1 Introduction - InterStat

Analysis of inhibitory point processes derived from a sequence of auto-Poisson lattice schemes J.M. Alberty J. Mateuz J.C. Pernasy yDepartment of Economics, Universitat Jaume

I

zDepartment of Mathematics, Universitat Jaume I

December 28, 1999

Abstract

It is known that almost any purely inhibitory pairwise interaction point process can be obtained as the limit of a suitable sequence of auto-Poisson lattice schemes. In this context, several authors have used the pseudo-likelihood parameter estimation procedure. However, there appears to be no extensive simulation study to analyze the behaviour of such parameter estimators as the number of grids increases. In this paper, we review the limit theorems concerned, present a computer software (SPPA) and analyze the results of a Monte Carlo simulation study based on a particular pairwise interaction model.

Key words: Auto-Poisson distribution, lattice data, pairwise interaction pro-

cesses, SPPA, Strauss point process.

1 Introduction A spatial point pattern is a collection of data f(xi ; yi ) i = 1; :::; ng consisting of n locations in an essentially planar region. Examples include the locations of cell nuclei in a microscopic tissue section, trees in a forest, or cases of disease in a geographical region. A fundamental assumption in the analysis of such data is that they can usefully be regarded as a partial realisation of a stochastic point process (Cox & Isham, 1980). There are many contexts in which the use of spatial point patterns can be of interest. For example, in the analysis of the human population's spatial distribution,  The

authors wish to thank nancial support by the Spanish Ministry of Education (CICYT: SEC96-1435-C03-03)

1

they can give us excellent information from both, the demographic and the economic points of view, since a close relationship exists between the economy and the spatial groupings of the human beings. Also, point processes theory plays a central rol in environmental epidemiology, see for example Diggle (1993). The interest in a pattern of points is found in that, with an appropriate choice of scale, even huge objects may be best represented by a point. Given suitable scales, the actual physical sizes of objects that may be represented that way are unbounded. On one extreme, microscopes are required, on the other extreme it is telescopes that are needed. The range of disciplines (pathology, geology, marine biology, zoology, physical and human geography, astronomy, economy,...) dealing with similar phenomena re ects the applicability of point processes techniques. Markov point processes were introduced in Ripley & Kelly (1977), whereas a similar concept of Gibbs distributions (Preston (1977), Ruelle (1969)) has been used in statistical physics for a longer time. Such processes are typically de ned by their densities with respect to a Poisson process. The latter serves as a reference distribution when deriving new models. Since the introduction of Markov point processes in spatial statistics attention has focused on the special case of pairwise interaction models in which each con guration of points interacts only via pairs of points from this con guration (Strauss, 1975; Besag et al., 1982; Diggle et al., 1987; Diggle et al., 1994). These provide a large variety of complex patterns starting from simple potential functions which are easily interpretable as attractive and/or repulsive forces acting among points. A great deal is understood about pairwise interaction models because they are very natural with respect to the derivation of conditional probabilities, Papangelou conditional intensities and Palm distributions. They are simple exponential families whose sucient statistics are often related to the popular K -function and they are very amenable to simulation and iterative statistical methods. However, pairwise interaction point processes do not seem to be able to produce clustered patterns in sucient variety. The original clustering model of Strauss (Strauss, 1975) turned out (Kelly & Ripley, 1976) to be non-integrable for parameter values  > 1 corresponding to the desired clustering. Pairwise interaction processes have been used widely as models for regular spatial patterns (Ripley, 1977; Ogata & Tanemura, 1981, 1984; Diggle, 1983; Penttinen, 1984; Tomppo, 1986 and Stoyan, Kendall & Mecke, 1995) and for aggregated patterns, but less widely and successfully for the reasons just noted. In this paper we focus on a particular pairwise interaction point process, the Strauss process, in view of its wide range of applications. This model is a demanding member of the exponential family for dependent samples and has been used in modelling of point patterns and as a prior in high level image analysis (Baddeley & Lieshout, 1993). Moreover, we can nd applications of exponential families in Markov random elds for lattice data (Besag, 1974; Geyer & Thompson, 1992), Markov random elds in image analysis (Geman & Geman, 1984) and modelling of random graphs and general interaction models (Strauss, 1986). On the other hand, Ripley & Kelly (1977) mentioned but did not exploit possible 2

connections between pairwise interaction processes and lattice models, though Besag (1977) outlined a parameter estimation procedure for the Strauss model based on the tting of an auto-Poisson lattice scheme to the cell counts obtained by imposing a ne grid over the point process realization. Then, Besag et al. (1982) proved that any purely inhibitory pairwise interaction point process can be obtained as the limit of a suitable sequence of auto-Poisson lattice schemes. Since then there seems to be no published extensive simulation study to analyze the meaning of the concept "limit" in practice nor to analyze the behaviour of the point process parameter estimators when such limit is reached, i.e., when the grid order increases. Due to this, the use of a pairwise interaction point process obtained as the limit of auto-Poisson lattice processes has been rarely used in literature despite the wide range of possible applications in such dierent elds as biology, geology, economics or nances. Within this context, in this paper we review the mathematical justi cation to prove that any purely inhibitory pairwise interaction point process can be obtained as the limit of a suitable sequence of auto-Poisson lattice schemes. Then, the goal of this paper is to present an extensive Monte Carlo simulation study to analyze the behaviour of the parameter estimators of a particular model, the Strauss process, derived using this technique. The pseudolikelihood estimation method (Jensen & Moller, 1991; Diggle et al., 1994) has been used to estimate the model parameters for each grid size. Model parameters are particularly important as they control the intensity and the spatial structure of the point process. We concentrate on the pseudolikelihood estimation method as it is a simple method to be used by those practitioners that are not expert in this eld and it provides (as this paper shows) reliable estimation results. However, it is a dense computationally time demanding method and in a general framework, it might be avoided in favour to a more ecient method of estimation, such as MCMC maximum likelihood. The simulation study has been carried out using our own computing software called SPPA (Spatial Point Pattern Analysis, 1997), which is also presented in this paper, and basically builds pairwise interaction processes by means of lattice-based point processes, then estimates the model parameters and test against the random pattern. The plan of the paper is as follows. Section 2 provides preliminaries and a background on lattice-based approximations and builds the pseudolikelihood function. Section 3 is devoted to develop the methodology for the Strauss point process. Then section 4 presents the SPPA software and the corresponding Monte Carlo simulation study is presented in section 5. Finally, the paper ends with a section of conclusions and further research.

