Some models for multitype spatial point processes, with remarks on analysing multitype patterns

Cellular networks have a great transformation during the last decade. The traditional cellular network system is often overlaid with well-deployed BSs of proper power. However, more and more complex elements such as picocells, femtocells and relays have been introduced to cellular network system. BSs in these elements are unplanned, user-installed and randomly deployed. All of these things compose heterogeneous cellular networks (HCNs). In order to evaluate the performance of HCNs, people usually model the distribution of BSs in HCNs as spatial stochastic point process model. One of the most popular spatial models for these HCNs is Poisson Point Process (PPP) model. However the PPP model is not exact for some cases where BSs in PPP model may be too close to each other. In this paper, we propose a more reasonable and practical model called Matern-like point process (MLPP) model which imposes a certain minimal distance between any two BSs to overcome this problem. Simulations show that the coverage probability of the proposed model noticeably outperforms the PPP model. This improvement depends on the value of minimal distance. We also evaluate the system efficiency defined by the ratio of coverage probability and total power. The results show that the MLPP model has a significant improvement compared to PPP model.

ABSTRACTIn this paper, we derive an exact formula for the covariance of two innovations computed from a spatial Gibbs point process and suggest a fast method for estimating this covariance. We show how this methodology can be used to estimate the asymptotic covariance matrix of the maximum pseudo‐likelihood estimator of the parameters of a spatial Gibbs point process model. This allows us to construct asymptotic confidence intervals for the parameters. We illustrate the efficiency of our procedure in a simulation study for several classical parametric models. The procedure is implemented in the statistical software R, and it is included in spatstat, which is an R package for analyzing spatial point patterns.

For a stationary long-range dependent point process N(.) with Palm distribution PO0, the Hurst index H =_sup{h : lim supt-o t-2h var N(0, t] = oo} is related to the moment index K - sup{k : EO(Tk) 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which K < 2 is necessarily long-range dependent with Hurst index greater than :. An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., EO(T2) <00oo.

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 Neerlandica54, 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.

In this paper, we study the problem of dynamic allocation of heterogeneous processors to parallel heterogeneous job traffic flows. Each traffic flow is a stationary ergodic random marked point process, with a parameter (traffic intensity rate) that depends on the job class. The service rates of the various job flows depend on both the job class and the processor class. This model captures the essential features of several practical systems, including flexible manufacturing ones, packet switches, distribution systems, etc. We first specify precisely the necessary and sufficient condition for stability of the system. We then identify a family of policies that achieves maximum throughput; i.e. they stabilize the system under the maximum possible input rates. The approach taken introduces the concept of virtual queueing, which proves powerful in establishing strong probabilistic results (convergence in distribution to a finite stationary regime) for queueing systems with complex dynamics due to sharing resources, under a very general stationary ergodic probabilistic structure.

Failure to account for time‐varying detection ranges when inferring space use of marine species from passive acoustic telemetry data can bias estimates and result in erroneous biological conclusions. This potential source of bias is widely acknowledged but often ignored in practice due to a lack of available statistical methods. Here, we describe and apply a spatial point process model for estimating individual centres of activity (COAs) from acoustic telemetry data that can be modified to account for both receiver‐ and time‐specific detection probabilities. We use simulation testing to evaluate the suitability of the proposed models for estimating COAs and compare their performance to that of the popular mean‐weighted COA method for a variety of scenarios. We illustrate how the approach can be applied to correct for variable detection ranges by integrating data from moored test tags and demonstrate how accounting for time‐varying detection probabilities can impact space use estimates by fitting the model to data collected from a black sea bass (Centropristis striata) on a receiver array off the east coast of the United States. The proposed model reduced bias in COA estimates, particularly when tagged individuals occurred along the periphery of the receiver array. The test tag‐integrated model largely corrected the bias associated with receiver‐ and time‐specific detection probabilities. When applied to the black sea bass detection data, the model revealed fine‐scale movements not apparent when detection ranges were assumed constant. Spatial management practices for coastal marine species are often based on trends in space use inferred from passive acoustic telemetry data, which can be misinterpreted when factors influencing detection ranges are not accounted for. Our approach provides a general framework for estimating individual COAs that can be modified on a study‐specific basis to ensure resulting patterns of space use reflect a species’ movements and behaviour, rather than variation in receiver detection ranges.

For a spatial point process model in which the intensity depends on spatial covariates, we develop graphical diagnostics for validating the covariate effect term in the model, and for assessing whether another covariate should be added to the model. The diagnostics are point-process counterparts of the well-known partial residual plots (component-plus-residual plots) and added variable plots for generalized linear models. The new diagnostics can be derived as limits of these classical techniques under increasingly fine discretization, which leads to efficient numerical approximations. The diagnostics can also be recognized as integrals of the point process residuals, enabling us to prove asymptotic results. The diagnostics perform correctly in a simulation experiment. We demonstrate their utility in an application to geological exploration, in which a point pattern of gold deposits is modeled as a point process with intensity depending on the distance to the nearest geological fault. Online supplementary materials include technical proofs, computer code, and results of a simulation study.

