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Type:	Package
Title:	Simulation, Estimation and Reliability of Semi-Markov Models
Version:	1.0.5
Date:	2025-10-20
Description:	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 for the sojourn times: Uniform, Geometric, Poisson, Discrete Weibull and Negative Binomial. The non-parametric estimation concerns the sojourn time distributions, where no assumptions are done on the shape of distributions. Moreover, the estimation can be done on the basis of one or several sample paths, with or without censoring at the beginning or/and at the end of the sample paths. Reliability indicators such as reliability, maintainability, availability, BMP-failure rate, RG-failure rate, mean time to failure and mean time to repair are available as well. The implemented methods are described in Barbu, V.S., Limnios, N. (2008) <doi:10.1007/978-0-387-73173-5>, Barbu, V.S., Limnios, N. (2008) <doi:10.1080/10485250701261913> and Trevezas, S., Limnios, N. (2011) <doi:10.1080/10485252.2011.555543>. Estimation and simulation of discrete-time k-th order Markov chains are also considered.
URL:	https://plmlab.math.cnrs.fr/lmrs/statistique/smmR,https://lmrs.pages.math.cnrs.fr/statistique/smmR
BugReports:	https://github.com/corentin-dev/smmR/issues
License:	GPL-3
Imports:	DiscreteWeibull, Rcpp, seqinr
Suggests:	utils, knitr, rmarkdown, testthat (≥ 3.0.0), roxytest
LinkingTo:	Rcpp, RcppArmadillo
VignetteBuilder:	knitr
Encoding:	UTF-8
RoxygenNote:	7.3.3
Config/testthat/edition:	3
NeedsCompilation:	yes
Packaged:	2025-11-07 07:55:43 UTC; corentin
Author:	Vlad Stefan Barbu [aut], Florian Lecocq [aut], Corentin Lothode [aut], Nicolas Vergne [aut, cre]
Maintainer:	Nicolas Vergne <nicolas.vergne@univ-rouen.fr>
Repository:	CRAN
Date/Publication:	2025-11-07 11:40:02 UTC

smmR : Semi-Markov Models, Markov Models and Reliability

Description

This package performs parametric and non-parametric estimationand simulation for multi-state discrete-time semi-Markov processes. For theparametric estimation, several discrete distributions are considered for thesojourn times: Uniform, Geometric, Poisson, Discrete Weibull and NegativeBinomial. The non-parametric estimation concerns the sojourn timedistributions, where no assumptions are done on the shape of distributions.Moreover, the estimation can be done on the basis of one or several samplepaths, with or without censoring at the beginning or/and at the end of thesample paths. Estimation and simulation of discrete-time k-th order Markovchains are also considered.

Semi-Markov models are specified by using the functionssmmparametric()andsmmnonparametric() for parametric and non-parametric specificationsrespectively. These functions return objects of S3 class (smm,smmparametric) and (smm,smmnonparametric) respectively (smm classinherits from S3 classessmmparametric orsmmnonparametric). Thus,smmis like a wrapper class for semi-Markov model specifications.

Based on a model specification (an object of classsmm), it is possible to:

• simulate one or several sequences with the methodsimulate.smm();

• plot conditional sojourn time distributions (methodplot.smm());

• computelog-likelihood,AIC andBIC criteria (methodslogLik(),AIC(),BIC());

• computereliability,maintainability,availability,failure rates (methodsreliability(),maintainability(),availability(),failureRate()).

Estimations of parametric and non-parametric semi-Markov models can be doneby using the functionfitsmm(). This function returns anobject of S3 classsmmfit. The classsmmfit inherits from classes(smm,smmparametric) or (smm,smmnonparametric).

Based on a fitted/estimated semi-Markov model (an object of classsmmfit),it is possible to:

• simulate one or several sequences with the methodsimulate.smmfit();

• plot estimated conditional sojourn time distributions(methodplot.smmfit());

• computelog-likelihood,AIC andBIC criteria (methodslogLik(),AIC(),BIC());

• compute estimatedreliability,maintainability,availability,failure rates and theirconfidence intervals(methodsreliability(),maintainability(),availability(),failureRate()).

Author(s)

Maintainer: Nicolas Vergnenicolas.vergne@univ-rouen.fr

Authors:

• Vlad Stefan Barbu (ORCID)

• Florian Lecocq

• Corentin Lothode (ORCID)

References

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

R.E. Barlow, A.W. Marshall, and F. Prochan. (1963). Properties of probabilitydistributions with monotone hazard rate. Ann. Math. Statist., 34, 375-389.

T. Nakagawa and S. Osaki. (1975). The discrete Weibull distribution.IEEE Trans. Reliabil., R-24, 300-301.

D. Roy and R. Gupta. (1992). Classification of discrete lives.Microelectron. Reliabil., 32 (10), 1459-1473.

I. Votsi & A. Brouste (2019) Confidence interval for the mean time tofailure in semi-Markov models: an application to wind energy production,Journal of Applied Statistics, 46:10, 1756-1773.

See Also

Useful links:

• https://plmlab.math.cnrs.fr/lmrs/statistique/smmR

• https://lmrs.pages.math.cnrs.fr/statistique/smmR

• Report bugs athttps://github.com/corentin-dev/smmR/issues

BMP-Failure Rate Function

Description

Consider a systemS_{ystem} starting to work at timek = 0. The BMP-failure rate at timek \in N is the conditionalprobability that the failure of the system occurs at timek, giventhat the system has worked until timek - 1.

Usage

.failureRateBMP(  x,  k,  upstates = x$states,  level = 0.95,  epsilon = 0.001,  klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.Denote byU = \{1,\dots,s_1\} the subset of operational states ofthe system (the up states) and byD = \{s_1 + 1,\dots,s\} thesubset of failure states (the down states), with0 < s_1 < s(obviously,E = U \cup D andU \cap D = \emptyset,U \neq \emptyset,\ D \neq \emptyset). One can think of the statesofU as different operating modes or performance levels of thesystem, whereas the states ofD can be seen as failures of thesystems with different modes.

We are interested in investigating the failure rate of a discrete-timesemi-Markov systemS_{ystem}. Consequently, we suppose that theevolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The system starts to work atinstant0 and the state of the system is given at each instantk \in N byZ_k: the event\{Z_k = i\}, for a certaini \in U, means that the systemS_{ystem} is in operating modei at timek, whereas\{Z_k = j\}, for a certainj \in D, means that the system is not operational at timekdue to the mode of failurej or that the system is under therepairing modej.

