Markov model: Difference between revisions

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{{Short description|Statistical tool to model changing systems}}

{{more citations needed|date=July 2017}}

In [[probability theory]], a '''Markov model''' is a [[stochastic model]] used to [[Mathematical model|model]] pseudo-randomly changing systems.<ref name=":0">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|___location=USA, NJ|pages=1–256}}</ref> It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the [[Markov property]]). Generally, this assumption enables reasoning and computation with the model that would otherwise be [[Intractability (complexity)|intractable]]. For this reason, in the fields of [[predictive modelling]] and [[probabilistic forecasting]], it is desirable for a given model to exhibit the Markov property.

==Introduction==

Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain.

There are four common Markov models used in different situations, depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:

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{{main|Markov chain}}

The simplest Markov model is the [[Markov chain]]. It models the state of a system with a [[random variable]] that changes through time.<ref name=":0" /> In this context, the Markov property suggestsindicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is [[Markov chain Monte Carlo]], which uses the Markov property to prove that a particular method for performing a [[random walk]] will sample from the [[joint distribution]].

==Hidden Markov model==

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==Hierarchical Markov models==

Hierarchical Markov models can be applied to categorize human behavior at various levels of abstraction. For example, a series of simple observations, such as a person's ___location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the [[Hierarchical hidden Markov model]]<ref name="HHMM">{{cite journal |first1=S. |last1=Fine |first2=Y. |last2=Singer |title=The hierarchical hidden markov model: Analysis and applications |journal=Machine Learning |volume=32 |issue=1 |pages=41–62 |year=1998 |doi=10.1023/A:1007469218079|doi-access=free }}</ref> and the Abstract Hidden Markov Model.<ref name="AHMM">{{cite journal |first1=H. H. |last1=Bui |first2=S. |last2=Venkatesh |first3=G. |last3=West |url=https://www.jair.org/index.php/jair/article/view/10316 |title=Policy recognition in the abstract hidden markov model |journal=Journal of Artificial Intelligence Research |volume=17 |pages=451–499 |year=2002 |doi=10.1613/jair.839|doi-access=free |hdl=10536/DRO/DU:30044252 |hdl-access=free |arxiv=1106.0672 }}</ref> Both have been used for behavior recognition.<ref name="HierarchicalLearningAndPlanningInPOMDPs">{{cite thesis |first=G. |last=Theocharous |url=http://dl.acm.org/citation.cfm?id=936140 |title=Hierarchical Learning and Planning in Partially Observable Markov Decision Processes |type=PhD |publisher=Michigan State University |year=2002}}</ref> and certain conditional independence properties between different levels of abstraction in the model allow for faster learning and inference.<ref name="AHMM" /><ref name="RecognitionOfHumanActivityThroughHierarchicalStochasticLearning">{{cite book |first1=S. |last1=Luhr |first2=H. H. |last2=Bui |first3=S. |last3=Venkatesh |first4=G. A. W. |last4=West |chapter-url=http://dl.acm.org/citation.cfm?id=826390 |chapter=Recognition of Human Activity through Hierarchical Stochastic Learning |title=PERCOM '03 Proceedings of the First IEEE International Conference on Pervasive Computing and Communications |pages=416–422 |year=2003 |doi=10.1109/PERCOM.2003.1192766|isbn=978-0-7695-1893-0 |citeseerx=10.1.1.323.928 |s2cid=13938580 }}</ref>

==Tolerant Markov model==

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==Markov-chain forecasting models==

Markov-chains have been used as a forecasting methods for several topics, for example price trends,<ref name="SLS">{{cite journal |first1=E.G. |last1=de Souza e Silva |first2=L.F.L. |last2=Legey |first3=E.A. |last3=de Souza e Silva |url=https://www.sciencedirect.com/science/article/pii/S0140988310001271 |title=Forecasting oil price trends using wavelets and hidden Markov models |journal=Energy Economics |volume=32 |year=2010|issue=6 |page=1507 |doi=10.1016/j.eneco.2010.08.006 |bibcode=2010EneEc..32.1507D |url-access=subscription }}</ref> wind power<ref name="CGLT">{{cite journal |first1=A |last1=Carpinone |first2=M |last2=Giorgio |first3=R. |last3=Langella |first4=A. |last4=Testa |title=Markov chain modeling for very-short-term wind power forecasting |journal=Electric Power Systems Research |volume=122 |pages=152–158 |year=2015|doi=10.1016/j.epsr.2014.12.025 |doi-access=free |bibcode=2015EPSR..122..152C }}</ref> and [[solar irradiance]].<ref name="MMW">{{cite journal |first1=J. |last1=Munkhammar |first2=D.W. |last2=van der Meer |first3=J. |last3=Widén |title=Probabilistic forecasting of high-resolution clear-sky index time-series using a Markov-chain mixture distribution model |journal= Solar Energy |volume=184 |pages=688–695 |year=2019|doi=10.1016/j.solener.2019.04.014 |bibcode=2019SoEn..184..688M |s2cid=146076100 }}</ref> The Markov-chain forecasting models utilize a variety of different settings, from discretizing the time-series<ref name="CGLT" /> to hidden Markov-models combined with wavelets<ref name="SLS" /> and the Markov-chain mixture distribution model (MCM).<ref name="MMW" />

== See also ==