9,085 research outputs found
Infinitely divisible nonnegative matrices, -matrices, and the embedding problem for finite state stationary Markov Chains
This paper explicitly details the relation between -matrices, nonnegative
roots of nonnegative matrices, and the embedding problem for finite-state
stationary Markov chains. The set of nonsingular nonnegative matrices with
arbitrary nonnegative roots is shown to be the closure of the set of matrices
with matrix roots in . The methods presented here employ nothing
beyond basic matrix analysis, however it answers a question regarding
-matrices posed over 30 years ago and as an application, a new
characterization of the set of all embeddable stochastic matrices is obtained
as a corollary
Exact finite approximations of average-cost countable Markov Decision Processes
For a countable-state Markov decision process we introduce an embedding which
produces a finite-state Markov decision process. The finite-state embedded
process has the same optimal cost, and moreover, it has the same dynamics as
the original process when restricting to the approximating set. The embedded
process can be used as an approximation which, being finite, is more convenient
for computation and implementation.Comment: Submitted to Automatic
Embeddability and rate identifiability of Kimura 2-parameter matrices
Deciding whether a Markov matrix is embeddable (i.e. can be written as the
exponential of a rate matrix) is an open problem even for matrices.
We study the embedding problem and rate identifiability for the K80 model of
nucleotide substitution. For these matrices, we fully characterize
the set of embeddable K80 Markov matrices and the set of embeddable matrices
for which rates are identifiable. In particular, we describe an open subset of
embeddable matrices with non-identifiable rates. This set contains matrices
with positive eigenvalues and also diagonal largest in column matrices, which
might lead to consequences in parameter estimation in phylogenetics. Finally,
we compute the relative volumes of embeddable K80 matrices and of embeddable
matrices with identifiable rates. This study concludes the embedding problem
for the more general model K81 and its submodels, which had been initiated by
the last two authors in a separate work.Comment: 20 pages; 10 figure
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