Here we discuss strategies for analysing spatial point pattern data, foreshadowing the main themes of the book.3
The choice of strategy for modelling and analysing a spatial point pattern depends on the re-search goals. Our attention may be focussed primarily on the intensity of the point pattern, or primarily on the interaction between points, or equally on the intensity and interaction. There is a choice concerning the scope of statistical inference, that is, the ‘population’ to which we wish to generalise.
5.7.1 Intensity
The intensity is the (localised) expected density of points per unit area. It is typically interpreted as the rate of occurrence, abundance or incidence of the events recorded in the point pattern. When the prevention of these events is the primary concern (e.g., defects in crystal, petty crimes, cases of infectious disease), the intensity is usually the feature of primary interest. The main task for analysis may be to quantify the intensity, to decide whether intensity is constant or spatially varying, or to map the spatial variation in intensity. If covariates are present, then the main task may be to investigate whether the intensity depends on the covariate, for example, whether the abundance of trees depends on the acidity of soil.
The intensity is a first moment quantity (related to expectations of counts of points). Hence it is possible to study the intensity by formulating a model for the intensity only, for example, a parametric or semiparametric model for the intensity as a function of the Cartesian coordinates.
See Chapter 6. In such analyses, stochastic dependence between points is a nuisance feature that complicates the methodology and inference.
Alternatively, we may formulate a complete stochastic model for the observed point pattern (i.e.
a spatial point process model) in which the main focus is the description of the intensity. The model should exhibit the right type of stochastic dependence, and the intensity should be a tractable func-tion of the model parameters. If points are independent, the correct model is a Poisson point process (Chapter 9). If there is positive association between points, useful models include cluster processes and Cox processes (Chapter 12). If there is negative association, Gibbs processes (Chapter 13) are appropriate, although the intensity is not a simple function of the model parameters.
5.7.2 Interaction
‘Interpoint interaction’is the conventional term for stochastic dependence between points. This covers a wide range of behaviour, since the only point processes which do not exhibit stochastic dependence are the Poisson processes. The term ‘interaction’ can be rather prejudicial. One possible cause of stochastic dependence is a direct physical interaction between the objects recorded in the point pattern. For example, if the spatial pattern of pine seedlings in a natural forest is found to exhibit negative association at short distances, this might be interpreted as reflecting biological interaction between the seedlings, perhaps due to competition for space, light, or water.
The main task for analysis may be to decide whether there is stochastic dependence, to determine the type of dependence (e.g., positive or negative association), or to quantify its strength and spatial range.
Whereas intensity is a ‘first moment’ property, interpoint interaction is measured by second-order moment quantities such as the K-function (Chapter 7), or by higher-second-order quantities such as
3Some of the material in this section was previously published in [30, Sec. 20.2].
in numerical data by carefully adjusting for changes in the mean, a rigorous analysis of interpoint interaction requires that we take into account any spatial variation in intensity.
A popular classical approach to spatial point pattern analysis was to assume that the point pat-tern is stationary. This implies that the intensity is constant. Analysis could then concentrate on investigating interpoint interaction. It was argued (e.g., [571]) that this approach was pragmatically justified when dealing with quite small datasets (containing only 30 to 100 points), or when the data were obtained by selecting a small subregion where the pattern appeared stationary, or when the assumption of stationarity is scientifically plausible.
Figure 5.24 shows the Swedish Pines dataset of Strand [642] presented by Ripley [575] as an example where the above-mentioned conditions for assuming stationarity were satisfied. There is nevertheless some suggestion of inhomogeneity. Contour lines represent the fitted intensity under a parametric model in which the logarithm of the intensity is a quadratic function of the Cartesian coordinates. Figure 5.25 shows the estimated K-function of the Swedish Pines assuming stationarity, and the inhomogeneous K-function which adjusts for the fitted log-quadratic intensity (see [46] and Section 7.10.2). The two K-functions convey a similar message, namely that there is inhibition between the saplings at distances less than one metre. They agree because gentle spatial variation in intensity over large spatial scales is irrelevant at shorter scales.
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Figure 5.24. Swedish Pines data (Left) with fitted log-quadratic intensity (Right) expressed in points per square metre.
5.7.3 Intensity and interaction
In some applications, intensity and interaction are both of interest. For example, a cluster of new cases of a disease may be explicable either by a localised increase in intensity due to aetiology (such as a localised pathogen), sampling effects (a localised increase in vigilance, etc.), or by stochastic dependence between cases (due to person-to-person transmission, familial association, genetics, social dependence, etc.). The spatial arrangement of galaxies in a galaxy cluster invites complex space-time models, in which the history of the early universe is reflected in the overall intensity of galaxies, while the observed local arrangement of galaxies involves gravitational interactions in recent history.