The BMP-failure rate at timek \in N is the conditional probabilitythat the failure of the system occurs at timek, given that thesystem has worked until timek - 1.

LetT_D denote the first passage time in subsetD, calledthe lifetime of the system, i.e.,

T_D := \textrm{inf}\{ n \in N;\ Z_n \in D\}\ \textrm{and}\ \textrm{inf}\ \emptyset := \infty.

For a discrete-time semi-Markov system, the failure rate at timek \geq 1 has the expression:

\lambda(k) := P(T_{D} = k | T_{D} \geq k)

We can rewrite it as follows :

\lambda(k) = 1 - \frac{R(k)}{R(k - 1)},\ \textrm{if } R(k - 1) \neq 0;\ \lambda(k) = 0, \textrm{otherwise}

The failure rate at timek = 0 is defined by\lambda(0) := 1 - R(0),withR being the reliability function (seereliability function).

The calculation of the reliabilityR involves the computation ofmany convolutions. It implies that the computation error, may be higher(in value) than the "true" reliability itself for reliability close to 0.In order to avoid inconsistent values of the BMP-failure rate, we use thefollowing formula:

\lambda(k) = 1 - \frac{R(k)}{R(k - 1)},\ \textrm{if } R(k - 1) \geq \epsilon;\ \lambda(k) = 0, \textrm{otherwise}

with\epsilon, the threshold, the parameterepsilon in thefunctionfailureRateBMP.

Value

A matrix withk + 1 rows, and with columns giving values ofthe BMP-failure rate, variances, lower and upper asymptotic confidencelimits (ifx is an object of classsmmfit).

Availability Function

Description

The pointwise (or instantaneous) availability of a systemS_{ystem} at timek \in N is the probability that the systemis operational at timek (independently of the fact that the systemhas failed or not in[0, k)).

Usage

availability(x, k, upstates = x$states, level = 0.95, klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.Denote byU = \{1,\dots,s_1\} the subset of operational states ofthe system (the up states) and byD = \{s_1 + 1,\dots,s\} thesubset of failure states (the down states), with0 < s_1 < s(obviously,E = U \cup D andU \cap D = \emptyset,U \neq \emptyset,\ D \neq \emptyset). One can think of the statesofU as different operating modes or performance levels of thesystem, whereas the states ofD can be seen as failures of thesystems with different modes.

We are interested in investigating the availability of a discrete-timesemi-Markov systemS_{ystem}. Consequently, we suppose that theevolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The state of the system is givenat each instantk \in N byZ_k: the event\{Z_k = i\},for a certaini \in U, means that the systemS_{ystem} is inoperating modei at timek, whereas\{Z_k = j\}, for acertainj \in D, means that the system is not operational at timek due to the mode of failurej or that the system is under therepairing modej.

The pointwise (or instantaneous) availability of a systemS_{ystem}at timek \in N is the probability that the system is operationalat timek (independently of the fact that the system has failed ornot in[0, k)).

Thus, the pointwise availability of a semi-Markov system at timek \in N is

A(k) = P(Z_k \in U) = \sum_{i \in E} \alpha_i A_i(k),

where we have denoted byA_i(k) the conditional availability of thesystem at timek \in N, given that it starts in statei \in E,

A_i(k) = P(Z_k \in U | Z_0 = i).

Value

A matrix withk + 1 rows, and with columns giving values ofthe availability, variances, lower and upper asymptotic confidence limits(ifx is an object of classsmmfit).

References

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

Discrete-time convolution product off andg(See definition 2.2 p. 20)

Description

Discrete-time convolution product off andg(See definition 2.2 p. 20)

Usage

convolution(f, g)

Arguments

Value

A vector giving the values of the discrete-time convolution off andg for eachk \in N.

Failure Rate Function

Description

Function to compute the BMP-failure rate or the RG-failure rate.

Consider a systemS_{ystem} starting to work at timek = 0. The BMP-failure rate at timek \in N is the conditionalprobability that the failure of the system occurs at timek, giventhat the system has worked until timek - 1.

The RG-failure rate is a discrete-time adapted failure rate, proposed byD. Roy and R. Gupta. Classification of discrete lives. MicroelectronicsReliability, 32(10):1459–1473, 1992. We call it the RG-failure rate anddenote it byr(k),\ k \in N.

Usage

failureRate(  x,  k,  upstates = x$states,  failure.rate = c("BMP", "RG"),  level = 0.95,  epsilon = 0.001,  klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.Denote byU = \{1,\dots,s_1\} the subset of operational states ofthe system (the up states) and byD = \{s_1 + 1,\dots,s\} thesubset of failure states (the down states), with0 < s_1 < s(obviously,E = U \cup D andU \cap D = \emptyset,U \neq \emptyset,\ D \neq \emptyset). One can think of the statesofU as different operating modes or performance levels of thesystem, whereas the states ofD can be seen as failures of thesystems with different modes.

We are interested in investigating the failure rate of a discrete-timesemi-Markov systemS_{ystem}. Consequently, we suppose that theevolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The system starts to work atinstant0 and the state of the system is given at each instantk \in N byZ_k: the event\{Z_k = i\}, for a certaini \in U, means that the systemS_{ystem} is in operating modei at timek, whereas\{Z_k = j\}, for a certainj \in D, means that the system is not operational at timekdue to the mode of failurej or that the system is under therepairing modej.

The BMP-failure rate at timek \in N is the conditional probabilitythat the failure of the system occurs at timek, given that thesystem has worked until timek - 1.

LetT_D denote the first passage time in subsetD, calledthe lifetime of the system, i.e.,

T_D := \textrm{inf}\{ n \in N;\ Z_n \in D\}\ \textrm{and}\ \textrm{inf}\ \emptyset := \infty.

For a discrete-time semi-Markov system, the failure rate at timek \geq 1 has the expression:

\lambda(k) := P(T_{D} = k | T_{D} \geq k)

We can rewrite it as follows :

\lambda(k) = 1 - \frac{R(k)}{R(k - 1)},\ \textrm{if } R(k - 1) \neq 0;\ \lambda(k) = 0, \textrm{otherwise}

The failure rate at timek = 0 is defined by\lambda(0) := 1 - R(0),withR being the reliability function (seereliability function).

The calculation of the reliabilityR involves the computation ofmany convolutions. It implies that the computation error, may be higher(in value) than the "true" reliability itself for reliability close to 0.In order to avoid inconsistent values of the BMP-failure rate, we use thefollowing formula:

\lambda(k) = 1 - \frac{R(k)}{R(k - 1)},\ \textrm{if } R(k - 1) \geq \epsilon;\ \lambda(k) = 0, \textrm{otherwise}

with\epsilon, the threshold, the parameterepsilon in thefunctionfailureRate.

