Interacting neighbour point processes: Some models for clustering

We introduce a class of spatial point processesinteracting neighbour point (INP) processes, where the density of the process can be written by means of local interactions between a point and subsets of its neighbourhood but where the processes may not be Ripley-Kelly Markov processes with respect to this neighbourhood. We show that the processes are iterated Markov processes defined by Hayat and Gubner (1996). Furthermore, we pay special attention to a subclass of interacting neighbour processes, where the density belongs to the exponential family and all neighbours of a point affect it simultaneously. A simulation study is presented to show that some simple processes of this subclass can produce clustered patterns of great variety. Finally, an empirical example is given.     

A law of large numbers for the scaled age distribution of linear birth-and-death processes

It is shown that certain measure-valued stochastic processes describing the age distribution of particles whose development is controlled by linear critical birth-and-death processes converge in distribution to a deterministic positive bounded measure.     

A Note on Continuous Spatial-Temporal Dynamics of Stochastic Processes

We propose a general form to analyze the space-time interdependency of continuous space-time stochastic processes. We present a new space-time approach based on the intensity function of the underlying point process. These formulations can be, to some extent, analytically solved to obtain explicit formulae of interest. We define a general function that controls the space-time interaction and allows for closed forms depending on the particular choice of several mathematical tools playing a role in this interaction function. In particular, we make use of copulas and Laplace transforms to provide interesting examples of the dynamics of the random intensity function and, in turn, of the number of points contained in a given region.     

Log Gaussian Cox Processes

Planar Cox processes directed by a log Gaussian intensity process are investigated in the univariate and multivariate cases. The appealing properties of such models are demonstrated theoretically as well as through data examples and simulations. In particular, the first, second and third-order properties are studied and utilized in the statistical analysis of clustered point patterns. Also empirical Bayesian inference for the underlying intensity surface is considered.     

Space-time Multi Type Log Gaussian Cox Processes with a View to Modelling Weeds

Log Gaussian Cox processes as introduced in Moller et al. (1998) are extended to space-time models called log Gaussian Cox birth processes. These processes allow modelling of spatial and temporal heterogeneity in time series of increasing point processes consisting of different types of points. The models are shown to be easy to analyse yet flexible enough for a detailed statistical analysis of a particular agricultural experiment concerning the development of two weed species on an organic barley field. Particularly, the aspects of estimation, model validation and intensity surface prediction are discussed.     

Stochastic comparison of processes generated by random interruptions of monotone functions and related results

Many stochastic processes considered in applied probability models, and, in particular, in reliability theory, are processes of the following form: Shocks occur according to some point process, and each shock causes the process to have a random jump. Between shocks the process increases or decreases in some deterministic fashion. In this paper we study processes for which the rate of increase or decrease between shocks depends only on the height of the process. For such processes we find conditions under which the processes can be stochastically compared. We also study hybrid processes in which periods of increase and periods of decrease alternate. A further result yields a stochastic comparison of processes that start with a random jump, rather than processes in which there is at the beginning some random delay time before the first jump.Supported by NSF Grant DMS 9303891.     

Non-parametric Bayesian Estimation of a Spatial Poisson Intensity

A method introduced by Arjas & Gasbarra (1994) and later modified by Arjas & Heikkinen (1997) for the non-parametric Bayesian estimation of an intensity on the real line is generalized to cover spatial processes. The method is based on a model approximation where the approximating intensities have the structure of a piecewise constant function. Random step functions on the plane are generated using Voronoi tessellations of random point patterns. Smoothing between nearby intensity values is applied by means of a Markov random field prior in the spirit of Bayesian image analysis. The performance of the method is illustrated in examples with both real and simulated data.     

Multivariate geometric anisotropic Cox processes

This paper introduces a new modeling and inference framework for multivariate and anisotropic point processes. Building on recent innovations in multivariate spatial statistics, we propose a new family of multivariate anisotropic random fields, and from them a family of anisotropic point processes. We give conditions that make the proposed models valid. We also propose a Palm likelihood-based inference method for this type of point process, circumventing issues of likelihood tractability. Finally we illustrate the utility of the proposed modeling framework by analyzing spatial ecological observations of plants and trees in the Barro Colorado Island data.     

Adaptive estimating function inference for nonstationary determinantal point processes

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this article, we establish asymptotic normality of estimating function estimators in a very general setting of nonstationary point processes. We then adapt this result to the case of nonstationary determinantal point processes, which are an important class of models for repulsive point patterns. In practice, often first‐ and second‐order estimating functions are used. For the latter, it is a common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance, which is usually specified in an ad hoc manner. We suggest instead a data‐driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non‐stationary determinantal point processes and by an application to a real dataset.     

The 1970 US draft lottery revisited: a spatial analysis

Summary.  We revise the result of the 1970 selective service draft lottery in the USA following an open question that was suggested by Fienberg in a paper published in Science in 1971. The result of the drawings can be viewed as a particular spatial pattern which can be analysed by using general spatial tools adapted to our context. Approaches for assessing the complete spatial randomness for this spatial process on a finite support are proposed. More specifically, these approaches involve the number of events in a square window and a k ( r )-based function used to analyse stationary spatial point processes.     

