Stochastic Point Field

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A Stochastic Point Field is a random variable set that is mapped to a topological space.

References

2018

  • (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/point_process Retrieved:2018-3-1.
    • 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.

      There are different mathematical interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying spaceon which it is defined, such as the real line or [math]\displaystyle{ n }[/math] -dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. Sometimes the term "point process" is not preferred, as historically the word "process" denoted an evolution of some system in time, so point process is also called a random point field.

      Point processes are well studied objects in probability theory[1] [2] and the subject of powerful tools in statistics for modeling and analyzing spatial data,[3] [4] which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others. Point processes on the real line form an important special case that is particularly amenable to study,[5] because the points are ordered in a natural way, and the whole point process can be described completely by the (random) intervals between the points. These point processes are frequently used as models for random events in time, such as the arrival of customers in a queue (queueing theory), of impulses in a neuron (computational neuroscience), particles in a Geiger counter, location of radio stations in a telecommunication network[6] or of searches on the world-wide web.

2016

  • (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/point_process Retrieved:2016-7-18.
    • In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces. For example, the occurrence of lightning strikes might be considered as a point process in both time and geographical space if each is recorded according to its location in time and space.

      Point processes are well studied objects in probability theory[1] [2] and the subject of powerful tools in statistics for modeling and analyzing spatial data,[3] [7] which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, [8] economics [9] and others. Point processes on the real line form an important special case that is particularly amenable to study,[5] because the points are ordered in a natural way, and the whole point process can be described completely by the (random) intervals between the points. These point processes are frequently used as models for random events in time, such as the arrival of customers in a queue (queueing theory), of impulses in a neuron (computational neuroscience), particles in a Geiger counter, location of radio stations in a telecommunication network[6] or of searches on the world-wide web.

  1. ↑ 1.0 1.1 Kallenberg, O. (1986). Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin. , . Cite error: Invalid <ref> tag; name "Kal86" defined multiple times with different content
  2. ↑ 2.0 2.1 Daley, D.J, Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. , . Cite error: Invalid <ref> tag; name "DVJ88" defined multiple times with different content
  3. ↑ 3.0 3.1 Diggle, P. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edition. Arnold, London. . Cite error: Invalid <ref> tag; name "Dig03" defined multiple times with different content
  4. ↑ Baddeley, A. (2006). Spatial point processes and their applications. In A. Baddeley, I. Bárány, R. Schneider, and W. Weil, editors, Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004, Lecture Notes in Mathematics 1892, Springer. , pp. 1–75
  5. ↑ 5.0 5.1 Last, G., Brandt, A. (1995).Marked point processes on the real line: The dynamic approach. Probability and its Applications. Springer, New York. , Cite error: Invalid <ref> tag; name "LB95" defined multiple times with different content
  6. ↑ 6.0 6.1 Gilbert, E.N. (1961) Random plane networks. SIAM Journal, Vol. 9, No. 4.
  7. ↑ Baddeley, A. (2006). Spatial point processes and their applications. In A. Baddeley, I. Bárány, R. Schneider, and W. Weil, editors, Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004, Lecture Notes in Mathematics 1892, Springer. ISBN 3-540-38174-0, pp. 1–75
  8. ↑ Brown, E. N., Kass, R. E., & Mitra, P. P. (2004). Multiple neural spike train data analysis: state-of-the-art and future challenges. Nature Neuroscience, 7, 456–461. doi:10.1038/nn1228.
  9. ↑ Robert F. Engle and Asger Lunde, 2003, "Trades and Quotes: A Bivariate Point Process". Journal of Financial Econometrics Vol. 1, No. 2, pp. 159–188

2016

  1. ↑ D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Stochastic geometry and its applications, Second Edition, Section 4.1, Wiley Chichester, 1995.
  2. ↑ M. Haenggi. Stochastic geometry for wireless networks. Chapter 2. Cambridge University Press, 2012.

2011

  • http://www.encyclopediaofmath.org/index.php/Stochastic_point_process
    • QUOTE: A stochastic process corresponding to a sequence of random variables , [math]\displaystyle{ \{t_i\} ... \lt t_{i-1} \lt t_0 \lte 0 \lt t_{i+1} \lt t_{i+2}... }[/math] , on the real line . Each value corresponds to a random variable called its multiplicity. In queueing theory a stochastic point process is generated by the moments of arrivals for service, in biology by the moments of impulses in nerve fibres, etc.

      The number of all points is called the counting process, , where is a martingale and is the compensator with respect to the -fields generated by the random points . Many important problems can be solved in terms of properties of the compensator .

1975

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