two-dimensional random point patterns: a review and an interpretation

T W O - D I M E N S I O N A L R A N D O M POINT PATTERNS: A REVIEW A N D A N INTERPRETATION

By MICHAEL. F. DACEY, Northwestern University

T H I S REPORT is largely a review of published mater ia l on random point patterns. One objective of this three-par t s u m m a r y is to formula te an informal in terpreta t ion of a random process in the plane. To make this exposit ion widely accessible, considerable detail is g iven in the mathemat ica l s ta tements , usually wri t ing down each step in a derivation.

Par t I contains no new mater ia l . Four conditions underlying a random spatial point process are s tated, and these are used to derive stat is t ical proper- t ies of random point pat terns . I t is shown that (1) this process obeys the Poisson probabi l i ty law and (2) the distance between neighboring points follows the g a m m a probabil i ty distribution.

Par t I I contains new numerical data. Tables 1 and 2 give low-order crude and central mom en t s and measures of skewness and kurtosis for distance f rom an a rb i t ra ry locus to (1) the j th nearest point and (2) the n nearest points. Tabula ted values are for j , n = 1(1)25(5)50(10)100.

Par t I I I contains the most interest ing mater ia l . L imi t ing propert ies of a random pa t t e rn are evaluated, and a fundamenta l relat ion is established between the two-dimensional random distribution, defined by the four assumpt ions of Par t I, and the fami l ia r rec tangular or uni form distribution. I t is not sur- pris ing that this random distr ibution is the formal , two-dimension equivalent of the rectangular distribution; possibly less obvious, is the high degree of areal un i fo rmi ty in the random point pat tern.

PART h QUALITATIVE PROPERTIES OF RANDOM POINT PATTERN

Three aspects of a random point pa t t e rn are reviewed: the assumpt ions of a random spatial point process are stated; it is shown that this is a Poisson process; and the Poisson process is used to show that distance relations in a random point pa t te rn are given by the g a m m a or chi-square probabi l i ty law.

A. Assumptions of Random Pattern

Feller (2, pp. 400-2) gives a clear s t a t ement of a random tempora l process. While he does not consider a random spatial process in equal detail, he does s ta te the modifications required to make the t ransi t ion f rom a one- to a two- dimensional random process. This s t a t ement s imply implements and amplifies his instructions.

A large plane contains a random a r r angemen t of points such that the average number of points per unit area is 2. In this plane, a region of area

The support of the Regional Science Research Institute and of the National Science Foundation is gratefully acknowledged by the author.

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42 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION

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DACEY: TWO-DIMENSIONAL RANDOM POINT PATTERNS 43

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A is selected by some a rb i t ra ry or random method. Four postulates about the occurrence of points in this region are stated. The first three describe a random point process and the fourth puts physical constraints on the random pat tern.

A. Stat ist ical Equil ibrium. The probabil i ty that a region of area A contains n points is the same regardless of (1) the location of tha t region and (2) the shape of that region. An al ternat ive s ta tement of this assumpt ion is tha t the average number of points in any region whatsoever of area A is 2A.

B. Independence of Events. The probabi l i ty that one point occurs in any region of area dA is 2dA, and the probabil i ty that more than one point occurs in tha t region is of smal ler magni tude than 2dA.

C. Differentiabili ty with respect to A. For all values of A, the probabil i ty that n points occur in a region of area A has a continuous der ivat ive with respect to A.

D. Boundary Conditions. Denote the probabil i ty that a region of area A contains exact ly n points by P(n, A). In the physical situation, it m a y be taken that A is not negat ive, and for n ~ 0

P(- - n, A) = 0

P(O, o) = 1

P(n, O) = 0

that is, no region contains a negat ive number of points, and a region with no area contains no points.

The propert ies of a random pa t te rn are derived f rom these four s t a t ements alone.

