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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 x—y, and if the
process is also isotropic, it further depends only on the length t = fix—ye.
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 = Axe—Xj~ 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') exit—A(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 1—h

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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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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
/ ( 1—Is ( 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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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 (t—z) 4t - i3u ~x)
where ¢(t) = E expel—tY). 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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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
^12ttJ—fit- = - (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

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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.
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