When a point pattern exhibits both spatial inhomogeneity and interpoint interaction, several strategies are possible. In an incremental or marginal modelling strategy we try to estimate spatial trend, then ‘subtract’ or ‘adjust’ for spatial trend, possibly in several stages, before looking for evidence of interpoint interaction. In a joint modelling strategy we try to fit one stochastic model
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Figure 5.25. Left: estimated K-function of Swedish Pines assuming stationarity. Right: estimated inhomogeneous K-function of Swedish Pines using fitted log-quadratic intensity.
that captures all relevant features of the point process and, in particular, allows the statistician to account for spatial inhomogeneity during the analysis of interpoint interaction.
These choices are familiar from time series analysis. Incremental modelling is analogous to seasonal adjustment of time series, while joint modelling is analogous to fitting a time series model that embraces both seasonal trend and autocorrelation. Incremental modelling is less prone to the effects of model misspecification, while joint modelling is less susceptible to analogues of Simp-son’s Paradox. Joint modelling would normally be employed in the final and more formal stages of analysis, while incremental modelling would usually be preferred in the initial and more exploratory stages.
For example, in the analysis of the Swedish Pines data above, we first fitted a parametric in-tensity model, then computed the inhomogeneous K-function which ‘adjusts’ for this fitted inten-sity. This is an incremental modelling approach. A corresponding joint modelling approach is to fit a Gibbs point process (Chapter 13) with non-stationary spatial trend. Again we assume a log-quadratic trend. Figure 5.25 suggests fitting a Strauss process model (Section 13.3.7) with in-teraction radius r between 4 and 15 units. The model selected by maximum profile pseudolikelihood has r = 9.5 and a fitted interaction parameter ofγ= 0.27, suggesting substantial inhibition between points.
5.7.4 Confounding between intensity and interaction
In analysing a point pattern, it may be impossible to distinguish between clustering and spatial inho-mogeneity. Bartlett [83] showed that a single realisation of a point process model that is stationary and clustered (i.e. exhibits positive dependence between points) may be identical to a single re-alisation of a point process model that has spatially inhomogeneous intensity but is not clustered.
Based on a single realisation, the two point process models are distributionally equivalent and hence unidentifiable. This represents a fundamental limitation on the scope of statistical inference from a spatial point pattern, assuming we do not have access to genuine replicate observations. The in-ability to separate trend and autocorrelation, within a single dataset, is also familiar in time series analysis.
This may be categorised as a form of confounding. In the theory of design and analysis of
if the columns of the design matrix X are not linearly independent, so that the parameter vector β is not identifiable. Bartlett’s examples show that a point process model involving both spatial inhomogeneity and interpoint interaction may be confounded, that is, unidentifiable, given only a single realisation of the spatial point process.
The potential for confounding spatial inhomogeneity and interpoint interaction is important in the interpretation of summary statistics such as the K-function. In Figure 5.26 the left panel shows a realisation of a spatially inhomogeneous Poisson process, its intensity a linear function of the Cartesian coordinates. The right panel is a plot of bL(r) − r against r, where bL is the estimate of L(r) =p
K(r)/π assuming the point process is stationary. The right-hand plot invites the incorrect interpretation that the points are clustered.
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Figure 5.26.Illusory clustering.Left: realisation of a nonstationary Poisson process. Right: plot of ˆL(r)−r against r for the same point pattern, inviting the interpretation that the pattern is clustered.
Formal tests such as theχ2test of CSR using quadrat counts (Section 6.4.2) are afflicted by a similar weakness (see Section 6.4.3).
5.7.5 Covariate effects
When a point pattern dataset is accompanied by covariate data (Sections 1.1.4 and 5.6.4), we typi-cally want to investigate whether the intensity depends on the covariates and to quantify this depen-dence. It may be enough to conduct a formal statistical test of the hypothesis that the point pattern does not depend on the covariate.
For a numerical spatial covariate Z(u), we could assume (in a simple case) that the intensity of the point process depends on Z through the relationshipλ(u) =ρ(Z(u)) whereρ is a function to be estimated. For example,ρ(z) could express the preference of a tree species for a particular habitat, or the likelihood of finding mineral deposits in a particular geochemical environment. The functionρcould be estimated using nonparametric methods such as kernel smoothing, or by fitting a parametric model.