B. Derivation of the Poisson Probability Law

The four postulates are used to obtain the probabi l i ty that a region of area A contains exact ly n points. I t is shown that P(n, A) is given by the Poisson probabil i ty law. If the region is chosen f rom a plane with an average of ,~ points per unit area, then

P(n, A) = e-~A(2A)~/n! .

The derivat ion of this result requires only e lementa ry probabil i ty methods and is given in some detail only for completeness.

The probabil i ty that n points occur in a region with area (A + dA) is obtained f rom the postulates of stat ist ical equil ibrium and independence in the occurrence of points. The required probabi l i ty is

P(n, A + dA) = P(n, A)P(O, dA) + P(n -- 1, A)P(1, dA)

+ P(n -- 2, A)P(2, dA) + . . . + P(O, A)P(n , dA) + . . . . ( 1 )

In a region, a point does not occur or at least one point occurs; so

P(O, dA) = 1 - - {P(1, dA) + P(2, dA) + . . . } . (2 )

Subst i tut ing (2) in (1) and dividing by dA gives


P(n , A + dA) -- P (n , A) _ P(1, dA) {P(n -- 1, A) -- P (n , A)} d A d A

-k P(2, dA) {P(n- -2 , A ) - - P ( n - - 1, A)} + . . . . ( 3 ) dA

By postulate B, all t e r m s but the first on the r ight-hand side of (3) are zero. Also by B, the quant i ty ~dA m a y be subst i tu ted for P(1, dA). So, (3) m a y be wr i t t en

P(n , A + dA) -- P (n , A) = 2{P(n -- 1, A) - P (n , A)}. (3a) d A

By postulate C, the left-hand side of (3) in the l imit becomes the der ivat ive of P(n , A ) with respect to area A. So, differentiat ing (3a) wi th respect to A gives

dP(n , A) + 2P(n, A) = 2 P ( n - - 1, A ) . (4 )

d A

In (4) put n = 0, and by postulate D, P ( - - 1, A) = 0; so

dP(0, A) + 2P(0, A) = 0 .

d A

This differential equation with boundary condition P(0, 0) = 1 has the solution

P(0, A) = e -x~ . ( 5 )

In (4) put n = 1 and the result (5) gives

dP(1, A ) + 2P(1, A) = 2e -~A .

d A

With the boundary condition P(1, 0 ) = 0, this differential equation has the solution

P(1, A) = 2Ae -~A . ( 6 )

In (4) put n = 2 and using (6) gives

dP(2, A) + 2P(2, A) = 22Ae -~A .

d A

Since P(2, 0 ) = 0, the solution is

P(2, A) - (2A)%-~A 2! (7 )

By continuing to operate recurs ively on (4) the general result is obtained

P(n , A ) = e-XA(,~A)~/n! . ( 8 )

This probabi l i ty is defined for ~ > 0, A > 0 and n = 0, 1, 2 . . . . . In all other cases P ( n , A ) = O.

T h e probabi l i ty (8) is the Poisson probabi l i ty density function with para- me te r hA.


C. Derivation of Distance between Neighboring Points

The four postulates and the result that a random pat te rn obeys the Poisson probabil i ty law are used to obtain the probabil i ty density function of distance f rom an a rb i t ra ry locus to the j nearest point. Th is random variable is denoted by Rj and a par t icular value of this distance var iable is indicated by r~-. The densi ty function of R~. is given by equation (17) and it has the g a m m a or Pearson T y p e I I I distribution. In part icular, for a random pat tern wi th theoretical density of points 2 per unit area, it is shown that the quant i ty 2~2r~ is a chi-square var iable with 2j degrees of f reedom.

The derivation of distance in a random point set has been a problem of continuing interest , and the same result has been obtained many t imes. Neares t neighbor distance, denoted by the variable R~, has a t t rac ted grea tes t interest . The earliest derivat ion I have been able to locate is a 1909 s tudy by Hertz. He obtained the nearest neighbor distance and gave numerical resul ts for Euclidean spaces of one through four dimensions. More general discussions of stochastic spatial processes, pert inent to nearest neighbor distances, include Chandrasekhar (1) and Skel lam (7). Morishita (5) and Thompson (8) have extended nearest neighbor relations to the more general j order neighbor relation.

