The Autocovariance Function for Marked Point Processes: A Comparison between Two Different Approaches

Exploratory analysis of marked point patterns has previously been conducted using two disjoint techniques, namely the mark correlation function and spectral analysis. Our purpose here is to present two alternative autocovariance estimators to the mark correlation function which not only apply in both planar and lattice situations, but which in the lattice case can also be considered in terms of the inverse Fourier transform of the spectrum. Moreover, they can be applied to isotropic or anisotropic marked point patterns. Various examples are presented to show how these estimators perform when applied to data sets possessing different kinds of mark structure, and a rank test procedure is proposed to enable the construction of empirical tests of hypothesis.     

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.     

Model diagnosis for parametric regression in high-dimensional spaces

We study tools for checking the validity of a parametric regressionmodel. When the dimension of the regressors is large, many ofthe existing tests face the curse of dimensionality or requiresome ordering of the data. Our tests are based on the residualempirical process marked by proper functions of the regressors.They are able to detect local alternatives converging to thenull at parametric rates. Parametric and nonparametric alternativesare considered. In the latter case, through a proper principalcomponent decomposition, we are able to derive smooth directionaltests which are asymptotically distribution-free under the nullmodel. The new tests take into account precisely the ‘geometryof the model’. A simulation study is carried through andan application to a real dataset is illustrated.     

On Variograms in Point Process Statistics

Variograms, which are frequently used in geostatistics, are of value also in the statistics of marked point processes. When the marks come from a random field which is independent of the point process, ideas of geostatistics suffice for the interpretation of point process variograms. When this model is not appropriate, interactions between the points lead to point process variograms having forms which are unusual in geostcistics. This is shown by three theoretical examples and one from forestry.     

A General Framework for Longitudinal Data through Marked Point Processes

This paper reviews a general framework for the modelling of longitudinal data with random measurement times based on marked point processes and presents a worked example. We construct a quite general regression models for longitudinal data, which may in particular include censoring that only depend on the past and outside random variation, and dependencies between measurement times and measurements. The modelling also generalises statistical counting process models. We review a non-parametric Nadarya-Watson kernel estimator of the regression function, and a parametric analysis that is based on a conditional least squares (CLS) criterion. The parametric analysis presented, is a conditional version of the generalised estimation equations of LIANG and ZEGER (1986). We conclude that the usual nonparametric and parametric regression modelling can be applied to this general set-up, with some modifications. The presented framework provides an easily implemented and powerful tool for model building for repeated measurements.