof: Assumption (4.1.7) implies that

LEA (e ,e )|P , J + N(-% h' r h, h' P h)
n n 0 n 60 60 60

80

By remark (4.1.1) it then follows that Pn’ and Pn,e are contiguous.

6D n

Remark 4.1.5: (i) Under the conditions of proposition 4.1.1, it follows

 

that

L(An(en,e P , ) + N(% h To h, h r h)

) l
0 n on 0 60

(ii) Condition (4.1.7) is known as the locally asymptotically normal

(LAN) conditions.

Example 4.1.1:

 

 

Take Pn’ E N(60,1)

DJ
3
D.
‘0
ll

N(en,l) w1th on
One can easily see that
An(en’90) = (en-90) X + 2

Take Tn(e) = -1—

/K 1

III
-—'
O

(X.

1 ~ 9) and r
1

"M:

6

It can be verified that 2.h.s of (4.1.7) (a) is identically zero and
the 2.h.s. of (4.1.7) (b) is exactly normal. 1

Thus, by proposition 4.1.1 the two probability measures Pn’e
O

-L
and Pn,e where on = 90 + n 2h (h bounded) are contiguous.
n

Remark 4.1.5: We describe below a typical problem that comes up in statis-

 

tical applications.
Suppose {Pn} and {0"} are two sequences of probability measures

and that On depends on an h in a specified way. Further suppose

81

that assumption (4.1.7) holds. The problem is that of finding the
asymptotic distribution of Tn under Qn'

What one does then is to first derive the asymptotic destribution
of (An’Tn) under Pn and use the contiguity of Pn and Qn to get
the asymptotic distribution of (An’Tn) under Qn’ From this, the

desired asymptotic distribution of Tn under Qn will follow.

4.2 Power of the test:

 

We are back in the particular subclass of auto-binary spatial
schemes. In Section 3.2, we discussed the asymptotic behaviour of the
log-likelihood ratio under the null hypothesis of complete randomness.
In this section, we attempt to discuss the asymptotic distribution of

the log-likelihood ratio under a specific alternative, Ha n: B = = n h

(h being bounded). Thus we have a set of hypothesis

-y
H : 3 = e = n 2h (h bounded)

So that B + O as n + m. Without any loss of generality one may

n
take h a 1 so that the hypotheses are

(4.2.1) <

1
-e

H : B = 8 = n + O as n + w

 

where B is the parameter describing the spatial interaction in the
auto-binary spatial models (both one-dimensional model of Section 3.2.2

and two-dimensional isotropic model of Section 3.2.4).

82

Following remark 4.1.5, in order to determine the asymptotic
distribution under the alternative, knowing the asymptotic distribution
under the null the first step is to show the contiguity of measures under
H0 and Ha,n' Also, proposition 4.1.1 says that contiguity will follow
if the LAN conditions given by (4.1.7) can be verified for our model.
However, unfortunately the LAN conditions are not satisfied in our

formulation (see Appendix C) so that the classical techniques of contiguity

fail.

4.2.1 Contiguity of measures under the hypotheses defined by (4.2.1)

Since the classical techniques of contiguity fail in our formulation
we approach the problem through basic principles of contiguity (de-
finition 4.1.1). Let (X,An) be a measurable space and {PB} be a
sequence of probability measures defined on (X,An). Let An 6 An.

Then we would like to show that

[P0(An) + 0] implies [P (An) + 0]

8n

One-dimensional model:

 

Let An 6 An and P0(An) + 0 would l1ke to show that

PBn(An)‘* O as n "*°°

The conditional probability distribution of an auto-binary spatial scheme
is given by

[a + B(X X

X .
e 1

i-l + i+1)J
“ + 8(Xi-l + Xi+11

dPB
3? [X1 IN-N] =

 

1 + e

83

(n being some dominating o-finite measure)

Consequently it follows that

dPBHEXilN-N] = e n 1+1 1(] + ea)
dP [X.TN-N] a + B (X._ + X. l)
0 1 [1 + e n 1 1 1+1 1

 

 

(4.2.2)

-4
where Xi's take values 0 or 1 and 8n = 2

converges to zero
as n + w.
Since xi 5 take only 0-1 values, X1(Xi_] + Xi+l) 15 at most 2.