3

2 Preliminaries 2.1 Set-up

A point process is a stochastic model governing the locations of events fsi g in some set X . Because our interest is in spatial point processes, we shall take X to be a bounded region in Rd or a torus, but more generaly X could be any locally compact Hausdor space whose topology has a countable base (Cressie, 1993). Let (X; F ;  ) be a measure space such that  (X ) < 1, X is a bounded subset of Rd , F its Borel - eld and  the Lebesgue measure on (X; F ) ; and let (Xe; Fe ;  = exp( )) the corresponding exponential space (Carter & Prenter, 1972). The measure  can be identi ed as the distribution of a Poisson process of unit intensity. A typical point in Xe will be denoted by x. A measurable map from a probability space into (Xe ; Fe ) is a point process and the measure induced by such a map on (Xe ; Fe ) is the distribution of the process. A point process is simple if its distribution is concentrated on the subset Xe of Xe , comprising all nite collections of distinct points from X . For any compact K  Rd ; x(K ) denotes the cardinality of x \ K: We are concerned with point processes P on (Xe ; Fe ) which are absolutely continuous with respect to  . Any such process can be identi ed by its Radon-Nikodym derivative f = dP=d with respect to the unit Poisson process. The derivative f is also considered to be hereditary (Preston, 1976; Ripley & Kelly, 1977): f (x) > 0 implies f (y) > 0 for all y  x. In this paper we restrict our attention to pairwise interaction point processes (Ripley, 1977), for which f has the special form Y Y f (x) / () (f; g) (1) 2x

f;gx;6=

where (:); for homogeneous stationary point processes, is usually a positive constant, say , related to the intensity of the process,  is an interaction function, i.e., a symmetric measurable function mapping Rd  Rd to [0; 1) such that (f;  g) = 1 for all  2 X . The normalizing constant in (1) can be written as f (;): Note that not every pair (; ) de nes a valid probability density f in that a suitable normalizing constant may fail to exist. Such processes for which f turns out to be integrable are called purely inhibitory. For a process de ned by a hereditary density f with respect to  , the conditional intensity (related to the Papangelou intensity (Papangelou, 1974; Daley & VereJones, 1988)) at  given x on X n is 8 < f (x [ f g) f (x) > 0 (jx) = : f (x) (2) 0 otherwise: Particularly, for any pairwise interaction point process of the form (1), Y (jx) = () (f; g): (3) 2x

4

An important and well known special class of processes is generated by interaction functions of the form (f; g) = (df; g), where d(:) denotes the usual Euclidean distance in Rd : Conditions to ensure integrability of f are given by Ruelle (1969), Preston (1976) or Ripley (1977). The interaction function (:) usually depends on a set of parameters, say , which have to be estimated. In such cases, the parametric interaction function is given by (:; ): In this paper we focus on the pseudolikelihood estimation method as it can be used routinely in applications and do not place arti cial restrictions on the parametric form of the interaction function. Other general estimating methods are also possible (Ogata & Tanemura, 1981, 1984, 1989; Penttinen, 1984; Takacs, 1986; Fiksel, 1984, 1988). A more detailed review can be found in (Diggle et al., 1994; Ripley, 1988; Geyer, 1998). If  and  are bounded, then for any nite point con guration x  X with f (x) > 0, the pseudolikelihood for pairwise interaction processes is de ned by (Besag, 1977; Jensen & Moller, 1991)

PL(x; ) = exp(;

Z

X

(jx)d)

Y

 2x

(jx n fg):

(4)

Usually, (4) is re-cast in terms of its logarithm. Maximization of (4) with respect to the parameter set  yields the maximum pseudolikelihood estimators.

2.2 Lattice-based approximations

In this section, we review a lattice-based method using auto-Poisson processes originally devised by Besag (1974) and Besag et al. (1982). Let X be a nite region of the plane. Then, overlay X with a grid of small square cells, each of area A and identi ed by integer Cartesian co-ordinates i = (r; s) with respect to a convenient origin. Let Ai represent both the intersection of X with cell i and the corresponding Lebesgue measure (area), assumed positive, of cell Ai . Let xi be the number of points in cell Ai with Ai > 0. We assign an auto-Poisson distribution of counts of points over all cells with positive area. Thus, for each i, the conditional probability of obtaining xi points in cell i, given all other counts, is n o P (xi jxj j 6= i) = e; (A ) [i (Ai)]x =xi!; xi = 0; 1; :::; (5) where the conditional mean has the form Y i (Ai ) = Ai (fi ; j g)x (6) i

i

i

j

f ; gx;i6=j i

j

with  > 0 and  an interaction function. The odds of a realization x, in the minimal sample space, to a realization of zeros, is given by Y P (x) = Y (Ai )x xx (7) P (0) i xi! f ; gx;i6=j (fi ; j g) : i

i

i

j

5

j

Now, for a given realization of this scheme, we generate a point pattern Pr , whose realizations are denoted by x, by distributing the points in Ai uniformly and independently, and this independently for each cell Ai . Note that if (f:; :g) = 1, in which case the xi `s are independent Poisson variables with means Ai , then the process so generated would be a homogeneous Poisson process on X with intensity . Taking account of the initial auto-Poisson distribution, it follows that the likelihood ratio fr of the density of such a process, with respect to an empty realization satis es,

fr (x) = fr (;)n(x)

Y

i

f ; gP ;i6=j i

(fi ; j g)x x

(8)

j

r

j

where ; denotes the empty realization, fr (;) can be considered a constant of proportionality, and n(x) is the number of points in x. Suppose now we have a sequence of processes P1 ; P2 ; :::for which r ! 0 as r ! 1, where r stands for the maximum diameter of the sets in the partition. Then, there exists a limit for the ratio fr (x)=fr (;) de ned by (Besag et al., 1982)

fr (x) ;! f (x) = f (;) n() Y (f; g): fr (;) r!1 f;g

(9)

The process de ned by f (x) is the corresponding pairwise interaction limit point process. We de ne the pseudolikelihood based on data X (r) = x by

PL(r)(x; ) =

Y

i

P (xi jxj j 6= i) :

(10)

Then using expression (5), (6) and (10), the pseudolikelihood for the r-th point pattern is given by

PL(r)(x; ) =

Y

i

fexp(;Ai 

Y

x x (fi ; j g)x ) Aix ! i

j

j :j 6=i

i

i

Y

j :j 6=i

(fji; kig)x x )g: (11) i

j

Finally, if we consider that  is bounded, then for any nite point con guration x  X with f (x) > 0 and since xi tends to 0 or 1 as i ! 1, (r)

PL (x; ) ! exp(; = exp(;

Z

Z



Y

X 2x

X

(f; g)d)

(jx)d)

Y

 2x

Y



Y

2x 2xnfg

(f; g)

(jx n fg) = PL(x; )

(12)

where ( jx) = f (x [f g)=f (x) is the conditional intensity given by (3) and PL(x; ) is the general expression for the pseudolikelihood of a stationary pairwise interaction point process (4). 6

3 Parameter estimation for the Strauss process The homogeneous Strauss process (Strauss, 1975), is characterized by having a constant ( ) =  and an interaction function (f; g) = 1[jj;jjr]: The total number of neighbours of the pattern of n points is given by X (13) s(x) = 21 1 [k ; k  r] ; ;

and then, the density function (1) depending on  = (; ) becomes

f (x) /

Y

2x

()

Y

f;gx;6=

(f; g) / n(x)s(x) :

(14)

