SMM: An R Package for Estimation and Simulation of Discrete-time semi-Markov Models

Abstract

Semi-Markov models, independently introduced by Lévy (1954), Smith (1955) and Takacs (1954), are a generalization of the well-known Markov models. For semi-Markov models, sojourn times can be arbitrarily distributed, while sojourn times of Markov models are constrained to be exponentially distributed (in continuous time) or geometrically distributed (in discrete time). The aim of this paper is to present the R package SMM, devoted to the simulation and estimation of discretetime multi-state semi-Markov and Markov models. For the semi-Markov case we have considered: parametric and non-parametric estimation; with and without censoring at the beginning and/or at the end of sample paths; one or several independent sample paths. Several discrete-time distributions are considered for the parametric estimation of sojourn time distributions of semi-Markov chains: Uniform, Geometric, Poisson, Discrete Weibull and Binomial Negative.Few R packages have been developed to handle semi-Markov models or hidden semi-Markov models. For semi-Markov models we have the recent semiMarkov R package (Król and Saint-Pierre, 2015) that performs maximum likelihood estimation for parametric continuous-time semi-Markov processes, where the distribution can be chosen between Exponential, Weibull or exponentiated Weibull. That package computes associated hazard rates; covariates can also be taken into account through the Cox proportional hazard model. Two R packages are also dedicated to hidden semi-Markov models, implementing estimation and prediction methods: the hsmm R package (Bulla et al., 2010) and the mhsmm R package (O'Connell and Højsgaard, 2011).Note that there is no R package developed for discrete-time multi-state semi-Markov models. Thus the purpose of this paper is to present an R package that we have developed, called SMM, which performs parametric and non-parametric estimation and simulation for multi-state discrete-time semi-Markov processes. For the parametric estimation, several discrete distributions are considered

Discrete Time Hybrid Semi-Markov Models in Manpower Planning

“…In our study, we have chosen to avoid left censoring issues: since the analyzed data contain only a limited number of data lines where left censoring is involved, we did not take into account the first observed state of an employee in case it was subjected to left censoring. We corrected for right censoring in computing the estimations of the parameters [41].…”

“…We now use the same data as in the previous section to estimate the empirical sojourn time distributions f ij according to Equation (17) with the aid of the R package SMM [41] and apply the test statistic S ij (Equation ( 16)) to each tuple of states (S i , S j ). The results are summarized in Table 2.…”

“…This means we have to reject the geometrical hypothesis for the sojourn time distributions f 15 and f 51 . Using the R package SMM[41], we estimated all f ij s as parametric distributions: f 15 and f 51 as Weibull distributions and the other f ij s as geometric distributions. We now consider three different models:…”

Estimation and Reliability for a Special Type of Semi-Markov Process

“…is the last censored sojourn time in the last visited state. We follow the methodology proposed in Barbu et al (2018) for parametric estimation of discrete-time semi-Markov processes. The contribution to the likelihood of a right censored time k, in state i is defined as:…”

Modelling the lifetime of banknotes with a semi-Markov chain model

“…Although some R packages with functions for semi-Markov models exist (e.g. Barbu et al (2018); Listwon & Saint-Pierre ( 2015)), we have built a model in R that does not depend on pre-existing packages 3 . The model and it's functions are built in away that works best for modeling the lifetime of banknotes.…”