Expressing the RG-failure rater(k) in terms of the reliabilityR we obtain that the RG-failure rate function for a discrete-timesystem is given by:

r(k) = - \ln \frac{R(k)}{R(k - 1)},\ \textrm{if } k \geq 1;\ r(k) = - \ln R(0),\ \textrm{if } k = 0

forR(k) \neq 0. IfR(k) = 0, we setr(k) := 0.

Note that the RG-failure rate is related to the BMP-failure rate by:

r(k) = - \ln (1 - \lambda(k)),\ k \in N

Value

A matrix withk + 1 rows, and with columns giving values ofthe BMP-failure rate or RG-failure rate, variances, lower and upperasymptotic confidence limits (ifx is an object of classsmmfit).

References

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

R.E. Barlow, A.W. Marshall, and F. Prochan. (1963). Properties of probabilitydistributions with monotone hazard rate. Ann. Math. Statist., 34, 375-389.

D. Roy and R. Gupta. (1992). Classification of discrete lives.Microelectron. Reliabil., 32 (10), 1459-1473.

Maximum Likelihood Estimation (MLE) of a k-th order Markov chain

Description

Maximum Likelihood Estimation of the transition matrix andinitial distribution of a k-th order Markov chain starting from one orseveral sequences.

Usage

fitmm(sequences, states, k = 1, init.estim = "mle")

Arguments

Details

LetX_1, X_2, ..., X_n be a trajectory of lengthn ofthe Markov chainX = (X_m)_{m \in N} of orderk = 1 withtransition matrixp_{trans}(i,j) = P(X_{m+1} = j | X_m = i). Themaximum likelihood estimation of the transition matrix is\widehat{p_{trans}}(i,j) = \frac{N_{ij}}{N_{i.}}, whereN_{ij}is the number of transitions from statei to statej andN_{i.} is the number of transition from statei to any state.Fork > 1 we have similar expressions.

The initial distribution of a k-th order Markov chain is defined as\mu_i = P(X_1 = i). Five methods are proposed for the estimationof the latter :

Maximum Likelihood Estimator:

The Maximum Likelihood Estimatorfor the initial distribution. The formula is:\widehat{\mu_i} = \frac{Nstart_i}{L}, whereNstart_i isthe number of occurences of the wordi (of lengthk) atthe beginning of each sequence andL is the number of sequences.This estimator is reliable when the number of sequencesL is high.

Stationary distribution:

The stationary distribution is used asa surrogate of the initial distribution. If the order of the Markovchain is more than one, the transition matrix is converted into asquare block matrix in order to estimate the stationary distribution.This method may take time if the order of the Markov chain is high(more than three (3)).

Frequencies of each word:

The initial distribution is estimatedby taking the frequencies of the words of lengthk for all sequences.The formula is\widehat{\mu_i} = \frac{N_i}{N}, whereN_iis the number of occurences of the wordi (of lengthk) inthe sequences andN is the sum of the lengths of the sequences.

Product of the frequencies of each state:

The initial distributionis estimated by using the product of the frequencies of each state(for all the sequences) in the word of lengthk.

Uniform distribution:

The initial probability of each state isequal to1 / s, withs, the number of states.

Value

An object of class S3mmfit (inheriting from the S3 classmm).The S3 classmmfit contains:

• All the attributes of the S3 classmm;

• An attributeM which is an integer giving the total length ofthe set of sequencessequences (sum of all the lengths of the listsequences);

• An attributelogLik which is a numeric value giving the valueof the log-likelihood of the specified Markov model based on thesequences;

• An attributesequences which is equal to the parametersequences of the functionfitmm (i.e. a list of sequences used toestimate the Markov model).

See Also

mm,simulate.mm

Examples

states <- c("a", "c", "g", "t")s <- length(states)k <- 2init <- rep.int(1 / s ^ k, s ^ k)p <- matrix(0.25, nrow = s ^ k, ncol = s)# Specify a Markov model of order 2markov <- mm(states = states, init = init, ptrans = p, k = k)seqs <- simulate(object = markov, nsim = c(1000, 10000, 2000), seed = 150)est <- fitmm(sequences = seqs, states = states, k = 2)

Maximum Likelihood Estimation (MLE) of a semi-Markov chain

Description

Maximum Likelihood Estimation of a semi-Markov chain startingfrom one or several sequences. This estimation can be parametric ornon-parametric, non-censored, censored at the beginning and/or at the endof the sequence, with one or several trajectories. Several parametricdistributions are considered (Uniform, Geometric, Poisson, DiscreteWeibull and Negative Binomial).

Usage

fitsmm(  sequences,  states,  type.sojourn = c("fij", "fi", "fj", "f"),  distr = "nonparametric",  init.estim = "mle",  cens.beg = FALSE,  cens.end = FALSE)

Arguments

Details

This function estimates a semi-Markov model in parametric ornon-parametric case, taking into account the type of sojourn time and thecensoring described in references. The non-parametric estimation concernssojourn time distributions defined by the user. For the parametricestimation, several discrete distributions are considered (see below).

The difference between the Markov model and the semi-Markov model concernsthe modeling of the sojourn time. With a Markov chain, the sojourn timedistribution is modeled by a Geometric distribution (in discrete time).With a semi-Markov chain, the sojourn time can be any arbitrary distribution.In this package, the available distribution for a semi-Markov model are :

• Uniform:f(x) = \frac{1}{n} for1 \le x \le n.n is the parameter;

• Geometric:f(x) = \theta (1-\theta)^{x - 1} forx = 1, 2,\ldots,n,0 < \theta < 1,\theta is the probability of success.\theta is the parameter;

• Poisson:f(x) = \frac{\lambda^x exp(-\lambda)}{x!} forx = 0, 1, 2,\ldots,n, with\lambda > 0.\lambda is the parameter;

• Discrete Weibull of type 1:f(x)=q^{(x-1)^{\beta}}-q^{x^{\beta}},x = 1, 2,\ldots,n,with0 < q < 1, the first parameter and\beta > 0 the second parameter.(q, \beta) are the parameters;

• Negative binomial:f(x)=\frac{\Gamma(x+\alpha)}{\Gamma(\alpha) x!} p^{\alpha} (1 - p)^x,forx = 0, 1, 2,\ldots,n,\Gamma is the Gamma function,\alpha is the parameter of overdispersion andp is theprobability of success,0 < p < 1.(\alpha, p) are the parameters;

• Non-parametric.

