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7 Spatial Statistics in Ecology Peter Guttorp University of Washington 7.1 Introduction Ecological theory is essentially spatial in character. Many methods for an- alyzing spatial data have been developed in an ecological context (Hertz, 1909; Greig-Smith, 1952; and Kershaw, 1957, are some important early ref- erences). Methods from spatial statistics have recently seen an increasing use in this field. Perhaps the most important data for quantitatively ori- ented plant ecologists are complete maps of the vegetation in an area at different times. While the construction of such maps used to be an incred- ibly time-consuming fieldwork task, modern digitization techniques enable an increased use of aerial photographs and satellite images. Here, as in many other fields, there has recently been a substantial increase in both the quantity and volume of data potentially available to the ecological mod- eler. Some overviews of the use of spatial methods in ecological analysis are Ripley (1987) and Legenctre and Fortin (1989~. Typically, a large number of factors interact in ecological processes, and the precise nature of these interactions is the subject of study. For example, in the study of forest growth, a limiting factor is availability of light (Ford and Diggle, 1981~. The death of a large tree yields sudden possibilities for growth of plants that would otherwise remain very small, and cart completely change the competitive advantages between species. The introduction of new species may eliminate many previously successful competitors (Ford, 1975, Linhart, 1976~. In order to evaluate forest resource management plans, it may prove important to develop adequate stochastic moclels for species growth and competition. The interactions take place at different scales: tile extent of a tree crown limits the availability of light, decreasing the potential 129

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130 for other growth beneath the crown, whereas the availability of nutrients in the local region can increase growth potential on a somewhat larger scale. In this chapter, we concentrate on one approach to stochastic modeling of ecological communities, namely, spatial point processes. Moclels for an- imal communities often need to include movement explicitly. The theory of branching diffusions (Dawson and {vanoff, 197S, KuIperger, 1979) can sometimes be applied to such situations. There is a plethora of predator- prey models in the applied probability literature, although so far most of them are not specifically spatial in nature. There is a need for more work on spatially nonhomogeneous competition models. Section 7.2 introduces the general concepts of point processes, discusses nonparametric estimation of second order parameters, and presents some particular models that have found use in the literature. Section 7.3 con- tains an outline of a point process approach to modeling single species for- est growth. It must be emphasized here that the efforts to date of using stochastic models (in particular point process models) and their attendant statistical analysis to aid ecological understanding has had only very limited success. This is due partly to oversimplifications (such as using only homo- geneous models or studying only one species rather than the interactions of several), partly to lack of high-quality data, and partly to the difficulty in interpreting interactions at vastly different scales. More work is also needed! on how to combine inference from the individual pieces that together make up a model of a complex system. 7.2 Point Processes A point process is a process of locations of events, taking place in some space X. Each event may have associated with it a mark, taking place in some mark space A. For example, an event may be a tree, and the mark may be the species of the tree, its crown length, crown angle, height, and diameter. An excellent description of point process theory is Daley ancl Vere-Jones (198S, especially ch. 7~. The random variable N(A) counts the number of events in the set A C X. A marked point process is a point process on X x y with the additional property that the marginal process of locations 1\~(A x Y), A C X is itself a point process. A case of particular interest is a multivariate point process, where y = {1, . . ., m) for some finite integer m. Harkness en cl Isham (1983) study a bivariate point process (i.e., m = 2) of ant nests for the species Cataglyphis

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131 bacolor and Messor wasmanni. Their main interest is in assessing whether the locations of Cataglyphis nests are dependent upon those of the Messor ants. This is suggested on biological grounds, since Cataglyphis ants eat dead insects, mainly Messor ants, whereas the latter collect seeds for food. An example of a trivariate point process is the data collected by Gerrard (1969) and analyzed by Besag (1977), Diggle (1983, sec. 7.1), and others, which contains locations of hickory, oak, and maple trees in Lansing Woods, Michigan. Of interest here is the interactions between the species. We return to these examples below. An important class of point processes consists of those whose distribution is invariant under translations; these are caned stationary or homogeneous. Those in the sub class of isotropic processes have distributions that adcli- tionally are invariant uncler rotation. The assumptions of homogeneity and isotropy are perhaps made more often than the various applications warrant. rm. ~ Me series analysis has benefittec! much from studying second order properties. These can be estimated nonparametricaDy, and for a Gaussian series completely determine the probabilistic structure of the series. But even in non-Gaussian cases, second-order parameter functions such as the spectrum or the correlogram convey interesting information. In the case of point processes, second-orcler parameters are perhaps less informative (Baddeley and Silverman, 1984), but are still an important aspect of the analysis of a point pattern. Diggle (1983, ch. 5; see also Ripley, 198S, ch. 3) presents second-order parameter estimation, and Brillinger (1978) discusses the relation between time series and point process analysis. In what follows, we concentrate on the spatial case where X = R2. The second-order product density of a point process is defined by E N(dx~) Eddy) Aim A2(x,y)= 11111 ~d ~~d . For a stationary process, )2 depends only on the vector xy, and if the process is also isotropic, it further depends only on the length t = fixye. A common variant of )2 for stationary isotropic processes is (Ripley, 1976) Aft' ;2~ ~2~y ~ ~p 0 0 ~ where ~ is the rate of the process. The parameter function I((t) measures the expected relative rate of events within distance t of an arbitrary event. For example, in a Poisson process (7.2.1 below) we have I((t) = art,

