perspectives on urban spatial systems || some properties of the superposition of a point lattice and...

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Some Properties of the Superposition of a Point Lattice and a Poisson Point ProcessAuthor(s): Michael F. DaceySource: Economic Geography, Vol. 47, No. 1, Perspectives on Urban Spatial Systems (Jan.,1971), pp. 86-90Published by: Clark UniversityStable URL: http://www.jstor.org/stable/143228 .

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RESEARCH NOTE

SOME PROPERTIES OF THE SUPERPOSITION OF

A POINT LATTICE AND A POISSON POINT PROCESS

MICHAEL F. DACEY*

Northwestern University

Let L denote a point lattice embedded on the euclidean plane E and U denote a poisson point process, with density X, on the euclidean plane. It is assumed that the density measure X and the trans- lation period of L are expressed in the same metric. A point pattern S with a regular lattice component and with a poisson component is obtained by the superposition of the patterns L and U. Medvedkov [5, 6] has proposed the measure of entropy of this pattern S as a basis for isolating the lattice and poisson components in an observed map pattern. Because it is not clear how Medvedkov obtained the results reported in his paper, the probability distributions of quadrat counts and nearest neighbor spacing measures are given in this paper for the pattern S. Some of these results are used to establish that his measure of entropy has limited applicability for the description and analysis of pattern.

Before commencing the analysis of properties of S, it is noted that a class of theoretical patterns more suitable for empirical analysis of map patterns is de- scribed in a series of papers by Dacey [2, 4] in which a pattern S' is obtained by the superposition of a pattern U and a pattern L' in which the location of each point in L' is a random deviation around the corresponding lattice position in L. While analysis of properties of S' has proved particularly difficult, in contrast, it is easy to obtain many properties of S, which may in part compensate for the superficiality of the underlying location process. One justification for study of S is that it yields properties of a degen- erate form of S'.

*The support of the National Science Foun- dation, Grant GS-1627, is gratefully acknowl- edged.

PRELIMINARIES

The point pattern S is obtained by the

superposition on the euclidean plane E of a point lattice L and a poisson point process U. Properties of the two under-

lying patterns that are assumed known include the density X of the poisson point process and the parameters of the lattice L. The lattice L is restricted to the

square and hexagonal point lattices, and

properties of these lattices needed for

subsequent analysis are the distance T from a lattice point to the nearest other lattice point and the radius R of the dirichlet region centered on each lattice

point. Consequently, each point of E is within distance R of a lattice point in L. The metric is selected so that the dirichlet region of L has unit area. This means that for the square lattice T = 1 and R = 1/V 2 while for the hexagonal point lattice T = 22/34 and R = 22/33/4 By putting the area of the dirichlet region equal to unity,

mean density of points of U mean density of points of L

Let o represent an arbitrary point of E and, similarly, let s, u and I represent arbitrary points of, respectively, the

point patterns S, U, and L. The expression "x is an arbitrary point

of a set X" means that x is selected in such a way that each point in X is equi- probable. In the probability structure of

Kolmogorov this random selection has no meaning for sets of unbounded measure, but it is acceptable within the more

general, conditional probability structure of Renyi.

The counting function V,(t) is the number of points of V, excluding x if xeV in a circle with radius t and center



SOME PROPERTIES OF THE SUPERPOSITION

at an arbitrary point x, where V is replaced by U, L, or S and x is replaced by u, 1, s, or co.

The nearest neighbor distance V, is the distance from an arbitrary point x to its nearest neighbor point in V, where V is replaced by U, L, or S and x is replaced by u, 1, s, or C).

The following properties of cell counts and nearest neighbor distance for L and U are stated without proof. These properties are used in the following sections to obtain corresponding properties for S.

The fundamental relation between nearest neighbor distance V$ from x to an element of V and the cell count V,(t) of number of elements of V in a circle with center x and radius t is

P\ V7_ < t = P V(t) < 1 = -P V,(t) 0 .

For t > 0, denote the mass function of the poisson distribution by

pk(t)= e -rXt2 (7rXt2)7'

k! '

k = 01,2, ...

The number of points of U in a circle with center at any arbitrary point has the poisson distribution so that for all t > 0,

P U, (t) = k = P U(t) = k ~ = PI Ul(t) = k \ = p,(t).

Hence, for all t > 0, Pg Uc < t = PI Uu < t = P Ui < t = 1 - po(t).