The case  = 1 de nes a homogeneous planar Poisson process, whilst  < 1 de nes a simple inhibition (regular) process with hard-core distance r. Strauss originally proposed (14) with  > 1 as a model for clustered patterns but, as subsequently pointed out by Kelly & Ripley (1976), this violates the requirement of a nite normalising constant f (;) in (14). On the other hand, the mean of the auto-Poisson distribution (6), is given by i(Ai ) = Ait ; (15) where ti denotes the total count in all other cells which lie wholly within a xed range r of any part of cell i. For a rst-order auto-Poisson scheme, if cell i = (r; s), then ti = xr;1;s + xr+1;s + xr;s;1 + xr;s+1 (16) and in a second-order scheme, ti = ti( rst-order) + xr;1;s;1 + xr+1;s+1 + xr;1;s+1 + xr+1;s;1 (17) The order schemes de ne the range of the pattern interaction structure. For the Strauss process, the likelihood ratio of the density of such a process (8), with respect to an empty realization, satis es PP fr (x) = fr (;)n(x) x x (18) where the double summation is over all pairs of cells within the range de ned by a rst or second-order scheme. Then, by letting r ! 1 and r ! 0; we generate a sequence of processes which converge to the corresponding Strauss process. Finally, the pseudolikelihood given by (10) and (11), and from now on denoted by PL, for the Strauss process becomes i

i

ln(PL) = ;Ai

n X i=1

ti + (lnAi + ln)

n X i=1

7

!

j

xi + ln

n X i=1

!

tixi ; ln

n Y i=1

!

xi!

(19)

Usually, in practice, Ai = A constant for all cells, and we then maximize (19). The gradient of ln(PL) is given by @ ln(PL) = ;A X t + P xi (20) i

@



@ ln(PL) = ;A X t t ;1 + P tixi i @ 

(21)

@ 2 ln(PL) = ; P xi @2 2

(22)

i

And the hessian is

@ 2 ln(PL) = ;A X t (t ; 1) t ;2 ; P tixi i i @2 2

(23)

i

@ 2 ln(PL) = ;A X t t ;1 (24) i @@ We can use the rst-order condition to maximize ln(PL) and concentrate the pseudolikelihood function with regard to : Setting expression (20) to zero and solving for  P  = A Pxit (25) i

i

Substituting (25) in (19) we obtain the concentrated pseudolikelihood function ln(PL) = ; and nally

X



P



xi + ln A + ln A Pxit

X

i

h

ln(PL) = constant ; ln

X

t

i

xi + ln 

iX

X

xi + ln 

tixi ; X

X

tixi

ln xi !

(26) (27)

Maximization of the pseudolikelihood function (27) has the restriction  > 0. However, it is easy to transform this maximization problem with restrictions into one without restrictions, de ning a new parameter  through  = e and then maximizing h

ln(PL) = constant ; ln

X

et

i

iX

xi + 

X

The two rst derivatives of this function are P d ln(PL) = ; Pti et X x + X t x i i i d et i

i

8

tixi

(28) (29)

and

P

 P



P



et ; tiet 2 X x d2 ln(PL) = ; t2i et (30) P i d2 [ et ]2 The estimation strategy can be based on maximization of (28), using the equations (29) and (30), by means of the Newton or the Brendt algorithm (Press et al., 1988). From the estimate of , the estimation of  is straightforward through  = e : Then, we can obtain the estimate of  through (25) and, nally, an estimate of the covariance matrix of both estimators,  and ; through the inverse of the hessian (22, 23). Once maximized the pseudolikelihood function, building a likelihood ratio test for the hypothesis  = 1 is simple. Under the null hypothesis, the likelihood function is given by i

i

i

i



ln PL ( = 1) = ln

P

Then, the statistic

xi  ; 1 X x ; X ln x ! i i n

(31)

LR = ;2 [ln PL ( = 1) ; ln PL (b)] (32) is distributed as a 2 with one degree of freedom, where b is the maximum likelihood estimate of . On the other hand, it is possible to build a Wald test using the second derivative of (27), 2 PL  W = ; (b ; 1)2 d ln (33) d2 where the second derivative is evaluated at the maximum likelihood estimate of . Then, W is distributed as a 2 with one degree of freedom. Finally, it is possible to obtain the Lagrange multipliers test for the null hypothesis  = 1 as "

#2 "





#

d2 ln PL  (34) LM = ; d lndPL  =1 d2 =1 where the rst an second dervivatives of ln PL are evaluated at  = 1, that is

d ln PL = X t x ; i i d =1

P

xi P ti n

P

(35) P

d2 ln PL = ; [ ti (ti ; 1)] n ; [ ti ]2 X x ; X t x i i i d2 =1 n2

(36)

The test statistic (34) is also distributed, under the null hypothesis, as a 2 with one degree of freedom. 9

4 The SPPA computer software We introduce here a computer software, SPPA, developed by the authors which has been used in the simulation study presented in the next section. This program is conceived to analyze the most important characteristics of spatial point models, using lattice-based approximations as explained in the preceeding sections. This program is divided into three modules: a) Simulation; b) Counts; c) Maximization and results.

Simulation module. It generates planar coordinates of points in a xed region

based on three dierent spatial structures. Random points can be generated according to a homogeneous Poisson process. The user has the chance to determine both, the mean of the Poisson process, and the area in which the simulation should be carried out. We can also generate regular or inhibitory point patterns by de ning a hard-core distance and then removing those points from a random pattern which lie within that distance from any other point. Finally, this module can also generate point patterns with one or several clusters. The user has the chance to pre x the variances and covariances of each cluster.

Counts module. Given a spatial point pattern in two dimensions, a grid of con-

tiguous quadrats can be generated (Ripley, 1981; Upton & Fingleton, 1994) where the size of the grid can be chosen by the user. There are no limits for the grid sizes, just those imposed by the computer memory. The aim with the quadrat counts is to investigate the pattern of the population under study by means of several indices reviewed in the following section. It is been a quite popular technique in environmental applications such as plant ecology, earthquake analysis and many other elds. The software gives the results based on rst and second-order counts.

Maximization and results module. Based on the counts obtained in the previous module, the following computer outputs are generated: { { { {

Denomination of the operative block. Date and hour in which the operations are runned. Name of the le which contains the counts. Information about the order ( rst or second) used in getting the counts and any results appeared after this prompt. { The grid size used for the counts. { The coordinates limits. { The basic cell area. 10

{ The number of points in the pattern. { The mean value of the cell counts,

x=

Pn i=1

n

xi

(37)

{ The standard deviation value, S , of the counts s

S=

Pn i=1

(xi ; x)2 n;1

(38)

{ The \index of dispersion", ID, value (Hoel, 1943) ID = (n ; 1) Sx

2

(39)

{ The \index of cluster size", ICS, value (David & Moore, 1954; Douglas, 1975) ICS = Sx ; 1 2

(40)

{ The \index of cluster frequency", ICF, value (Douglas, 1975)

x ICF = ICS

{ The \index of mean crowding", IMC, value (Lloyd, 1967) IMC = x + ICS

(41) (42)

Then, using the contiguous quadrats information, the program maximizes a pseudolikelihood function obtained from an Auto-Poisson distribution function as outlined in previous sections. The maximization, for the Strauss process, is carried out as a function of two parameters,  and ; which inform us about the intensity and the spatial arrangement of the points, respectively. The software gives the following outputs: { The value of the logarithm of the pseudolikelihood function. { Estimates of both parameters with their corresponding standard errors, the t;Student statistic, p-value, and the gradient. { Covariance matrix between  and . { Hypothesis tests of the random or Poisson pattern based on the likelihood ratio, Wald and Lagrange multipliers tests. For each one of them both the statistical value and the corresponding p-value are obtained. 11

5 A simulation study For this section we have chosen the Strauss model as a parametric family of point process models. This model has been widely used in spatial statistics and its parameters give a clear information about the spatial structure of the pattern. Then, conclusions drawn from this model can be extended to other more particular models. The goal of this study is to analyze the behaviour of the parameter estimators of  and ; obtained through the pseudolikelihood estimation method applied to the lattice system. Then, in turn to analyze how this lattice-based approximation system works in practice. Recall that  de nes the intensity of the point pattern and  its spatial structure ( = 1 de nes a random pattern and  < 1 a regular or inhibitory pattern). Of particular importance is the detection of cluster in terms of values of  > 1. Though this condition makes the density function (14) not to be integrable, we are able to simulate cluster patterns in such conditions and then also present the corresponding results.

