Ecological information from spatial patterns of plants: insights from point process theory

Authors


*Correspondence author. E-mail: RL1@york.ac.uk

Summary

  • 1This article reviews the application of some summary statistics from current theory of spatial point processes for extracting information from spatial patterns of plants. Theoretical measures and issues connected with their estimation are described. Results are illustrated in the context of specific ecological questions about spatial patterns of trees in two forests.
  • 2The pair correlation function, related to Ripley's K function, provides a formal measure of the density of neighbouring plants and makes precise the general notion of a ‘plant's-eye’ view of a community. The pair correlation function can also be used to describe spatial relationships of neighbouring plants with different qualitative properties, such as species identity and size class.
  • 3The mark correlation function can be used to describe the spatial relationships of quantitative measures (e.g. biomass). We discuss two types of correlation function for quantitative marks. Applying these functions to the distribution of biomass in a temperate forest, it is shown that the spatial pattern of biomass is uncoupled from the spatial pattern of plant locations.
  • 4The inhomogeneous pair correlation function enables first-order heterogeneity in the environment to be removed from second-order spatial statistics. We illustrate this for a tree species in a forest of high topographic heterogeneity and show that spatial aggregation remains after allowing for spatial variation in density. An alternative method, the master function, takes a weighted average of homogeneous pair correlation functions computed in subareas; when applied to the same data and compared with the former method, the spatial aggregations are smaller in size.
  • 5Synthesis. These spatial statistics, especially those derived from pair densities, will help ecologists to extract important ecological information from intricate spatially correlated plants in populations and communities.

Introduction

Spatial patterns of plants in natural communities hold an enduring interest for plant ecologists (Watt 1947; Pielou 1968; Greig Smith 1983). They carry information about the processes which operated in the past, and they form the template on which processes will take place in the future. There are, however, some special difficulties in extracting and drawing inferences from this information. For instance, there are no strong grounds for assuming that past processes leave their own unique, identifiable footprint in a spatial pattern. Also, plants in a spatial pattern cannot be treated as independent sample units: their locations and properties are likely to have been determined, at least in part, by the neighbourhood in which they disperse and share with other plants. Therefore, standard methods that assume independence cannot be used, especially when the interest is in understanding the spatial dependence itself (Diggle 2003).

Analysis of spatial patterns is a subject of current statistical research (Cressie 1991; Stoyan & Stoyan 1994; Stoyan et al. 1995; van Lieshout 2000; Møller & Waagepetersen 2004; Diggle 2003; Illian et al. 2008), driven by the need to make rigorous inferences about objects in continuous and discrete spaces, and about objects represented as points and finite areas. However, the literature is large and technical, and there is relatively little exchange between the mathematicians developing the theory and the ecologists working on spatial patterns. The purpose of this essay review is to show some ways in which ecologists can apply summary statistics from current spatial statistical theory to draw more information from spatial patterns of plants.

We focus on just one part of the statistical literature, statistics of spatial point processes in continuous space, on the grounds that point pattern data are more and more frequently collected in plant communities. This applies especially to plots in tropical rainforest communities, from which there is now spatial information on about three million tropical trees of about 6000 species (approximately 10% of the known tropical tree flora) (http://www.ctfs.si.edu). A large amount of ecological information can potentially be extracted from such data sets if appropriate statistical tools are used.

The kinds of ecological insights that can be gained from point process theory are best seen in the context of specific ecological issues. We therefore illustrate the use of spatial statistics in the context of three ecological questions, taking data on spatial patterns of trees in two forests. The first question is, how plants sense the community in which they are embedded, sometimes referred to as the ‘plant's-eye’ view (Turkington & Harper 1979; Mahdi & Law 1987; Law et al. 2001). Are neighbours close to the spatial average of the density? If not, how do the neighbourhoods depart from randomness? Statistics for answering the first question have already received some attention in the ecological literature, but provide the foundation for subsequent work, and are therefore reviewed briefly.

The second question is, to what extent the spatial pattern of biomass is uncoupled from the spatial locations of plants. In answering this, it is possible to see the extent to which area-based ecosystem processes, such as productivity, can be separated from the spatial birth–death processes of population and community ecology.

The third question is, how inhomogeneities in the external environment can be detected and allowed for. This question is motivated by the common occurrence of clustering in spatial patterns (e.g. Condit et al. 2000; Gunatilleke et al. 2006; Wiegand et al. 2007a), which could be due to either local dispersal, or habitat specialization, or both. When, if at all, can different causes of clustering be distinguished?

Data sets

We consider the questions above in the context of two data sets that contrast in their environmental heterogeneity. The first is a 1-ha plot of mixed beech-spruce forest in Rothwald, Austria, in which trees with diameter at breast height (d.b.h.) ≥ 1 cm were censused in 2001 (Splechtna et al. 2005; G. Gratzer et al., unpublished data). Rothwald is ecologically interesting because, unlike most European forests, it has not been managed for timber production, and its spatial structure is largely an outcome of natural processes of birth, growth and death. Most trees in the 1-ha plot studied were either beech (Fagus sylvatica) or Norway spruce (Picea abies); a small number of stems of silver fir (Abies alba) were also present. The locations of trees of these species are shown in Fig. 1.

Figure 1.

Spatial patterns of (a) beech (Fagus sylvatica), (b) Norway spruce (Picea abies) and (c) silver fir (Abies alba) in a 1-ha plot at Rothwald, Austria. Diameters of circles are proportional to d.b.h.

