Second-order analysis of inhomogeneous spatial point processes using case-control data

Summary Methods for the statistical analysis of stationary spatial point process data are now well established, methods for nonstationary processes less so. One of many sources of nonstationary point process data is a case–control study in environmental epidemiology. In that context, the data consist of a realization of each of two spatial point processes representing the locations, within a specified geographical region, of individual cases of a disease and of controls drawn at random from the population at risk. In this article, we extend work by Baddeley, Møller, and Waagepetersen (2000, Statistica Neerlandica 54 , 329–350) concerning estimation of the second‐order properties of a nonstationary spatial point process. First, we show how case–control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second‐order properties. Second, we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls. Our primary focus is estimation, but we also propose a new test for spatial clustering that we show to be competitive with existing tests. We describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases.     

A multiscale maximum entropy moment closure for locally regulated space–time point process models of population dynamics

The prevalence of structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based only on mean densities (local or global). Individual-based models (IBMs) were introduced during the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite useful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space–time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. This approach has shown that simplified representations of fecundity, local dispersal and density-dependent mortality weighted by the local competitive environment are sufficient to generate spatial patterns that mimic field observations. Continuum approximations of these stochastic processes try to distill their fundamental properties, and they keep track of not only mean densities, but also higher order spatial correlations. However, due to the non–linearities involved they result in infinite hierarchies of moment equations. This leads to the problem of finding a ‘moment closure’; that is, an appropriate order of (lower order) truncation, together with a method of expressing the highest order density not explicitly modelled in the truncated hierarchy in terms of the lower order densities. We use the principle of constrained maximum entropy to derive a closure relationship for truncation at second order using normalisation and the product densities of first and second orders as constraints, and apply it to one such hierarchy. The resulting ‘maxent’ closure is similar to the Kirkwood superposition approximation, or ‘power-3’ closure, but it is complemented with previously unknown correction terms that depend mainly on the avoidance function of an associated Poisson point process over the region for which third order correlations are irreducible. This domain of irreducible triplet correlations is found from an integral equation associated with the normalisation constraint. This also serves the purpose of a validation check, since a single, non-trivial domain can only be found if the assumptions of the closure are consistent with the predictions of the hierarchy. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the truncated hierarchy to predict equilibrium values for mildly aggregated spatial patterns. However, the maxent closure performs comparatively poorly in segregated ones. Although the closure is applied in the context of point processes, the method does not require fixed locations to be valid, and can in principle be applied to problems where the particles move, provided that their correlation functions are stationary in space and time.     

On statistical inference for the random set generated Cox process with set-marking

Cox point process is a process class for hierarchical modelling of systems of non-interacting points in Rd under environmental heterogeneity which is modelled through a random intensity function. In this work a class of Cox processes is suggested where the random intensity is generated by a random closed set. Such heterogeneity appears for example in forestry where silvicultural treatments like harvesting and site-preparation create geometrical patterns for tree density variation in two different phases. In this paper the second order property, important both in data analysis and in the context of spatial sampling, is derived. The usefulness of the random set generated Cox process is highly increased, if for each point it is observed whether it is included in the random set or not. This additional information is easy and economical to obtain in many cases and is hence of practical value; it leads to marks for the points. The resulting random set marked Cox process is a marked point process where the marks are intensity-dependent. The problem with set-marking is that the marks are not a representative sample from the random set. This paper derives the second order property of the random set marked Cox process and suggests a practical estimation method for area fraction and covariance of the random set and for the point densities within and outside the random set. A simulated example and a forestry example are given.     

Bayesian Wombling for Spatial Point Processes

Summary In many applications involving geographically indexed data, interest focuses on identifying regions of rapid change in the spatial surface, or the related problem of the construction or testing of boundaries separating regions with markedly different observed values of the spatial variable. This process is often referred to in the literature as boundary analysis or wombling. Recent developments in hierarchical models for point‐referenced (geostatistical) and areal (lattice) data have led to corresponding statistical wombling methods, but there does not appear to be any literature on the subject in the point‐process case, where the locations themselves are assumed to be random and likelihood evaluation is notoriously difficult. We extend existing point‐level and areal wombling tools to this case, obtaining full posterior inference for multivariate spatial random effects that, when mapped, can help suggest spatial covariates still missing from the model. In the areal case we can also construct wombled maps showing significant boundaries in the fitted intensity surface, while the point‐referenced formulation permits testing the significance of a postulated boundary. In the computationally demanding point‐referenced case, our algorithm combines Monte Carlo approximants to the likelihood with a predictive process step to reduce the dimension of the problem to a manageable size. We apply these techniques to an analysis of colorectal and prostate cancer data from the northern half of Minnesota, where a key substantive concern is possible similarities in their spatial patterns, and whether they are affected by each patient's distance to facilities likely to offer helpful cancer screening options.     

