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.     

Assessing isotropy for spatial point processes

A common assumption while analyzing spatial point processes is direction invariance, i.e., isotropy. In this article, we propose a formal nonparametric approach to test for isotropy based on the asymptotic joint normality of the sample second-order intensity function. We derive an L(2) consistent subsampling estimator for the asymptotic covariance matrix of the sample second-order intensity function and use this to construct a test statistic with a chi(2) limiting distribution. We demonstrate the efficacy of the approach through simulation studies and an application to a desert plant data set, where our approach confirms suspected directional effects in the spatial distribution of the desert plant species.     

Detection of spatial change points in the mean and covariances of multivariate simultaneous autoregressive models

In this paper, we propose a test procedure to detect change points of multidimensional autoregressive processes. The considered process differs from typical applied spatial autoregressive processes in that it is assumed to evolve from a predefined center into every dimension. Additionally, structural breaks in the process can occur at a certain distance from the predefined center. The main aim of this paper is to detect such spatial changes. In particular, we focus on shifts in the mean and the autoregressive parameter. The proposed test procedure is based on the likelihood‐ratio approach. Eventually, the goodness‐of‐fit values of the estimators are compared for different shifts. Moreover, the empirical distribution of the test statistic of the likelihood‐ratio test is obtained via Monte Carlo simulations. We show that the generalized Gumbel distribution seems to be a suitable limiting distribution of the proposed test statistic. Finally, we discuss the detection of lung cancer in computed tomography scans and illustrate the proposed test procedure.     

A KPSS test for stationarity for spatial point processes

Summary .   We propose a formal method to test stationarity for spatial point processes. The proposed test statistic is based on the integrated squared deviations of observed counts of events from their means estimated under stationarity. We show that the resulting test statistic converges in distribution to a functional of a two-dimensional Brownian motion. To conduct the test, we compare the calculated statistic with the upper tail critical values of this functional. Our method requires only a weak dependence condition on the process but does not assume any parametric model for it. As a result, it can be applied to a wide class of spatial point process models. We study the efficacy of the test through both simulations and applications to two real data examples that were previously suspected to be nonstationary based on graphical evidence. Our test formally confirmed the suspected nonstationarity for both data.     

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.     

Spatial analysis of two-species interactions

Summary In this paper, I present and discuss some methods for the analysis of univariate and bivariate spatial point pattern data. Examples of such data in ecology include x-y coordinates of organisms in mapped field plots. I illustrate the methods with analyses of data from mapped field plots on Mount St. Helens, Washington state, USA. The statistical methods I emphasize are graphical methods that rely on analysis of distances between organisms. Hypothesis testing for methods like these is easily done using Monte Carlo methods, which I also discuss. For both univariate and bivariate analyses, I find that second-order methods such as K-function plots are often preferable to first-order methods (i.e., QQ-plots). However, for multivariate analyses, these second-order methods are more sensitive to small sample sizes than first-order analyses.     

Modeling a Poisson forest in variable elevations: a nonparametric Bayesian approach

A nonparametric Bayesian formulation is given to the problem of modeling nonhomogeneous spatial point patterns influenced by concomitant variables. Only incomplete information on the concomitant variables is assumed, consisting of a relatively small number of point measurements. Residual variation, caused by other unmeasured influential factors, is modeled in terms of a spatially varying baseline intensity function. A Markov chain Monte Carlo scheme is proposed for the simultaneous nonparametric estimation of each unknown function in the model. The suggested method is illustrated by reanalysing a data set in Rathbun (1996, Biometrics 52, 226-242), and the estimated models are compared with those obtained by Rathbun.     

Inference for clustered inhomogeneous spatial point processes

Summary .  We propose a method to test for significant differences in the levels of clustering between two spatial point processes (cases and controls) while taking into account differences in their first-order intensities. The key advance on earlier methods is that the controls are not assumed to be a Poisson process. Inference and diagnostics are based around the inhomogeneous K -function with confidence envelopes obtained from either resampling events in a nonparametric bootstrap approach, or simulating new events as in a parametric bootstrap. Methods developed are demonstrated using the locations of adult and juvenile trees in a tropical forest. A simulation study briefly examines the accuracy and power of the inferential procedures.     

Investigation of spatial clustering from individually matched case-control studies

The paper demonstrates how existing theory to assess spatial clustering based on second-moment properties of a labelled point process can be adapted to matched case-control studies. The null hypothesis that cases are a random sample from the superposition of cases and controls is replaced by the hypothesis that each case is a random sample from the set consisting of itself and its k matched controls. We compare the proposed test with other tests of spatial clustering, and describe an application to data on childhood diabetes in Yorkshire, England.     

神农架天然针阔混交林乔木更新的空间格局

为深入了解神农架天然针阔混交林群落乔木树种的更新特征,于神农架国家公园设置1 hm²森林动态监测样地对树种更新进行调查,利用Ripley的L函数分析优势更新树种的空间分布格局、种间空间关联性以及与大树的空间分布关系。结果显示：（1）更新个体共有35 752株,隶属于19科29属45种。更新优势种基本是乔木层优势树种,物种更新数量表现为：巴山冷杉（Abies fargesii Franch.） > 华山松（Pinus armandii Franch.） > 四蕊槭（Acer stachyophyllum subsp.tetramerum（Pax）A.E.） > 华中山楂（Crataegus wilsonii Sarg.）,混交林群落更新良好。（2）随着尺度增加,巴山冷杉、华山松更新空间格局由聚集分布向随机分布和均匀分布发展;华中山楂、四蕊槭更新的空间格局在整个尺度上均为聚集分布,表明针叶、阔叶树种更新在空间生态利用策略上出现分化。（3）巴山冷杉与其余3个优势更新树种、华山松与华中山楂更新呈显著正相关,表现出对微生境的共同喜好;四蕊槭分别与华山松、华中山楂更新的空间关系表现为显著负相关,种间竞争激烈。（4）巴山冷杉、华中山楂、四蕊槭的更新个体与大树的空间分布总体为显著正相关,这可能与种子传播限制相关;华山松幼苗、幼树与大树的空间分布在小尺度上呈负相关,存在资源不对称竞争。     