Further, the likelihood ratio is given by

dPBnEXilN-N]

 

n
1n(8na0) = E

 

 

 

 

 

i 1 3p [XilN‘NJ
1.e.,
( ) ( ) n eBn Xi(Xi-l I xi+l) ( a)
4.2.3 1n8.0=11{ 1+e}
n .= a+BTX. +X.)
1 l 1 + e n 1+1
Now,
dPBn
P (A ) = f dP
8n n A dP0 0
n
f dPBn
' X —-*dP
An dPO 0
f n eBnX1(Xi-1+Xi+l) ( a)
= x H l + e dP
A _ a+B(X +X.) 0
n 1-1 1 + e n 1+1
n 628
f( (1 + ea )f XA H dP0
An 1 1 1+e
2
= (l + ea )f xA (e 8” )n dP
a 0

84

 

 

 

 

 

1 e 28" n
OI e
(4.2.4) P§n(An) 5 (l + e ) 1+ a P0(An)
e
28n n
. e .»
Claim (for B = n 2) + 0 as n + w
l + ea n
for any a > 0
_L
e28n n e2n 2n
Now <
l + ea ' on
2n%
=_._1____
eon - 2n%

+ 0 as n + m for a > 0.

Since P0(An) + 0 (by hypothesis), therefore it follows from (4.2.4)

that

thus establishing the contiguity of measures under H and Ha

0 ,n

defined by (4.2.1).

Remark 4.2.1: In case of a 2-dimensional isotropic: auto-binary model

 

also, the lines of proof (as given above) give the contiguity of

measures under HO and Ha n’ defined by (4.2.1).

3

85

Remark 4.2.2: We have not yet managed to establish the asymptotic

 

distribution of the log-likelihood under the alternative. A good con-
jecture is that it is a non-central chi-square distribution, which we

hope to establish in the near future.

 

APPENDIX A

As in Section 1.1, let S denote the d-dimensional Euclidean
space and 8(5) the family of borel subsets of S. Let 95 be the
family of point processes {E(A,m): A 6 8(5)} that satisfy conditions
a), b) and c) of definition 1.1.1.

Let D be a countable subfamily of 8(5) that generates

8(3) and F

0 denote the field generated by elements of D.

 

Definition A-l: Finite dimensional distributions generated by a point

 

process g(A,w) are the distributions

Pr0b{€(A]aw) = r -: €(A w) = rm}

1"“ m,
where A1,...,Am 6 8(5) and r],...,rm are non-negative integers.

Let

q(A],...,Am; r1,...,rm) = Prob{g(A],w) = r],...,g(Am,w) = rm}

denote the finite dimensional distributions generated by a point process

€(Aaw)'

Definition A-2: Let A1,...,Am be sets in F0' Then a set in as

 

determined by conditions on 5(A],w),...,g(Am,m) is called a cylinder

set in as.

86

87

i.e. a set of point processes determined by its finite dimensional
distributions is a cylinder set in 95.

Let C be the family of cylinder sets and C* be the borel
extension of C.

Let 9' be the family of non-negative integer valued set functions
€(A,w) for A in FD.

Let C(Q') be the family of cylinder sets in 9' and C*(o')
its borel extension.

Similarly define Q” to be the family of those set functions
;n 9‘ that satisfy b(i) for Ai E FD namely €(igl Ai’w) =
Z €(Ai’w) a.e. for A1,...,Am disjoint sets in FD and Q"' to be
lhe family of those set functions of 9' that satisfy both b(i) and

b(ii) for A1 6 F .

0
namely
m m
5(U A.,w) = 2 E(A.,w), A. 6 F and disjoint
1 1 1 1 1 D
and g(An,w) + O for A]:: A2:: ... 1n FD such that n An = 4

n

'k
Similar to C(9') and C (9') we define C(Q"), C*(n”), C(Q”') and C*(o”').

Converse implication of theorem l.l.l:

Given a point process {5(A,w): A 6 8(5)} that satisfies
conditions a), b) and c) of definition l.l.l we need to show that g
is an (M+(S), M(S)) - valued random variable.

i.e., Given the finite dimensional distributions q(A],...,Am; r],...,rm)

88

there existsaiunique probability measure Q that is determined by these

q-functions.

Conditions A-l: (n is any positive integer; A1,...,Am are sets in

 

8(8) and r1,...,rm are non-negative integers):
(i) q(A],...,Am; r1,...,rm) is a probability distriubtion on m-tuples

of non-negative integers r],...,rm.

A150: Q(A]:A2; r13r2) = Q(A2,A]; r29r1)°

(ii) The functions q are "consistent" i.e. for example

 

X Q(A],A2; rI’rZ) = q(A],r]).