Basic methods of explorative statistical analysis for stationary and isotropic planar point processes are briefly and informally reviewed. At the explorative level, planar point patterns may be characterized in terms of the intensity, the K-function and the pair correlation function. These second-order functions enable one to classify a given point process as completely random, clustering or repulsive. The repulsive behaviour may be quantified by an estimate of the hard-core distance. In the exploratory approach, the statistics are essentially free from model assumptions. Second-order spatial functions have been estimated to characterize genuine planar point processes in the macroscopic domain, for example in forestry, geography and epidemiology. For light microscopy and transmission electron microscopy, two situations are distinguished, which may be summarized as the genuine planar case and the stereological case. In the genuine planar case, a direct interpretation of the results of spatial statistics is feasible. Here, monolayers in cell culture, intramembranous particles on freeze fracture specimens and amacrine cells of the retina are mentioned as examples. In the stereological case, point patterns are generated by sections through 3D structures. Here the observed point patterns may arise as the centres of sectional profiles of particles, or as centres of sectional profiles of spatial fibre processes. In both situations, exploratory spatial point process statistics allow a quantitative characterization of sectional images for the purposes of group comparisons and classification. Moreover, for spatial fibre processes it has recently been shown that the observed pair correlation function of the centres of the fibre profiles is an estimate of the reduced pair correlation function of the fibre process in 3D. Hence for fibre processes a stereological interpretation of point process statistics obtained from sections is an additional option.

Binary-valued Markov random fields may be used as models for point processes with interactions (e.g. repulsion or attraction) between their points. This paper aims to provide a simple nontechnical introduction to Markov random fields in this context. The underlying spaces on which points occur are taken to be countable (e.g. lattice vertices) or continuous (Euclidean space). The role of Markov random fields as equilibrium processes for the temporal evolution of spatial processes is also discussed and various applications and examples are given.

The positive association between alcohol outlet density and assault rates is well established, but little is known about how this association differs across victim subpopulations. We use spatial point process models on police data from Flint, Michigan, to test how the link between alcohol outlet density and assault rates changes as a function of three victim characteristics: age, gender, and race. We found that, although both on-premises and package outlet densities consistently emerge as risk factors for victimization, their relative effects are markedly larger in Whites than in African Americans. No analogous age- or gender-based differences were found. These results suggest the racial effects arise more from relative differences in the atmosphere in and around alcohol outlets than differences in drinking behavior alone.

We answer two open questions concerning the existence of universal schemes for classification and regression estimation from stationary ergodic processes. It is shown that no measurable procedure can produce weakly consistent regression estimates from every bivariate stationary ergodic process, even if the covariate and response variables are restricted to take values in the unit interval. It is further shown that no measurable procedure can produce weakly consistent classification rules from every bivariate stationary ergodic process for which the response variable is binary valued. The results of the paper are derived via reduction arguments and are based in part on recent work concerning density estimaton from ergodic processes.

How is the progression of a virus influenced by properties intrinsic to individual cells? We address this question by studying the susceptibility of cells infected with two strains of the human respiratory syncytial virus (RSV-A and RSV-B) in an in vitro experiment. Spatial patterns of infected cells give us insight into how local conditions influence susceptibility to the virus. We observe a complicated attraction and repulsion behavior, a tendency for infected cells to lump together or remain apart. We develop a new spatial point process model to describe this behavior. Inference on spatial point processes is difficult because the likelihood functions of these models contain intractable normalizing constants; we adapt an MCMC algorithm called double Metropolis-Hastings to overcome this computational challenge. Our methods are computationally efficient even for large point patterns consisting of over 10,000 points. We illustrate the application of our model and inferential approach to simulated data examples and fit our model to various RSV experiments. Because our model parameters are easy to interpret, we are able to draw meaningful scientific conclusions from the fitted models.

A joint spatial marked point process model for dengue and severe dengue in Medellin, Colombia

In a previous study, Foufoula‐Georgiou (1989a) investigated a stochastic storm transposition (SST) approach as a possible methodology of assessing the probability of exceedance of extreme precipitation depths over a catchment. The purpose of this paper is to report further methodological advances on the SST approach and present the results of a pilot implementation study in the Midwest. Under the assumptions of an elliptical storm shape and spread function of a given functional form, extreme storms have been described by the joint probability distribution of seven storm parameters (five specifying the magnitude, orientation, shape, and within‐storm spatial variability, and two specifying the location of the storm center). A nonhomogeneous spatial multivariate point process model has been postulated for the joint probability distribution of the storm center depth and storm location and it has been fitted to 65 extreme storms in the nine‐state mid‐western region. The method has been used to estimate the tails of the probability distribution of the average catchment depths over several hypothetical catchments in the Midwest and to describe the spatial variability of the estimates over the studied region.

It is shown that whenever a stationary random field (Z/sub n,m/)/sub n,m/spl isin/z/ is given by a Borel function f:/spl Ropf//sup z/ /spl times/ /spl Ropf//sup z/ /spl rarr/ /spl Ropf/ of two stationary processes (X/sub n/)/sub n/spl isin/z/ and (Y/sub m/)/sub m/spl isin/z/ i.e., then (Z/sub n, m/) = (f((X/sub n+k/)/sub k/spl epsi/z/, (Y/sub m + /spl lscr// )/sub /spl lscr/ /spl epsi/z/)) under a mild first coordinate univalence assumption on f, the process (X/sub n/)/sub n/spl isin/z/ is measurable with respect to (Z/sub n,m/)/sub n,m/spl epsi/z/ whenever the process (Y/sub m/)/sub m/spl isin/z/ is ergodic. The notion of universal filtering property of an ergodic stationary process is introduced, and then using ergodic theory methods it is shown that an ergodic stationary process has this property if and only if the centralizer of the dynamical system canonically associated with the process does not contain a nontrivial compact subgroup.