We define :

• the semi-Markov kernelq_{ij}(k) = P( J_{m+1} = j, T_{m+1} - T_{m} = k | J_{m} = i );

• the transition matrix(p_{trans}(i,j))_{i,j} \in states of theembedded Markov chainJ = (J_m)_m,p_{trans}(i,j) = P( J_{m+1} = j | J_m = i );

• the initial distribution\mu_i = P(J_1 = i) = P(Z_1 = i),i \in 1, 2, \dots, s;

• the conditional sojourn time distributions(f_{ij}(k))_{i,j} \in states,\ k \in N ,\ f_{ij}(k) = P(T_{m+1} - T_m = k | J_m = i, J_{m+1} = j ),f is specified by the argumentparam in the parametric caseand bydistr in the non-parametric case.

The maximum likelihood estimation of the transition matrix of the embeddedMarkov chain is\widehat{p_{trans}}(i,j) = \frac{N_{ij}}{N_{i.}}.

Five methods are proposed for the estimation of the initial distribution :

Maximum Likelihood Estimator:

The Maximum Likelihood Estimatorfor the initial distribution. The formula is:\widehat{\mu_i} = \frac{Nstart_i}{L}, whereNstart_i isthe number of occurences of the wordi (of lengthk) atthe beginning of each sequence andL is the number of sequences.This estimator is reliable when the number of sequencesL is high.

Limit (stationary) distribution:

The limit (stationary)distribution of the semi-Markov chain is used as a surrogate of theinitial distribution.

Frequencies of each state:

The initial distribution is replacedby taking the frequencies of each state in the sequences.

Uniform distribution:

The initial probability of each state isequal to1 / s, withs, the number of states.

Note thatq_{ij}(k) = p_{trans}(i,j) \ f_{ij}(k) in the general case(depending on the present state and on the next state). For particular cases,we replacef_{ij}(k) byf_{i.}(k) (depending on the presentstatei),f_{.j}(k) (depending on the next statej) andf_{..}(k) (depending neither on the present state nor on the nextstate).

In this package we can choose different types of sojourn time.Four options are available for the sojourn times:

• depending on the present state and on the next state (fij);

• depending only on the present state (fi);

• depending only on the next state (fj);

• depending neither on the current, nor on the next state (f).

Iftype.sojourn = "fij",distr is a matrix of dimension(s, s)(e.g., if the 1st element of the 2nd column is"pois", that is to say wego from the first state to the second state following a Poisson distribution).Iftype.sojourn = "fi" or"fj",distr must be a vector (e.g., if thefirst element of vector is"geom", that is to say we go from (or to) thefirst state to (or from) any state according to a Geometric distribution).Iftype.sojourn = "f",distr must be one of"unif","geom","pois","dweibull","nbinom" (e.g., ifdistr is equal to"nbinom", that isto say that the sojourn time when going from one state to another statefollows a Negative Binomial distribution).For the non-parametric case,distr is equal to"nonparametric" whatevertype of sojourn time given.

If the sequence is censored at the beginning and/or at the end,cens.begmust be equal toTRUE and/orcens.end must be equal toTRUE.All the sequences must be censored in the same way.

Value

Returns an object of S3 classsmmfit (inheriting from the S3classsmm andsmmnonparametric class ifdistr = "nonparametric"orsmmparametric otherwise). The S3 classsmmfit contains:

• All the attributes of the S3 classsmmnonparametric orsmmparametric depending on the type of estimation;

• An attributeM which is an integer giving the total length ofthe set of sequencessequences (sum of all the lengths of the listsequences);

• An attributelogLik which is a numeric value giving the valueof the log-likelihood of the specified semi-Markov model based on thesequences;

• An attributesequences which is equal to the parametersequences of the functionfitsmm (i.e. a list of sequences used toestimate the Markov model).

References

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

See Also

smmnonparametric,smmparametric,simulate.smm,simulate.smmfit,plot.smm,plot.smmfit

Examples

states <- c("a", "c", "g", "t")s <- length(states)# Creation of the initial distributionvect.init <- c(1 / 4, 1 / 4, 1 / 4, 1 / 4)# Creation of the transition matrixpij <- matrix(c(0, 0.2, 0.5, 0.3,                 0.2, 0, 0.3, 0.5,                 0.3, 0.5, 0, 0.2,                 0.4, 0.2, 0.4, 0),               ncol = s, byrow = TRUE)# Creation of the distribution matrixdistr.matrix <- matrix(c(NA, "pois", "geom", "nbinom",                          "geom", NA, "pois", "dweibull",                         "pois", "pois", NA, "geom",                          "pois", "geom", "geom", NA),                        nrow = s, ncol = s, byrow = TRUE)# Creation of an array containing the parametersparam1.matrix <- matrix(c(NA, 2, 0.4, 4,                           0.7, NA, 5, 0.6,                           2, 3, NA, 0.6,                           4, 0.3, 0.4, NA),                         nrow = s, ncol = s, byrow = TRUE)param2.matrix <- matrix(c(NA, NA, NA, 0.6,                           NA, NA, NA, 0.8,                           NA, NA, NA, NA,                           NA, NA, NA, NA),                         nrow = s, ncol = s, byrow = TRUE)param.array <- array(c(param1.matrix, param2.matrix), c(s, s, 2))# Specify the semi-Markov modelsemimarkov <- smmparametric(states = states, init = vect.init, ptrans = pij,                             type.sojourn = "fij", distr = distr.matrix,                             param = param.array)seqs <- simulate(object = semimarkov, nsim = c(1000, 10000, 2000), seed = 100)# Estimation of simulated sequencesest <- fitsmm(sequences = seqs, states = states, type.sojourn = "fij",               distr = distr.matrix)

Function to compute the value of the sojourn time cumulative distributionH

Description

Function to compute the value ofH (See equation (3.4) p.46).

Usage

get.H(q)

Arguments

Value

An array giving the value ofH(k) at each time between 0andk.

Method to get the number of parameters of a Markov or semi-Markov chain

Description

Method to get the number of parameters of a Markov orsuch as AIC and BIC.

Usage

get.Kpar(x)

Arguments

Value

A positive integer giving the number of parameters.