The second data set is taken from a 25-ha forest plot at the Sinharaja World Heritage site in Sri Lanka (Gunatilleke et al. 2004), an evergreen lowland rainforest dominated by the Dipterocarpaceae. The plot is centred on a steep sided valley encompassing a range in elevation of approximately 150 m (Fig. 2). All trees in the plot with d.b.h. ≥ 1 cm have been mapped, amounting to 205 373 stems of 205 species when censused from 1994 to 1996. The data shown in Fig. 2 are the locations of trees of just one species, Shorea affinis (Dipterocarpaceae), including the locations of some large individuals emergent from the canopy (Gunatilleke et al. 2006). Previous spatial analyses at this site include a detailed analysis of Shorea congestiflora (Wiegand et al. 2007b), and an analysis of the multispecies spatial pattern of the trees (Wiegand et al. 2007a).

Figure 2.

Spatial pattern of Shorea affinis at Sinharaja, Sri Lanka, showing locations of stems with d.b.h. ≥ 4 cm. Contours of altitude are shown in 10 m intervals. Boxed subwindows were used for constructing a master pair correlation function.

Theoretical summary statistics of spatial point processes

A spatial point process N is a random mechanism that generates a set of points at a single discrete point in time in a physical space which, in ecological applications, is usually a two-dimensional plane of finite area (Diggle 2003: 42). The simplest of the many possible spatial point process models is the Poisson process, which describes complete spatial randomness (CSR). Intuitively this says that a random number of individuals are located independently following a uniform distribution in region A. In applying spatial point processes to plant ecology, individual plants are envisaged as points, the locations of individuals usually being given by their Cartesian coordinates. In addition, individuals may have properties that can be either qualitative (such as species identity) or quantitative (such as mass or height). Such properties are referred to as marks in the statistical literature and the random process is called a marked point process.

In reality, spatial patterns studied by ecologists are generated by stochastic processes which operate over time as well as space. In such spatio-temporal processes, birth, dispersal, growth and death of plants take place potentially under the influence of neighbouring plants and environment. Point pattern processes used in this article are a substitute for spatio-temporal processes when information over time is not available and cannot be included explicitly in a statistical model.

We focus here on the use of summary statistics of spatial point patterns for exploratory analysis of spatial data. Spatial point process models (such as those described in van Lieshout 2000; Møller & Waagepetersen 2004) and spatio-temporal processes (Daley & Vere-Jones 2007) are major subject areas in their own right and beyond the scope of this article. In this section we discuss several basic theoretical summary statistics to describe spatial properties of spatial point processes (see Ripley (1976), Stoyan et al. (1995), Møller & Waagepetersen (2004), Illian et al. (2008) for details). In the following section we describe estimators of these theoretical statistics, which can be understood in the same way as the sample mean may be interpreted as an estimator for a population mean. In subsequent sections we show how these measures can be used to address questions of ecological interest.

theoretical first-order statistics

These describe the expected density of individuals in the plane, and in ecological terms reflect the probability of an individual being found at a given location in the sample space. Density may either be constant over space, resulting in a stationary (or homogeneous) pattern, or vary over space, for instance having a spatial trend, resulting in an inhomogeneous pattern. The density (or intensity) function of individuals near location x = (x1, x2) is denoted by λ(x), where λ(xdx is the probability that there is a point of point process N in a disc with centre x and infinitesimal area dx (Diggle 2003: 43). The density is given as a function of location, because it is quite conceivable that the density of individuals is affected by environmental conditions at location x.

theoretical second-order statistics

These relate to the relative positions of pairs of individuals. The most primitive version is a pair density (or second-order product density) ρ(x, y) obtained in a similar way as above by considering ρ(x, ydx dy, that is, the probability that there is a point of point process N in each of two disjoint discs b1 and b2 with centres x and y and infinitesimal areas dx and dy (Diggle 2003: 43). This expression holds information about spatial structure because the density of pairs may depend on the distance || y − x || between them.

Individuals in a pair may have different qualitative marks, such as species identity or life stage. For a point pattern with qualitative marks i and j one may consider a pair density ρij(x, y), similar to the one above, where ρij(x, ydx dy is the probability that there is a point of type i in a disc b1 with centre x and a point of type j in a disc b2 with centre y, the discs again having infinitesimal areas dx and dy, respectively. Note that, if i = j, ρij(x, y) = ρii(x, y) = ρ(x, y). Another possibility is that the individuals have quantitative marks, such as plant size. The mark pair density ρmm(x, y) is appropriate for this. This makes use of the extra piece of information, m, contained in the quantitative mark. The expression ρmm(x, ydx dy is the expected value of the product of the marks in discs b1 and b2 with centres x and y and infinitesimal areas dx and dy. Functions other than the product of the marks may be considered here, depending on the questions of interest (Illian et al. 2008).

higher-order theoretical statistics

There is no intrinsic reason why the spatial analysis should not continue beyond second order to the density of triplets, quadruplets, etc. (Schladitz & Baddeley 2000). However, as yet there has been little exploration of higher-order spatial moments and the interpretation of these is not straightforward.

Estimation of theoretical summary statistics

null models

Null models are used to determine whether observed properties of a spatial pattern, such as the extent of clustering, can be explained by making an explicit hypothetical assumption. To compare empirical spatial patterns to those expected from a null model, summary statistics are usually compared with a distribution of the statistics obtained from the null model. In the case of second-order statistics which are functions, the distribution is often summarized as a confidence envelope for the summary statistics, computed from repeated simulated realizations of an appropriate point process model, under conditions specified by the null model. Sometimes the choice of null model is self-evident. For instance, if the question is whether the locations of plants depart from CSR, the null model is that of a Poisson process.

