Point process

In statistics and probability theory, a point process or point field is a set of a random number of mathematical points randomly located on a mathematical space such as the real line or Euclidean space.[1][2]

Point processes on the real line form an important special case that is particularly amenable to study,[3] 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[4] or of searches on the world-wide web.

General point processes on a Euclidean space can be used for spatial data analysis,[5][6] which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience,[7] economics[8] and others.

  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. ^ 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
  4. ^ Gilbert E.N. (1961). "Random plane networks". Journal of the Society for Industrial and Applied Mathematics. 9 (4): 533–543. doi:10.1137/0109045.
  5. ^ Diggle, P. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edition. Arnold, London. ISBN 0-340-74070-1.
  6. ^ 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
  7. ^ 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)
  8. ^ 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.