Li(t) represents the number of points of L, excluding 1, in a circle with center I and radius t. If there are any lattice points on the circumference of Li(t), they are treated as a single point in the circle LI(t). Since the distance between nearest lattice points is T, this conven- tion and leL imply P I L (T) = 1 = 1. Conversely, leL and t < T imply P L1(t) = 0 = 1. Hence, the entire mass of L1 is concentrated at a single point and Pt L, = T [ = 1. Further, for every positive integer n there is an

integer kn so that P X Ll(nT) = kcn = 1. The probability distribution of L,(t)

= L (t) is not readily available. Though it is a simple task to derive this distribution for t < T, the distribution is not known for large values of t. For the pres- ent purposes, it is adequate to note that

0 < t < 2T implies P0 < L,(t) < 1 = 1,

1T < t < R implies P 0 < L,(t) <2 = 1.

and that for t > R, P L (t) = k ~ 1 for all integer k.

The probability distribution of L, is

given by Dacey [3]. For the square lattice

(1) P X L , < t \ = F(t) = 7(t/T)2,

O < t < ,T

= 7(t/T)2 + (4t2 - T2)L/IT

4(t/T)2 cos-1 T-, T < t < R/V2 2t

-

For the hexagonal lattice

(2) P L, < t = F(t)= 27rt2/V3T2,

O < t < TT

= 27t2/ V3T2 + (12t2 - 3T2)S/T

T - 4[3(t/T)2] cos-1'-, T < t < R.

Distance from an arbitrary point in E to the second nearest lattice point in L is also needed. Let G(t) denote the cum- ulation probability function of this second order distance, which is given by Dacey [3]. The function G(t) is subse- quently used only for t < R, which for the square lattice is G(t) = 0 for t < ST and

G(t) = 4(t/T)2 cos-1 t

-_~2 (4t _ TT t R

and for the hexagonal lattice is G(t) = 0 for t < 1T and

G(t) - 4V3(t/T)2 cos-1 t - (4t2 - T2)//2T, 1T < t < R.

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ECONOMIC GEOGRAPHY

CELL COUNTS

Some properties are given of cell counts for the pattern S obtained by superposition of L and U. Clearly, for each point x

(3) S,(t) = U(t) + L,(t).

Sz(t), t < T. First, consider the quadrat count for a circle centered on an arbitrary lattice point IeL. Since Li(t) has, with probability 1, the value 0 for t < T and the value 1 for t = T,

Pe S/(t) = k = = pk(t), t < T,

PI Ut(t) - k

which is the poisson distribution with parameter 7rXt2, and

(4) k

P SI(T) k - 1 . = pk l(T),

P= Ut(T) = k= 1,2, ...,

which is a shifted poisson distribution. Su(t), t <T. Next consider a circle of

radius t < %T that is centered on an arbitrary point of the plane cWE. Since for t < 1T, L,(t) has the value 0 or 1, use of (3) gives

PL S (t) = kO = P\ U (t) = k LJ,(t)= 0 + P\ UW(t) k - 1, L,,(t) -1 I

(5) = Pt

P\ Lc(t) = 0 \ + k - 1 P Lo(t) =

U,(t)= k k P U(t) = 1 .

Since F(t) = P L < t = P Lc(t) > 1 = 1 - P Lw,(t) = 0 , then for t < iT PI L,(t) = O = 1 - F(t);

while L ;(t) < 1 for t < i2T implies PI L (t) = 1 = F(t), t < ,T,

where F(t) is given by (1) for the square lattice and by (2) for the hexagonal lattice. Substituting these results into (5) gives

P S (t) = k = P\ Jo(t)= k[ [1-F(t)] + P Uo(t) k-1 [ F(t), t < rT,

p7jt) [1 - F(t)] -p pk (t)F(t),

which is a mixture of a poisson and a shifted poisson distribution.

Further,

P- Su(t) k PISc(t)= k .

The probability distribution of S,(t) was obtained, in a different context, by Dacey [1] for a county seat location model.

The probability distribution of the random variable S ,(t) = S,(t) is not identified for t > R simply because the probability distribution of L,(t) is not known for large values of t. However, it is clear that t > R implies that, for every integer k, P Lo(t) = k ) < 1.

Ss(t), t < 2T. The density of points in U relative to the density of points in L is X, and an arbitrary point seS also be- longs to U with probability X(1+ X) and to L with probability j(l+ X). This im-

plies

(6)

P Ss,(t) - k x

1+X 1

P Su(t) = k + + P S(t) - k.

Hence,

PI Ss(t) - k p7(t) + +

[pc-1(t) - pk(t)]F(t), t < %T.

S, (t), t < R. Before considering Ss(t) for t <R, S, (t) is studied for t < R. For t in this range, L, (t) may take on the value 0, 1, or 2 so that

2 P S,(t) = k = - P U (t) -

i-O k-i iPj LW(t) - i .

