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The dsmmR R package allows the user to estimate, simulate and define different Drifting semi-Markov model (DSMM) specifications.


# Install the released version from CRAN
# Or the development version from GitHub
# install.packages("devtools")

High-level documentation

The main functions of dsmmR are the following:

Theory overview

Drifting semi-Markov models are best suited to capture non-homogeneities which evolve in a linear (or polynomial) way. For example, through this approach we account for non-homogeneities that occur from the intrinsic evolution of the system or from the interactions between the system and the environment.

For a detailed introduction in Drifting semi-Markov models consider the documentation through ?dsmmR.

For an extensive description of this approach, consider visiting the complete documentation of the package on the official CRAN page.


The easiest way to use dsmmR is through the main function dsmm_fit() in the non-parametric case. This function can estimate a Drifting semi-Markov model from a sequence of states (i.e. a character vector in R). Example data is included in the package, defined in the DNA sequence lambda. Also some parameters need to be specified before using dsmm_fit(), most notably the polynomial degree and the model of our choice. The model is chosen by defining whether the sojourn times f and the transition matrices p are drifting or not.

# Loading the package

# Obtaining the sequence
data("lambda", package = "dsmmR")
sequence <- c(lambda)

# Obtaining the states
states <- sort(unique(sequence))

# Defining the polynomial degree
degree <- 1 # we define a linear evolution in time (state jumps of the embedded Markov chain)

# Defining the model 
f_is_drifting <- TRUE # sojourn time distributions are drifting in time (state jumps of the EMC)
p_is_drifting <- FALSE # transition matrices are not drifting in time (state jumps of the EMC)
# When both f and p are drifting, we have Model 1.

# Fitting the Drifting semi-Markov model
fitted_model <- fit_dsmm(sequence = sequence,
                         states = states,
                         degree = degree,
                         f_is_drifting = f_is_drifting,
                         p_is_drifting = p_is_drifting)

For more details about the estimation, consider viewing the extended documentation through ?fit_dsmm.

Defining drifting semi-Markov models

When defining a DSMM object we need to input parameters like the polynomial degree, the state space, the DSMM size (length of the embedded Markov chain), the sojourn times f, the transition matrices p and more.

For more information, consider the documentation through ?parametric_dsmm and ?nonparametric_dsmm.


After fitting a DSMM (or defining it through nonparametric_dsmm() or parametric_dsmm()), we can simulate a sequence from that DSMM. This is pretty straightforward:

sim_seq <- simulate(fitted_model)

Since we follow an object oriented approach, providing the previous object fitted_model is the only necessary attribute.

For more information, consider the documentation through ?simulate.dsmm.

Drifting semi-Markov kernel

In order to account for the large dimension of the DSM kernel, a separate function was necessary. You can obtain the DSM kernel through the command:

kernel <- get_kernel(fitted_model)

The dimensionality of the DSM kernel can be reduced further through the attributes of the function.

For more information, consider the documentation through ?get_kernel.

Further reading

Regarding semi-Markov models, the book Semi-Markov Chains and Hidden Semi-Markov Models toward Applications gives a good overview of the topic and also combines the flexibility of the semi-Markov chain with the known advantages of hidden semi-markov models.

If you are not familiar with Drifting Markov models, they were first introduced in Drifting Markov models with Polynomial Drift and Applications to DNA Sequences, while a comprehensive overview is provided in Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation.

Community Guidelines

For third parties wishing to contribute to the software, or to report issues or problems about the software, they can do so directly through the development github page of the package.


Automated tests are in place in order to aid the user with any false input made and, furthermore, to ensure that the functions used return the expected output. Moreover, through strict automated tests, it is made possible for the user to properly define their own dsmm objects and make use of them with the generic functions of the package.

If you are in need of support, please contact the maintainer at


Barbu, V. S., Limnios, N. (2008). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications - Their Use in Reliability and DNA Analysis. New York: Lecture Notes in Statistics, vol. 191, Springer.

Vergne, N. (2008). Drifting Markov models with Polynomial Drift and Applications to DNA Sequences. Statistical Applications in Genetics Molecular Biology 7 (1).

Barbu V. S., Vergne, N. (2019). Reliability and survival analysis for drifting Markov models: modelling and estimation. Methodology and Computing in Applied Probability, 21(4), 1407-1429.


We acknowledge the DATALAB Project (financed by the European Union with the European Regional Development fund (ERDF) and by the Normandy Region) and the HSMM-INCA Project (financed by the French Agence Nationale de la Recherche (ANR) under grant ANR-21-CE40-0005).