A test for spatial relationships between neighbouring plants in plots of heterogeneous plant density

Abstract. Maps of plant individuals in (x, y) coordinates (i.e. point patterns) are currently analysed through statistical methods assuming a homogeneous distribution of points, and thus a constant density within the study area. Such an assumption is seldom met at the scale of a field plot whilst delineating less heterogeneous subplots is not always easy or pertinent. In this paper we advocate local tests carried out in quadrats partitioning the plot and having a size objectively determined via a trade‐off between squared bias and variance. In each quadrat, the observed pattern of points is tested against complete spatial randomness (CSR) through a classical Monte‐Carlo approach and one of the usual statistics. Local tests yield maps of p‐values that are amenable to diversified subsequent analyses, such as computation of a variogram or comparison with co‐variates. Another possibility uses the frequency distribution of p‐values to test the whole point pattern against the null hypothesis of an inhomogeneous Poisson process. The method was demonstrated by considering computer‐generated inhomoge‐neous point patterns as well as maps of woody individuals in banded vegetation (tiger bush) in semi‐arid West Africa. Local tests proved able to properly depict spatial relationships between neighbours in spite of heterogeneity/clustering at larger scales. The method is also relevant to investigate interaction between density and spatial pattern in the presence of resource gradients.     

植物空间分布格局中邻体距离的概率分布模型及参数估计

最近邻体法是一类有效的植物空间分布格局分析方法,邻体距离的概率分布模型用于描述邻体距离的统计特征,属于常用的最近邻体法之一。然而,聚集分布格局中邻体距离(个体到个体)的概率分布模型表达式复杂,参数估计的计算量大。根据该模型期望和方差的特性,提出了一种简化的参数估计方法,并利用遗传算法来实现参数优化,结果表明遗传算法可以有效地估计的该模型的两个参数。同时,利用该模型拟合了加拿大南温哥华岛3个寒温带树种的空间分布数据,结果显示:该概率分布模型可以很好地拟合美国花旗松(P.menziesii)和西部铁杉(T.heterophylla)的邻体距离分布,但由于西北红柏(T.plicata)存在高度聚集的团簇分布,拟合结果不理想;美国花旗松在样地中近似随机分布,空间聚集参数对空间尺度的依赖性不强,但西北红柏和西部铁杉空间聚集参数具有尺度依赖性,随邻体距离阶数增加而变大。最后,讨论了该模型以及参数估计方法的优势和限制。     

A practical approach to the study of spatial structure in simple cases of heterogeneous vegetation

Abstract. Spatial heterogeneity is a characteristic of most natural ecosystems which is difficult to handle analytically, particularly in the absence of knowledge about the exogenous factors responsible for this heterogeneity. While classical methods for analysis of spatial point patterns usually require the hypothesis of homogeneity, we present a practical approach for partitioning heterogeneous vegetation plots into homogeneous subplots in simple cases of heterogeneity without drastically reducing the data. It is based on the detection of endogenous variations of the pattern using local density and second‐order local neighbour density functions that allow delineation of irregularly shaped subplots that could be considered as internally homogeneous. Spatial statistics, such as Ripley's K‐function adapted to analyse plots of irregular shape, can then be computed for each of the homogeneous subplots. Two applications to forest ecological field data demonstrate that the method, addressed to ecologists, can avoid misinterpretations of the spatial structure of heterogeneous vegetation stands.     