Sometimes the choice of a null model needs more careful thought (Goreaud & Pélissier 2003). For instance, suppose the question is about spatial locations of individuals with different qualitative marks. The null model could be that there are independent spatial point processes responsible for the locations of plants with different marks (random superposition or population independence). This would be appropriate in testing whether two types of individuals, such as young and old, are independently distributed in space. The null model could alternatively be that, given fixed locations of plants, quantitative or qualitative marks are randomly distributed over these locations (referred to as random labelling in the case of qualitative marks). This would be appropriate in testing, for instance, whether the size or survival of plants is independent of the neighbourhood in which they are growing.

Some authors advocate the use of more complex models such as Neyman–Scott processes (e.g. Thomas processes or Matern cluster processes) or other cluster processes as a null model (Wiegand & Moloney 2004; Wiegand et al. 2007b). This article, however, is primarily concerned with the exploratory analysis of spatial point patterns, which may be regarded as the first step in a thorough analysis of a pattern to aid the choice of a suitable model class in subsequent steps. Choosing an appropriate point-process model requires detailed knowledge of available modelling approaches such as those discussed in van Lieshout (2000) and Møller & Waagepetersen (2004). Furthermore, the estimation of parameters for a specific cluster model is often done by a minimum-contrast approach based on a suitable summary statistic (Diggle 2003). Using simulation envelopes derived from a model with the parameters estimated in this way constitutes assessing the suitability of a specific model rather than an exploratory data analysis. In addition, assessing the suitability of a model based on the characteristics that were chosen for the estimation is not a suitable statistical approach, but assesses the quality of the estimation rather than the suitability of a model.

If hypothesis testing entails a comparison between an observed function and its confidence envelope from a null model, it is important to realize that comparisons made at more than one point on the function have the effect of inflating Type I error (Loosmore & Ford 2006). Envelope tests should not be thought of as formal tests of significance, although they can still be instructive, and we make use of them in this article. An alternative approach is to construct a scalar quantity from the function and to examine the deviation it has from its distribution under the null model. The quantity could for instance be the accumulated deviation of the observed function from that of the theoretical statistic (e.g. Plotkin et al. 2000). In the case of an observed pair correlation function and a null model of CSR, the scalar
quantity would be inline image over all distances r; in
practical applications the integral is replaced by a sum over a finite number of distances.

edge corrections

In practice, data sets collected in the field are taken from an observation window W and each plant k in the window W is characterized by its location xk = (x1, x2)k and any additional marks mk = (m1, ... ) such as species identity and size. Together, the plants form a set {[xk, mk]}, where xk ∈ W. Often no information is available on the pattern outside W and it is impossible to determine the neighbours of plants close to the edge.

Several approaches to edge corrections have been discussed and are recommended in both the statistical and ecological literature (Haase 1995; Goreaud & Pélissier 1999). One possibility is to make no edge correction, which is suitable for the estimation of indices such as Pielou's index or the Clark-Evans index when the window and the number of points are large (Pommerening & Stoyan 2006). Another possibility is a form of plus sampling when information on points outside the window is available either in situ or by artificially generating additional points based on periodic edge correction (sometimes termed a toroidal edge correction, see Wiegand & Moloney 2004) or reflection. However, information outside the window is often unavailable, and toroidal edge corrections can produce spurious point configurations that differ substantially from those in the empirical pattern. Also, in some applications, when the edges of the observation window are ‘real’ edges, one might want to not correct for edge effect as the spatial behaviour towards the edges of a plot might be truly different from the behaviour in the centre, say as a result of different light conditions (Lancaster & Downes 2004).

For the second-order summary statistics discussed here, second-order edge correction methods that weight inter-point distances proportional to the size of the distance between points are most suitable (Ohser 1983; Ripley 1988). An important consequence of the spatial displacement inherent to second-order statistics like those discussed above is that the loss of information about pairs of plants close to the boundary of W is greater the greater the displacement ξ = y – x between the plants. An edge correction is therefore used, based on a window, translated by an amount ξ, Wξ ={x + ξ: x ∈ W}. The inverse of the area of overlap Aξ of the windows

Aξ = (W ∩ Wξ)(eqn 1)

is taken as a weight for a pair displaced by ξ.

kernel functions for smoothing

Second-order statistics are continuous functions of spatial displacement, whereas pairs of individuals in a point spatial pattern occur at discrete displacements. Non-cumulative pair densities thus appear as a set of spikes in their raw state (one spike for each pair). When estimating second-order radial statistics that assume stationarity (i.e. that the pattern is translation invariant) and isotropy (i.e. that the pattern is rotation invariant), it is usual to apply some kernel function k(z) over a distance z = || ξ || – r, to make the statistics smooth functions of distance. Such kernel functions have dimensions
1/length and the property that inline image Many different
kernel functions have been applied in the literature, most notably the Epanechnikov kernel (Stoyan et al. 1995) and the simple box kernel with a bandwidth parameter h

image(eqn 2)

The Epanechnikov kernel has sometimes been recommended, but recent statistical research has shown that the box kernel is preferable as it minimizes the variance of the estimators of pair densities (Illian et al. 2008: 4.3.3). Note that the appropriate choice of the bandwidth h is crucial to the quality of the estimator. If the bandwidth is too small the estimate becomes too noisy resulting in (spurious) local features, and if it is too large important information might be lost. Since there are no rigorous mathematical results on the optimal choice of h, it has to be chosen by trial and error. This involves choosing several values of h and comparing the results starting from a value close to h ≈ 0.1/√λ. In some cases it may be advisable to use different bandwidths for different distances. The qualities of the estimator can be improved by the reflection method introduced in Stoyan and Stoyan (1992).