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132 and for a Poisson cluster process of Neyman-Scott type (7.2.2) I~ (t) = arts + E 5~5 - 1~02(t)/(pCE25), where S is the number of points in a cluster, H2 is the cUf of the vector difference between two points in the same cluster, and Pc is the rate of the cluster process. Bartlett (1964) stressed the inferential importance of the distribution of nearest-neighbor distances (which is equivalent to the I(-function introduced above). Ripley (1976) proposed to estimate I~ (t) from points A, . . . ,xr, in a set A by I~ (t) = ,, , . where uij = AxeXj~ and wij is the proportion of the area of the sphere of radius uij about xi inside A. This nonparametric estimator can be used to fit a parametric model by minimizing the distance between the estimate and the parametric form of the function. When comparing: to a Poisson process ~ ~7 ~ , it is a common practice to use a square root transformation to stabilize the variance of the plotted function. Harkness and Isham (1983) found that this plot for Messor nests (Figure 7.1) lay below the envelope for simulated values from a Poisson process for distances below 50 feet, indicating an inhi- bition between nests, presumably due to the foraging practices of these ants (similar findings for other ant species are reported by Lerings and Franks, 1982~. On the other hand, the Cataglyphas nests were consistent with spatial randomness. Further analysis indicated a tendency for Cataglyphis nests to be located at or near the mean foraging path for a Messor nest (Figure 7.2~. Harrison and Gentry (1981) discussed biological and statistical aspects of foraging paths for a single species. The study area consisted of about half scrub anti half field, and the Cataglyphas nests were located mostly in the field region. Stationarity over the entire study region did not seem to be a reasonable assumption for these nests, and Harkness and Isham separated out the field region in their stu(ly. In the case of anisotropic stationary point processes one can estimate >2 directly using the obvious empirical counterpart; essentially a histogram estimator (cf. BriDinger, 1978; Ohser and Stoyan, 1981~. Standard error for the estimators can often be developed under the Poisson hypothesis (Baddeley, 1980), whereas for more complicated processes, one may have to use Monte Carlo methods to assess the variability (see the examples in

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133 {K(~)/~} 100 80 60 40 20 - , ', /, , An, , '/,/ . '/,/ // / ,,>/,/ ,~" "it"' ,';//"' , ... . , .. . 'I"' data ---- envelope of simulations ,,,~ ,/ , , . . i_ , , , 20 40 60 80 100 120 t (ft) ~ 1 FIGURE 7.~: (~(t)/;r): for the Messor nests, calculated at 5-ft inter- vals, together with the envelope of 19 curves from simulated Poisson data. Reprinted, by permission, from Harkness and Isham (1983). Copyright ~ 1983 by the Royal Statistical Society. Besag and Diggle, 1977). Bootstrap and other resampling methods have been proposed in the ecological literature by Solow (1989~. For multivariate processes a cross intensity can be defined. The corre- sponding I(-function is ]~$j(t)=2~)iA;) ii Nij(u)udu, o where Ai is the rate of points of type i, and Aij is the cross-intensity function defined by ~ (I 1) ii ENt(d~)~(dy) . Corresponding quantities can be defined for more general marked point pro- cesses, and their estimation is discussed by Hanisch and Stoyan (1979~.