I t has been established that the probabil i ty of finding exact ly n points in an a rb i t ra ry region is given by the Poisson probabi l i ty law. Let the a rb i t ra ry region be a disk or circle of area a = ~r ~, and assume the center of this disk is located at random in a Euclidean plane containing an average of ~ points per unit area. I t follows f rom the derivat ion in B that the probabi l i ty that this disk contains exact ly n points is

P ( n , a) - (a~)~e-~ ( 9 ) n!

The probabil i ty of finding no points in a disk wi th area a is obtained f rom (9) by put t ing n = 0. Tha t probabil i ty is

P(0, a) = e - ~ . (10)

So, the probabil i ty of finding at least one point in the disk is

P ( n , a) = 1 -- e -~x n >__ 1. (11)

The probabil i ty of finding exact ly one point in a disk with area a is obtained f rom (9) by put t ing n = 1. T h a t probabil i ty is

P(1, a) = a,~e - ~ �9 (12)

So, the probabi l i ty of finding at least two points within the disk is

P ( n , a) = 1 -- (P(0, a) + P(1, a)} n >_ 2 . (13)

Continuing in this fashion, the probabil i ty of finding at least j points in a disk of area a is derived. T h a t probabil i ty is


P ( n , a) = 1 - - P ( n , a) n > j .

= 1 - - {e - ~ + a2e -~x + (a2)2e-~X/2! + . . . + (a2)~- le -~a/ ( j - - 1)! }. (14)

Let i denote the center of an arb i t rar i ly located disk. Then the probabi l i ty tha t at least j points are found within distance r of the centroid i is given by (14) wi th a = 7or 2.

The probabi l i ty that the j nearest point to i is found in the distance interval r to r t is evident ly equal to the probabi l i ty that the j nearest point is found in the annulus with inner radius r and outer radius rq This probabil i ty is the difference be tween P ( n , rcr 2) and P ( n , ~ r ~2) and is expressed in symbols as

P ( j , r ~ r ' ) = P ( n , ~rr t2) - - P ( n , ~rr 2) n >_ j . (15)

The probabi l i ty tha t the j nearest point is found in the small distance interval r to r + Ar is obtained by le t t ing r I tend to r, which gives

P ( j , r ,.~ r + Ar) = P ( j , 0 ..~ r ' ) - - P ( j , 0 ~ r) r ' ~ r . (16)

Assumpt ion C s ta ted that probabil i t ies of occurrence of points were dif- ferent iable wi th respect to area of a region over all possible values of A. Since the area of a disk is a function of its radius, postulate C is equivalent to assuming tha t probabil i t ies of occurrence of points in a disk are differenti- able wi th respect to radius for all possible values of r. Hence, the der ivat ive of P ( j , r N r + zJf) exists. Accordingly, the probabi l i ty densi ty function for distance to j order point is obtained by let t ing (r I -- r) go to the infinitesimal interval d r and then differentiat ing (16) wi th respect to r. The result is

f ( r j ) d r j = f ~ ( r ) d r - d P ( j , 0 ~.. r ) _ 2(~2) ~ r2~_le_~X,2dr j = 1, 2, . . . d r ( j - - 1)! r > 0 .

= 0 elsewhere. (17)

T h e f ( r 3 ) is the probabi l i ty densi ty function for distance f rom the centroid i to the j nearest point. Because the selection of i is random, f ( r j ) is the probabil i ty densi ty function for both distance f rom (1) an a rb i t r a ry point in the random pa t te rn to the j neares t point and (2) an a rb i t r a ry location in the plane, which m a y or m a y not happen to contain a point, to the j nearest point.