5.1 Design of the simulation study

The process of simulations has been carried out for three qualitatively dierent spatial structures: random, cluster and regularity patterns, corresponding to the three dierent ranges of parameter . We have used several total number of points per pattern: for random structures, n = 400; 1000; 2500; for cluster structures, n = 1000; 2500; for regular patterns, n = 300; 800; 2500: Cluster pattern simulation is based on cluster Poisson processes with several number of fathers, ranging from 1 to 4. The bivariate normal distribution is used. Inhibitory pattern simulation is based on the simulation of a sequential spatial inhibition pattern (SSI) with inhibition radius ir = 0:01; 0:03; 0:05. For each combination, we simulated r = 2000 realizations in the unit square, (0; 1) x(0; 1). For each pattern, and to apply the lattice approximation system, we have used several grid sizes shown in Table 1. Each line of Table 1 gives us information about the type of simulated pattern, the number of points in each pattern (Points ), the number of replications for each experiment (Repl.), the number of fathers in the cluster simulation (Clust.), the inhibition value used in the regular pattern simulations (Inhib.), and grid order values. All grid orders are magnitudes to the square, i.e., (10  10; 15  15; 20  20, etc), but they are shown simpli ed (10; 15; 20, etc). For each simulated combination of pattern, replication and grid size we have evaluated  and  parameter estimators for rst and second-order schemes using the pseudolikelihood estimation method as described in section 3. Each combination of pattern, replicate, grid size and order schemes yielded 2000 estimates of both parameters which are summarised by box-plots and tables. Also, for each sample we have tested the null hypothesis of  = 1 based on the likelihood ratio test, the Wald test and the Lagrange multipliers test. It is known that these 12

Points Repl. Clust. Inhib. 400 1000 2500

2000 2000 2000

1000 1000 1000 1000 2500 2500 2500 2500

2000 2000 2000 2000 2000 2000 2000 2000

300 800 2500

2000 2000 2000

1 2 3 4 1 2 3 4

Grid order

Random pattern 10, 15, 20, 25, 30, 35, 40 : : : 95, 100 10, 15 : : : 35, 40, 50, : : : 220, 230 10, 15, 35, 40, 60, 80, 90, 110 : : : 380, 400 Cluster pattern 10, 20, 30, 40, 50, 60 : : : 180, 190, 200 20, 40, 60, 80, 100 : : : 160, 180, 200 20, 40, 60, 80, 100 : : : 160, 180, 200 20, 40, 60, 80, 100 : : : 160, 180, 200 20, 60, 100, 140, 180 : : : 340, 380, 400 20, 60, 100, 140, 180 : : : 340, 380, 400 20, 60, 100, 140, 180 : : : 340, 380, 400 20, 60, 100, 140, 180 : : : 340, 380, 400 Regularity pattern 0.05 10, 20, 30, 35 0.03 10, 20, 30, 40, 50, 60 0.01 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 Table 1: Simulation design.

statistical tests require normal distribution. Then, we carried out KolmogorovSmirnov (KS) tests to evaluate the gaussianity assumption of both parameters.

5.2 Results

Table 2 gives the results of the Kolmogorof-Smirnof test to evaluate the normal assumption of both parameter estimators. The sub-index 1 and 2 refer to the rst and second-order scheme, respectively. If a column has, for example, 10, this indicates the starting grid order from which the parameter estimator behavior is normal. If it has the word \all", it means that in any analyzed grid order the parameter estimator normality has not been rejected. The simulation results of  and  for each combination are shown in the gures and tables of the appendix. The corresponding tables show, for each pattern, grid order and order scheme, the mean value and the standard deviation, S (
), of the parameters obtained through the 2000 simulations per case. The gures show, again for each pattern and combination, the parameters box-plots (quartiles). The 13

Points Clust. Inhib. 1 400 1000 2500 1000 1000 1000 1000 2500 2500 2500 2500 300 800 2500

1 2 3 4 1 2 3 4

1

Random pattern 10 50 10 25 15 25 Cluster pattern all 40 20 40 20 60 20 40 all all 60 all all all all 60 Regularity pattern 0.05 all all 0.03 all all 0.01 all 10

2

2

25 15 15

70 60 40

all 40 20 20 all all all all

all all all 20 all all all all

all all all

all all all

Table 2: Results of Kolmogorov-Smirnov tests.

14

vertical axes measure the parameter estimator magnitude, and the horizontal axes the corresponding grid order.

5.2.1 Random patterns

(See tables 3a{3c and gures 1a{1c in the appendix). The convergence of  parameter estimator to the number of points (the intensity) for both order schemes and dierent number of points is clear as the number of grids increase. Also the standard deviation rapidly decreases as the number of grids increases. With respect to  parameter estimator, its value always oscillates around the unit for both order schemes. All the three statistical tests can not reject the null hypothesis H0 :  = 1. However, the standard deviation seems to be growing up as the number of grids increases though this increment is not signi cantly important to reject the null hypothesis. Note that a bigger order scheme reduces the standard deviation. The increment of the standard deviation with the grid order can be easily explained in the following terms. Recall expressions (29) andP(30). NotePthat both P t expressions have in the denominator e = 1  (grid order ; xi )+ e xi : Then, P t as the grid order increases, e increases and d2 ln(PL) =d2 decreases. Taking into account that the variance is the inverse of the hessian, the result holds. This explanation is valid for all the three pattern structures. i

i

5.2.2 Clustered patterns

(See tables 4a{4d and gures 2a{2d in the appendix). For shortness, we only show the gures corresponding to 1 and 2 clusters with 1000 points and 3 and 4 clusters with 2500 points in the unit square, though we report on the whole set of results. Patterns with 1000 points in the unit square: { The value of  parameter estimator clearly increases with the grid size until convergence is reached at around 900 indicating the importance of the grid size in a correct speci cation of this parameter in a cluster setting. The corresponding standard deviation keeps within the same range for all grid size and order scheme. { The value of  parameter estimator keeps at all time above the unit and increases with the number of grids until reaching a comfortable value signi cantly dierent from 1 which is indicative of clustered patterns. The magnitude of this parameter estimator in the rst-order scheme is bigger than those for the second-order. However, the latter order scheme gives lower standard deviations. { We have also analyzed the combination of both parameter estimators in terms of their product,   . This synthetic indicator, particularly important in the economical context, has an erratic behavior in small grid orders, but it converges to what economists call the total cluster index of the point pattern. 15

These results are still valid for those patterns with 2500 points in the unit square. However, we observe that we need a bigger grid order to get the same information as with 1000 points.

