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 than movement between partitions. Particularly efficient algorithms exist to compute the stationary distribution of Markov chains with this property.
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.
Example A Markov chain with transition matrix
P = ( 1 2 1 2 0 0 1 2 1 2 0 0 0 0 1 2 1 2 0 0 1 2 1 2 ) + ϵ ( − 1 2 0 1 2 0 0 − 1 2 0 1 2 1 2 0 − 1 2 0 0 1 2 0 − 1 2 ) {\displaystyle 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}}}
is nearly completely decomposable if ε is small (say 0.1).
Stationary distribution algorithms Special-purpose iterative algorithms have been designed for NCD Markov chains though the multi–level algorithm, a general purpose algorithm, has been shown experimentally to be competitive and in some cases significantly faster.
See also Lumpability
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