Put t = ~2r 2 and subst i tu t ing t into (17) gives

f ( t i ) = f l - l e - t / ( j - 1)!. (18)

This is the g a m m a distr ibution wi th shape p a r a m e t e r ( j -- 1). Al ternat ively, put s = 2~2r 2 and subst i tu t ing s into (17) gives

f ( s ) = J - ~ e - ' / 2 / 2 3 ( j - - 1)! . (19)

Th i s is the chi-square distr ibution wi th 2j degrees of f reedom. Probabi l i ty densi ty functions for the sum of squares to distances to the n


nearest points f rom an arb i t ra ry centroid i are readily derived f rom the established relations. Because of the addit ive proper ty of g a m m a variates, the random variable

T* = (T1 + T~ + . . . + T~) (20)

obeys the g a m m a distr ibution wi th shape p a r a m e t e r

( j - - 1) + n - - 1 = (n 2 + n - - 2)/2.

Because of the addit ive proper ty of chi-square variates, the random variable

S* = (81 + S~ + . . . + S~) (21)

obeys the chi-square distr ibution with degrees of f reedom equal to

~. 2 j = n ~ § j = l

PART Ih MOMENTS AND MOMENT CONSTANTS OF ORDER DISTANCES

T h e densi ty functions for j order and n mean distances were obtained in Section C of Par t I. Because standardized distances have well-known and complete ly tabulated forms, the hypothesis of a random pa t te rn may be tes ted by s tandard procedures, and, for these tests, it is not necessary to have momen t s of near neighbor distances. These momen t s and resul t ing m o m e n t constants are useful, in displaying propert ies of a random pat tern .

Propert ies of j order distance are given in Section A; propert ies of the mean of the n smal les t j order distances are given in Section B. For each type of distance, the crude and central momen t s and measures of skewness and kurtosis are tabulated for j , n = 1(1)25(5)50(10)100. Also, recursion fo rmulas are derived, and these may be used to extend the range of these tables or for interpolation of exact values.

For evaluation of moments , it is convenient to use a s tandardized distance. In the remainder of this paper, it is assumed tha t the met r ic is selected so tha t the average number of points per unit area is unity; that is, ,~ = 1. In wri t ing probabi l i ty density functions for s tandardized distances, the scale para- me te r or measure of point densi ty is dropped. I t does not seem necessary to distinguish the s tandard distance f rom other distance measures by change of symbols . Accordingly, the density function for s tandardized j order distance is obtained by wri t ing (17) wi thout the pa rame te r

f (r j ) = 2=Jr2J-le-~/(j -- 1)!. (17a)

The density function for mean of the n smal les t j order distances is

f(gn,) = n -I ~ f ( r j ) . (22) j=: t

A. ~ Order Distance

The probability density funct ion for standardized j order distance is (17a).

DACEY: TWO-DIMENSIONAL RANDOM POINT PATTERNS 4 9

So, the c rude m o m e n t of s t anda r id ized d i s tance is

, 2~fl f : r 2 j + ~ _ l e _ ~ d r F ( j + z / 2 ) (23) ta~(R~)= F ( j ) = T,(j)~#2

Compu ta t i ona l f o r m u l a s and usefu l r ecu r s ive re la t ions for the odd m o m e n t s ( i > 1) a re

1 . 3 . 5 . . . . . ( 2 j - - 1) _ ( 2 j - - 1) , /~(R~.) = j 2 . 4 . 6 . . . . . ( 2 j ) (2 j - - 2------~ tt~(RJ-~)"

~i(R~) = J . 7C

, j 3 . 5 . 7 . . . . . ( 2 j + l ) 2 j + l , v3(Rj) = 2~r 2 . 4 . 6 . . . . . ( 2 j ) -- 2 j - - 2 ,a3(Rs-~).

tt~(R~ ) = j ( j + 1) (24) 2

Values of these c rude m o m e n t s and the cen t ra l m o m e n t s ob ta ined f r o m t h e m a re g iven in T a b l e 1 for j = 1(1)25(5)50(10)100. Also l is ted are the Pearson m e a s u r e s of skewness and kur tos i s defined, r e spec t ive ly , as

fll = t~(R~)/I~(Rj) , fi2 = v4(Rj)/tt~(Rs) �9 (25)

I t is obse rved tha t as j increases , the j o rder s t a t i s t i c approaches the n o r m a l f o r m , defined by fl~ = 0 and f12 = 3.