5.2.3 Regular patterns

(See tables 5a{5c an gures 3a{3c in the appendix). { The value of  parameter estimator seems to increase when the number of grids increases. Dierences are clearly found between both order schemes, the second-order scheme giving higher values of the intensity. In general, this parameter does not give a good estimation of the intensity of the process, in fact, it overestimates the real intensity of the process. { On the other hand,  values are signi cantly dierent from 1 from grid size 20x20 (the null hypothesis H0 :  = 1 is rejected) clearly detecting a lack of randomness in favour of regularity or inhibition. The standard deviations are kept within the same range for all grid sizes but the value of  decreases without apparent convergence with the grid size suggesting that there may be an optimum value for the grid size. Also, there seems to be an inverse relationship between both parameter estimators, if one of them does not converge to a xed value, incresing or decreasing at all times, the other one will not do. However, this parameter seems to agree with the strength of the inhibition, indicating we have a stronger interaction for bigger inhibition radius.

6 Conclusions and further research We have presented a methodology based on stochastic point processes obtained as the limit of lattice-based point processes. A simulation study has been conducted to analyze the behaviour of  and  parameter estimators for a particular pairwise interaction point process model with varying grid sizes. Using this methodology,  parameter estimator seems to detect all three possible pattern structures, even under aggregation, for which traditionally pairwise interaction models have been rejected. However, if the researcher is also looking for an adequate parameter estimator value, the grid size plays an important role. Analyzing the simulations it seems that there is an optimum grid size or a set of grid sizes which should be found. One possibility is to use goodness-of- t tests. It seems quite natural the existence of such optimum grid size as the bigger the grid size, the larger number of zeros are introduced in the lattice system which in turn aects the nature of the spatial structure. We have also showed that as the number of points in the point pattern increases, we need to increase the grid size to detect the same properties. So, the optimum grid size depends on the number of points. In any case, we do believe that much more research in this direction should be carried out. 16

It is important to stress here the practical limitations of the Strauss process in terms of detecting clustering. However, we still have obtained good results under this condition. In this paper we have not considered marked point processes. The authors are actually working on this kind of approximation for marked models. Also, it might be interesting to explore how this approximation behaves with other models which are not pairwise interaction models, such as nearest-neighbour or area-interaction processes. Then, our conclusions would be much more general as we might obtain adecuate cluster processes. We think it is worth pointing out here that there are lots of applications coming from dierent applied elds that could bene t from applying this methodology. Finally, the SPPA computer program has been a useful tool to work with this kind of spatial statistical techniques, since it allows us to reach the appropriate theoretical limits in terms of grid sizes. Those interested users may ask for this software to the corresponding authors.

Acknowledgements Don Edwards is gratefully acknowledged for his comments and suggestions on an earlier version of the paper.

References Baddeley, A.J. & Lieshout, M.N.M. (1993). Stochastic geometry in high-level vision. In K.V. Mardia and G.K. Kanji, (eds.), Statistics and images, Vol. 1 of Advances in Applied Statistics, 231-256. Baddeley, A.J. & Lieshout, M.N.M. (1995). Area-interaction point processes. Annals of Institute of Statistical Mathematics, 46, 601-619. Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, B 36, 192-236. Besag, J. (1977). Some methods of statistical analysis for spatial data. Bulletin of the International Statistical Institute, 47, 77-92. Besag, J., Milne, R. & Zachary, S. (1982). Point process limits of lattice processes. Journal of Applied Probability, 19, 210-216. Carter, D.S. & Prenter, P.M. (1972). Exponential spaces and counting processes. Z. Wahrscheinlichkeitsth, 21, 1-19. Cox, D.R. & Isham, V. (1980). Point processes. Chapman & Hall, London. Cressie, N. (1993). Statistics for Spatial Data. New York, John Wiley & Sons. Daley, D.J. & Vere-Jones, D. (1988). An introduction to the theory of point processes. Springer, New York. David, F.N. & Moore, P.G. (1954). Notes on contagious distributions in plant populations. Annals of Botany, 18, 47-53. 17

Diggle, P.J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London. Diggle, P.J., Gates, D.J. & Stibbard, A. (1987). A nonparametric estimator for pairwise interaction point processes. Biometrika, 74, 763-770. Diggle, P.J. (1993). Point Process Modelling in Environmental Epidemiology. In Vic Barnett & K. Feridun Turkman, (eds.), Statistics for the Environment, John Wiley & Sons Ltd., 89-110 Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. & Tanemura, M. (1994). On parameter estimation for pairwise interaction point processes. International Statistical Review, 62, 99-117. Douglas, J.B. (1975). Clustering and aggregation. Sankhya, 37 B, 398-417. Fiksel, T. (1984). Estimation of parameterized pair potentials of marked and non-marked Gibbsian point processes. Elektron. Inform. Kybernet., 20, 270-278. Fiksel, T. (1988). Estimation of interaction potentials of gibbsian point processes. Math. Operationsf. Statist. Ser. Statist., 19, 77-86. Gates, D.J. & Westcott, M. (1986). Clustering estimates for spatial point distributions with unstable potentials. Ann. Inst. Statist. Math., 38, 123-135. Geman, S. & Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6, 721-741. Geyer, C.J. & Thompson, E.A. (1992). Constrained Monte Carlo maximum likelihood for dependent data (with discussion). Journal of the Royal Statistical Society, Series B, 54, 657-699. Geyer, C.J. (1998). Likelihood inference for spatial point processes. In: Proccedings Seminaire Europeen de Statistique, "Stochastic geometry, likelihood and computation ". O. Barndor-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds), London, Chapman and Hall. Hoel, P.G. (1943). On indices of dispersion. Annals of Mathematical Statistics, 14, 155-162. Jensen, J.L. & Moller, J. (1991). Pseudo-likelihood for exponential family models of spatial point processes. Annals of Applied Probability, 1, 445-461. Kelly, F.P. & Ripley, B.D. (1976). On Strauss model for clustering. Biometrika, 63, 357-360. Lieshout, M.N.M. & Molchanov, I.S. (1997). Shot-noise-weighted processes: a new family of spatial point processes. Stochastic Models, 14, to appear. Lloyd, M. (1967). Mean crowding. Journal of Animal Ecology, 36, 1-30. Moller, J. (1998). Markov Chain Monte Carlo and spatial point processes. In: Proccedings Seminaire Europeen de Statistique, "Stochastic geometry, likelihood and computation ". O. Barndor-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds), London, Chapman and Hall. Moyeed, R.A. & Baddeley, A.J. (1991). Stochastic approximation of the MLE for a spatial point pattern. Scandinavian Journal of Statistics, 18, 39-50. 18

