Estimating the Parameters of a Continuous-Time Markov Chain from Discrete-Time Data with ctmcd
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**Keywords**: Embedding Problem, Generator Matrix, Continuous-Time Markov Chain, Discrete-Time Markov Chain

**Webpages**: https://CRAN.R-project.org/package=ctmcd

The estimation of the parameters of a continuous-time Markov chain from discrete-time data is an important statistical problem which occurs in a wide range of applications: e.g., with the analysis of gene sequence data, for causal inference in epidemiology, for describing the dynamics of open quantum systems in physics, or in rating based credit risk modeling to name only a few.

The parameters of a continuous-time Markov chain are called generator matrix (also: transition rate matrix or intensity matrix) and the issue of estimating generator matrices from discrete-time data is also known as the embedding problem for Markov chains. For dealing with this missing data situtation, a variety of estimation approaches have been developed. These comprise adjustments of matrix logarithm based candidate solutions of the aggregated discrete-time data, see (Israel, Rosenthal, and Wei 2001) or (Kreinin and Sidelnikova 2001). Moreover, likelihood inference can be conducted by an instance of the expectation-maximization (EM) algorithm and Bayesian inference by a Gibbs sampling procedure based on the conjugate gamma prior distribution (Bladt and Sørensen 2005).

The*R* package **ctmcd** (Pfeuffer 2016) is the first publicly available implementation of the approaches listed above. Besides point estimates of generator matrices, the package also contains methods to derive confidence and credibility intervals. The capabilities of the package are illustrated using Standard & Poor’s discrete-time credit rating transition data. Moreover, methodological issues of the described approaches are discussed, i.e., the derivation of the conditional expectations of the E-Step in the EM algorithm and the sampling of endpoint-conditioned continuous-time Markov chain trajectories for the Gibbs sampler.

References Bladt, M., and M. Sørensen. 2005. “Statistical Inference for Discretely Observed Markov Jump Processes.”*Journal of the Royal Statistical Society B*.

Israel, R. B., J. S. Rosenthal, and J. Z. Wei. 2001. “Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings.”*Mathematical Finance*.

Kreinin, A., and M. Sidelnikova. 2001. “Regularization Algorithms for Transition Matrices.”*Algo Research Quarterly*.

Pfeuffer, M. 2016. “ctmcd: An R Package for Estimating the Parameters of a Continuous-Time Markov Chain from Discrete-Time Data.”*In Revision (the R Journal)*.

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The estimation of the parameters of a continuous-time Markov chain from discrete-time data is an important statistical problem which occurs in a wide range of applications: e.g., with the analysis of gene sequence data, for causal inference in epidemiology, for describing the dynamics of open quantum systems in physics, or in rating based credit risk modeling to name only a few.

The parameters of a continuous-time Markov chain are called generator matrix (also: transition rate matrix or intensity matrix) and the issue of estimating generator matrices from discrete-time data is also known as the embedding problem for Markov chains. For dealing with this missing data situtation, a variety of estimation approaches have been developed. These comprise adjustments of matrix logarithm based candidate solutions of the aggregated discrete-time data, see (Israel, Rosenthal, and Wei 2001) or (Kreinin and Sidelnikova 2001). Moreover, likelihood inference can be conducted by an instance of the expectation-maximization (EM) algorithm and Bayesian inference by a Gibbs sampling procedure based on the conjugate gamma prior distribution (Bladt and Sørensen 2005).

The

References Bladt, M., and M. Sørensen. 2005. “Statistical Inference for Discretely Observed Markov Jump Processes.”

Israel, R. B., J. S. Rosenthal, and J. Z. Wei. 2001. “Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings.”

Kreinin, A., and M. Sidelnikova. 2001. “Regularization Algorithms for Transition Matrices.”

Pfeuffer, M. 2016. “ctmcd: An R Package for Estimating the Parameters of a Continuous-Time Markov Chain from Discrete-Time Data.”

MariusPfeuffer
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