A s imple re la t ion holds b e t w e e n the j and ( j - 1) o rder c rude m o m e n t s . Define

T h e q u a n t i t y A~ is found. m o m e n t g ives

Accord ing ly

ai(Rj) = z/~.zt~(R~-l) �9 (26)

R e a r r a n g i n g (26) and subs t i t u t i ng (23) for the crude

,dj~ - F ( j + z/2) F ( j -- 1 + z/2) / F ( j + z/2) F ( j - - 1)

F ( j - - 1 + z/2) F ( j )

( j - - 1 + z / 2 ) F ( j - - 1 + z/2) F ( j - - 1) l ' ( j - - 1 + z/2) ( j - - 1)F( j -- 1)

_ j - - l + z / 2 z - 1 + - - (27)

j -- 1 2 ( j -- 1) "

/~P~(Rj) = /~(Rj -1 ) + 2 ( j -- 1)

and the r a t e of increase b e t w e e n ( j - - 1 ) and j o rder expec t ed d is tances is � 8 9 - - 1 ) .

T h i s r e su l t is useful for i n t e rpo la t ing in T a b l e 1. Crude m o m e n t s of o rders not l i s ted in T a b l e 1 m a y be ob ta ined by success ive appl ica t ion of the

5 0 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION

opera tor zl~ to the neares t lower order tabula ted value. However , for mos t pract ica l applicat ions, l inear in terpolat ion m a y be used on both the crude and cen t ra l momen t s .

T h e j order c rude m o m e n t s m a y also be expressed in t e rms of the neares t ne ighbor crude momen t s . Define

v~(Rj) = (zlj~)l/2~(RJ. (29)

By repea ted appl icat ion of (26) it is found tha t

( ~ ) ! = z l J j _ i . . . . . ~.~

: I~[ (1 + z - - - - -L- - ) o:~ 2 ( a - 1) " (30)

This result , however , does not provide addit ional in format ion . For example , it m a y be shown tha t for the first m o m e n t

(zli~)i = F(2 j)/[2J-~F(j)] 2 .

Neares t ne ighbor dis tance has the value �89 so

E(Rj) = (,13~)! E(R~) = F(2j)/[2JF(j)]2,

which is s imply an a l te rna t ive express ion for the value of the in tegra l (23) wi th z = 1.

B. n Order Mean Distance

T h e dens i ty funct ion for s tandard ized n order mean dis tance is (22). So, the z c rude m o m e n t is

~(M~)'- = -~2'~1--y--~ z~ jot~m2J+~-~e-~dm

_ 4 ff~+i 17 z + 2 F(n + 1) oo m2~+'+le-~2drn

2 F(n + 1 + z/2) = z + ~ F(n + 1)Td/2 (3,1)

Computa t ion fo rmulas for the first four m o m e n t s and recurs ive relat ions for the odd m o m e n t s are

, - 1 . 3 . 5 . 7 . . . . . ( 2 n + 1) 2n + 1 , - /2~(M~) = 3 . 2 . 4 . 6 . . . . . ( 2 n ) - 2n z~(M~_~)

~ ( ~ r ~ ) - n + 1

2~

r - (n + 1) 3 .5 .7 . . .(2n + 3) 2n + 3 /28(M~) = "" -

57: 2 - 4 . 6 . . . . . ( 2 n + 2) 2n /~(3~_~)

, - ( n + l ) ( n + 2 ) t t 4 ( M ~ ) = 3 ~ (32)

Tab le 2 lists m o m e n t s and m o m e n t cons tants for n - - 1(1)25(5)50(10)100. T h e dis t r ibut ion is a Pearson T y p e I or J-shaped curve and for n > 7 is skewed to

DACEY: TWO-DIMENSIONAL RANDOM POINT PATTERNS 5]

the left.