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134 1 ~ q _ ~ . _ _ ~ _ _ ~ e: or as, A. ~ 1 0 c, ~ ~ ~ FIGURE 7.2: Mean foraging path for seven Messor nests. The solid dots correspond to Cataglyphas nests, and the open dots denote Messor nests. The field is between 330 and 340 ft wide. Reprinted, by permission, from Harkness and Isham (1983~. Copyright ~ 1983 by the Royal Statistical Society. 7.2.1 The Poisson Process The simplest mode! for point processes is the completely random, or Poisson process. To define it, assume that there is a finite measure A, such that for all finite families of disjoint intervals Al, . . ., Ak we have P(~N(~Ai) = n', i = 1, . . ., k) = II ~ i') exitA(Ai)) . i=! I. In particular, the counts in disjoint sets are independent, and hence one can- not improve the prediction of the number of points in a set from information about numbers of points in, say, surrounding sets. This is what constitutes the complete randomness of the Poisson process. There are many equivalent ways of describing the distribution of point processes. For example, one may be able to specify the zero probability function ((A) = P(N(A) = 0), or the probability generating functional (pgfl) G(h) = E (exp i/ log h(X)N(1X)] defined for all real measurable functions h with O < h 1 such that 1h

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135 vanishes outside a bounded set. The pgfl of a Poisson process is G(h) = exp (-do (1- h~x))A(dxjJ The Poisson process is often taken as a nub hypothesis, to be rejected in favor of some more structured ecologically relevant process. This was com- mon practice in the nineteenth century (Darwin, ISSI, Hensen, 1884) and is still a very common hypothesis in ecological models. Besag and Dig- gle (1977) discuss how to assess such a pattern (as wed ones) using Monte Cario testing, which enables a researcher to test specific hypotheses by simulating the assumed process, and then to check whether the observed statistic of interest is extreme among the simulations. Among other examples, the authors applied this to the locations of 65 Japanese black pine saplings (Numata, 1961; cf. Bartlett, 1964~. More specifically, they used Monte Cario testing on a X2-statistic comparing observed in- tertree distances to what would be expected under spatial randomness. The observed X2-statistic, which would have been deemed significant were the intertree distances independent, was in fact found consistent with a Pois- son process. Much confusion has arisen in the ecological literature (and elsewhere) from a failure to appreciate the statistical dependence present in inter-event distances of a Poisson process. A more detailed analysis of spatial patterns of ponderosa pines was per- formed by Getis and Franklin (1987) who found that, while the overall pat- tern of locations was consistent with spatial randomness, nearest neighbor distances for individual trees showed evidence of clustering on relatively large scales (about 20 m), and inhibition (presumably due to competition) on smaller scales (about 6 m). Here the mapping was done from aerial photographs, and the smallest resolvable distance was 2.4 m. 7.2.2 Cluster Processes as more complex The concepts of clustering and regularity are important ecological concepts, describing deviations from the completely random process. On an intuitive level, clustering describes the phenomenon of an ecological niche, or local regions with higher than average density, separated by regions of Tow density, while regularity indicates a tendency towards spacing between individuals. A cluster point process is a two-tiered process, defined in a conditional fashion. Given a point process Nc of cluster centers, one associates with each of its events a secondary point process Ns(-~X), centered at an event

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136 at x. The cluster process is the superposition of these secondary processes. Formally, N(A) N~(A~x)Nc(dx). x Usually the secondary processes are assumed independent pgfl takes the simple form G(h) = GC(G3(h~-~), , in which case the where GC is the pgfl of the process of cluster centers and Gs(~x) is the peg of a secondary process centered at x. A necessary and sufficient condition for the existence of a cluster process is that / ( 1Is ( A ~ x ) ) 1Vc tax ) < oo a. s. [Nc] . A special case which has found many applications is the Poisson cluster process, where Nc is a Poisson process. The pgfl for a Poisson cluster process iS G(h) = exp (|x(Gs(hlx)l)~(dx)) It is easily shown that the Poisson cluster process is overdispersec3 with respect to a Poisson process with the same mean measure, i.e., that the cluster process shows greater variability in the number of events in a set. This overdispersion has often been taken as a definition of clustering in both ecological and engineering literature. However, it is easy to construct clustering processes (where Nc is non-Poisson) which are unclerdispersed relative to a Poisson process. For example, the findings (described above) of Getis and Franklin (1987), as wed as the similar earlier results of Besag (1977), may be described by a cluster process (on larger scales) driven by a primary process that is more regular (on small scales) than a Poisson process. The most common Poisson cluster process is the Neyman-Scott process, in which a random number of points are laid out in an i.i.~. fashion around the cluster center. This mode} was introduced by Neyman (1939) to describe the dispersion of larvae in a field. It has since found important applications in astronomy to describe the distribution of clusters of galaxies (Neyman and Scott, 1959; Peebles, 1980), and in hydrology, where it has been used to describe precipitation (Kavvas and Delleur, 1981, Rodriguez-Iturbe et al., 1984; cf. ch. 4).