Inversion of the Renewal Density with Dead-time

Stationary renewal point processes are defined by the probability distribution of the distances between successive points (lifetimes) that are independent and identically distributed random variables. For some applications it is also interesting to define the properties of a renewal process by using the renewal density. There are well-known expressions of this density in terms of the probability density of the lifetimes. It is more difficult to solve the inverse problem consisting in the determination of the density of the lifetimes in terms of the renewal density. Theoretical expressions between their Laplace transforms are available but the inversion of these transforms is often very difficult to obtain in closed form. We show that this is possible for renewal processes presenting a dead-time property characterized by the fact that the renewal density is zero in an interval including the origin. We present the principle of a recursive method allowing the solution of this problem and we apply this method to the case of some processes with input dead-time. Computer simulations on Poisson and Erlang (2) processes show quite good agreement between theoretical calculations and experimental measurements on simulated data.     

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

We consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.     

Statistical methods for regular monitoring data

Summary.  Meteorological and environmental data that are collected at regular time intervals on a fixed monitoring network can be usefully studied combining ideas from multiple time series and spatial statistics, particularly when there are little or no missing data. This work investigates methods for modelling such data and ways of approximating the associated likelihood functions. Models for processes on the sphere crossed with time are emphasized, especially models that are not fully symmetric in space–time. Two approaches to obtaining such models are described. The first is to consider a rotated version of fully symmetric models for which we have explicit expressions for the covariance function. The second is based on a representation of space–time covariance functions that is spectral in just the time domain and is shown to lead to natural partially nonparametric asymmetric models on the sphere crossed with time. Various models are applied to a data set of daily winds at 11 sites in Ireland over 18 years. Spectral and space–time domain diagnostic procedures are used to assess the quality of the fits. The spectral-in-time modelling approach is shown to yield a good fit to many properties of the data and can be applied in a routine fashion relative to finding elaborate parametric models that describe the space–time dependences of the data about as well.     

Algorithms for Point Processes Analysis

A time point process can be defined either by the statistical properties of the time intervals between successive points or by those of the number of points in arbitrary time intervals. There are mathematical expressions to link up these two points of view, but they are in many cases too complicated to be used in practice. In this article, we present an algorithmic procedure to obtain the number of points of a stationary point process recorded in some time intervals by processing the values of the distances between successive points. We present some results concerning the statistical analysis of these numbers of points and when analytical calculations are possible the experimental results obtained with our algorithms are in excellent agreement with those predicted by the theory. Some properties of point processes in which theoretical calculations are almost impossible are also presented.     

Random spectral measure for non Gaussian moving averages

We study the distribution of phases and amplitudes for the spectral representation of weighted moving averages of a general noise measure. The simple independent structure, known for the Gaussian case, and involving Rayleigh amplitude and uniform phase distributions, is lost for the non Gaussian noise case. We show that the amplitude/phase distributions exhibit a rich and more complex structure depending not just on the covariance of the process but specifically on the form of the kernel and the noise distribution. We present a theoretical tool for studying these distributions that follows from a proof of the spectral theorem that yields an explicit expression for the spectral measure. The main interest is in noise measures based on second-order Lévy motions since such measures are easily available through independent sampling. We approximate the spectral stochastic measure by independent noise increments which allows us to obtain amplitude/phase distributions that is of fundamental interest for analyzing processes in the frequency domain. For the purpose of approximating the moving average process through sums of trigonometric functions, we assess the mean square error of discretization of the spectral representation. For a specified accuracy, the approximation is explicitly given. We illustrate the method for the moving averages driven by the Laplace motion.     

Gamma Kernel Intensity Estimation in Temporal Point Processes

In this article, we propose a nonparametric approach for estimating the intensity function of temporal point processes based on kernel estimators. In particular, we use asymmetric kernel estimators characterized by the gamma distribution, in order to describe features of observed point patterns adequately. Some characteristics of these estimators are analyzed and discussed both through simulated results and applications to real data from different seismic catalogs.     

On Comparison of Stopping Times in Sequential Procedures for Exponential Families of Stochastic Processes

For curved ( k + 1), k -exponential families of stochastic processes a natural and often studied sequential procedure is to stop observation when a linear combination of the coordinates of the canonical process crosses a prescribed level. For such procedures the model is, approximately or exactly, a non-curved exponential family. Subfamilies of these stopping rules defined by having the same Fisher (expected) information are considered. Within a subfamily the Bartlett correction for a point hypothesis is also constant. Methods for comparing the durations of the sampling periods for the stopping rules in such a subfamily are discussed. It turns out that some stopping times tend to be smaller than others. For exponential families of diffusions and of counting processes the probability that one such stopping time is smaller than another can be given explicity. More generally, an Edgeworth expansion of this probability is given     

The Modelling of Ethernet Data and of Signals that are Heavy‐tailed with Infinite Variance*

We provide an overview of some of the research of the last ten years involving computer network data traffic. We describe the original Ethernet data study which suggested that computer traffic is inherently different from telephone traffic and that in the context of computer networks, self‐similar models such as fractional Brownian motion, should be used. We show that the on–off model can physically explain the presence of self‐similarity. While the on–off model involves bounded signals, it is also possible to consider arbitrary unbounded finite‐variance signals or even infinite‐variance signals whose distributions have heavy tails. We show that, in the latter case, one can still obtain self‐similar processes with dependent increments, but these are not the infinite‐variance fractional stable Lévy motions which have been commonly considered in the literature. The adequate model, in fact, can either have dependent or independent increments, and this depends on the respective size of two parameters, namely, the number of workstations in the network and the time scale under consideration. We indicate what happens when these two parameters become jointly asymptotically large. We conclude with some comments about high frequency behaviour and multifractals.