PART III. LIMITING PROPERTIES OF ORDER DISTANCE

Essential properties of a random point pat tern are clarified by considering the behavior of order distance for large j. In Section A, it is shown that as j increases, the area of the annulus defined by radii of lengths E(Rj-1)and E(RA approaches unity. In Section B, it is shown that the ratio E(RA/E(R,), j _< n, has the l imiting v a l u e (j/n) ~/2. These two results establish that the annuli formed between concentric disks with radii defined by successive order distances tend to equal area. Tabulated values indicate that the asymptot ic propert ies are approached rapidly.

An informal summary of a random point pa t tern would not be entirely misleading if it contained the observation that to the three-dimensional creature viewing a random point pat tern f rom above there is no discernable regulari ty, but the sesile inhabitant of A.E . Abbott 's Flatland notices a tendency for a point to be located on each r im of concentric annuli of unit area. While his vantage point displays a uniform spacing of points, he, however, does not discern a regular i ty in the angles formed by pairs of points.

A. Limiting Value of ~ { E 2 ( R ~ ) - - E 2 ( R j _ ~ ) }

It is shown that for standardized distances

~{E2(R~) -- E2(R3_l)} --~ 1 . (33)

j - - . co

This property may be proved by considering

~{E2(RJ) -- E~(RJ-')} = t L ~ J - LF( j - 1)~/-~-J J

= [ (j -- �89 -- �89 2 -- [ F( j -- �89

(j �89 1) t_ - - - - F ( j 1)

F ( j - 1 ) [ ( ' ( J ( j - - 1]

and showing that the l imiting value is unity. This result is obtained if and only if as j--* co

c~(/_ /j-�89 + F 2 ( j - - • j -- 1 ] / ~ 2 ( j _ 1)

or, by rearranging terms, as j--* oo

F~(j- 1) ( j - - � 8 9 / '~(j -- �89 --~ \ j -- 1 / "

The expression to the right of --> has 0 for the l imit ing value; so, it is required to show

F2(x) ~ 0 as x ~ c o . (34) r~(x + �89

,52 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION

This is shown by use of a theorem by Nielsen (6, p. 288) and at t r ibuted to Binet:

/'2(x) 1 F(�89 �89 x + 1, ! ) , x > 0 (34a) r~(x + �89 - x

where F(�89 �89 x + 1, 1) is the hypergeometr ic function. Whi t taker and Watson (9, p. 282) show that

l im F(�89 �89 x, 1) = 1 ;

so, the l imit ing value of (34a) is 1Ix and (34) follows directly. Hence, the area of the annulus (33) with radii given by E(Rj_~) and E(R~) approaches unity in the limit. The asympto te is approached rapidly; for example , with outer radius defined by j = 10 the area of the annulus is .99958 and by j = 25 the area is .99989.

B. Upper- and Lower-Bounds on E(RyE(R,d

A more precise s t a tement for the l imit ing proper ty (33) is obtained by set t ing upper- and lower-bounds on the ratio E(Rj)/E(R,~), for j < n. These bounds in turn provide a sharp intuit ive notion of the essence of a random pat tern.

