Estimation and Simulation of Drifting Semi-Markov Models.
README
Developer Version
1.0.5
dsmmR
The dsmmR R package allows the user to estimate, simulate and define different Drifting semi-Markov model (DSMM) specifications.
Installation
# Install the released version from CRAN
install.packages('dsmmR')
# Or the development version from GitHub
# install.packages("devtools")
devtools::install_github("Mavrogiannis-Ioannis/dsmmR")
High-level documentation
The main functions of dsmmR are the following:
fit_dsmm(): estimate a DSMM (parametric or non-parametric estimation is possible).parametric_dsmm(): define a parametric DSMM.nonparametric_dsmm(): define a non-parametric DSMM.simulate(): simulate a sequence from a DSMM.get_kernel(): obtain the Drifting semi-Markov kernel.
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.
Estimation
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
library(dsmmR)
# 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 f is drifting and p is not drifting, we have Model 3.
# Fitting the drifting semi-Markov model on the sequence.
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.
Simulation
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 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.
Defining drifting semi-Markov models
We can put together all the previous concepts in the showcase of parametric estimation. First, we will define the drifting transition matrices and the drifting sojourn time distributions. Then, we will create a dsmm_parametric object, we will simulate a sequence from it and then finally we will estimate a drifting semi-Markov model from that simulated sequence.
For more information, consider the documentation through ?parametric_dsmm and ?nonparametric_dsmm.
First of all we load the package,
library(dsmmR)
and then we define the states and we set the degree equal to 1.
states <- c("a", "b", "c")
s <- length(states)
degree <- 1
Since degree is equal to 1, we then define the 2 drifting transition matrices:
p_dist_1 <- matrix(c(0, 0.4, 0.6,
0.5, 0, 0.5,
0.3, 0.7, 0 ), ncol = s, byrow = TRUE)
p_dist_2 <- matrix(c(0, 0.55, 0.45,
0.25, 0, 0.75,
0.5, 0.5, 0 ), ncol = s, byrow = TRUE)
p_dist <- array(c(p_dist_1, p_dist_2), dim = c(s, s, degree + 1))
Let us also consider the case where only the parameters of the distributions modeling the sojourn times are drifting across the sequence. Note that distributions like the Negative Binomial and the Discrete Weibull require two parameters, which we define in two matrices for each distribution.
f_dist_1 <- matrix(c(NA, "nbinom", "unif",
"geom", NA, "pois",
"pois", "dweibull", NA ), nrow = s, ncol = s, byrow = TRUE)
f_dist_1_pars_1 <- matrix(c(NA, 4, 3,
0.7, NA, 5,
3, 0.6, NA), nrow = s, ncol = s, byrow = TRUE)
f_dist_1_pars_2 <- matrix(c(NA, 0.5, NA,
NA, NA, NA,
NA, 0.8, NA), nrow = s, ncol = s, byrow = TRUE)
f_dist_2 <- f_dist_1
f_dist_2_pars_1 <- matrix(c(NA, 3, 5,
0.3, NA, 2,
5, 0.3, NA), nrow = s, ncol = s, byrow = TRUE)
f_dist_2_pars_2 <- matrix(c(NA, 0.4, NA,
NA, NA, NA,
NA, 0.5, NA), nrow = s, ncol = s, byrow = TRUE)
f_dist <- array(c(f_dist_1, f_dist_2), dim = c(s, s, degree + 1))
f_dist_pars <- array(c(f_dist_1_pars_1, f_dist_1_pars_2,
f_dist_2_pars_1, f_dist_2_pars_2),
dim = c(s, s, 2, degree + 1))
Then, defining a dsmm_parametric object is done simply through the function parametric_dsmm():
dsmm_model <- parametric_dsmm(
model_size = 10000,
states = states,
initial_dist = c(0.6, 0.3, 0.1),
degree = degree,
p_dist = p_dist,
f_dist = f_dist,
f_dist_pars = f_dist_pars,
p_is_drifting = TRUE,
f_is_drifting = TRUE
)
We can then simulate a sequence from this parametric object like-so:
sim_seq <- simulate(dsmm_model, klim = 30, seed = 1)
To fit this sequence with a drifting semi-Markov model, one can use:
fitted_model <- fit_dsmm(sequence = sim_seq,
states = states,
degree = degree,
f_is_drifting = TRUE,
p_is_drifting = TRUE,
estimation = 'parametric',
f_dist = f_dist)
Finally, the drifting transition matrix is estimated as:
print(fitted_model$dist$p_drift, digits = 2)
with output:
, , p_0
a b c
a 0.00 0.40 0.60
b 0.51 0.00 0.49
c 0.27 0.73 0.00
, , p_1
a b c
a 0.00 0.54 0.46
b 0.23 0.00 0.77
c 0.51 0.49 0.00
and the parameters for the drifting sojourn time distributions are:
print(fitted_model$dist$f_drift_parameters, digits = 2)
with output:
, , 1, fpars_0
a b c
a NA 3.66 3.0
b 0.65 NA 4.8
c 3.09 0.62 NA
, , 2, fpars_0
a b c
a NA 0.46 NA
b NA NA NA
c NA 0.84 NA
, , 1, fpars_1
a b c
a NA 2.74 5.0
b 0.31 NA 2.1
c 5.02 0.29 NA
, , 2, fpars_1
a b c
a NA 0.38 NA
b NA NA NA
c NA 0.50 NA
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.
Notes
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 [email protected].
References
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.
Acknowledgements
We acknowledge the DATALAB Project https://lmrs-num.math.cnrs.fr/projet-datalab.html (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).