Point process

About: Point process is a research topic. Over the lifetime, 5312 publications have been published within this topic receiving 149169 citations.

Stochastic Geometry and Its Applications

Abstract: Mathematical Foundation. Point Processes I--The Poisson Point Process. Random Closed Sets I--The Boolean Model. Point Processes II--General Theory. Point Processes III--Construction of Models. Random Closed Sets II--The General Case. Random Measures. Random Processes of Geometrical Objects. Fibre and Surface Processes. Random Tessellations. Stereology. References. Indexes.

A spatial scan statistic

Abstract: The scan statistic is commonly used to test if a one dimensional point process is purely random, or if any clusters can be detected. Here it is simultaneously extended in three directions:(i) a spatial scan statistic for the detection of clusters in a multi-dimensional point process is proposed, (ii) the area of the scanning window is allowed to vary, and (iii) the baseline process may be any inhomogeneous Poisson process or Bernoulli process with intensity pro-portional to some known function. The main interest is in detecting clusters not explained by the baseline process. These methods are illustrated on an epidemiological data set, but there are other potential areas of application as well.

The statistics of peaks of Gaussian random fields

Abstract: A set of new mathematical results on the theory of Gaussian random fields is presented, and the application of such calculations in cosmology to treat questions of structure formation from small-amplitude initial density fluctuations is addressed. The point process equation is discussed, giving the general formula for the average number density of peaks. The problem of the proper conditional probability constraints appropriate to maxima are examined using a one-dimensional illustration. The average density of maxima of a general three-dimensional Gaussian field is calculated as a function of heights of the maxima, and the average density of 'upcrossing' points on density contour surfaces is computed. The number density of peaks subject to the constraint that the large-scale density field be fixed is determined and used to discuss the segregation of high peaks from the underlying mass distribution. The machinery to calculate n-point peak-peak correlation functions is determined, as are the shapes of the profiles about maxima.

Stochastic Geometry for Wireless Networks

Abstract: Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.

Spectra of some self-exciting and mutually exciting point processes

Abstract: SUMMARY In recent years methods of data analysis for point processes have received some attention, for example, by Cox & Lewis (1966) and Lewis (1964). In particular Bartlett (1963a,b) has introduced methods of analysis based on the point spectrum. Theoretical models are relatively sparse. In this paper the theoretical properties of a class of processes with particular reference to the point spectrum or corresponding covariance density functions are discussed. A particular result is a self-exciting process with the same second-order properties as a certain doubly stochastic process. These are not distinguishable by methods of data analysis based on these properties.