Ogata, Y. & Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Annals of the Institute of Statistical Mathematics, 33, 315-338. Ogata, Y. & Tanemura, M. (1984). Likelihood analysis of spatial point patterns. Journal of the Royal Statistical Society, B 46, 496-518. Ogata, Y. & Tanemura, M. (1989). Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns. Annals of the Institute of Statistical Mathematics, 41, 583-600. Papangelou, F. (1974). The conditional intensity of general point processes and an application to line processes. Z. Wahrscheinlichkeitsth, 28, 207-26. Penttinen, A. (1984). Modelling interaction in spatial point patterns: parameter estimation by the maximum likelihood method. Jyvaskyla Studies in Computer Science, Economics and Statistics, 7. Press, W., Flannery, B., Teulovsky, S. & Vetterling, W. (1986). Numerical Recipes in C: The Art of Scienti c Computing. Cambridge University Press, 2nd edition. Preston, C.J. (1976). Random elds. Lecture Notes in Mathematics 534, Springer-Verlag, Berlin. Preston, C.J. (1977). Spatial birth-and-death processes. Bull. Inst. Intern. Statist., 46, 371-391. Ripley, B.D. (1977). Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society,, B 39, 172-212. Ripley, B.D. & Kelly, F.P. (1977). Markov point processes. Journal of the London Mathematical Society, 15, 188-192. Ripley, B.D. (1981). Spatial Statistics, Wiley, New York. Ripley, B.D. (1988). Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge, New York. Ruelle, D. (1969). Statistical Mechanics, Wiley, New York. SPPA (1997). Spatial Point Pattern Analysis. Computer Software developed by Albert, Albert, Mateu, Pernias, Universitat Jaume I, Castellon. Stoyan, D., Kendall, W. & Mecke, J. (1995). Stochastic Geometry and its Applications. Akademie-Verlag, Berlin. Strauss, D.J. (1975). A model for clustering. Biometrika, 63, 467-475. Strauss, D.J. (1986). On a general class of models for interaction. SIAM Review, 28, 513-527. Takacs, R. (1986). Estimator for the pair-potential of a Gibbsian point process. Math. Oprationsf. Statist. Ser. Statist., 17, 429-433. Tomppo, E. (1986). Models and methods for analysing spatial patterns of trees. Communicationes Instituti Forestalis Fenniae, 138, 1-65. Upton, G. & Fingleton, B. (1994). Spatial Data Analysis by Example. Vol. 1, Wiley, New York.

19

Appendix

Grid 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

1

S (1 )

1

S (1 )

2

S (2)

2

S (2 )

423.2 100.1 0.9977 0.0152 456.4 2154.7 0.9953 0.0571 415.0 74.4 0.9967 0.0259 423.5 88.0 0.9970 0.0154 407.8 56.9 0.9973 0.0351 416.9 74.6 0.9963 0.0230 405.0 48.1 0.9976 0.0442 412.0 63.0 0.9961 0.0299 404.2 41.9 0.9968 0.0533 409.4 54.7 0.9955 0.0366 403.7 36.9 0.9958 0.0614 407.6 47.9 0.9950 0.0427 403.0 34.0 0.9957 0.0713 406.4 43.8 0.9945 0.0498 402.5 31.8 0.9955 0.0801 405.2 40.3 0.9946 0.0563 402.1 29.4 0.9956 0.0884 405.1 37.4 0.9930 0.0630 402.0 28.1 0.9946 0.0975 403.8 34.8 0.9940 0.0701 401.6 27.2 0.9957 0.1066 402.9 33.0 0.9951 0.0749 401.3 25.7 0.9956 0.1145 402.7 30.6 0.9943 0.0805 401.0 25.4 0.9977 0.1253 402.4 30.0 0.9946 0.0879 400.8 24.4 0.9990 0.1312 402.2 28.6 0.9941 0.0941 401.1 24.0 0.9947 0.1406 401.9 27.8 0.9947 0.0998 400.7 23.6 0.9986 0.1512 401.9 27.3 0.9938 0.1086 400.7 23.5 0.9985 0.1600 401.4 26.5 0.9960 0.1123 400.7 23.0 0.9977 0.1703 401.5 25.6 0.9946 0.1188 400.6 22.6 0.9983 0.1762 401.3 25.1 0.9948 0.1251 Table 3a: Random pattern with 400 points in the unit square.

20

Grid 10 15 20 25 30 35 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230

1

1038.7 1027.5 1012.9 1009.3 1008.5 1006.7 1002.4 1001.6 1001.9 1002.2 1000.8 1000.1 1000.4 1000.2 999.8 999.7 999.9 999.5 999.7 999.3 999.5 999.5 999.6 999.6 999.7 999.2

S (1 ) 192.9 159.2 130.4 112.0 96.0 84.9 76.1 63.5 56.2 50.4 47.5 44.2 42.4 40.6 39.0 38.0 37.5 36.8 36.0 35.8 35.4 34.9 34.7 34.1 34.1 34.0

1

0.9993 0.9990 0.9994 0.9993 0.9988 0.9986 0.9997 0.9995 0.9987 0.9975 0.9988 0.9995 0.9985 0.9985 0.9996 0.9998 0.9986 1.0002 0.9990 1.0020 1.0003 1.0003 0.9990 0.9982 0.9978 1.0038

S (1 )

0.0048 0.0091 0.0132 0.0173 0.0211 0.0249 0.0286 0.0354 0.0429 0.0488 0.0570 0.0635 0.0708 0.0767 0.0845 0.0899 0.0967 0.1071 0.1123 0.1196 0.1293 0.1347 0.1426 0.1504 0.1570 0.1647

2

S (2 )

2

S (2)

1073.5 1893.5 0.9987 0.0310 1030.0 150.7 0.9993 0.0044 1026.9 150.8 0.9990 0.0076 1024.0 140.2 0.9987 0.0109 1019.9 125.1 0.9984 0.0138 1015.8 112.7 0.9982 0.0169 1008.4 98.9 0.9990 0.0193 1006.3 82.7 0.9987 0.0244 1006.1 71.2 0.9978 0.0293 1004.1 63.6 0.9980 0.0342 1001.6 57.9 0.9991 0.0391 1001.1 54.1 0.9992 0.0448 1001.1 50.8 0.9989 0.0491 1000.8 47.5 0.9988 0.0538 1000.6 45.5 0.9987 0.0591 999.8 43.9 1.0002 0.0636 1000.2 42.6 0.9991 0.0694 999.4 41.2 1.0009 0.0747 999.8 40.0 0.9996 0.0794 999.4 39.7 1.0011 0.0846 999.7 38.5 0.9999 0.0889 999.3 37.7 1.0013 0.0946 999.6 37.4 1.0002 0.1003 999.4 36.7 1.0012 0.1052 999.8 36.3 0.9986 0.1081 999.4 36.3 1.0010 0.1145

Table 3b: Random pattern with 1000 points in the unit square.