Function giving the value of the counting process Niuj used in theestimation of the kernel and the transition matrix of censored andnon-parametric semi-markov chains (cf. article Exact MLE and asymptoticproperties for nonparametric semi-Markov models)

Description

Function giving the value of the counting process Niuj used in theestimation of the kernel and the transition matrix of censored andnon-parametric semi-markov chains (cf. article Exact MLE and asymptoticproperties for nonparametric semi-Markov models)

Usage

get.Niuj(sequences)

Arguments

Value

An array giving the values of the counting processN_{iuj}.

Method to compute the value ofP

Description

Method to compute the value ofP(See equation (3.33) p.59).

Usage

get.P(x, k, states = x$states, var = FALSE, klim = 10000)

Arguments

Value

An array giving the value ofP_{i,j}(k) at each time between 0andk ifvar = FALSE. Ifvar = TRUE, a list containing thefollowing components:

x:

an array giving the value ofP_{ij}(k) at each timebetween 0 andk;

sigma2:

an array giving the asymptotic variance of the estimator\sigma_{P}^{2}(i, j, k).

Method to compute the value ofP_{Y}

Description

Method to compute the value ofP_{Y}(See Proposition 5.1 p.105-106).

Usage

get.Py(x, k, upstates = x$states)

Arguments

Value

An array giving the value ofP_{Y}(k) at each time between 0andk.

Method to get the conditional sojourn time distribution f

Description

Computes the conditional sojourn time distributionf(k),f_{i}(k),f_{j}(k) orf_{ij}(k).

Usage

get.f(x, k)

Arguments

Value

A vector, matrix or array giving the value off at each timebetween 1 andk.

Method to get the limit (stationary) distribution

Description

Method to get the limit (stationary) distribution of asemi-Markov chain.

Usage

get.limitDistribution(x, klim = 10000)

Arguments

Value

A vector of length\textrm{card}(E) giving the values of thelimit distribution.

Function to compute the value of the matrix-valued function\psi

Description

Function to compute the value of\psi, the matrix-valuedfunction (See equation (3.16) p.53).

Usage

get.psi(q)

Arguments

Value

An array giving the value of\psi(k) at each time between 0andk.

Method to get the semi-Markov kernelq_{Y}

Description

Computes the semi-Markov kernelq_{Y}(k)(See proposition 5.1 p.106).

Usage

get.qy(x, k, upstates = x$states)

Arguments

Value

An array giving the value ofq_{Y}(k) at each time between 0andk.

Method to get the stationary distribution

Description

Method to get the stationary distribution of a Markov chain.If the order of the Markov chain is higher than one, a block matrix iscomputed to get the stationary distribution.

Usage

get.stationaryDistribution(x)

Arguments

Value

A vector of length\textrm{card}(E) giving the values of thestationary distribution.

Method to get the semi-Markov kernelq

Description

Computes the semi-Markov kernelq_{ij}(k).

Usage

getKernel(x, k, var = FALSE, klim = 10000)

Arguments

Value

An array giving the value ofq_{ij}(k) at each time between 0andk ifvar = FALSE. Ifvar = TRUE, a list containing thefollowing components:

x:

an array giving the value ofq_{ij}(k) at each timebetween 0 andk;

sigma2:

an array giving the asymptotic variance of the estimator\sigma_{q}^{2}(i, j, k).

Function to compute processes based on a list of sequences

Description

Function to compute processes based on a list of sequences

Usage

getProcesses(sequences)

Arguments

Value

A list giving:

s:

Cardinal of the states space;

states:

Vector of state space of lengths;

M:

The total length (sum of the length of each sequence);

L:

The number of sequences;

kmax :

Maximum length of the sojourn times;

Ym:

List of semi-Markov chains;

Jm:

List of successively visited states for each sequence;

Tm:

List of successive time points when state changes for each sequence;

Lm:

List of sojourn time for each sequence;

Um:

List of backward recurrence time for each sequence;

counting:

List giving the counting processes such asN_{ij}(k),N_{i}(k),N_{j}(k)...

Function to check if an object is of classmm

Description

is.mm returnsTRUE ifx is an object ofclassmm.

Usage

is.mm(x)

Arguments

Value

is.mm returnsTRUE orFALSE depending on whetherx is anobject of classmm or not.

Function to check if an object is of classmmfit

Description

is.mmfit returnsTRUE ifx is an object ofclassmmfit.

Usage

is.mmfit(x)

Arguments

Value

is.mmfit returnsTRUE orFALSE depending on whetherx is anobject of classmmfit or not.

Function to check if an object is of classsmm

Description

is.smm returnsTRUE ifx is an object of classsmm.

Usage

is.smm(x)

Arguments

Value

is.smm returnsTRUE orFALSE depending on whetherx is anobject of classsmm or not.

Function to check if an object is of classsmmfit

Description

is.smmfit returnsTRUE ifx is an object of classsmmfit.

Usage

is.smmfit(x)

Arguments

Value

is.smmfit returnsTRUE orFALSE depending on whetherx is anobject of classsmmfit or not.

Function to check if an object is of classsmmnonparametric

Description

is.smmnonparametric returnsTRUE ifx is an object ofclasssmmnonparametric.

Usage

is.smmnonparametric(x)

Arguments

Value

is.smmnonparametric returnsTRUE orFALSE depending on whetherx is an object of classsmmnonparametric or not.

Function to check if an object is of classsmmparametric

Description

is.smmparametric returnsTRUE ifx is an object ofclasssmmparametric.

Usage

is.smmparametric(x)

Arguments

Value

is.smmparametric returnsTRUE orFALSE depending on whetherx is an object of classsmmparametric or not.

Maintainability Function

Description

For a reparable systemS_{ystem} for which the failureoccurs at timek = 0, its maintainability at timek \in N isthe probability that the system is repaired up to timek, given thatit has failed at timek = 0.

Usage

maintainability(x, k, upstates = x$states, level = 0.95, klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.Denote byU = \{1,\dots,s_1\} the subset of operational states ofthe system (the up states) and byD = \{s_1 + 1,\dots, s\} thesubset of failure states (the down states), with0 < s_1 < s(obviously,E = U \cup D andU \cap D = \emptyset,U \neq \emptyset,\ D \neq \emptyset). One can think of the statesofU as different operating modes or performance levels of thesystem, whereas the states ofD can be seen as failures of thesystems with different modes.

We are interested in investigating the maintainability of a discrete-timesemi-Markov systemS_{ystem}. Consequently, we suppose that theevolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The system starts to fail atinstant0 and the state of the system is given at each instantk \in N byZ_k: the event\{Z_k = i\}, for a certaini \in U, means that the systemS_{ystem} is in operating modei at timek, whereas\{Z_k = j\}, for a certainj \in D, means that the system is not operational at timekdue to the mode of failurej or that the system is under therepairing modej.