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137 7.2.3 The Cox Process The doubly stochastic Poisson process (often called a Cox process) arises when the mean measure A of a Poisson process is taken as a realization of a nonnegative stochastic process. A detailed discussion can be found in Grandell (1976) and Karr (1986~. The pgfl of a Cox process is G(h) = L~1 - h), where [A iS the Laplace functions of the stochastic process (random mea- sure) A. It follows that Var N(A) = E N(A) + Var (~(A)), so that a Cox process is also overdispersed relative to a Poisson process. As an example, consider the shot noise process, used by Vere-Jones and Davies (1966) to model earthquake sequences (including aftershocks). It is a Cox process with A given by 1~(A) = ~ Yi / flub flu, i A+~ where Ti are the locations of events in a temporal Poisson process of con- stant rate v, which triggers stresses of random amplitude Y', assumed i.i.~. These can give rise to major earthquakes. The intensity then decays accord- ing to some nonnegative integrable function f on [O,oo), possibly yielcling aftershocks. Consequently, (/x jR+ ~] hit f (tz) 4t - i3u ~x) where (t) = E expeltY). Comparing this to the Poisson cluster process pgfl given above, we see that it is of the same form. Hence, the shot noise process (so named since the moments agree with the moments derived by Campbell, 1909, for shot noise in vacuum tubes) is a Poisson cluster process (in fact, a Neyman-Scott process), and the two different mechanisms for constructing the process are indistinguishable from data. However, from an ecological point of view the two mechanisms are very different, and need to be distinguished from each other. In order to do so, more complex descrip- tions (perhaps involving more factors or species) are required. Although the Cox process is overdispersed (clustered) relative to a Pois- son process, a multivariate version can be constructed to model extreme inhibition between patterns. Let N = GNU, N2) be a bivariate point process driven by a bivariate nonnegative stationary stochastic process A(x), such

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138 that given A, the two components No and N2 are independent Poisson pro- cesses, but A~(x) + A2(x) = a, where ~ is a positive constant. Then (Diggle, 1983, sec. 6.6.2) >12(U) =C22(U) + >~2, where c22(u) is the covariance density for A2(x). Consequently, *' ~ ax .9 ^12ttJfit- = - (2~1~2~-l ~~jj(~)Its. A plot of a nonparametric estimate of the left-hand side of this equation against a similar estimate of the right-hand side may indicate the adequacy of this model. Die:~:le (1983 sec. 7.7) applied this model to the Lansing: Woods data. ~~ ~ , , i,: ~ ~ As demonstrated by Besag (1977), there is a strong negative dependence be- tween maples and hickories. The diagnostic plot mentioned above indicates that the fit of the competing Cox model is reasonable. However, the su- perposition of maples and hickories, which under this model should exhibit spatial randomness, does not follow a Poisson process. When adding the oaks, the Poisson fit for the superposition is adequate (although there still is some indication of clustering in the superposition process, possibly due to the other kinds of trees that are left out of the analysis). The oaks exhibit much less overdispersion than the other two species. A nonparametric esti- mate of (local) intensity confirms that a compensatory mechanism may be operating, but does on the other hand cast some doubt over the stationarity assumption. On the whole, this analysis, while providing a nice description of the observed spatial pattern, fails to produce an ecological explanation of it. 7.2.4 Markovian Point Processes Markovian models, which are defined through a local dependence structure, have found much use in biology. In the spatial context, Markovian point processes were introduced by Strauss (1975) and by Ripley and Kelly (1977~. A point process on a finite region A is Markov of range ~ if the conditional density of a point at x, given all the points in A \ {x3, only depends on the points in the sphere of radius ~ around x (excluding x itself). Call two points neighbors if their distance is less than I, and define a clique as a set of mutual neighbors. It is convenient to describe the distribution of a

