Point Processes

Last updated

Point Processes is a book on the mathematics of point processes, randomly located sets of points on the real line or in other geometric spaces. It was written by David Cox and Valerie Isham, and published in 1980 by Chapman & Hall in their Monographs on Applied Probability and Statistics book series. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries. [1]

Contents

Topics

Although Point Processes covers some of the general theory of point processes, that is not its main focus, and it avoids any discussion of statistical inference involving these processes. Instead, its aim is to present the properties and descriptions of several specific processes arising in applications of this theory, [2] [3] [4] [5] which had not been previously collected in texts in this area. [3]

Three of its six chapters concern more general material, while the final three are more specific. The first chapter includes introductory material on standard processes: Poisson point processes, renewal processes, self-exciting processes, and doubly stochastic processes. The second chapter provides some general theory including stationarity, orderliness (meaning that the probability of multiple arrivals in short intervals is sublinear in the interval length), Palm distributions, Fourier analysis, and probability-generating functions. [6] Chapter four (the third of the more general chapters) concerns point process operations, methods of modifying or combining point processes to generate other processes. [5] [6]

Chapter three, the first of the three chapters on more specific models, is titled "Special models". [5] The special models that it covers include non-stationary Poisson processes, compound Poisson processes, and the Moran process, along with additional treatment of doubly stochastic processes and renewal processes. Until this point, the book focuses on point processes on the real line (possibly also with a time dimension), but the two final chapters concern multivariate processes and on point processes for higher dimensional spaces, including spatio-temporal processes and Gibbs point processes. [6]

Audience and reception

The book is primarily a reference for researchers. [2] It could also be used to provide additional examples for a course on stochastic processes, or as the basis for an advanced seminar. Although it uses relatively little advanced mathematics, readers are expected to understand advanced calculus and have some familiarity with probability theory and Markov chains. [3]

Writing some ten years after its original publication, reviewer Fergus Daly of The Open University writes that his copy has been well used, and that it "still is a very good book: lucid, relevant and still not matched in its approach by any other text". [6]

Related Research Articles

Stochastic process A collection of random variables

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Many stochastic processes can be represented by time series. However, a stochastic process is by nature continuous while a time series is a set of observations indexed by integers. A stochastic process may involve several related random variables.

Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains.

Mathematical statistics

Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times.

In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on some underlying mathematical space such as the real line, the Cartesian plane, or more abstract spaces. Point processes can be used as mathematical models of phenomena or objects representable as points in some type of space.

In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the intensity that varies across the underlying mathematical space is itself a stochastic process. The process is named after the statistician David Cox, who first published the model in 1955.

Stochastic geometry

In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.

Valerie Susan Isham is a British applied probabilist and former President of the Royal Statistical Society. Isham's research interests in include point processes, spatial processes, spatio-temporal processes and population processes.

In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem, also known as Campbell's formula, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving moment measures and factorial moment measures that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the theory of point processes and queueing theory as well as the related fields stochastic geometry, continuum percolation theory, and spatial statistics.

In queueing theory, a discipline within the mathematical theory of probability, Ross's conjecture gives a lower bound for the average waiting-time experienced by a customer when arrivals to the queue do not follow the simplest model for random arrivals. It was proposed by Sheldon M. Ross in 1978 and proved in 1981 by Tomasz Rolski. Equality can be obtained in the bound; and the bound does not hold for finite buffer queues.

In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space. More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs.

In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.

In probability and statistics, point process notation comprises the range of mathematical notation used to symbolically represent random objects known as point processes, which are used in related fields such as stochastic geometry, spatial statistics and continuum percolation theory and frequently serve as mathematical models of random phenomena, representable as points, in time, space or both.

In probability and statistics, a point process operation or point process transformation is a type of mathematical operation performed on a random object known as a point process, which are often used as mathematical models of phenomena that can be represented as points randomly located in space. These operations can be purely random, deterministic or both, and are used to construct new point processes, which can be then also used as mathematical models. The operations may include removing or thinning points from a point process, combining or superimposing multiple point processes into one point process or transforming the underlying space of the point process into another space. Point process operations and the resulting point processes are used in the theory of point processes and related fields such as stochastic geometry and spatial statistics.

In probability and statistics, a spherical contact distribution function, first contact distribution function, or empty space function is a mathematical function that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process. This function can be contrasted with the nearest neighbour function, which is defined in relation to some point in the point process as being the probability distribution of the distance from that point to its nearest neighbouring point in the same point process.

Poisson point process

In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications.

In probability and statistics, a nearest neighbor function, nearest neighbor distance distribution, nearest-neighbor distribution function or nearest neighbor distribution is a mathematical function that is defined in relation to mathematical objects known as point processes, which are often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. More specifically, nearest neighbor functions are defined with respect to some point in the point process as being the probability distribution of the distance from this point to its nearest neighboring point in the same point process, hence they are used to describe the probability of another point existing within some distance of a point. A nearest neighbor function can be contrasted with a spherical contact distribution function, which is not defined in reference to some initial point but rather as the probability distribution of the radius of a sphere when it first encounters or makes contact with a point of a point process.

References

  1. ↑ "Point Processes (not yet reviewed)", MAA Reviews, Mathematical Association of America, retrieved 2020-12-13
  2. 1 2 Biggins, J. D. (June 1981), "Review of Point Processes", The Mathematical Gazette , 65 (432): 153, doi:10.2307/3615757, JSTOR   3615757
  3. 1 2 3 Holmes, Paul T. (June 1983), Journal of the American Statistical Association, 78 (382): 500–501, doi:10.2307/2288675, JSTOR   2288675 CS1 maint: untitled periodical (link)
  4. ↑ Daley, D. J., "Review of Point Processes", zbMATH, Zbl   0441.60053
  5. 1 2 3 Vere-Jones, David (1982), "Review of Point Processes", Mathematical Reviews , MR   0598033
  6. 1 2 3 4 Daly, Fergus (1991), "Review of Point Processes", Journal of the Royal Statistical Society, Series A , 154 (2): 358–359, doi:10.2307/2983051, JSTOR   2983051
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.