Statistical analysis of spatial patterns of the heliozoanActinophrys sol

Summary Cell-to-cell interaction and spatial distribution of the heliozoanActinophrys sol was analyzed with computer-aided video microscopy. By means of goodness-of-fit statistics (² analysis) and a quadrat-count analysis (I-curve analysis), the spatial point pattern of the cells was shown to be of regular distribution, which implies that a regulating mechanism is operating to encourage an even spatial distribution of the cell centers ofActinophrys. An attempt was further made to define a unified model which fitsActinophrys cell distribution observed at different cell densities. For this purpose, the fitting of a parameterized potential function (r)=(/r)¹² was carried out, wherer is the distance between cell centers of two neighboring cells. The scaling parameter a was estimated from the maximum likelihood procedure for obtaining the best fit for the data, which was found to be a decreasing function of the cell density; we obtained = 0.44 mm at a low cell density (0.5 cell/mm²) and =0.10 mm at the highest cell density (6.5 cells/mm²). These results suggest that (1) the possible nearest distance between two neighboring cells is primarily defined by the axopodial length, and (2) at lower cell densities,Actinophrys can recognize the presence of distant neighboring cells by some unknown means.     

A Model-Based Approach for Making Ecological Inference from Distance Sampling Data

Summary .  We consider a fully model-based approach for the analysis of distance sampling data. Distance sampling has been widely used to estimate abundance (or density) of animals or plants in a spatially explicit study area. There is, however, no readily available method of making statistical inference on the relationships between abundance and environmental covariates. Spatial Poisson process likelihoods can be used to simultaneously estimate detection and intensity parameters by modeling distance sampling data as a thinned spatial point process. A model-based spatial approach to distance sampling data has three main benefits: it allows complex and opportunistic transect designs to be employed, it allows estimation of abundance in small subregions, and it provides a framework to assess the effects of habitat or experimental manipulation on density. We demonstrate the model-based methodology with a small simulation study and analysis of the Dubbo weed data set. In addition, a simple ad hoc method for handling overdispersion is also proposed. The simulation study showed that the model-based approach compared favorably to conventional distance sampling methods for abundance estimation. In addition, the overdispersion correction performed adequately when the number of transects was high. Analysis of the Dubbo data set indicated a transect effect on abundance via Akaike's information criterion model selection. Further goodness-of-fit analysis, however, indicated some potential confounding of intensity with the detection function.     

Woody vegetation spatial patterns in a semi-arid savanna of Burkina Faso,West Africa

Spatial patterns of woody individuals were studied in a semi-arid savanna of West Africa located in Burkina Faso at and around 14° 12 N and 2° 27 W. The study was based upon a 10.24 ha plot within which individuals were mapped. Spatial pattern analysis was carried out using second order characteristics of point processes as K functions and pair correlations. The overall density amounted to 298 individuals ha^-1. The most abundant species were Combretum micranthum G. Don., Grewia bicolor Juss. and Pterocarpus lucens Lepr. Anogeissus leiocarpus (D.C.) G. et Perr. was also an important constituant of this vegetation type, owing to its taller stature. Clumped spatial distributions were identified for all species except for two, for which complete spatial randomness (CSR) was found (including P. lucens, a dominant woody plant). No regular pattern was found even when tall individuals were considered alone. Aggregation dominates interspecific relationships, resulting in multispecific clumps and patches. The overall aggregation pattern was constituted by two different structures. A coarse-grain pattern of ca. 30–40 m was based on edaphic features, and expresses the contrast between sparse stands on petroferric outcrops and denser patches on less shallow soils. A finer-grain pattern made of clumps ca. 5–10 m wide, with no obvious relation to pre-existing soil heterogeneity. There was no overall pattern for saplings (between 0.5 m and 1.5 m in height) irrespective of species, and thus no obvious common facilitation factor. For species with a high recruitment level there was no significant relationship between mature adult and saplings. The only case of clumped saplings with randomly distributed adults was found in P. lucens. However, this cannot be unequivocally interpreted as density dependent regulation since the existence of such a process was not consistent with the spatial distribution of dead P. lucens individuals (victims of the last drought). The mean density around dead P. lucens was lower than around surviving ones, indicating that the last drought tended to reinforce clumping rather than promote a regular pattern of trees. Spatial pattern analysis yielded no evidence supporting a hypothesis of stand density regulation through competition between individuals. Other processes, as surface sealing of bare soils or insufficient recruitment, may play a more important role in preventing a savanna-like vegetation from turning into denser woodlands or thickets.     

On Second Order Neighbourhood Analysis of Mapped Point Patterns

Getis and Franklin (1987), introduced a technique based on second order methods, called second order neighbourhood method, which is used to quantify clustering at various spatial scales. Variants of this method are introduced for testing whether a spatial point pattern is consistent with the hypothesis of a Poisson process. These variants are applied to point location data for a sample of Ponderosa pine (Pinus ponderosa) trees.