When several covariates are present, we could focus on one covariate Z(u) and assume that it has a multiplicative effect on the point process intensity, λ(u) =ρ(Z(u))B(u) where B(u) is a
‘baseline’ or ‘reference’ intensity that takes the other covariates into account. Againρ could be estimated nonparametrically or parametrically. When many covariates are present, or when the effect of a covariate is not so simple, a parametric model is usually preferable.
One reason for investigating covariate effects is to adjust for them when studying interaction between points. For example, the standard analysis of correlation between points using Ripley’s K-function is sensitive to spatial variation in the underlying intensity. The inhomogeneous K-K-function [46] adjusts for this spatial variation: it requires an accurate estimate of the intensity function.
5.7.6 Multitype point patterns
A multitype point pattern (Section 1.1.2) is a pattern of points of several different types. It is usually represented as a marked point pattern where the marks are categorical (factor) values.
Multitype point patterns introduce many new scientific questions. Under the heading of ‘inten-sity’ we may want to know whether the intensity functions of the points of each type are proportional to each other, implying that the relative proportions of each type of point are constant over the study region. If not, the most extreme alternative is that the different types of points are ‘segregated’, tending to occupy different parts of the study region.
Under the heading of ‘interaction’ we can investigate dependence between points of the same type or different types. Dependence between points of the same type has the usual interpretations of clustering, inhibition and so on. Independence between points of the same type implies that they form a Poisson process. However, dependence between points of different types has a completely different interpretation. Independence between points of types i and j means that the two point processes, consisting of points of type i and points of type j, respectively, are independent point processes, but does not imply anything about their spatial pattern.
Chapter 14 discusses the issues and techniques involved in analysing multitype point patterns.
5.7.7 Scope of inference
There is a choice concerning the scope of statistical inference, that is, the ‘population’ to which we wish to generalise from the data.
At the lowest level of generalisation, we are interested only in the region that was actually sur-veyed. In applying precision agriculture to a particular farm, we might use the observed spatial point pattern of tree seedlings, which germinated in a field sown with a uniform density of seed, as a means of estimating the unobservable, spatially varying, fertility of the soil in the same field.
Statistical inference here is a form of interpolation or prediction. The modelling approach is influ-enced by the prediction goals: to predict soil fertility it may be sufficient to model the point process intensity only, and ignore interpoint interaction.
At the next level, the observed point pattern is treated as a ‘typical’ sample from a larger pattern which is the target of inference. To draw conclusions about an entire forest from observations in a small study region, we treat the forest as a spatial point process X, effectively extending throughout the infinite two-dimensional plane. In order to draw inferences based only on a sample of X in a fixed bounded window W , we might assume that X is stationary and/or isotropic, meaning that sta-tistical properties of the point process are unaffected by vector translations (shifts) and/or rotations, respectively. This implies that our dataset is a typical sample of the process, and supports nonpara-metric inference about distributional properties of X such as its intensity and K-function. It also supports parametric inference, for example about the interaction parameterγ of a Strauss process model for the spatial dependence between trees.
At a higher level, we seek to extract general ‘laws’ or ‘relationships’ from the data. This involves generalising from the observed point pattern to a hypothetical population of point patterns which are governed by the same ‘laws’ but which may be very different from the observed point pattern.
One important example is modelling the dependence of the point pattern on a spatial covariate (such as terrain slope). This is a form of regression. We might assume that the intensityλ(u) of the point process at a location u is a functionλ(u) =ρ(Z(u)) of the spatial covariate Z(u). The regression functionρis the target of inference. The scope of inference is a population of experiments where the same variables are observed and the same regression relationship is assumed to hold. A model for ρ (parametric, non-, or semi-parametric) is formulated and fitted. More detailed inference requires either replication of the experiment, or an assumption such as joint stationarity of the covariates and the response, under which a large sample can be treated as containing sufficient replication.
At the highest level, we seek to capture all sources of variability that influence the spatial point
covariate, and also ‘random effects’ such as regression on an unobserved, random spatial covariate.
For example, a Cox process (Section 5.5 and Chapter 12) is defined by starting with a random inten-sity function Λ(u) and, conditional on the realisation of Λ, letting the point process be Poisson with intensity Λ. In forestry applications, Λ could represent the unobserved, spatially inhomogeneous fertility of soil, modelled as a random process. Thus Λ is a ‘random effect’. Whether soil fertility should be modelled as a fixed effect or random effect depends on whether the main interest is in inferring the value of soil fertility in the study region (fixed effect) or in characterising the variability of soil fertility in general (random effect).