Editing Stochastic matrix (section)

== History ==
[[File:Andrej Markov.jpg|thumb|[[Andrey Markov]] in 1886]]
The stochastic matrix was developed alongside the Markov chain by [[Andrey Markov]], a [[List of Russian mathematicians|Russian mathematician]] and professor at [[Saint Petersburg State University|St. Petersburg University]] who first published on the topic in 1906.<ref name=":0">{{cite journal | last1 = Hayes | first1 = Brian | year = 2013 | title = First links in the Markov chain | journal = American Scientist | volume = 101 | issue = 2| pages = 92–96 | doi = 10.1511/2013.101.92 }}</ref> His initial intended uses were for linguistic analysis and other mathematical subjects like [[Shuffling|card shuffling]], but both Markov chains and matrices rapidly found use in other fields.<ref name=":0" /><ref>''Charles Miller Grinstead; James Laurie Snell (1997). Introduction to Probability. American Mathematical Soc. pp. 464–466. {{isbn|978-0-8218-0749-1}}.''</ref>

Stochastic matrices were further developed by scholars such as [[Andrey Kolmogorov]], who expanded their possibilities by allowing for continuous-time Markov processes.<ref>{{cite journal | last1 = Kendall | first1 = D. G. | last2 = Batchelor | first2 = G. K. | last3 = Bingham | first3 = N. H. | last4 = Hayman | first4 = W. K. | last5 = Hyland | first5 = J. M. E. | last6 = Lorentz | first6 = G. G. | last7 = Moffatt | first7 = H. K. | last8 = Parry | first8 = W. | last9 = Razborov | first9 = A. A. | last10 = Robinson | first10 = C. A. | last11 = Whittle | first11 = P. | year = 1990 | title = Andrei Nikolaevich Kolmogorov (1903–1987) | journal = Bulletin of the London Mathematical Society | volume = 22 | issue = 1| page = 33 | doi = 10.1112/blms/22.1.31 }}</ref> By the 1950s, articles using stochastic matrices had appeared in the fields of [[econometrics]]<ref>{{Cite journal|last=Solow|first=Robert|date=1 January 1952|title=On the Structure of Linear Models|journal=Econometrica|volume=20|issue=1|pages=29–46|doi=10.2307/1907805|jstor=1907805}}</ref> and [[Network analysis (electrical circuits)|circuit theory]].<ref>{{Cite journal|last=Sittler|first=R.|date=1 December 1956|title=Systems Analysis of Discrete Markov Processes|journal=IRE Transactions on Circuit Theory|volume=3|issue=4|pages=257–266|doi=10.1109/TCT.1956.1086324|issn=0096-2007}}</ref> In the 1960s, stochastic matrices appeared in an even wider variety of scientific works, from [[Behavioural sciences|behavioral science]]<ref>{{Cite journal|last=Evans|first=Selby|date=1 July 1967|title=Vargus 7: Computed patterns from markov processes|journal=Behavioral Science|language=en|volume=12|issue=4|pages=323–328|doi=10.1002/bs.3830120407|issn=1099-1743}}</ref> to geology<ref>{{Cite journal|last=Gingerich|first=P. D.|date=1 January 1969|title=Markov analysis of cyclic alluvial sediments|journal=Journal of Sedimentary Research|language=en-US|volume=39|issue=1|pages=330–332|doi=10.1306/74d71c4e-2b21-11d7-8648000102c1865d|issn=1527-1404|bibcode=1969JSedR..39..330G}}</ref><ref>{{Cite journal|last1=Krumbein|first1=W. C.|last2=Dacey|first2=Michael F.|date=1 March 1969|title=Markov chains and embedded Markov chains in geology|journal=Journal of the International Association for Mathematical Geology|language=en|volume=1|issue=1|pages=79–96|doi=10.1007/BF02047072|bibcode=1969MatG....1...79K |issn=0020-5958}}</ref> to [[Residential area|residential planning]].<ref>{{Cite journal|last=Wolfe|first=Harry B.|date=1 May 1967|title=Models for Conditioning Aging of Residential Structures|journal=Journal of the American Institute of Planners|volume=33|issue=3|pages=192–196|doi=10.1080/01944366708977915|issn=0002-8991}}</ref> In addition, much mathematical work was also done through these decades to improve the range of uses and functionality of the stochastic matrix and [[Markov chain|Markovian processes]] more generally.

From the 1970s to present, stochastic matrices have found use in almost every field that requires formal analysis, from [[Structural engineering|structural science]]<ref>{{Cite journal|title=A Markov matrix for fatigue load simulation and rainflow range evaluation |language=en|doi=10.1016/0167-4730(89)90025-8|volume=6|issue=2–4|journal=Structural Safety|pages=247–258 | last1 = Krenk | first1 = S.|date=November 1989 }}</ref> to [[medical diagnosis]]<ref>{{Cite journal|last1=Beck|first1=J.Robert|last2=Pauker|first2=Stephen G.|date=1 December 1983|title=The Markov Process in Medical Prognosis|journal=Medical Decision Making|language=en|volume=3|issue=4|pages=419–458|doi=10.1177/0272989X8300300403|pmid=6668990|issn=0272-989X}}</ref> to [[Human resource management|personnel management]].<ref>{{Cite journal|last1=Gotz|first1=Glenn A.|last2=McCall|first2=John J.|date=1 March 1983|title=Sequential Analysis of the Stay/Leave Decision: U.S. Air Force Officers|journal=Management Science|volume=29|issue=3|pages=335–351|doi=10.1287/mnsc.29.3.335|issn=0025-1909}}</ref> In addition, stochastic matrices have found wide use in [[land change modeling]], usually under the term Markov matrix.<ref>{{Cite journal|last1=Kamusoko|first1=Courage|last2=Aniya|first2=Masamu|last3=Adi|first3=Bongo|last4=Manjoro|first4=Munyaradzi|date=1 July 2009|title=Rural sustainability under threat in Zimbabwe – Simulation of future land use/cover changes in the Bindura district based on the Markov-cellular automata model|journal=Applied Geography|volume=29|issue=3|pages=435–447|doi=10.1016/j.apgeog.2008.10.002|bibcode=2009AppGe..29..435K }}</ref>