Thus, we take(\alpha_{i} := P(Z_{0} = i))_{i \in U} = 0 and wedenote byT_U the first hitting time of subsetU, called theduration of repair or repair time, that is,

T_U := \textrm{inf}\{ n \in N;\ Z_n \in U\}\ \textrm{and}\ \textrm{inf}\ \emptyset := \infty.

The maintainability at timek \in N of a discrete-time semi-Markovsystem is

M(k) = P(T_U \leq k) = 1 - P(T_{U} > k) = 1 - P(Z_{n} \in D,\ n = 0,\dots,k).

Value

A matrix withk + 1 rows, and with columns giving values ofthe maintainability, variances, lower and upper asymptotic confidence limits(ifx is an object of classsmmfit).

References

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

Discrete-time matrix convolution product(See definition 3.5 p. 48)

Description

Discrete-time matrix convolution product(See definition 3.5 p. 48)

Usage

matrixConvolution(a, b)

Arguments

Value

An array of dimension(S, S, k + 1) giving the discrete-timematrix convolution product for eachk \in N.

Method to get the mean recurrence times\mu

Description

Method to get the mean recurrence times\mu.

Usage

meanRecurrenceTimes(x, klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.

We are interested in investigating the mean recurrence times of adiscrete-time semi-Markov systemS_{ystem}. Consequently, we supposethat the evolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The state of the system is givenat each instantk \in N byZ_k: the event\{Z_k = i\}.

LetT = (T_{n})_{n \in N} denote the successive time points whenstate changes in(Z_{n})_{n \in N} occur and let alsoJ = (J_{n})_{n \in N} denote the successively visited states atthese time points.

The mean recurrence of an arbitrary statej \in E is given by:

\mu_{jj} = \frac{\sum_{i \in E} \nu(i) m_{i}}{\nu(j)}

where(\nu(1),\dots,\nu(s)) is the stationary distribution of theembedded Markov chain(J_{n})_{n \in N} andm_{i} is the meansojourn time in statei \in E (seemeanSojournTimes function forthe computation).

Value

A vector giving the mean recurrence time(\mu_{i})_{i \in [1,\dots,s]}.

Mean Sojourn Times Function

Description

The mean sojourn time is the mean time spent in each state.

Usage

meanSojournTimes(x, states = x$states, klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.

We are interested in investigating the mean sojourn times of adiscrete-time semi-Markov systemS_{ystem}. Consequently, we supposethat the evolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The state of the system is givenat each instantk \in N byZ_k: the event\{Z_k = i\}.

LetT = (T_{n})_{n \in N} denote the successive time points whenstate changes in(Z_{n})_{n \in N} occur and let alsoJ = (J_{n})_{n \in N} denote the successively visited states atthese time points.

The mean sojourn times vector is defined as follows:

m_{i} = E[T_{1} | Z_{0} = j] = \sum_{k \geq 0} (1 - P(T_{n + 1} - T_{n} \leq k | J_{n} = j)) = \sum_{k \geq 0} (1 - H_{j}(k)),\ i \in E

Value

A vector of length\textrm{card}(E) giving the values of themean sojourn times for each statei \in E.

Markov model specification

Description

Creates a model specification of a Markov model.

Usage

mm(states, init, ptrans, k = 1)

Arguments

Value

An object of classmm.

See Also

simulate.mm,fitmm

Examples

states <- c("a", "c", "g", "t")s <- length(states)k <- 1init <- rep.int(1 / s, s)p <- matrix(c(0, 0, 0.3, 0.4, 0, 0, 0.5, 0.2, 0.7, 0.5,               0, 0.4, 0.3, 0.5, 0.2, 0), ncol = s)# Specify a Markov model of order 1markov <- mm(states = states, init = init, ptrans = p, k = k)

Mean Time To Failure (MTTF) Function

Description

Consider a systemS_{ystem} starting to work at timek = 0. The mean time to failure (MTTF) is defined as the meanlifetime.

Usage

mttf(x, upstates = x$states, level = 0.95, klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.Denote byU = \{1,\dots,s_1\} the subset of operational states ofthe system (the up states) and byD = \{s_1 + 1,\dots,s\} thesubset of failure states (the down states), with0 < s_1 < s(obviously,E = U \cup D andU \cap D = \emptyset,U \neq \emptyset,\ D \neq \emptyset). One can think of the statesofU as different operating modes or performance levels of thesystem, whereas the states ofD can be seen as failures of thesystems with different modes.

We are interested in investigating the mean time to failure of adiscrete-time semi-Markov systemS_{ystem}. Consequently, we supposethat the evolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The system starts to work atinstant0 and the state of the system is given at each instantk \in N byZ_k: the event\{Z_k = i\}, for a certaini \in U, means that the systemS_{ystem} is in operating modei at timek, whereas\{Z_k = j\}, for a certainj \in D, means that the system is not operational at timekdue to the mode of failurej or that the system is under therepairing modej.

LetT_D denote the first passage time in subsetD, calledthe lifetime of the system, i.e.,

T_D := \textrm{inf}\{ n \in N;\ Z_n \in D\}\ \textrm{and}\ \textrm{inf}\ \emptyset := \infty.

The mean time to failure (MTTF) is defined as the mean lifetime, i.e., theexpectation of the hitting time to down setD,

MTTF = E[T_{D}]

Value

A matrix with\textrm{card}(U) = s_{1} rows, and with columnsgiving values of the mean time to failure for each statei \in U,variances, lower and upper asymptotic confidence limits (ifx is anobject of classsmmfit).

References

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

I. Votsi & A. Brouste (2019) Confidence interval for the mean time tofailure in semi-Markov models: an application to wind energy production,Journal of Applied Statistics, 46:10, 1756-1773

Mean Time To Repair (MTTR) Function

Description

Consider a systemS_{ystem} that has just failed at timek = 0. The mean time to repair (MTTR) is defined as the mean of therepair duration.

Usage

mttr(x, upstates = x$states, level = 0.95, klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.Denote byU = \{1,\dots,s_1\} the subset of operational states ofthe system (the up states) and byD = \{s_1 + 1,\dots,s\} thesubset of failure states (the down states), with0 < s_1 < s(obviously,E = U \cup D andU \cap D = \emptyset,U \neq \emptyset,\ D \neq \emptyset). One can think of the statesofU as different operating modes or performance levels of thesystem, whereas the states ofD can be seen as failures of thesystems with different modes.