21

Grid 10 15 20 25 30 35 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

1

S (1 )

1

S (1 )

2

S (2 )

2

S (2 )

2916.7 15798.7 0.9998 0.0021 5067.1 51375.3 0.9976 0.0449 2546.6 327.2 0.9997 0.0030 2534.7 261.2 0.9999 0.0012 2541.4 298.5 0.9996 0.0048 2540.2 280.0 0.9998 0.0023 2523.4 262.5 0.9997 0.0067 2534.2 288.7 0.9997 0.0037 2524.6 233.8 0.9995 0.0084 2533.5 276.9 0.9996 0.0050 2514.5 199.4 0.9996 0.0097 2529.3 247.3 0.9995 0.0061 2515.8 178.7 0.9994 0.0113 2533.5 225.1 0.9992 0.0072 2507.6 126.6 0.9993 0.0171 2518.0 170.9 0.9991 0.0120 2502.8 98.3 0.9997 0.0223 2509.9 132.4 0.9991 0.0162 2503.8 85.6 0.9990 0.0281 2506.3 109.7 0.9992 0.0198 2501.9 76.3 0.9995 0.0334 2503.7 94.2 0.9994 0.0232 2501.8 70.9 0.9992 0.0394 2503.0 86.0 0.9993 0.0274 2500.3 66.1 1.0003 0.0453 2501.4 78.0 0.9998 0.0314 2500.3 63.6 1.0003 0.0512 2500.7 73.6 1.0001 0.0356 2499.9 60.1 1.0010 0.0572 2500.7 69.5 1.0000 0.0394 2500.7 60.0 0.9997 0.0636 2500.4 67.7 1.0003 0.0446 2500.6 58.7 0.9997 0.0705 2501.1 65.5 0.9995 0.0484 2500.7 56.1 0.9993 0.0732 2499.9 61.9 1.0008 0.0509 2500.9 55.2 0.9983 0.0808 2500.5 60.8 1.0000 0.0561 2500.8 55.3 0.9984 0.0847 2500.6 60.1 0.9998 0.0598 2501.1 54.7 0.9972 0.0910 2500.7 59.2 0.9997 0.0649 2500.5 53.3 0.9992 0.0944 2501.0 57.2 0.9987 0.0666 2500.4 53.7 0.9998 0.1039 2500.3 56.8 1.0003 0.0708 2500.3 53.6 1.0002 0.1116 2500.5 56.3 0.9998 0.0765 2500.6 52.9 0.9985 0.1159 2500.6 55.8 0.9994 0.0811 Table 3c: Random pattern with 2500 points in the unit square.

22

1

Grid 10 20 40 60 80 100 120 140 160 180 200

331.8 395.3 543.2 668.8 756.7 818.6 860.6 890.7 912.2 928.2 940.5

S (1 ) 15.9 21.6 25.6 26.1 25.7 24.2 22.9 20.9 18.8 17.4 16.1

1

1.0230 1.0759 1.2095 1.3250 1.4130 1.4732 1.5184 1.5492 1.5734 1.5914 1.6035

S (1 )

0.0010 0.0041 0.0145 0.0285 0.0463 0.0655 0.0878 0.0107 0.1250 0.1451 0.1646

2

333.9 365.8 459.5 562.5 650.0 720.5 774.2 816.3 848.3 872.8 892.1

S (2) 12.2 15.1 19.9 19.6 20.9 21.9 22.3 21.9 21.1 20.1 19.2

2

1.0119 1.0410 1.1293 1.2234 1.3071 1.3743 1.4287 1.4693 1.5021 1.5300 1.5526

S (2 )

0.0004 0.0016 0.0053 0.0111 0.0192 0.0299 0.0427 0.0562 0.0703 0.0839 0.0988

Table 4a: 1-cluster pattern with 1000 points in the unit square.

Grid 10 20 40 60 80 100 120 140 160 180 200

1

250.4 344.9 464.6 571.1 658.6 725.9 778.1 818.3 848.7 873.1 891.9

S (1 ) 11.4 19.9 23.2 23.7 23.1 21.4 20.9 20.1 19.9 18.2 17.6

1

1.0242 1.0702 1.2140 1.3776 1.5308 1.6651 1.7771 1.8680 1.9468 2.0076 2.0615

S (1 )

0.0007 0.0038 0.0134 0.0254 0.0391 0.0535 0.0731 0.0945 0.1229 0.1416 0.1697

2

208.8 311.7 397.5 478.7 554.8 620.6 676.9 723.2 761.7 794.1 820.5

S (2 ) 8.1 13.9 17.7 18.0 17.2 17.4 18.2 18.9 18.9 18.5 19.0

2

1.0143 1.0382 1.1230 1.2328 1.3484 1.4609 1.5639 1.6576 1.7403 1.8111 1.8742

S (2 )

0.0002 0.0017 0.0063 0.0117 0.0167 0.0223 0.0306 0.0416 0.0536 0.0665 0.0849

Table 4b: 2-cluster pattern with 1000 points in the unit square.

23

Grid 20 60 100 140 180 220 260 300 340 380 400

1

812.2 1154.4 1448.8 1686.5 1866.9 1999.4 2097.8 2171.6 2228.8 2272.8 2291.3

S (1 ) 26.3 38.0 36.0 32.6 30.5 29.0 27.7 25.4 24.9 22.9 22.1

1

1.0279 1.1809 1.3894 1.5942 1.7717 1.9203 2.0413 2.1397 2.2157 2.2778 2.3037

S (1 )

0.0008 0.0073 0.0164 0.0267 0.0392 0.0538 0.0710 0.0869 0.1098 0.1265 0.1358

2

740.1 1022.1 1237.9 1442.8 1618.1 1763.9 1881.6 1976.7 2053.5 2116.3 2143.4

S (2 ) 17.7 31.5 28.1 25.8 24.0 23.9 24.3 23.8 24.3 23.8 23.2

2

1.0153 1.1004 1.2341 1.3834 1.5307 1.6658 1.7867 1.8922 1.9839 2.0621 2.0963

S (2 )

0.0003 0.0035 0.0076 0.0119 0.0160 0.0212 0.0285 0.0369 0.0486 0.0603 0.0657

Table 4c: 3-cluster pattern with 2500 points in the unit square. Grid 20 60 100 140 180 220 260 300 340 380 400

1

813.4 1223.1 1555.5 1804.8 1979.2 2102.6 2189.0 2252.8 2299.5 2334.5 2348.2

S (1 ) 23.7 37.2 36.3 34.7 33.2 30.1 27.8 26.2 24.2 21.6 21.6

1

1.0309 1.1910 1.3816 1.5451 1.6729 1.7663 1.8378 1.8868 1.9239 1.9533 1.9691

S (1 )

0.0008 0.0075 0.0167 0.0287 0.0440 0.0585 0.0750 0.0944 0.1115 0.1246 0.1373

2

726.5 1062.8 1320.5 1548.6 1734.9 1881.1 1994.7 2082.5 2151.3 2205.3 2228.6

S (2 ) 13.7 25.5 26.4 26.8 27.4 27.5 27.8 27.3 26.0 25.2 24.6

2

1.0177 1.1098 1.2420 1.3764 1.4964 1.5977 1.6804 1.7482 1.8025 1.8475 1.8646

S (2 )

