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We generalize Ripley’s

Investigating the spatial structure of point patterns has been a long-time challenge for ecologists. Pielou [

However, identifying interactions under the assumption of nonhomogeneity of space is still an open question. Twenty years ago, Cuzick and Edwards [

Other tools were also developed by economists [

In this study, we introduce a relative measure of spatial structure, namely, the

The paper is organized as follows: first, we derive the

The theoretical framework is a point process whose realization is observed in a window of area

Points are denoted

Ripley’s

An unbiased estimator of univariate

Points located close to the window borders are problematic because a part of the circle inside which points are supposed to be counted is outside the window. Various answers have been proposed to correct for this [

Besag [

Equation (

We transpose

The average weighted ratio of neighbor points around reference points is

In the whole window, the same ratio is

We define the univariate

A particular attention must be paid to case-control designs. In practical terms, all points of interest (called

The

The first-order property (intensity) of the process must be controlled to allow the detection of the second-order property (nonindependence of points, that is to say interactions between the objects they represent). Thus, a point distribution generated according to the null hypothesis must respect, on the one hand, the local values of the density of the process the point distribution is a realization of and, on the other hand, its points must be distributed independently from each other.

The practical difficulty comes from the lack of knowledge of the point process that gave the point distribution, which is its unique available realization. Its first-order property is consequently widely unknown. We can only assume that the actual set of point locations is a good approximation of it, following Duranton and Overman [

The intertype function must support two null hypotheses [

The tests based on Monte Carlo simulations are actually not correct because they are repeated at each step of the function (see [

If

Three theoretical examples are given. Two of them illustrate very simple point patterns on a homogeneous space for a comparison of

We overall want to provide evidence of the interest of relative spatial structure in ecology. Trees are considered in a 25 ha plot of tropical rainforest in Paracou field station in French Guiana [

In what follows, we generate a point pattern (“black points,” represented by closed circles in the figures) to investigate its spatial structure with the

All confidence intervals are computed at 1% risk level generated from 10,000 simulations.

200 grey points are completely randomly distributed. Black points are generated by a Matérn process [

Aggregates, Point map. Grey circles are drawn from a homogenous Poisson process. Black disks are generated by a Matérn (radius = 0.5) process.

Aggregates, univariate

The

200 grey points are drawn from a homogenous Poisson process again. 100 black points have a regular distribution around a square, 1-by-1 grid, with a perturbation: each point is randomly moved horizontally and vertically within a 0.4 interval around the grid nodes (Figure

Regular point set, Point map. Grey circles are from a Poisson process. Black disks are located close to a

The first part of the univariate

Regular point set, univariate

Negative peaks of both the univariate

We generated two completely random point sets in a 10-by-10 window. Then, we transformed the point coordinates: after having calculated the polar coordinates

Inhomogeneous point set, Point map. All points are drawn from an inhomogeneous Poisson point process.

It can be seen (Figure

Inhomogeneous point set, univariate

The case-control

Childhood Leukemia epidemiology [

In the discussion of [

Both methods suffer here a severe lack of power due to the very little number of controls. The confidence envelopes are computed at 10% levels (from 1000 simulations). The GoF test applied to

The dataset (map in Figure

Map of trees.

Aggregation of both species is detected up from 4–6 meters (Figures

Spatial structure of

These results suggest competition if our null hypothesis is correct: we suppose that both species could locate anywhere if the other did not impede it. Of course, it might be wrong so further work is necessary to test alternate hypotheses: the environment may be different and niche preferences may be the reason for segregation, or else the spatial distribution of populations may not be in equilibrium, and we may be observing the contact of two colonization fronts.

The density of

Regeneration of

Jansen et al. [

The theoretical examples illustrate that

Although the

Figure

Distance-based measures of spatial concentration can be classified into two main categories [

The topographic toolbox is already well furnished, with Baddeley et al.’s [

The

To allow the effective use of the

The authors thank the editor and an anonymous referee for useful suggestions. This work has benefited from an “Investissement d’Avenir” grant managed by Agence Nationale de la Recherche (CEBA, ref. ANR-10-LABX-0025). This paper partially incorporates earlier unpublished work written by E. Marcon and F. Puech (generalizing Ripley's