We are interested in investigating the mean time to repair of adiscrete-time semi-Markov systemS_{ystem}. Consequently, we supposethat the evolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The system has just failed atinstant 0 and the state of the system is given at each instantk \in N byZ_k: the event\{Z_k = i\}, for a certaini \in U, means that the systemS_{ystem} is in operating modei at timek, whereas\{Z_k = j\}, for a certainj \in D, means that the system is not operational at timekdue to the mode of failurej or that the system is under therepairing modej.

LetT_U denote the first passage time in subsetU, called theduration of repair or repair time, i.e.,

T_U := \textrm{inf}\{ n \in N;\ Z_n \in U\}\ \textrm{and}\ \textrm{inf}\ \emptyset := \infty.

The mean time to repair (MTTR) is defined as the mean of the repairduration, i.e., the expectation of the hitting time to up setU,

MTTR = E[T_{U}]

Value

A matrix with\textrm{card}(U) = s_{1} rows, and with columnsgiving values of the mean time to repair for each statei \in U,variances, lower and upper asymptotic confidence limits (ifx is anobject of classsmmfit).

References

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

I. Votsi & A. Brouste (2019) Confidence interval for the mean time tofailure in semi-Markov models: an application to wind energy production,Journal of Applied Statistics, 46:10, 1756-1773

Plot function for an object of class smm

Description

Displays the densities for the conditional sojourn timedistributions depending on the current statei and on the next statej.

Usage

## S3 method for class 'smm'plot(x, i, j, klim = NULL, ...)

Arguments

Value

None.

Plot function for an object of class smmfit

Description

Displays the densities for the conditional sojourn timedistributions depending on the current statei and on the next statej.

Usage

## S3 method for class 'smmfit'plot(x, i, j, klim = NULL, ...)

Arguments

Value

None.

References

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

Examples

states <- c("a", "c", "g", "t")s <- length(states)# Creation of the initial distributionvect.init <- c(1 / 4, 1 / 4, 1 / 4, 1 / 4)# Creation of the transition matrixpij <- matrix(c(0, 0.2, 0.3, 0.4, 0.2, 0, 0.5, 0.2, 0.5,                 0.3, 0, 0.4, 0.3, 0.5, 0.2, 0), ncol = s)# Creation of the distribution matrixdistr.vec <- c("pois", "geom", "geom", "geom")parameters <- matrix(c(2, 0.6, 0.8, 0.8, NA, NA, NA, NA),                      ncol = 2, byrow = FALSE)# Specify the semi-Markov modelsmm <- smmparametric(states = states, init = vect.init, ptrans = pij,                      type.sojourn = "fi", distr = distr.vec, param = parameters)seqs <- simulate(object = smm, nsim = rep(5000, 100), seed = 10)est <- fitsmm(sequences = seqs, states = states, type.sojourn = "fi", distr = distr.vec)class(est)plot(x = est, i = "a", col = "blue", pch = "+")

Reliability Function

Description

Consider a systemS_{ystem} starting to function at timek = 0. The reliability or the survival function ofS_{ystem}at timek \in N is the probability that the system has functionedwithout failure in the period[0, k].

Usage

reliability(x, k, upstates = x$states, level = 0.95, klim = 10000)

Arguments

Details

Consider a system (or a component)S_{ystem} whose possiblestates during its evolution in time areE = \{1,\dots,s\}.Denote byU = \{1,\dots,s_1\} the subset of operational states ofthe system (the up states) and byD = \{s_1 + 1,\dots, s\} thesubset of failure states (the down states), with0 < s_1 < s(obviously,E = U \cup D andU \cap D = \emptyset,U \neq \emptyset,\ D \neq \emptyset). One can think of the statesofU as different operating modes or performance levels of thesystem, whereas the states ofD can be seen as failures of thesystems with different modes.

We are interested in investigating the reliability of a discrete-timesemi-Markov systemS_{ystem}. Consequently, we suppose that theevolution in time of the system is governed by an E-state spacesemi-Markov chain(Z_k)_{k \in N}. The system starts to work atinstant0 and the state of the system is given at each instantk \in N byZ_k: the event\{Z_k = i\}, for a certaini \in U, means that the systemS_{ystem} is in operating modei at timek, whereas\{Z_k = j\}, for a certainj \in D, means that the system is not operational at timekdue to the mode of failurej or that the system is under therepairing modej.

LetT_D denote the first passage time in subsetD, calledthe lifetime of the system, i.e.,

T_D := \textrm{inf}\{ n \in N;\ Z_n \in D\}\ \textrm{and}\ \textrm{inf}\ \emptyset := \infty.

The reliability or the survival function at timek \in N of adiscrete-time semi-Markov system is:

R(k) := P(T_D > k) = P(Zn \in U,n = 0,\dots,k)

which can be rewritten as follows:

R(k) = \sum_{i \in U} P(Z_0 = i) P(T_D > k | Z_0 = i) = \sum_{i \in U} \alpha_i P(T_D > k | Z_0 = i)

Value

A matrix withk + 1 rows, and with columns giving values ofthe reliability, variances, lower and upper asymptotic confidence limits(ifx is an object of classsmmfit).

References

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

Set the RNG Seed from within Rcpp

Description

Set the RNG Seed from within Rcpp

Usage

setSeed(seed)

Arguments

Value

A set RNG scope.

Examples

set.seed(10)x <- rnorm(5, 0, 1)setSeed(10)y <- rnorm(5, 0, 1)all.equal(x, y, check.attributes = FALSE)

Simulates k-th order Markov chains

Description

Simulates k-th order Markov chains.

Usage

## S3 method for class 'mm'simulate(object, nsim = 1, seed = NULL, ...)

Arguments

Details

Ifnsim is a single integer then a chain of that length isproduced. Ifnsim is a vector of integers, thenlength(nsim)sequences are generated with respective lengths.

Value

A list of vectors representing the sequences.

See Also

mm,fitmm

Examples

states <- c("a", "c", "g", "t")s <- length(states)k <- 2init <- rep.int(1 / s ^ k, s ^ k)p <- matrix(0.25, nrow = s ^ k, ncol = s)# Specify a Markov model of order 1markov <- mm(states = states, init = init, ptrans = p, k = k)seqs <- simulate(object = markov, nsim = c(1000, 10000, 2000), seed = 150)

Simulates Markov chains

Description

Simulates sequences from a fitted Markov model.

Usage

## S3 method for class 'mmfit'simulate(object, nsim = 1, seed = NULL, ...)

Arguments

Details

Ifnsim is a single integer then a chain of that length isproduced. Ifnsim is a vector of integers, thenlength(nsim)sequences are generated with respective lengths.