0.0003 0.0031 0.0068 0.0112 0.0173 0.0253 0.0356 0.0469 0.0574 0.0703 0.0760

Table 4d: 4-cluster pattern with 2500 points in the unit square. Grid 10 20 30 35

1

601.9 599.7 742.8 774.2

S (1 ) 80.7 47.6 22.8 24.1

1

0.9360 0.7703 0.3439 0.1244

S (1 )

2

S (2 )

2

S (2)

0.0123 480.3 44.8 0.9764 0.0047 0.0249 768.1 108.5 0.8372 0.0233 0.0234 1449.2 101.4 0.4660 0.0178 0.0224 1409.9 47.1 0.2966 0.0142

Table 5a: Regularity pattern, inhibition 0.05, with 300 points in the unit square. 24

Grid 10 20 30 40 50 60

1

S (1 )

1

S (1 )

2

S (2 )

2

1251.1 113.7 0.9849 0.0031 1042.5 53.7 0.9953 1834.2 149.0 0.8959 0.0100 1697.4 137.2 0.9502 1590.8 99.7 0.8153 0.0162 1994.6 208.0 0.8722 1827.1 45.5 0.6146 0.0121 3308.8 243.5 0.6761 2026.2 37.1 0.3331 0.0142 4069.0 169.5 0.4564 2096.6 39.4 0.0848 0.0121 3888.0 67.4 0.2504

S (2)

0.0009 0.0053 0.0141 0.0152 0.0104 0.0085

Table 5b: Regularity pattern, inhibition 0.03, with 800 points in the unit square.

Grid 10 20 30 40 50 60 70 80 90 100

1

2541.9 2657.2 2861.3 3068.9 3227.7 3356.5 3467.5 3578.0 3683.8 3784.8

S (1 ) 258.7 282.6 245.0 201.4 170.0 144.8 120.9 103.0 89.4 76.6

1

0.9999 0.9977 0.9878 0.9669 0.9360 0.8946 0.8422 0.7763 0.6968 0.6031

S (1 )

0.0011 0.0045 0.0079 0.0107 0.0131 0.0152 0.0165 0.0174 0.0183 0.0185

2

2543.1 2622.9 2811.9 3064.7 3310.6 3543.3 3762.3 3986.8 4236.7 4504.9

S (2 ) 181.2 244.9 273.4 276.2 260.2 239.8 220.5 203.6 193.3 176.3

2

0.9999 0.9991 0.9946 0.9835 0.9645 0.9369 0.9003 0.8530 0.7926 0.7185

S (2)

0.0004 0.0020 0.0046 0.0074 0.0100 0.0121 0.0140 0.0155 0.0171 0.0174

Table 5c: Regularity pattern, inhibition 0.01, with 2500 points in the unit square.

25

900

900

800

800

700

700

600

600

500

500

400

400

300

300

200

200 10

20

30

40

50

60

70

80

90

100

10

20

30

40

(a) 1

50

26

60

70

80

90

100

60

70

80

90

100

(b) 2

2.0

2.0

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4 10

20

30

40

50

(c) 1

60

70

80

90

100

10

20

30

40

50

(d) 2

Figure 1a: Random pattern with 400 points in the unit square.

2000

2000

1800

1800

1600

1600

1400

1400

1200

1200

1000

1000

800

800

600

600

400

400 50

100

150

200

50

(a) 1

100

27

150

200

150

200

(b) 2

2.0

2.0

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 50

100

(c) 1

150

200

50

100

(d) 2

Figure 1b: Random pattern with 1000 points in the unit square.

3400

3400

3200

3200

3000

3000

2800

2800

2600

2600

2400

2400

2200

2200

2000

2000

1800

1800 50

100

150

200

250

300

350

400

50

100

150

(a) 1

200

28

250

300

350

400

250

300

350

400

(b) 2

1.5

1.5

1.4

1.4

1.3

1.3

1.2

1.2

1.1

1.1

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 50

100

150

200

(c) 1

250

300

350

400

50

100

150

200

(d) 2

Figure 1c: Random pattern with 2500 points in the unit square.

1000

1000

900

900

800

800

700

700

600

600

500

500

400

400

300

300

200

200 0

50

100

150

200

0

50

(a) 1

100

29

150

200

150

200

(b) 2

2.4

2.4

2.2

2.2

2.0

2.0

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0 0

50

100

(c) 1

150

200

0

50

100

(d) 2

Figure 2a: 1-cluster pattern with 1000 points in the unit square.

1000

1000

900

900

800

800

700

700

600

600

500

500

400

400

300

300

200

200 0

50

100

150

200

0

50

(a) 1

100

30

150

200

150

200

(b) 2

2.8

2.8

2.6

2.6

2.4

2.4

2.2

2.2

2.0

2.0

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0 0

50

100

(c) 1

150

200

0

50

100

(d) 2

Figure 2b: 2-cluster pattern with 1000 points in the unit square.

2400

2400

2200

2200

2000

2000

1800

1800

1600

1600

1400

1400

1200

1200

1000

1000

800

800

600

600 0

50

100

150

200

250

300

350

400

0

50

100

(a) 1

150

200

31

250

300

350

400

250

300

350

400

(b) 2

3.0

3.0

2.5

2.5

2.0

2.0

1.5

1.5

1.0

1.0 0

50

100

150

200

(c) 1

250

300

350

400

0

50

100

150

200

(d) 2

Figure 2c: 3-cluster pattern with 2500 points in the unit square.

2500

2500

2000

2000

1500

1500

1000

1000

500

500 0

50

100

150

200

250

300

350

400

0

50

100

(a) 1

150

200

32

250

300

350

400

250

300

350

400

(b) 2

2.6

2.6

2.4

2.4

2.2

2.2

2.0

2.0

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0 0

50

100

150

200

(c) 1

250

300

350

400

0

50

100

150

200

(d) 2

Figure 2d: 4-cluster pattern with 2500 points in the unit square.

2000

2000

1800

1800

1600

1600

1400

1400

1200

1200

1000

1000

800

800

600

600

400

400

200

200 5

10

15

20

25

30

35

40

5

10

15

(a) 1

20

33

25

30

35

40

25

30

35

40

(b) 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 5

10

15

20

(c) 1

25

30

35

40

5

10

15

20

(d) 2

Figure 3a: Regularity pattern, inhibition 0.05, 300 points in the unit square.

5000

5000

4500

4500

4000

4000

3500

3500

3000

3000

2500

2500

2000

2000

1500

1500

1000

1000

500

500 0

10

20

30

40

50

60

70

0

10

20

(a) 1

30

34

40

50

60

70

40

50

60

70

(b) 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 0

10

20

30

(c) 1

40

50

60

70

0

10

20

30

(d) 2

Figure 3b: Regularity pattern, inhibition 0.03, 800 points in the unit square.

5500

5500

5000

5000

4500

4500

4000

4000

3500

3500

3000

3000

2500

2500

2000

2000

1500

1500 0

20

40

60

80

100

0

20

(a) 1

40

35

60

80

100

60

80

100

(b) 2

1.1

1.1

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5 0

20

40

(c) 1

60

80

100

0

20

40

(d) 2

Figure 3c: Regularity pattern, inhibition 0.01, 2500 points in the unit square.