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139 . point process in terms of its likelihood ratio (Radon-Nikodym derivative) with respect to a unit rate Poisson process. In general this can be written i(X1, ~ An) C I| 9i(xi) I| 9ij(xi, xj) 912...n(XI ~ ~ An) i j>i Ripley and Kelly proved that if a process is Markovian, it must have all of the g-functions identically 1, except when the arguments constitute a clique. This generalizes to other neighborhood systems, not necessarily distance- based. The simplest nontrivial conditionally specified point process (also called a Gibbs point process) is one in which only pairwise interactions are allowed. Then (x1, . . , an) or exp ~ tb1 (xi) + ~ 2(Xi, xj) ~i=1 i A. If the point process is stationary, 2 depends only on the distance between its arguments, and ~ is a constant. Writing boxy) = Vexye), we can specify the type of interaction by specifying V. These models are most commonly used to mode} repulsive interactions, leading to what is often called a regular point pattern (Strauss, 1975, Ogata and Tanemura, 1984). Examples include V(r) _ 0, the Poisson process; V(r) = - log(l - exp( - (r/a)2), a soft core repulsive model; V(r) = (cr/r)k, an intermediate case; and V(r) = oo if r < a, and 0 otherwise, a hard core rejection mode! (where no points are closer than a). Bartlett (1975, sec. 3.2.2) applied a simple inhibitory mode] to the spatial distribution of guns' nests. The idea was to regard the distribution of nests as following a Poisson process, but to allow for the association with each point of a random cutoff, within which radius no other nests can be found. This combined the hard-core rejection model above with features of the Cox process of 7.2.3. It is Circuit to estimate the parameters of Markovian point process models, mainly because the normalizing constant in the likelihood is very hard to evaluate. The two most common approaches are to use approxima- tions developed in statistical physics for the normalizing constant (Ogata and Tanemura, 1984, discuss some of these approximations; see also Ripley, 198S, ch. 4, and recent work using stochastic approximation techniques by .. .. . . . ..

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140 Moyeed and Baddeley, 1989), or to use Besag's method of pseudolikelihood (Besag 1975, 1977; some recent theoretical results are in Jensen and Metier, 1989, and Sarkka, 1989~. 7.3 A Spatio-Temporal Point Process Model for Tree Growth Most situations where spatial point process models can be useful include a temporal aspect. In this section, we discuss a possible approach to modeling tree growth in a pristine forest, with a view toward use for regenerative policies in national parks following major natural disasters. The intent of this section is to indicate how a physically based model may be used to suggest facets of a stochastic model of forest growth. This is different from the statistical (or descriptive) models that have been the main emphasis in the past for such efforts. In order to construct such a model, it seems reasonable to separate out the occurrence of new growth, the process of growth itself, and the process of tree death, as suggested by Rathbun and Cressie (1989~. We will call the three components the birth, growth, and death processes. For simplicity, we will only consider a single species and will use a discrete time scale of, say, a year. We separate trees into adult individuals that are well established and juveniles that are struggling to succeed. Foresters tend to make this classification based on simple measurements such as base diameter. It is assumed that the forest under study is mapped completely (with regard to the species of interest) at regular intervals. Sterner et al. (19861 developed , v - - - ~---I ~~ -r -A ~ ~ . .~ . . ~ ~ . ~ ~ ~ _ ~ . . _ ~ _ models similar to those discussed below lor the interaction of four tropical tree species. The birth process, at any given time t, will be constructed conditional upon the location of mature adults. Potential sites for new juveniles are obtained from a cluster process of Neyman-Scott type with cluster centers given by the mature adult locations. This represents the spread of seeds from the adult trees. The germination of seeds, or more precisely, germination and subsequent establishment of a juvenile plant, is modeled by thinning the potential sites, i.e., by deletion of each cluster point independently with . . . probability depending on the configuration of adults aro',nt1 t.h~ Line If ~ 1 ~ 1 1. ~ , ~ ad- - ~ c~ ~ Ine IleareSI aGU11 IS Tar away, so the seeollug IS in a relatively open area, it would have a comparatively high probability of germination, whereas if the seedling is located next to an adult, this probability would be low.