(iii) If A],A2,...',Am are d1sgoint sets, and A = A1 U A2 U ... U Am
then q(A,A],...,Am; r1,r1,...,rm) = 0 unless r = r1 + ... + rm
and
q(A,A],...,Am; r1 +...+ rm,r],...,rm) = q(A],...,Am; r],...,rm)
m
i.e. Prob{§(A,m) = Z €(Ai’w)} = 1.
1

(Corresponds to b(i) of definition l.l.l)

(iv) If A,:3 A223 ... such that n An = p, then
n

11m q(An; 0) = 1

n

i.e. Prob {€(An,w) + O} = 1

(Suggested by condition b(ii) of definition l.l.l).

89

Remark A-l: It is sometimes convenient to be able to define the q-

 

functions by prescribing their values only when the sets A],A2,...,Am

)

are disjoint. Suppose, we have a set of functions qO(A],...,Am; r1,...,rm
defined whenever the sets A1,...,Am are disjoint so that qo can be
regarded as defining the joint distribution of the random variables
g(A],w),...,g(Am,w) whenever the sets Ai's are disjoint. At this
point it is not clear how condition A-l(iii) is defined.

Suppose that condition A~l(i) and (ii) are satisfied for dis-
joint Ai's and also that condition A-1(iv) is satisfied. Suppose
further that if A],A2,...,Am are disjoint sets, each being a union
of a finite number of disjoint sets i.e., Ai = A1] U A1.2 U ... then
the joint distrubition of g(A],-),...,5(Am,o) is the same as that of

{em .,.),...,2 eat ...).
.1 ‘3 .1' "'3

For example, if A, B and C are disjoint sets then we require

(A-l-l) qO(A,B U C; r].r2) = Z qO(A,B,C; r],r3,r4).

r3+r4=r2

With this definition the functions qO can be extended in a unique
manner to functions q that agree with qo when the sets A1 are
disjoint. For completion of the proof, interested readers are referred

to Harris [1963].

 

Theorem A-l: Let q(A],...,Am; r],...,rm) be given defined whenever
A1,...,Am 6 F0 and satisfying conditions A-1(i-iv) when the sets in-
volved are in F0. Then there exists a unique probability measure Q

*
on C such that

90

Q{E(A]9w) = r1sA-os€(Amaw) = rm} = q(A],...,Am,r1,...,rm),r],...,rm = 0,1,...

whenever the A's 6 F0.

Proof: The fundamental theorem of Kolmogorov [1956, p. 29] implies that

*
the q-functions determine a unique probability measure Q1 on C (o').

it
Claim: 9" e C (9') and 01(9") = 1

Now 9” consists of those E(A,w) of 9' that satisfy

m m
E( U Aiaw) = Z E(Aiaw)
1=1 1

 

where Al""’Am 6 F0 and are disjoint. By condition A-1(iii) each
such relation has 01-measure l and there are only denumerably many of
them.

Thus it follows that
*
o" e C (o') and 01(9") = 1.

Further, now a cylinder set 8 in o“ is the intersection of Q"

with a cylinder set in 9' so that
*
B = 81 0 Q" where B] E C (9')

Thus, having a probability measure 01 on C*(Q'), we can define a
*
probability measure 0' on C (9“) by putting: Q'(B) = 01(81) for
*
B E C (9“). Conversely if Q' is a probability measure on C*(Q")

then we can define a probability measure Q1 on C*(Q') by putting

01(81) = Q'(B1 n o“) for B1 6 C*(o')

91

Also, the measure Q' is unique since if not, there would be two different
Q.I measures on C*(n') contradicting the uniqueness of Kolmogorov
theorem.

Thus, we have a unique probability measure 0' on C*(n") and
Q'(n") = 1. If now a e n“ is such that it does not satisfy b(ii)

for A1 6 F0 namely

A13A23... w1th nAn=¢

g(An,w) +. 0 for A1,... 6 F0

 

then the set nO of all such 5's has a measurable subset whose Q1
measure is zero (follows from condition A-l(iv). This implies that,
01*(n0) = inner measure of nO = 0
clearly we have n"' = n" - n0 so that Q:(n”') = 1.
(0: denotes the outer probability measure)

As discussed before, having a unique probability measure on
C*(n”) we can have a unique probability measure Q on C*(n”') by
putting Q(B n n"') = Q'(B): B E C*(n"). Consequently it follows that
there exists a unique probability measure Q determined by the given

q-functions.