Pair correlation functions: the plant's-eye view

The pair density and related functions have a natural interpretation as the average plant's-eye view of the community (Turkington & Harper 1979; Mahdi & Law 1987). For illustration, we compute these functions for the trees at Rothwald. For now, assume that all individuals within a spatial pattern are equivalent (this assumption is relaxed in the section on the mark correlation function, below). Assume also that the spatial pattern is stationary across space, which means that the density of individuals λ(x) does not depend on location x. This means that λ(x) can be replaced by the spatial average density λ, and estimated by dividing the number of points by the area A of the window W. The pair density is then independent of location and determined simply by the spatial displacement between individuals ξ = y − x. Furthermore, assume that there is no directionality in the pattern (i.e. isotropy applies); all that matters is the distance r = || ξ || between the pairs. Anisotropy can arise if plants are more likely to survive in some directions than in others, for instance along contours, decaying logs, river margins, or road margins.

With these assumptions, the estimator of the pair density at a distance r is given by

image(eqn 3)

Essentially, the expression on the right-hand side considers each pair of individuals k and l within the window W, with coordinates xk, yl, except self pairs (the exclusion denoted by ≠). The equation uses a kernel function k(|| ξ || – r), where ξ = yl – xk, to give each pair a weight according to how close the distance || ξ || between them is to r (see above), and applies an edge correction Aξ (see above). Summing this weighted measure over all pairs yields an estimate inline image(r) of ρ(r). Lastly, a dimensionless measure, the pair correlation function, is obtained by dividing the estimate by the square of the average density of individuals (intensity)

image(eqn 4)

where inline image is an estimator for the spatial average density. If the densities of individuals at x and y in eqn 4 are independent, then ?(r) ~ 1. If ?(r) > 1, then pairs of plants are more abundant than the spatial average at a distance r. Conversely, if ?(r) < 1, then pairs of plants are less abundant than the spatial average at a distance r. In effect, the function ?(r) is a normalized measure of how, on average, a plant perceives the density of other plants as it ‘looks out’ over increasing distances r into the community in which it is living. An alternative normalization would be to divide inline image(r) by l yielding inline image?(r), which measures directly the average density of plants at distance r and is referred to as the O-ring statistic (Wiegand & Moloney 2004).

In the literature, cumulative measures, although having a less obvious physical interpretation, are often discussed. The relationship of the pair correlation function to a cumulative statistic is similar to the relationship of a probability density function to its distribution function. As in classical statistics, where histograms rather than cumulative empirical distribution functions are easier to interpret, second-order spatial structure is more easily understood when expressed in the form of pair correlation functions. This is notwithstanding the advantage that the cumulative statistics are estimated without binning and avoid the need for a kernel function such as that in eqn 2. However, in line with conventions in classical statistics, cumulative statistics are sometimes preferred for statistical tests, just as the distribution function is typically used for statistical tests rather than the associated probability density function.

The best known of the cumulative statistics is ‘Ripley's K function’K(r) (Ripley 1976, 1977; Haase 1995), related to the pair correlation function as the integral over radii up to r

image(eqn 5)

which has dimensions of area. A variant on this theme is the function

image(eqn 6)

which stabilizes the variance of the estimator (Besag 1977). This has dimensions of length and is often plotted as L(r) − r, which gives it a value 0 if there is a homogeneous Poisson process. Despite its appearance, L(r) − r is still a cumulative statistic and needs careful interpretation. The effect of clustering at short distances, for instance, is only gradually diluted out by randomness at larger distances, and can lead to a superficial impression of clustering over longer distances than operate in reality (Wiegand & Moloney 2004).

The results of analysing the same data set using the pair correlation function, the K function and the L function are illustrated in Fig. 3 using the spruce and beech data from Rothwald forest. It is clear from the pair correlation functions that individuals of both species experience an excess of conspecific neighbours in their immediate vicinity. Over longer distances the aggregations diminish such that, by a distance of about 20 m, the pair correlation is close to that given by the spatial averages (g(r) ∼ 1). In this way, the pair correlation function provides a formal measure of both the intensity and scale of the spatial structure in the community which is evidently quite different from the average over space. These features are also evident in the K(r) and L(r) − r functions, although the distances at which aggregation is most evident are not so easy to interpret, because these functions use the cumulative density of pairs up to a given radius r.

Figure 3.

Spatial statistics for beech trees (Fagus sylvatica) (column 1) and spruce trees (Picea abies) (column 2) at Rothwald. Horizontal axes are the distances r between pairs of individuals. Heavy lines show the observed statistic, calculated with an edge correction as in eqn 1. (a) and (b) show pair correlation functions, estimated using eqn 4, and a box kernel function as in eqn 2, based on point estimates of the functions, spaced such that each pair was counted once; the horizontal dotted line shows the pair correlation function expected from a Poisson process. (c) and (d) show cumulative K(r) functions. (e) and (f) show normalised versions of the L(r) functions. Dashed lines are approximate 99% confidence envelopes for the hypothesis of complete spatial randomness, obtained from 200 independent randomizations of the locations of the trees.