Distances to the set of n nearest points are considered. These distances are mapped onto a line of unit length by the projection

E(Rjn) = E(Rj) /E(R~) j <_ n . (35)

The value of * E(Rj,~) is obtained by subst i tut ing the values f rom (27) for E ( R y E ( R , d , and the result is

. j ( 2 j + 2)(2j + 4) 2n E(Rj~) = - - "'"

n(2j + 1)(2j + 3) . . . (2n-- 1) "

Bounds may be set on the value of E(R~*,). I t is shown that as j - ~ n and f t ----~ o o

E(Rj* ) --~ ] ~ ( j / n ) . (36~

Subst i tut ing from (27) for the ratio (35) gives

E ( R L ) - r ( j + � 8 9 + �89

Y ( j + �89 F(n) (35a) = V(j) P ( n + � 8 9 "

A fundamental property of the gamma function is, for c a real number

lira F (x + c) _ x o. �9 -= F(x)

One derivation of this result is given by Lbsch (4, p. 31). (37) gives

lira F(x + �89 = 1 / } - , (38a) ~ F(x)

(37)

Put t ing c = �89 in

DACEY" TWO-DIMENSIONAL RANDOM POINT PATTERNS 53

and p u t t i n g c = --�89 gives

lira F(x - �89 1 (38b) ~-~ y ( ~ ) - ~ - .

Using (38a) and (38b) in (35a) gives

r(g + �89 ;j �89 lira (39)

s . . . . F ( j ) F ( n + � 8 9 - t - -J \n/ '

so that E(Rs) /E(R~)~ (j/n) ~p as j-->n and n ~ oo.

Tab l e 3 l ists the decile vales of E(Rj*) for collections of size 10 and t00 and the l imi t ing values. The va lues for n = 10 are close to (j/n) v2, and for n = 100 the difference be tween actual and l imi t ing values is nowhere greater than �9

TABLE 3 DECILES OF ORDER STATISTICS FROM THE RANDOM DISTRIBUTION

ON THE UNIT CIRCLE FOR SAMPLE SIZES n=10,100, AND %-->co

n=10 E(R*~)

.28377

.42566

.53207

.62075

.69835

.76818

.83220

.89164

.94737 1.00000

n=lO0 E(R~)

.31269

�9 .54613 �9

�9

�9 �9

�9

�9 1.00000

E(RS) �9 31623

�9 44721

�9 54772 �9 63246

.70711

�9 77460 �9 83666

�9 89443

�9 94868 1.00000

�9

.2

�9

�9

.5

�9

.7

.8

.9 1.0

C. Relat ionship of the Random Distr ibut ion

to an Area l Uni form Distr ibut ion

I t is shown tha t the d i s t r i bu t ion of Rs*~ is a sympto t i ca l l y equ iva len t to a two-d imens iona l or areal u n i f o r m dis t r ibut ion �9

T h e areal u n i f o r m d i s t r i bu t i on is defined for a circle wi th finite radius D. I t is a s s u m e d tha t each smal l area in the circle has an equal p robab i l i ty of r ece iv ing a point . So, the p robab i l i ty tha t a r a n d o m l y located point lies w i t h i n d i s tance Y of the cen te r of the circle is g iven by the d i s t r ibu t ion func t i on

F(y) = ~y2/~D2 0 < y < D . (40)

To allow the compar i son wi th the t r a n s f o r m e d order d i s tance s ta t i s t i cs f rom the r a n d o m d i s t r ibu t ion , it is c o n v e n i e n t to consider a c i rcle wi th u n i t radius, For this circle, the d i s t r i bu t ion func t ion (40) reduces to


F ( y ) = y2 O < y < D

and the probabil i ty density function of Y is

f ( y ) d y = dF(y ) = 2ydy .

