點過程

喺統計學同概率論當中，點過程或者點場（英文：point process, point field）係啲數學點嘅集合，啲隨機噉落喺一柵數學空間（譬如實線或者歐氏空間）度嘅。點過程可作爲數學模型畀啲現象或者對象，衹要佢哋表示得為某類空間入邊嘅一啲點。

對於點過程有嘸同嘅數學解釋，譬如係隨機計數度量（random counting measure）或者隨機集（random set）。[1][2]一啲作者將點過程同隨機過程睇作係兩個嘸同嘅對象，所以點過程係由隨機過程產生或者關聯埋隨機過程嘅隨機對象，[3][4]儘管有人指出點過程之間嘅差異同隨機過程嘸清楚。其他人將點過程睇作係一種隨機過程，其中有個潛在空間（underlying space）[a]，譬如實線或者維歐幾里得空間，空間啲集作爲過程索引（index）。[7][8]其他隨機過程喺點過程理論當中都得研究埋，譬如啲更新同計數過程。[9]有時啲人嘸傾向優先使個術語「點過程」，因為歷史上「過程」表示某種系統演變係喺時間上嘅，係噉點過程都喊做隨機點場。[10]

點過程係概率論有深入研究到嘅對象，亦都係整統計學嘅有力工具嘅主題畀空間數據建模同分析嗰陣[11][12]，後者喺好多啲學科都好受關注，似喺林業、植物生態學、流行病學、地理學、地震學、材料科學、天文學、電信、計算神經科學[13]、經濟學[14]等。

點過程喺實線上嘅例構成咗啲重要特例係特別適合研究嘅，因為啲點係以自然嘅方式排序，並且成個點過程可以完全由啲點之間嘅（隨機）間隔描述得到。呢啲點過程經常得用作時間隨機事件嘅模型，譬如啲客入隊（排隊論）、一粒神經元入便嘅脈衝（計算神經科學）、啲粒子入到蓋革計數器、電信網絡啲無線電台嘅位置[15]或者喺萬維網上嘅搜索行爲。

考

[編輯]
  1. ↑ 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.
  2. ↑ Martin Haenggi (2013). Stochastic Geometry for Wireless Networks. Cambridge University Press. p. 10. ISBN 978-1-107-01469-5.
  3. ↑ 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.
  4. ↑ Cox, D. R.; Isham, Valerie (1980). Point Processes. CRC Press. p. 3. ISBN 978-0-412-21910-8.
  5. ↑ J. F. C. Kingman (17 December 1992). Poisson Processes. Clarendon Press. p. 8. ISBN 978-0-19-159124-2.
  6. ↑ 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.
  7. ↑ Samuel Karlin; Howard E. Taylor (2 December 2012). A First Course in Stochastic Processes. Academic Press. p. 31. ISBN 978-0-08-057041-9.
  8. ↑ Volker Schmidt (24 October 2014). Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms. Springer. p. 99. ISBN 978-3-319-10064-7.
  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. ISBN 978-0-387-21564-8.
  10. ↑ 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.
  11. ↑ Diggle, P. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edition. Arnold, London. ISBN 0-340-74070-1.
  12. ↑ 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
  13. ↑ 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.{{cite journal}}: CS1 maint: multiple names: 作者名單 (link)
  14. ↑ 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.
  15. ↑ Gilbert E.N. (1961). "Random plane networks". Journal of the Society for Industrial and Applied Mathematics. 9 (4): 533–543. doi:10.1137/0109045.

註

[編輯]
  1. ↑ 喺點過程背景下，術語「狀態空間」可以指埋定義個點過程嘅空間（譬如實線）[5][6]，相當於隨機過程術語中嘅index集。