Cross pair correlation functions

The analysis above uses only spatial data within species and treats all individuals as equivalent. A more nuanced spatial analysis would take advantage of information on the relative locations of pairs of individuals with different properties (marks). There are different approaches to the spatial analysis of marks, depending on whether the marks are treated as qualitative (e.g. species, life stages) or quantitative (e.g. plant size).

Qualitative marks can be dealt with as a straightforward extension to the pair correlation function above. We cover them briefly here to distinguish them from quantitative marks considered in the next section. Indexing now has to be used to distinguish different marks i and j at locations x and y, respectively. In estimating the pair density, only those individuals in the windows W, Wξ (ξ = y − x) with the correct marks are picked. An indicator function Iij (mk, ml) is used for this purpose, that is,

image(eqn 7)

for each pair of individuals k, l. The marks i and j might, for instance, denote species, or discrete size classes, or both. An estimator for the cross pair density function of individuals with marks i and j is

image(eqn 8)

As in eqn 3, the right-hand side picks out and adds up the contribution of pairs, but it does so only when individual k has mark i and individual l has mark j; the kernel function k(|| ξ || – r) and area Aξ remain as in eqn 3. The cross pair density function is normalized and made dimensionless by means of the spatial average densities of the two types of individuals,

image(eqn 9)

inline imagei, inline imagej being the estimators of these spatial averages. The interpretation of the function depends on the choice of the appropriate null model. If random labelling is deemed the suitable model, ?ij(r) ∼ ?(r); random superposition holds for types i and j if ?ij(r) ~ 1. Cumulative measures related to Ripley's K function extend to qualitative marks in a similar way but are again not recommended for an exploratory analysis.

Note that the difference gij(r) − gii(r) has sometimes been used to assess how, on the average, a plant senses the density of other individuals of a different type. However, δij(r) =λjgij(r) – λigii(r) is more appropriate as it takes into account both the relative locations and the respective potentially strongly differing densities of the two types (Illian et al. 2008, 5.3).

Mark correlation functions: biomass distributions of plants

To illustrate the use of the mark pair density and related functions, consider the distribution of biomass over space at Rothwald, and the extent to which it is decoupled from the spatial pattern of tree locations. The pair correlation functions of beech and spruce in Fig. 3 already show the existence of clusters of individuals within species. Beech, spruce and fir are now combined, because together they make up the majority of plant biomass in the community. Aggregating the three species, however, does not change the picture: strong spatial clustering of individuals is still evident (Fig. 4a).

Figure 4.

Correlation functions at Rothwald, combining beech (Fagus sylvatica), spruce (Picea abies) and fir (Abies alba) trees. Protocol for calculations as in Fig. 3, with heavy lines showing the observed statistic and dashed lines showing 99% confidence envelopes. (a) Pair correlation function g(r), with a random superposition null model. (b) Multiplicatively-weighted pair correlation function inline image with a null model of random labelling of marks across fixed locations of trees. (c) Mark correlation function gmm(r), also based on a null model of random labelling of marks across fixed locations of trees. Marks for (b) and (c) are ln(biomass + 1). For beech and spruce biomass was obtained from tree height and d.b.h. (Zianis et al. 2005: Appendix A), using formula (86) for beech, (140) for spruce with d.b.h. from 1 to 13.5 cm, and (147) for spruce from 14 cm onwards. Zianis et al. (2005: Appendix C, formula (1)) give a conversion to wood volume of fir, here transformed to biomass using the figure 419.5 kg m−3 from worldagroforestrycentre.org.

The degree to which the spatial pattern of biomass reflects this clustering of individuals can be found from a spatial analysis in which biomass is treated as a quantitative mark. To obtain a measure of above-ground biomass, d.b.h. and tree height were transformed as described in Fig. 4, giving quantitative marks from which a mark pair density could be computed. Assuming stationarity, isotropy and that the points in pairs are of the same qualitative type (i = j), an estimator of the mark pair density (eqn 3) is:

image(eqn 10)

where mk and mlare scalar marks for individuals k and l (Illian et al. 2008: 5.3.3). This differs from eqn 8 in that it adds up the product of the marks of all pairs of individuals, and it therefore has squared dimensions of the marks, in addition to the squared dimensions of density. The kernel function k(|| ξ || – r) and area Aξ keep the same meaning as in eqn 3.

Two normalizations of the mark pair density have been suggested resulting in two rather different summary statistics with different interpretations. The first, termed the multiplicatively weighted pair correlation function, divides the mark pair density by the product of the mean densities and makes no allowance for the pair density at distance r

image(eqn 11)

The extra term here, inline image, is the estimator for the mean mark of the individuals and is needed to make the function dimensionless. With this normalization, a departure from inline image could be caused either by spatial location of plants, or by the distribution of marks across plants, or by both. The second normalization, the mark correlation function, separates the effect of the marks from the effect of plant locations, by measuring the mark pair density relative to the pair density at distance r

image(eqn 12)

A value of inline image would indicate that the mark pair products at a distance r are explicable just in terms of the spatial location of individuals. In contrast, a value of inline image would indicate mutual stimulation, that is, an excess of mark pair products beyond a value explicable just in terms of locations of individuals. Conversely, a value of inline image would indicate mutual inhibition.