Remark A-l: The easy implication of theorem l.l.l says that every Q

 

on (M+(S), M(S)) can be regarded as a P for some point process

6
g, and the converse implication as proved here shows that given a point
process 5 there exists a unique probability measure Q determined

by the finite dimensional distributions of E(A,w) given by the q-functions.

APPENDIX B

Theorem 8-1: A sequence {Xiz i 3 l} of independent identically dis-

 

tributied random variables is strictly stationary and ergodic.

Proof: a) To show that {Xiz i > 1} is stationary we need to show

that (X1,X2,...) has the same joint distribution as (X2,X3,...)
This follows clearly from the identically distributed property of the
Xi's.
b) Need to establish the ergodicity of the stationary sequence
{Xiz i 3 1}. For each m = (x1,x2,...,xn,...) define a new process
by
Y1(w) = Xi the ith~coordinate the {Yi: i 3 1} defines a

stochastic process known as the co-ordinate representation of
{Xiz i 3 1} clearly both {Xiz i 3 l} and {Yi: i 3 l} have the
same distribution. Define a transformation S by

S(x],x2,...,xn,...) = (x2,x3,...,xn+1,...) (S is the so-called
shift transformation).

In what follows have a probability space (Rm,Cm,P) where P

is defined by
P(B) = P[(x1,x2,...) 6 B].

(i) S is measurable for:
Let C be a measurable finite dimensional product cylinder then S'IC

is also such a cylinder set. These cylinder sets generate Cco implies

92

93
that S is Cm-measurable
(ii) fi(s'1c) = P(C) v c 6 cm

For: Let C E Cm.

6(5'1c) PEUISw e C]

P[x1,x2,...) [ (x2,x3,...) E C]
= P[(x2,x3,...) e C]

= P[(x],x2,...) e C] (by stat.)
= P(C)

(i) and (ii) together imply that S is measure-preseving.

(iii) Need to show that P(A) = 0 or 1 for invariant events A.

Let A be an invariant event and let Gn = (Xm: m 3 n).
Let G = n Gn’

n
A invariant and S measure-preserving implies that S']A = A
implies [w|(X2,X3,...) 6 A] = [w|(X],X2,...) E A].

Continuing this gives:
[m|(xk,xk+1,...) e A] = [wl(x],x2,...) e A] V k 3 1

implies A e Gk V k 3 1 implies A e G, the tail o-field. Since the

Xi's are independent, by Kolmogorov D-l law it follows that

P(A) = 0 or 1

(i), (ii), and (iii) imply (b)

94

Theorem B-2: [Anderson - 1971, p. 427]

 

Let y1,y2,... be a stationary stochastic process such that

for every integer n and integers t1,...,tn (0 < t1 < ... < tn)

yt ,...,yt is distributed independently of'.y1,... and

.y _ _
1 n t1 m 1

—4~4—4

If Eyt = 0 and Eyi < m then yt has a limiting

1
y 1 I , o o o o 7"—
tn+ 1 f1

normal distribution with mean zero and variance

2
Ey1 + 2E y1y2 + .... + 2E y1ym+1

Theorem 8-3: [Hannan - 1970, p. 203]

 

If {x(n)} is stationary and ergodic and E{|xj(n)|} < m
N

then 1im 1- Z x(n) = E{x(n)} a.s.
N+m N 1
Also if E{(xj(n)2} < m
then
1 N
1im 'N X x(m)x(m+2) = E{x(m)x(m+2)}
N+m 1

Remark B-l: Proof of the fact that

 

— 1 ll 0 II *
6n - fi-[Ln ( ) - Ln (8 )1 converges

a.s. to zero as n + m.

Proof: Let 6 > 0 be given.

Let the parameter space 8 be an open interval of the real line.

Assume that V B E 8 there corresponds an n(8) > 0 such that

95

32h(X0,X1,X2; 8*) . *
(*) Egtsupil 2 I: e e 8. 18-8 1 < n(8)1]
e as

 

is finite where EB denotes the expectation when 8 is the true para-
meter.
Let U0 = {8: [8| 5 no = n(0)} be a neighbourhood of B = 0

Choose a o: 0 < o < "0'

 

 

 

 

Now
a+B(X0 + X2)
h(X0,X1,X2; s) = [a + 3(XO + X2)1 X11h £1 + e 1
2
ah(x ,X ,X ; B) a h(X ,X ,X ; 8)
clearly 0 a] 2 and 0 I 2
8 2
38
exist and are continuous
2 .
Further, the continuity of a h 58) implies the lower semi-continuity
as
of
82h(X0.X1.X2; s) 32h(X0,X1,X2; 0)
sule 2 - 2 I: IS] < 6}
38 88

