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Point process

In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space.[1][2] Point processes can be used for spatial data analysis,[3][4] which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience,[5] economics[6] and others.

There are different mathematical interpretations of a point process, such as a random counting measure or a random set.[7][8] 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,[9][10] though it has been remarked that the difference between point processes and stochastic processes is not clear.[10] Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying space[a] on which it is defined, such as the real line or -dimensional Euclidean space.[13][14] Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.[15][10] 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.[16]

Point processes on the real line form an important special case that is particularly amenable to study,[17] 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[18] or of searches on the world-wide web.

  1. ^ Kallenberg, O. (1986). Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin. ISBN 0-12-394960-2, MR854102.
  2. ^ Daley, D.J, Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. ISBN 0-387-96666-8, MR950166.
  3. ^ Diggle, P. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edition. Arnold, London. ISBN 0-340-74070-1.
  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. ISBN 3-540-38174-0, pp. 1–75
  5. ^ 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 (5): 456–461. doi:10.1038/nn1228. PMID 15114358. S2CID 562815.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  6. ^ Engle Robert F., Lunde Asger (2003). "Trades and Quotes: A Bivariate Point Process" (PDF). Journal of Financial Econometrics. 1 (2): 159–188. doi:10.1093/jjfinec/nbg011.
  7. ^ Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (27 June 2013). Stochastic Geometry and Its Applications. John Wiley & Sons. p. 108. ISBN 978-1-118-65825-3.
  8. ^ Martin Haenggi (2013). Stochastic Geometry for Wireless Networks. Cambridge University Press. p. 10. ISBN 978-1-107-01469-5.
  9. ^ D.J. Daley; D. Vere-Jones (10 April 2006). An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Springer Science & Business Media. p. 194. ISBN 978-0-387-21564-8.
  10. ^ a b c Cox, D. R.; Isham, Valerie (1980). Point Processes. CRC Press. p. 3. ISBN 978-0-412-21910-8.
  11. ^ J. F. C. Kingman (17 December 1992). Poisson Processes. Clarendon Press. p. 8. ISBN 978-0-19-159124-2.
  12. ^ Jesper Moller; Rasmus Plenge Waagepetersen (25 September 2003). Statistical Inference and Simulation for Spatial Point Processes. CRC Press. p. 7. ISBN 978-0-203-49693-0.
  13. ^ Samuel Karlin; Howard E. Taylor (2 December 2012). A First Course in Stochastic Processes. Academic Press. p. 31. ISBN 978-0-08-057041-9.
  14. ^ Volker Schmidt (24 October 2014). Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms. Springer. p. 99. ISBN 978-3-319-10064-7.
  15. ^ D.J. Daley; D. Vere-Jones (10 April 2006). An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Springer Science & Business Media. ISBN 978-0-387-21564-8.
  16. ^ Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (27 June 2013). Stochastic Geometry and Its Applications. John Wiley & Sons. p. 109. ISBN 978-1-118-65825-3.
  17. ^ Last, G., Brandt, A. (1995).Marked point processes on the real line: The dynamic approach. Probability and its Applications. Springer, New York. ISBN 0-387-94547-4, MR1353912
  18. ^ Gilbert E.N. (1961). "Random plane networks". Journal of the Society for Industrial and Applied Mathematics. 9 (4): 533–543. doi:10.1137/0109045.


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