The potential of these correlation functions based on marks has yet to be taken up in ecology; the only case we are aware of is an example of the mark correlation function in forest ecology (Stoyan & Penttinen 2000). In the case of Rothwald, it turns out that the spatial pattern of biomass is quite different from the aggregated spatial pattern of tree locations (Fig. 4). In analysing the data, the appropriate null model is that the biomasses are independently and randomly assigned to the fixed locations of individuals. The function inline image stays close to 1 as distance r increases (Fig. 4b), which means that there is compensation for aggregations of individuals by low biomass within these aggregations. The confidence envelope for inline image reflects these aggregations, and shows significant compensation up to about 20 m. The mark correlation function inline image goes a step further, removing the signal of spatial aggregation and showing directly the mutual inhibition in spatial correlations of the tree biomasses at short distances (Fig. 4c).

The ecological message from these spatial correlations at Rothwald is that biomass is much more uniformly distributed over space than the trees are themselves (see also Stoyan & Penttinen 2000). Tree growth is plastic and space filling (Purves et al. 2007). In itself, this is not surprising: what the spatial correlations do, is make the measure of the spatial pattern of biomass precise. However, there is an important implication that the processes associated with biomass per unit area, such as energy flux, are at least partly decoupled from the fine details of birth and death processes which determine the locations of individuals. Such decoupling provides some justification for treating energy flux separately from the turnover of individuals which underlie population and community dynamics at Rothwald. Environmental events such as storms which cause extensive mortality of large trees do, however, act to thwart such neat ecological separations.

Inhomogeneous pair correlation functions: spatial patterns in heterogeneous environments

The assumption of spatial homogeneity is an important limitation of the methods above, and there are at least two reasons why this assumption could be violated. First, the species responsible for the pattern could be contracting or expanding in range, leaving large areas of low or zero density. Second, external environmental features could vary from one location to another: spatial patterns of plants can depend as much on this environmental template as on internal processes of reproduction, local dispersal and death of individuals. Here we focus on removing effects of large scale spatial heterogeneity in the external environment, to uncover the remaining signal at a smaller spatial scale likely to be influenced by internal processes. In fact, a clear-cut separation of environment from local internal processes may not be possible. If all that is known is that there are local aggregations of plants, these could be due to limited dispersal, to patchy herbivory or pathogens, or to locally suitable environments. But with knowledge about the system and the spatial scales at which internal and external factors operate, some separation of the signals from these factors may be possible.

The dipterocarp S. affinis at Sinharaja, Sri Lanka (Fig. 2) illustrates the issues. In this plot, there is pronounced variation in altitude and this species tends to occur at higher elevations. A naïve treatment of the spatial pattern of S. affinis would ignore this inhomogeneous distribution of trees, would take the pair correlation function under an assumption of stationarity (eqn 3), and would conclude that the species is stationary and has an aggregated spatial pattern (Fig. 5a). However, the assumption of stationarity is obviously unjustified in this instance. Even in the absence of a clear correlation with the environment, there would be doubt about the assumption of stationarity, because the correlation function shows aggregation at distances as great as 50 m at which internal processes such as dispersal are unlikely to be acting. How to deal with spatially inhomogeneous patterns like this is a matter of current statistical research. The inhomogeneous pair correlation function and the related inhomogeneous K function have been used in a number of studies and we illustrate several versions below. However, other approaches are also possible and we describe one alternative, using a master function obtained from aggregating local homogeneous pair correlation functions.

Figure 5.

Pair correlation functions g(r) and corresponding estimated intensity surfaces for Shorea affinis at Sinharaja. Pair correlation functions in (a) are based on: assumption of homogeneous intensity (dotted line); nonparametric intensity surface shown in (b) (continuous line); polynomial intensity surface, eqn 14, shown in (c) (long dashed line); harmonic polynomial intensity surface, eqn 15, shown in (d) (short dashed line).

inhomogeneous versions of the pair correlation function and ripley's k function

These statistics have been suggested as a solution to the problem of first-order non-stationarity (Baddeley et al. 2000; Waagepetersen 2007) and are being applied in plant ecology (Couteron et al. 2003; Shimatani & Kubota 2004; Perry et al. 2006; Wiegand et al. 2007a). They use the product of local densities at locations x and y, λ(x) and λ(y), rather than a product of constant densities λ2 averaged over the whole space. The corresponding estimate of the inhomogeneous pair correlation function comes from eqn 3, now with estimates of these local densities taken into the summation

image( eqn 13)

In effect, this factors out first-order inhomogeneities, leaving behind the residual local spatial structure at second order. Notice that there could still be second-order inhomogeneity. Strictly speaking, ginhom, being a function of the distance between points, is only appropriate for data where the second-order structure does not vary in space. Non-stationarity at second order can be assessed by comparing pair correlation functions obtained from smaller subareas.

A key issue in using eqn 13 is the choice of an estimator for inline image(x); several possibilities, both nonparametric and parametric, are given below.

A nonparametric estimate for λ(x) can be obtained by applying a smoothing function for density to the area. To illustrate this, Fig. 5b shows a surface of density obtained from a Gaussian smoothing function, with a standard deviation of 75 m. Some judgement is needed in choosing the smoothing function and the appropriate bandwidth; 75 m was used here as this is larger than the likely scale of processes internal to the dynamics of S. affinis. λ(x) was then interpolated to the location of each tree for use in eqn 13 and ginhom(r) was computed. The inhomogeneous pair correlation function shows a much more rapid decay than the homogeneous function (see Fig. 5a), but evidence of aggregation at short distances remains. We did not compute a confidence envelope in Fig. 5a, because this would require additional assumptions about the distribution of trees over altitude.

