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In [[probability theory]], a '''nearly completely decomposable (NCD) Markov chain''' is a [[Markov chain]] where the state-space can be partitioned in such a way that movement within a partition occurs much more frequently that movement between partitions.<ref>{{cite jstor|1427937}}</ref> Particularly efficient algorithms exist to compute the [[stationary distribution]] of Markov chains with this property.<ref name="kms">{{cite doi|10.1137/0605019}}</ref>
 
==Definition==
 
[[Albert Ando|Ando]] and [[Franklin M. Fisher|Fisher]] define a completely decomposable matrix as one where "an identical rearrangement of rows and columns leaves a set of square [[submatrices]] on the [[principal diagonal]] and zeros everywhere else." A nearly completely decomposable matrix is one where an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and ''small nonzeros'' everywhere else.<ref>{{cite doi|10.2307.2F2525455}}</ref><ref>{{cite jstor|1913078}}</ref>
 
==Example==
 
A [[Markov chain]] with [[transition matrix]]
::<math>P =
\begin{pmatrix}
\frac{1}{2} & \frac{1}{2} & 0 & 0 \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & \frac{1}{2} & \frac{1}{2} \\
0 & 0 & \frac{1}{2} & \frac{1}{2} \\
\end{pmatrix} + \epsilon \begin{pmatrix}
-\frac{1}{2} & 0 & \frac{1}{2} & 0 \\
0 & -\frac{1}{2} & 0 & \frac{1}{2} \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 \\
0 & \frac{1}{2} & 0 & -\frac{1}{2} \\
\end{pmatrix}</math>
is nearly completely decomposable if ''ε'' is small (say 0.1).<ref>Example 1.1 from {{cite book|page=8|title=Discrete-time Markov chains: two-time-scale methods and applications|first1=George|last1=Yin|first2=Qing|last2=Zhang|publisher=Springer|year=2005|isbn= 0-387-21948-X}}</ref>
 
==Stationary distribution algorithms==
 
Special-purpose iterative algorithms have been designed for NCD Markov chains<ref name="kms" /> though the multi–level algorithm, a general purpose algorithm,<ref>{{cite doi|10.1145/183019.183040}}</ref> has been shown experimentally to be competitive and in some cases significantly faster.<ref>{{cite techreport|title=On the Utility of the Multi-Level Algorithm for the Solution of Nearly Completely Decomposable Markov Chains (ICASE Report No. 94-44)|url=http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA284423|last1=Leutenegger |first1=Scott T.|last2=Horton |first2=Graham |month=June | year=1994 |quote=We present experimental results indicating that the general- purpose Multi-Level algorithm is competitive, and can be significantly faster than the special-purpose KMS algorithm when Gauss-Seidel and Gaussian Elimination are used for solving the individual blocks. Markov chains, Multi- level, Numerical solution.|institution=NASA|id=Contractor Report 194929}}</ref>
 
==See also==
 
* [[Lumpability]]
 
==References==
 
{{Reflist}}
 
[[Category:Markov processes]]
 
{{Probability-stub}}

Revision as of 07:59, 13 January 2013

In probability theory, a nearly completely decomposable (NCD) Markov chain is a Markov chain where the state-space can be partitioned in such a way that movement within a partition occurs much more frequently that movement between partitions.[1] Particularly efficient algorithms exist to compute the stationary distribution of Markov chains with this property.[2]

Definition

Ando and Fisher define a completely decomposable matrix as one where "an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and zeros everywhere else." A nearly completely decomposable matrix is one where an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and small nonzeros everywhere else.[3][4]

Example

A Markov chain with transition matrix

P=(121200121200001212001212)+ϵ(−1201200−12012120−1200120−12)

is nearly completely decomposable if ε is small (say 0.1).[5]

Stationary distribution algorithms

Special-purpose iterative algorithms have been designed for NCD Markov chains[2] though the multi–level algorithm, a general purpose algorithm,[6] has been shown experimentally to be competitive and in some cases significantly faster.[7]

See also

References

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Template:Probability-stub

  1. ↑ Template:Cite jstor
  2. ↑ 2.0 2.1 Template:Cite doi
  3. ↑ Template:Cite doi
  4. ↑ Template:Cite jstor
  5. ↑ Example 1.1 from 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  6. ↑ Template:Cite doi
  7. ↑ Template:Cite techreport
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