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141 The growth process takes into account the amount of sunlight available to a tree by using data on crown angle and height. This process determines the development of marks from year to year, rather than the points themselves. The death process needs to have several factors. The process of To- cations is thinned using a probability proportional to size (and thereby, approximately, to age). In addition, competition between juveniles affects their survival probabilities. The effect of large windstorms can be thought of as a constant force (this would usually have a preferred direction) whose mortality effect on a given tree depends on its size and on the configuration and sizes of its neighbors. Large isolated trees have the highest mortality from windstorms, whereas sheltered trees in the middle of a tight cluster have the smallest. Major disasters, such as fires, can be modeled using de- pendent thinning, where nearby trees have a very high probability of death, conditional upon a given tree to have succumbed to fire. Each year the prob- ability of a major disaster is very small. It can be estimated from tree-ring data. The probability of death from storms is comparatively higher, and may vary from year to year, based on meteorological factors. The combination of these forces yields an anisotropic process, for which one can determine, at least qualitatively, the behavior of second-order inten- sities. Since many of the subprocesses are observable, it is possible to assess these aspects of the mode! using data. The combination of the subprocesses into a complex mechanism and the detailed fitting and inference yields many challenging theoretical problems. The main use for this type of mode! is to assess effects of changes in the driving forces of the ecological process, and evaluate various possible reseeding policies. It is straightforward to include modest amounts of harvesting in the model, which can then be used to assess various recruitment policies. For assessment purposes, computer simulation is likely to be necessary. 7.4 Conclusion and Further Directions The use of point process models in ecology to date has perhaps not reaped the expected benefits. While the models sometimes have men ages! to cle- scribe a complex data set in a relatively compact form, there have been very few instances where data-analytic findings have found proper explanation in ecological/biological terms. With the increased quality of aerial maps, the requisite data for certain types of vegetation ecology studies will be made more readily available, and the quality of the analysis should improve.

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142 The interaction between statisticians and subject area scientists is al- ways the key to relating data-analytic findings to scientific explanations. Increased awareness in the ecological community of the methods made avail- able by improved methods of spatial statistical analysis wiD undoubtedly benefit both statisticians and ecologists. There is a substantial need for more theoretical research into statistical inference based on interacting com- ponents of complex systems, and into the comparison of mode] data (be it the result of simulation, mathematical, or stochastic anaylsis) to "ground truth" measurements. Bibliography [1] Baddeley, A. J., A limit theorem for some statistics of spatial data, Adv. Appl. Prob. 12 (1980), 447-461. [2] Baddeley, A. J., and B. W. Silverman, A cautionary example of the use of second-order methods for analyzing point patterns, Biometrics 40 (1984), 1089-1093. [3] Bartlett, M. S.,The spectral analysis of two-dimensional point pro- cesses, Biometrika 51 (1964), 299-311. [4] Bartlett, M. S., The Statistical Analysis of Spatial Pattern, Chapman and Hall, London, 1975. [5] Besag, J., Statistical analysis of non-lattice data, Statistician 24 (1975), 179-195. [6] Besag, J., Some methods of statistical analysis for spatial data, Bull. Int. Stat Inst. 47(2~) (1977), 77-92.

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143 [7] Besag, J., and P. J. Diggle, Simple Monte Carlo tests for spatial pattern, Appl. Stat. 26 (1977), 327-333. [8] BriDinger, D. R., Comparative aspects of the study of ordinary time series and of point processes, pp. 33-133 in Developments in Statistics 1, P. R. Krishnaiah, ea., Academic Press, Orlando, 1978. [9] Campbell, N. R., The study of discontinuous phenomena, Proc. Camb. Philos. Soc. Math. Phys. Sci. 15 (~1909), 117-136. [10] Daley, D. J., and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer-Veriag, New York, 1988. [11] Darwin, C., The Formation of Vegetable Mould Through the Action of Worms, John Murray, London, 1881. t12] Dawson, D., and G. {vanoff, Branching diffusions and random measures, in Branching Processes Advances in Probability and Related Topics, A. Joke and P. Ney, eds., Dekker, New York, 1978. [13] Diggle, P. J., Statistical Analysis of Spatial Point Patterns, Academic Press, London, 1983. t14] Ford, E. D., Competition and stand structure in some even-aged plant monocultures, IT. Ecol. 63 (1975), 311-333. tI5] Ford, E. D., and P. J. Diggle, Competition for light as a spatial stochas- tic process, Ann. Bot. 48 (1981), 481-500. [16] Gerrard, D. J., Competition quotient: A new measure of the competi- tion affecting individual forest trees, Res. Bull. 20 (1969), Agricultural Experiment Station, Michigan State University. [17] Getis, A., and J. Franklin, Second-order neighborhood analysis of mapped point patterns, Ecology 68 (1987), 473-477. [18] GrandeD, J., Doubly Stochastic Poisson Processes, Lecture Notes in Math 529, Springer, Berlin, 1976. [19] Greig-Smith, P., The use of random and contiguous quadrants in the study of the structure of plant communities, Ann. Bot. 16 (1952), 293- 316.

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