The z crude m om en t of y is

(41)

(42)

/2~(Y) = 2 - (43) z j 2 "

I t can be shown that the random distribution mapped onto the unit circle by the t ransformat ion (35) in its l imit ing form has the moment s (43). The z crude momen t of the mean distance to the n nearest neighbors in a random distribution is g iven by (31), which may be wr i t ten

r ~ 2 F ( n + l + z / 2 ) / z , ~ ( ~ ) - (31a) z + 2 F(n + l)= #2

Projecting the n nearest points onto the unit circle gives the z crude m o m e n t of

2 F(n + 1 + z/2) / F(n + z/2)

= - z + 2 I F(n)

2 (n + z/2)F(n+z/2) F(n) z + 2 nF n I '(n + z/2)

_ ~ n + z/2 2 ( l + z / 2 n ) z + 2 n z + 2

(44)

For any fixed z, the l imit ing value is 2/(z + 2), as given by (43) for the areal uni form distribution. For the lower order moments wi th n at all large, the correspondence between (43) and (44) is high.

D. Summary

This review has s ta ted assumpt ions describing a random point process. Several quali tat ive and quant i ta t ive propert ies of the random point pa t te rn were derived f rom these postulates. Limi t ing propert ies of the random point pa t te rn were also examined and it was shown that these propert ies are asymptot ica l ly equivalent to propert ies of a two-dimensional uniform distribution. The re are at least two impor tant implications to this correspondence.

1. The equivalence of l imit ing propert ies of the discrete random point pa t tern with propert ies of the continuous two-dimensional uniform distribution varifies that the random pat tern is a uniform pat te rn in which each small area has an equal probabil i ty of receiving a point. Since postulates of the random point pa t tern were concerned with equally probable point locations, this correspondence s imply establishes that the stated postulates for a random pat te rn do in fact tend to produce an even distribution of points. Thus, the discrete point pa t te rn wi th density function (8) is the discrete equivalent of the continuous uniform distr ibution with density function (42).

2. T h e s tudy of l imit ing propert ies of a random point pa t te rn has practical


implications to map description and analysis. The correspondence between the random point pattern and the uniform distribution is based largely upon asymptot ic properties. While it was shown that asymptotic values are approached quite rapidly, the mathematical analysis and tabulated values establish that (a) for fixed n the correspondence increases as j approaches n and (b) for fixed j the correspondence increases as n becomes large. The description and analysis of map patterns is conventionally based upon relations between near neighbors in a pattern. Because asymptotic properties hold for small j only for n very large or, equivalently, with a high density of point per unit area, for values of greatest interest to map pattern interpretation, the correspondence between the discrete and continuous distributions is weakest. Accordingly, if a map analysis requires accurate theoretica! statistics the values in Tables 1 and 2, obtained from the definition of a random pattern as a discrete stochastic process, should be used. However, if only a general qualitative description of a random pattern is required, the use of the uniform distribution may facilitate the interpretation of spatial relations. The continuous uniform distribution does provide the geographer with a simple description of the spatial essence of a random point pattern.

REFERENCE5

1. Chandrasekhar, S. "Stochastic Problems in Physics and Chemistry," Review of Modern Physics, XV, 1943, pp. 1-89.

2. Feller, W. An I~troduction to Probability Theory and Its Applications, I (2nd ed). New York: Wiley & Sons, Inc., 1957.

3. Hertz, P. "Uber den geigerseitigen durchschnittlichen Abstand von Punkten, die mit bekannter mittlerer Dichte im Raume angeordnet sind," Mathematiche Annalen, LXVII, 1909, pp. 387-98.

4. LSsch, F. Die Fakultat (Gammafunktion). Leipzig: Tuebner, 1951. 5. Morishita, M. "Estimation of Population Density by Spacing Methods," Memoirs of the

Faculty of Science (Kyushu University, Series E), I, 1954, pp. 187-97. 6. Nielsen, N. Handbueh der Theorie der Gammafunktion. Leipzig: Teubner, 1905. 7. Skellam, J. G. "Random Dispersal in Theoretical Populations," Biometrilca, XXXVIII,

1951, 196-218. 8. Thompson, H. R. "Distribution of Distance to nth Neighbour in a Population of

Randomly Distributed Individuals," Ecology, XXXVII, 1956, pp. 391-94. 9. Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis, Cambridge:

University Press, 1962.

two-dimensional random point patterns: a review and an interpretation

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