A parametric estimate for λ(x) can be obtained by modelling density as a function of environmental variables; Fig. 2 shows that elevation is a likely contender for being such a variable. Fitting density to altitude using a second-order polynomial function, estimates parameters b0, ... , b6 in

image(eqn 14)

The surface is shown in Fig. 5c and the corresponding inhomogeneous pair correlation function is shown in Fig. 5a. There is a spurious decrease in density at the extreme right-hand side of the plot; this can be avoided by a harmonic polynomial which attains its global extremum only on the edge of a bounded area. Fitting density to altitude using a harmonic polynomial, estimates parameters b0, ... , b5 in

image(eqn 15)

The surface is shown in Fig. 5d and the corresponding inhomogeneous pair correlation function is shown in Fig. 5a. Formally, a likelihood ratio test indicates that the harmonic polynomial provides a significantly better fit of density to altitude than eqn 14 (p < 0.001, χ2 = 450.455 (d.f. = 3)). However, altitude does not explain all the large-scale inhomogeneity in the data, because the inhomogeneous function calculated from eqn 15 remains above the one calculated from the nonparametric estimate of density and gets close to 1 at about 40 m as opposed to 30 m. Further environmental variables could be introduced as appropriate.

a master function from local homogeneous pair correlation functions

An alternative approach, appropriate for large data sets such as that of S. affinis in Fig. 2, is to take homogeneous pair correlation functions from several homogeneous subplots (Couteron et al. 2003; Pélissier & Goreaud 2001). These can then be combined into a single ‘master’ pair correlation function using a suitable aggregation method (Illian et al. 2008: 4.7) assuming that the subpatterns are independent homogeneous samples from the larger pattern. This approach is of particular interest if the aim is to understand spatial autocorrelation at small scales. In this context it is noticeable in Fig. 2 that there are two large clusters on the right-hand side of the plot separated by a gap at much the same altitude, with trees at a lower density (perhaps as a result of varying soil conditions generated by landslides). Whatever the cause, the inhomogeneous pair correlation function interprets the signal of this gap as clustering at larger spatial scales. This signal cannot be separated from clustering at smaller scales unless further covariates can be identified that explain these two large clusters. However, taking subplots within these two larger clusters which are approximately homogeneous and aggregating the respective pair correlation functions, provides information on clustering at smaller scales without the effect of the gap.

The local pair correlation functions are combined by weighting each of the original functions at each distance to allow for subpatterns collected in subplots of different sizes and shapes. This takes into account the fact that patterns in smaller or narrower plots have fewer points at longer distances than patterns in larger or square plots, resulting in less reliable estimates at these long distances. If m pair correlation functions ?1, ... , ?m have been collected in m different subwindows W1, ... ,Wm, a master pair correlation function ?(r) may be calculated as

image(eqn 16)

(Ohser & Mücklich, 2000). Here γi(r) describes the rotation average of the area of the overlap of window Wi and a window of the same dimensions translated over all directions at distance r (called the isotropised set covariance); γ(r) is the sum of this quantity over all windows, ∑ γi(r). The isotropised set covariance gives a larger weight to estimates of the pair correlation function in square windows than in rectangular windows at larger distances. This reflects the fact that a larger number of points with larger distances can be expected in a square window than in a rectangular window yielding a better-quality estimation at these distances. Closed formulae for windows of a variety of specific shapes are available in the literature (Ohser & Mücklich 2000). In the example below, a formula for a rectangular area i with sides of length ai, bi (ai ≤ bi) and r ≤ ai is used:

image(eqn 17)

To illustrate the master function, we take three subwindows, as indicated in Fig. 2. Different sizes and shapes are used deliberately to illustrate the different weights associated with correlation functions (windows of the same size and shape would get equal weighting at every distance). Figure 6 shows the pair correlation functions estimated for each subwindow, together with the master function. Subwindow (a) is small, contains relatively little information and has a low weight in the master function; its contribution on the shape of the function is therefore small. Subwindows (b) and (c) both contribute a lot. The bottom-right window (b) makes a slightly greater contribution at short distances because it is somewhat larger and this difference is accentuated at longer distances because (c) is rectangular in shape. It is notable that, in contrast to the inhomogeneous pair correlation function (Fig. 5), there is little evidence of spatial clustering in the master function beyond the first few metres. In part, this could be because the master function uses less information. But, as noted earlier, an appearance of clustering over intermediate distances in the inhomogeneous function could be an artefact of spatial averaging of density over inhomogeneities such as the gap on the right-hand side of the plot.

Figure 6.

Homogeneous pair correlation functions for Shorea affinis at Sinharaja. Local functions labelled (a), (b) and (c) correspond to the subwindows in Fig. 2. The heavy line is the master function obtained from weighting the local functions as in eqn 16.

Discussion

The analysis of spatial patterns to gain understanding of ecological processes is essentially a substitution of space for time and is controversial in plant ecology (Lepš 1990). On the one hand, the link between pattern and process which emerged in the middle of the last century (Watt 1947), has led to a lot of interest in formal measures of spatial pattern to achieve precise measures of spatial structure (Pielou 1968; Greig Smith 1979, 1983; Kershaw & Looney 1985; Dale 1999). On the other hand, because past processes do not leave unique identifiable signals in spatial patterns, many plant ecologists have preferred to adopt the approach of animal ecologists of going directly to birth, death and growth processes in a largely non-spatial context (Harper 1977). Yet plant community dynamics is about the dynamics of multispecies spatial patterns, driven by spatially-dependent birth, death and growth processes embedded in a heterogeneous landscape. The subject is incomplete without the spatial dimensions.