Natural regeneration of forest related to the spatial structure of trees: A study of two forest communities in Western Carpathians,southern Poland

The existence of a relationship between the spatial pattern of trees and the distribution of young individuals beneath the canopy has been tested in the beech (Fagus sylvatica) and spruce (Picea abies) — fir (Abies alba) forests in the mountainous region, using two different methods. The first method was the analysis of spatial pattern of individuals, the second one was based on calculating sums of influences of all trees occurring within analysed plot on a given point on the forest floor. Results of spatial pattern analyses were surprisingly consistent: almost all mature trees and seedlings didplayed a random pattern of spatial arrangement. However, there is a clear, although statistically insignificant tendency towards uniformity of spatial pattern with increasing sizes of analysed trees. Results of comparing sums of influences on regularly distributed points with sums of influences on seedlings or saplings revealed no tendency in forest regeneration to concentrate in places, where the sums were smaller than the average for a plot. This, coupled with the dominance of random spatial pattern of trees, suggests, that viewed on a small spatial scale, influence of competition among forest trees on their spatial arrangement is obscured by other factors, which are not closely related to the distribution of individuals.     

Spatial patterns on the sagebrush steppe/Western juniper ecotone

Analysis of the spatial patterns of woody plants is important to better understand the ecological processes that govern the worldwide expansion of woody plants across semi-arid ecosystems. Second-order characteristics of a marked spatial point pattern of western juniper (Juniperus occidentalis subsp. occidentalis) were analyzed using Ripley’s K-functions and the pair-correlation function g. The marked point process of crown diameters was produced via two-dimensional wavelet analysis of a fine scale aerial photograph at the woodland-steppe ecotone in the Reynolds Creek watershed in the Owyhee Mountains, southwestern Idaho. Colonization of J. occidentalis stems from mature juniper trees growing in rocky, fire resistant areas. Although these areas introduce components of natural heterogeneity within the landscape, the selected study area is situated within a single soil type, and we modeled the expansion of juniper plants into previously juniper-free sagebrush steppe as a homogeneous point process with constant intensity. Through this research we have identified two statistically significant spatial scales characteristic of J. occidentalis on the woodland/steppe ecotone: (1) We observed inhibition between J. occidentalis plants at distances <15 m, resulting in a regular pattern, rather than clumped or random. This short-distance inhibition can be attributed to competition for water and other resources. Recruitment of young J. occidentalis occurs significantly more often in a direction away from older plants, maximizing the utilization of water and light resources, and perpetuating the spread of the species into previously juniper-free shrub-steppe. (2) J. occidentalis on the ecotone exhibits significant clustering within a 30–60 m radius. Bivariate point pattern analyses provide evidence that, within a distance of 50–70 m, there is a spatial dependence in tree size such that medium trees are more likely than small trees to be close to large trees. We attribute these phenomena to the fact that juniper seeds are commonly dispersed by berry-eating birds with small territories (0.3–1 ha). Beyond a distance of 50–70 m, juniper plants are randomly distributed, suggesting that additional long-range seed dispersal processes are at work. We further acknowledge the importance of including a reference to spatial scale when formulating hypotheses in statistical analysis of spatio-temporal point patterns.     

Temporal Analogues to Spatial K Functions

In this article, the spatial statistic known as the K function is adapted for temporal processes and patterns. The (optimal) K-function estimator is used in a testing procedure to determine whether behavior patterns of exposed rats versus control rats are different. Specifically, the temporal analogue to the K function is given and an approximately optimal estimator is developed. Next, a testing procedure, to determine whether a group of point patterns is generated from complete temporal randomness, is given. Finally, a testing procedure, to compare pairwise two groups of point patterns to each other, is given. The testing procedures are illustrated with rat-behavior data from both a control-control experiment as well as an exposed-control experiment, where in the latter case a difference in behavior is known to exist.     