It was established that P L (t) = 0 = 1 - F(t). For t < R, when L o, is the

distance from an arbitrary point O)EE to the second nearest lattice point in L, then

G(t) P\ L < t - P Lw(t) >2 1 - P\ Lw(t)= 0 - P L (t)= 1[ = F(t)- Pi L,(t) 1

implies

88



SOME PROPERTIES OF THE SUPERPOSITION

P L,(t) = 1 \ = F(t) - G(t), 2T < t < R.

For t < R, L,(t) < 2 so that P L,(t) = 2 = G(t).

Hence, P S (t) = k = p,(t) [1 - F(t)]

q- p,1-l(t) [F(t)- G(t)] + pk-2G(t), O < t < R.

Moreover,

PI S ,(t) = k = P\ S,(t)= k .

Ss(t), t < R. Use of the identity (6) and a few algebraic manipulations yield

PI S,(t) = k = p(t) + + [F(t)

pk-l(t)- pk(t) \ + G(t) I pk-2(t) - pk-l(t) I ]

for t < R.

NEAREST NEIGHBOR DISTANCE

The cumulative probability distribution of nearest neighbor distance follows immediately from the fundamental relation between nearest neighbor order distance from a point x and the cell count for a circle with center x.

PI S,< t [ = P\ SC (t) 1 = I - P s (t) = o

1 - po(t) [1 - F(t)], t< R

=1, t = R. P SI, < t t = P, S w < t PI Si < t = P Sl(t) > 1 =

-P si(t) = O 1- po(t), t < T

=1, t = T. PI S, < t\ = P S,(t) 2 1 =

1-P S,(t)= 0 = 1 - po(t) [1 - F(t)-

x 1 1+x t<R

1 = 1- X+ po(R), t- R.

Further, it follows from (3) and (4) that

P. S, < t \ 1 1 - po(R),

R < t < T, 1 t = T.

MEASURE OF ENTROPY

If x is an arbitrary point in a set X and S,(t) is a counting function for a point pattern S, the entropy of S with respect to X is

oo

Hx(S) - P P S,(t) k=o

log2 P , S(t) = k .

k -

The entropy of the patterns L and U with respect to X is

00

H,(L) =- 2 P I L(t) k=o

log2 P Lx(t)= 7k and

00

Hx(U) --=- : P Ux(t) k=O

log2 P U(t) = k .

k *

k \-

If X is replaced by L, for all t > 0 H1(L)= 0

and

Hz(S)= Hz(U). However, if X is replaced by U, S, or E, H,(L) > 0 and H( )S #7 H,(U).

Medvedkov [5, 6] considers the entropy of the pattern S, which he symbol- ized by :, with respect to the probability distribution of points within square cells. To express his results in terms of the notation used in this paper, let V * (t) represent the number of points of V, excluding x if xeV, within a square with sides of length t and centered at an arbitrary point x, where V is replaced by U, L, or S and x is replaced by u, 1, s, or o. In Medvedkov's study, L is restricted to the hexagonal lattice. The entropy measure used by Medvedkov is

c00

H,(V) = - Y P Vx(t) =- k k=o

log2 Pq V *(t) = k [. Medvedkov's results pertain to a set of X for which xeX implies H,(L*) = 0 and, hence, H,(S*) = H,(U*). By the meth- ods leading to the results of the preced- ing paragraph, it may be shown that the

89



ECONOMIIC GEOGRAPHY

entropy measure H,(L*) = 0 only when X corresponds to L. In other words, when entropy is defined with respect to the probability distribution of the number of points of S within square cells of equal area then the entropy of pattern S equals the entropy of pattern U only if each cell is centered on a point in L. This implies that the technique of pattern analysis suggested by Medvedkov identifies the lattice and poisson components of the pattern S only when the points in L are known, for otherwise the cells are almost surely not centered on lattice points.

LITERATURE CITED

1. Dacey, M. F. "Modified poisson probability law for point patterns more regular than random," Annals of the Association American Geographers, 54 (1964), pp. 559-65.

2. Dacey, M. F. "A probability model for central place locations," Annals of the Associa- tion American Geographers, 56 (1966), pp. 549-68.

3. Dacey, M. F. "Tabulations of Geometric Probabilities I. Lattice Measures," Special Publication No. 2, Department of Geog- raphy, Northwestern University, Evanston, 1967.

4. Dacey, M. F. "Regularity in spatial distributions: a stochastic model of the imperfect central place plane." Paper presented at International Symposium on Statistical Ecol-

ogy held at New Haven, (August, 1969). Forthcoming in proceedings of the symposium.

5. Medvedkov, Y. V. "The regular component in settlement patterns as shown on a map," Soviet Geography, 8 (1967), pp. 150-68.

6. Medvedkov, Y. V. "The concept of entropy in settlement pattern analysis," Papers, Re- gional Science Association, 18 (1967), pp. 165-9.

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