Clearly, the results of purely spatial analyses as described in this article have to be interpreted with caution. For instance, one ought not to equate a measure of spatial proximity of plants, such as the area under a pair density function, with a dynamical notion of competitive interaction strength. Such an equation would fail simply on dimensional grounds, because the pair density does not include a dimension 1/time needed for an interaction strength as used in community dynamics. The equation would also fail because it does not allow for the asymmetry which often accompanies interactions between plants of different types (e.g. Freckleton & Watkinson 2001; Bauer et al. 2004).

At the same time, there is a danger in being too dismissive about information in spatial patterns (Lepš 1990). The effort required to gain knowledge of processes over time is so great that the best information available may be a single census in many instances. Species-abundance curves, for instance, are based just on single-census, first-order measures of species abundance, and have been widely used as a test of the neutral theory of biodiversity (Hubbell 2001). It turns out to be hard to distinguish the neutral and niche-structured models from this information (Purves & Pacala 2005; Illian & Burslem 2007). Multispecies spatial patterns carry a great deal more than first-order spatial information, including the locations of plants relative to one another, and can potentially give more sensitive tests of neutral and related theories using second-order spatial information. Ecologists are learning how to use spatial information in multispecies assemblages to tackle basic questions such as this, and we expect the methods described in this article to help in extracting some of the important ecological signals when this work is done (e.g. Wiegand et al. 2007a).

Fundamentally, plant community dynamics deals with the unfolding of multispecies spatial patterns over time and as a result involves both time and space. In this context, spatial point-process models which underpin point pattern processes are not mechanistic: for mechanisms one has to turn to spatio-temporal models, which incorporate rates of birth, death and growth. When this is done, it turns out that much of the dynamic behaviour can be captured quite well through the coupled dynamics of two spatial statistics (Bolker & Pacala 1999; Murrell & Law 2003). The first statistic is the average density of individuals, the traditional state variable of population dynamics. The second statistic is the pair density function, which is new in population dynamics, but familiar to spatial statisticians as the function on which the second-order measures in this article are based. This brings the pair density centre stage of spatiotemporal dynamics, as well as of spatial pattern analysis.

As in any formal representation of ecological systems, some abstraction is used in point pattern statistics. Plants are obviously not points: they have a finite area in the plane, and volume in three dimensions. To some extent, the use of marks such as biomass can deal with this, because a point is simply a centre of location and the mark can be the measure of its size. But it would be unsafe to make inferences about pattern at distances so short that the physical size prevents space being occupied by other individuals. Moreover, some communities (such as trees in forests) lend themselves to point pattern representations better than others (such as diffusely-growing clonal plants in grasslands). We have concentrated on forests, because of the wide availability of point pattern data from them, but other statistical approaches would be more appropriate in other kinds of communities. The assumptions of stationarity and isotropy are also important but can be relaxed within the framework of point pattern statistics. Anisotropy can be measured using angular information on second-order spatial statistics (e.g. Llambi et al. 2004), and non-stationarity at first order can be dealt with using second-order statistics which allow for inhomogeneity, as described above.

Second-order spatial statistics are intricate and time-consuming to compute. Those which make use of binning, like the pair correlation function, involve an order of n2 pair distances (where n is the number of individuals), and cumulative statistics like the K function, in which pair distances are also to be ranked, are still more computer intensive. Careful planning of computations is therefore needed when working with large data sets such as ones from tropical rain forest plots where n can be of the order 105. Spatial statisticians have an open-source statistical package ‘spatstat’ in the R programming environment for carrying out many of the computations, and this may be of help to the ecological user (Baddeley & Turner 2005). New functions are being added to this package as techniques are developed, and spatstat is likely to be a standard toolbox for this subject area in the foreseeable future. In this article, most of the computations were done using C code written for the purpose because not all summary statistics, in particular those for marked point patterns, are currently implemented in spatstat. However, a new version of spatstat is currently being developed that is likely to contain additional features (Baddeley, personal communication).

conclusions

Multispecies spatial patterns, such as those becoming available from analysis of tropical rainforest data, carry a wealth of ecological information. However, this information is tied up in a Gordian knot from which it is hard to draw inferences about ecological processes. We have written this article because we believe that the developing theory of spatial point processes can potentially help ecologists in untying the knot. The pair density function and its allies are especially informative and are easier to interpret than their cumulative counterparts (the K and L functions). However, there should be no illusion about what can be learned from spatial pattern information: strong inferences about ecological processes require knowledge of the way in which the patterns change over time.

Acknowledgements

This work was developed in a working group ‘Spatial analysis of tropical forest biodiversity’ funded by the Natural Environment Research Council and English Nature through the NERC Centre for Population Biology and UK Population Biology Network. We thank the participants and Drew Purves for sharing their ideas; discussions with Antti Penttinen greatly helped our understanding of technical matters. The Rothwald data was recorded as part of project P14583, funded by the Austrian Science Fund (FWF). Establishment of the Sinharaja plot was funded by The John D. and Catherine T. MacArthur Foundation, the Smithsonian Tropical Research Institute, the U.S. National Science Foundation, Arnold Arboretum of Harvard University and the National Institute for Environmental Studies of Japan.

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