From assumption (*) we have for o > no

32h( - 0) 32h( . 8*)
E0{sup| é - g 13
as as

 

 

*
ls l < 6}<:e
2
Let us choose such a 5. Now under H0, én éééé-O so that there
exists an integer N1 such that V n 3 N1 [8"] < 5
(N1 possibly depending on the sample).
Thus for n 3 N1:

96

 

 

 

 

 

 

 

_ *
lenl = n ‘1Ln"101 - Ln"(p )1
< -1 " 32h(_; 0) 32h( ; 3*)
- n 2 1 2 ' 2 I
1 as as
-1 n 32h( . 0) 82h( o 8*) * A
S n I sup{l é - 5 l : 1s 1 < lsnl}
1 as as
_ n 2 . 2 . * *
5 n 1 Z sup{1a h( 5 0) _ a h( 5 p)1 : Is 1 < 6}.
1 as as
By assumption (*)
32h( - 0) 32h( - s*) *
E {supl ’ - ’ 1= 18 l < 6}
0 as2 as2

is finite.
Since under H0 the sequence {X1: 1 3 l} is stationary and ergodic

theorem 2.1[2] of Borwanker et a1 implies that
- n 2 . 2 . * ,
as 38

converges a.s. to

2 . 2 .
EOtSUpila h(.201 _ a “(2’ s) 1
38 38

= |s|<61

Therefore there exists an integer N2 such that for n > N2

 

 

 

 

-1 " 32h( - 0) 32h( ° 8*) *
n ) supII ’2 - g 1: Is 1 < a}
1 38 38
2 . 2 . * *
< 2 Eotsupila h‘ ’20) - 3 “( 5 B ) |= Is 1 < 511

38 88

97

N2 again possibly depending on the sample

Take N = max(N1,N Thus for n > N

2)
lenl < e

6 being arbitrary, it follows that lenl + 0 a.s.

APPENDIX C

The conditional probability distribution of a l-dimensional
near—neighbour auto-binary spatial secheme is given by

i-l i+l)]xi
“+3(Xi-1 + Xi+11

[0+B(X + X

 

_ e
(c-1) PBEXilN-NJ -
l + e

where 8 .describes the spatial interaction between near-neighbour
particles.

The hypotheses of interest as given by (4.2.1) are

-%
: = = n + + .
a,n 6 8n 0 as n m

We need to show that in this formulation the LAN conditions given

H

by (4.1.7) are not satisfied.

Now

An(sn.0) = L116") - an>

Using Taylor's series expansion we have
= I __r_1_ II ___n_ "I *
An(sn.0) sIn Ln (0) + 2 Ln (01+ 6 Ln (8)
* -s
where [8 | < Isnl = n 2

Following the argument of Section 3.2, we get

98

99

 

 

 

 

 

a
1 " e(i1 i+l)
(s .0) —{ X.(X. +x. )- 1
711121 1 "1 1” 1+e
n (x +x )2 -1/
1 1'1 1+1 6 n 2 1 n1 *
l (l + e )
. . 1 * . * -L .
By cla1m 3 (of Sect1on 3.2) fi-Ln"'(s ), w1th Is I < lsnl = n 2, 1s
asymptotically bounded (under H0) as n + m.
By claim 2 (of Section 3.2) the 2nd term on the right Eééé- ~k2 (under
H0)
Further,
a
_1 " e(Xi-1+Xi+l)
Tn(0) -'E: Z{ +Xi+l) - a }
/n 1 l + e
e2“(3 + e“)
is asymptotically normal with mean zero and variance 0 = 4
5 (1+e“)

(under H0).
Thus, clearly (4.1.7) (b) is satisfied with

 

n,0 2 2

-k + k 0

20 a 20 a
£.h.s. = -k2 + g 52 = -e (1 + 2: 1+ % e (32+ e4)
5 (1 + e“) (1 + e3)

2a<1 _ 3ea)
(l + 60.14

 

# D for all a

100

Thus,
Pn,0
An(8n,0) - Tn(0) + P2 to —+—+ 0 V 0.
Consequently, it follows that LAN conditions are not satisfied in our
l-dimensional auto-binary spatial scheme. In a similar manner, it can
be shown that the LAN conditions are not satisfied in our 2-dimensional
isotropic model either. Thus, the classical techniques of contiguity

fail in our formulation.

10.

11.

12

13.

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