A new non‐parametric method for analyzing replicated point patterns in ecology

Most ecological studies that involve point pattern analyses are based on a single plot, which prevent the separation of the effects of various processes that could act simultaneously, as well as limiting the conclusions that can be extracted from these studies. However, considering the spatial distribution of individuals in several plots as replicates of the same process could help to differentiate its specific effects from those of other confounding processes. Thus, we introduce a new method for analyzing spatial point patterns that are replicated according to a two‐factorial design. By summarizing the spatial patterns as K‐functions, the proposed method computes the average K‐functions for each level of the two factors (i.e. predictors) and for each combination of levels, before estimating the sum of squared deviations from the overall mean K‐function. Inferences of the strength of the relationship between the predictors, their interaction, and the spatial structure are made based on a non‐parametric bootstrap procedure, which considers the dependency among spatial scales. We illustrate the proposed approach based on an analysis of the effects of altitude (with two levels: low and high) and slope (with two levels: flat and steep slopes) on the spatial pattern of Croton wagneri, a dominant shrub in an Andean dry scrubland. Our method detected a significant effect of the interaction between slope and altitude, which could not have been detected using current point pattern analysis methodology. The prevalence of single‐plot analysis in ecological studies may be due to a lack of familiarity with appropriate methods for replicated point patterns, as well as the greater complexity of these methods and the absence of appropriate software. Our approach can be applied to a significant number of ecological questions while maintaining a simple, understandable, and easily reportable methodological framework.     

基于不同零模型的三江平原沼泽湿地主要物种小尺度点格局分析

在三江平原沼泽湿地,基于不同零模型的点格局方法,研究了常年积水环境条件下沼泽湿地主要优势种毛苔草、漂筏苔草、狭叶甜茅以及小叶章种群的空间分布格局特征。结果表明:在0—200cm尺度范围内4种物种基本都偏离完全随机模型,表现为聚集分布,但偏离的程度不同;除了数量最少的小叶章种群外,其他3种物种都在一定尺度上偏离了泊松聚块模型;毛苔草和狭叶甜茅种群在极小尺度上偏离嵌套双聚块模型,但不显著,而漂筏苔草种群在所有尺度上符合嵌套双聚块模型。种群的实测值偏离完全随机模型的程度越大,越有可能符合符合泊松聚块模型,偏离泊松聚块模型的程度越大,越有可能符合嵌套双聚块模型。4种物种在0—200cm尺度范围内形成的聚块是由营养繁殖引起的,多个分株系统组成大聚块,而每个分株系统形成一个小聚块,聚块的形成加剧了种内竞争,使得种群的聚集强度降低。     

Bias-corrected variance estimation and hypothesis testing for spatial point and marked point processes using subsampling

Summary We introduce novel regression extrapolation based methods to correct the often large bias in subsampling variance estimation as well as hypothesis testing for spatial point and marked point processes. For variance estimation, our proposed estimators are linear combinations of the usual subsampling variance estimator based on subblock sizes in a continuous interval. We show that they can achieve better rates in mean squared error than the usual subsampling variance estimator. In particular, for n×n observation windows, the optimal rate of n^?2 can be achieved if the data have a finite dependence range. For hypothesis testing, we apply the proposed regression extrapolation directly to the test statistics based on different subblock sizes, and therefore avoid the need to conduct bias correction for each element in the covariance matrix used to set up the test statistics. We assess the numerical performance of the proposed methods through simulation, and apply them to analyze a tropical forest data set.     

Modelling Tree Roots in Mixed Forest Stands by Inhomogeneous Marked Gibbs Point Processes

The aim of the paper is to apply point processes to root data modelling. We propose a new approach to parametric inference when the data are inhomogeneous replicated marked point patterns. We generalize Geyer's saturation point process to a model, which combines inhomogeneity, marks and interaction between the marked points. Furthermore, the inhomogeneity influences the definition of the neighbourhood of points. Using the maximum pseudolikelihood method, this model is then fitted to root data from mixed stands of Norway spruce (Picea abies (L.) Karst.) and European beech (Fagus sylvatica L.) to quantify the degree of root aggregation in such mixed stands. According to the analysis there is no evidence that the two root systems are not independent.     

A new digital method of data collection for spatial point pattern analysis in grassland communities

A major objective of plant ecology research is to determine the underlying processes responsible for the observed spatial distribution patterns of plant species. Plants can be approximated as points in space for this purpose, and thus, spatial point pattern analysis has become increasingly popular in ecological research. The basic piece of data for point pattern analysis is a point location of an ecological object in some study region. Therefore, point pattern analysis can only be performed if data can be collected. However, due to the lack of a convenient sampling method, a few previous studies have used point pattern analysis to examine the spatial patterns of grassland species. This is unfortunate because being able to explore point patterns in grassland systems has widespread implications for population dynamics, community‐level patterns, and ecological processes. In this study, we developed a new method to measure individual coordinates of species in grassland communities. This method records plant growing positions via digital picture samples that have been sub‐blocked within a geographical information system (GIS). Here, we tested out the new method by measuring the individual coordinates of Stipa grandis in grazed and ungrazed S. grandis communities in a temperate steppe ecosystem in China. Furthermore, we analyzed the pattern of S. grandis by using the pair correlation function g(r) with both a homogeneous Poisson process and a heterogeneous Poisson process. Our results showed that individuals of S. grandis were overdispersed according to the homogeneous Poisson process at 0–0.16 m in the ungrazed community, while they were clustered at 0.19 m according to the homogeneous and heterogeneous Poisson processes in the grazed community. These results suggest that competitive interactions dominated the ungrazed community, while facilitative interactions dominated the grazed community. In sum, we successfully executed a new sampling method, using digital photography and a geographical information system, to collect experimental data on the spatial point patterns for the populations in this grassland community.     

Fine‐scale spatial distribution of murundus structures in the semi‐arid region of Brazil

Murundus are earth mounds widespread in most landscapes in the semi‐arid region of Brazil. Evidence obtained from predictive modelling has suggested a termite origin for these structures, opening up new opportunities for further research. Distribution of densely packed murundus at larger spatial scales is most related to climatic regime and soil nutrient availability. However, factors and processes underlying their distribution and density at smaller spatial scales are not yet fully understood. In this study, we adopted an approach based on mapping point data using high‐resolution satellite imagery, multi‐scale second‐order analysis and general linear models to examine the fine‐scale spatial distribution and density of murundus. Our results suggest that the distribution of those structures within densely packed areas is regulated by more than one process acting or interacting across multiple spatial scales. All densely packed murundus showed a significant regular distribution at the distance scale of up to 50 m radially and a completely random distribution across all other upper distance scales. We interpret the regular pattern as a result of competition for foraging territories between different termite colonies during the process formation of densely packed murundus. The random pattern at larger distance scales (above 50 m radially) can be attributed to habitat selection preferences by termite species builders of murundus mediated by local environmental resources and conditions (i.e. availability of food resources and nesting and open habitat), which would be randomly distributed in space. Thus, at finer spatial scales murundus distributions are associated with biotic interactions acting on an abiotic template. On the basis of significant linear correlations, we suggest that the density of murundus is strongly related to local temperature regime with soil‐type influencing its effect on the murundus densities. Our findings provide novel evidences that mound‐building termites are involved in the formation of murundus in the semi‐arid region of Brazil.     

PCR method for generating multiple mutations at adjacent sites

Procedures to introduce point mutations, restriction sites and insert or delete DNA fragments are very important tools to study protein function. We describe here two-step PCR-based method for generating single or multiple mutations, insertions and delections in a small region of the sequence. In the first step, a unique restriction site is introduced near the part of DNA sequence to be changed, without changing the amino acid sequence. For this step, one of the methods already described can be used. In the second step, mutations are introduced using mutagenic primers containing the unique restriction site from the first step at the 5′ end, paired with a universal primer crossing another unique restriction site present originally in the sequence. The method is very simple, economic and rapid. In comparison with the traditionalin vitro mutagenesis methods, one can generate large numbers of mutated plasmids in hours.     

呼中林区火烧点格局分析及影响因素

林火是森林生态系统景观格局、动态和生态过程的重要自然驱动力,理解林火发生空间格局与影响因素对于林火安全管理具有重要的作用。采用点格局分析方法,以黑龙江大兴安岭呼中林区1990-2005年火烧数据为研究案例,分析了火烧点空间格局及其影响因素。结果表明,火烧点在空间上的分布是不均匀的,呈现聚集分布,存在一些火烧高发区和低发区。呼中林区火烧概率是0.004-0.012次/(km² · a),平均火烧概率为0.0077次/(km² · a)。人类活动因子、地形因子和植被因子对林火的发生均具有重要作用。应用空间点格局分析方法表明,距离居民点和道路的距离、高程、坡度和林型是影响林火发生的显著因子。因此在进行森林防火管理时,仅仅通过控制人类活动对于降低林火火险的效果是有限的,地形和林型也是林火防控时重点要考虑的因素。