Value

A list of vectors representing the sequences.

See Also

mm,fitmm

Simulates semi-Markov chains

Description

Simulates sequences from a semi-Markov model.

Usage

## S3 method for class 'smm'simulate(object, nsim = 1, seed = NULL, ...)

Arguments

Details

Ifnsim is a single integer then a chain of that length isproduced. Ifnsim is a vector of integers, thenlength(nsim)sequences are generated with respective lengths.

Value

A list of vectors representing the sequences.

References

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

See Also

smmparametric,smmnonparametric,fitsmm

Simulates semi-Markov chains

Description

Simulates sequences from a fitted semi-Markov model.

Usage

## S3 method for class 'smmfit'simulate(object, nsim = 1, seed = NULL, ...)

Arguments

Details

Ifnsim is a single integer then a chain of that length isproduced. Ifnsim is a vector of integers, thenlength(nsim)sequences are generated with respective lengths.

Value

A list of vectors representing the sequences.

References

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

See Also

smmnonparametric,smmparametric,fitsmm

Non-parametric semi-Markov model specification

Description

Creates a non-parametric model specification for a semi-Markov model.

Usage

smmnonparametric(  states,  init,  ptrans,  type.sojourn = c("fij", "fi", "fj", "f"),  distr,  cens.beg = FALSE,  cens.end = FALSE)

Arguments

Details

This function creates a semi-Markov model object in thenon-parametric case, taking into account the type of sojourn time and thecensoring described in references. The non-parametric specification concernssojourn time distributions defined by the user.

The difference between the Markov model and the semi-Markov model concernsthe modeling of the sojourn time. With a Markov chain, the sojourn timedistribution is modeled by a Geometric distribution (in discrete time).With a semi-Markov chain, the sojourn time can be any arbitrary distribution.

We define :

• the semi-Markov kernelq_{ij}(k) = P( J_{m+1} = j, T_{m+1} - T_{m} = k | J_{m} = i );

• the transition matrix(p_{trans}(i,j))_{i,j} \in states of the embedded Markov chainJ = (J_m)_m,p_{trans}(i,j) = P( J_{m+1} = j | J_m = i );

• the initial distribution\mu_i = P(J_1 = i) = P(Z_1 = i),i \in 1, 2, \dots, s;

• the conditional sojourn time distributions(f_{ij}(k))_{i,j} \in states,\ k \in N ,\ f_{ij}(k) = P(T_{m+1} - T_m = k | J_m = i, J_{m+1} = j ),f is specified by the argumentdistr in the non-parametric case.

In this package we can choose different types of sojourn time.Four options are available for the sojourn times:

• depending on the present state and on the next state (f_{ij});

• depending only on the present state (f_{i});

• depending only on the next state (f_{j});

• depending neither on the current, nor on the next state (f).

Let definekmax the maximum length of the sojourn times.Iftype.sojourn = "fij",distr is an array of dimension(s, s, kmax).Iftype.sojourn = "fi" or"fj",distr must be a matrix of dimension(s, kmax).Iftype.sojourn = "f",distr must be a vector of lengthkmax.

If the sequence is censored at the beginning and/or at the end,cens.begmust be equal toTRUE and/orcens.end must be equal toTRUE.All the sequences must be censored in the same way.

Value

Returns an object of classsmm,smmnonparametric.

References

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

See Also

simulate,fitsmm,smmparametric

Examples

states <- c("a", "c", "g", "t")s <- length(states)# Creation of the initial distributionvect.init <- c(1 / 4, 1 / 4, 1 / 4, 1 / 4)# Creation of the transition matrixpij <- matrix(c(0, 0.2, 0.5, 0.3,                 0.2, 0, 0.3, 0.5,                 0.3, 0.5, 0, 0.2,                 0.4, 0.2, 0.4, 0),               ncol = s, byrow = TRUE)# Creation of a matrix corresponding to the # conditional sojourn time distributionskmax <- 6nparam.matrix <- matrix(c(0.2, 0.1, 0.3, 0.2,                           0.2, 0, 0.4, 0.2,                           0.1, 0, 0.2, 0.1,                           0.5, 0.3, 0.15, 0.05,                           0, 0, 0.3, 0.2,                           0.1, 0.2, 0.2, 0),                         nrow = s, ncol = kmax, byrow = TRUE)semimarkov <- smmnonparametric(states = states, init = vect.init, ptrans = pij,                                type.sojourn = "fj", distr = nparam.matrix)semimarkov

Parametric semi-Markov model specification

Description

Creates a parametric model specification for a semi-Markov model.

Usage

smmparametric(  states,  init,  ptrans,  type.sojourn = c("fij", "fi", "fj", "f"),  distr,  param,  cens.beg = FALSE,  cens.end = FALSE)

Arguments

Details

This function creates a semi-Markov model object in the parametriccase, taking into account the type of sojourn time and the censoringdescribed in references. For the parametric specification, several discretedistributions are considered (see below).

The difference between the Markov model and the semi-Markov model concernsthe modeling of the sojourn time. With a Markov chain, the sojourn timedistribution is modeled by a Geometric distribution (in discrete time).With a semi-Markov chain, the sojourn time can be any arbitrary distribution.In this package, the available distribution for a semi-Markov model are :

• Uniform:f(x) = 1/n fora \le x \le b, withn = b-a+1;

• Geometric:f(x) = \theta (1-\theta)^x forx = 0, 1, 2,\ldots,n,0 < \theta < 1, withn > 0 and\theta is the probability of success;

• Poisson:f(x) = (\lambda^x exp(-\lambda))/x! forx = 0, 1, 2,\ldots,n, withn > 0 and\lambda > 0;

• Discrete Weibull of type 1:f(x)=q^{(x-1)^{\beta}}-q^{x^{\beta}}, x=1,2,3,\ldots,n, withn > 0,q is the first parameter and\beta is the second parameter;

• Negative binomial:f(x)=\frac{\Gamma(x+\alpha)}{\Gamma(\alpha) x!} p^{\alpha} (1 - p)^x,forx = 0, 1, 2,\ldots,n,\Gamma is the Gamma function,\alpha is the parameter of overdispersion andp is theprobability of success,0 < p < 1;

• Non-parametric.

We define :

• the semi-Markov kernelq_{ij}(k) = P( J_{m+1} = j, T_{m+1} - T_{m} = k | J_{m} = i );

• the transition matrix(p_{trans}(i,j))_{i,j} \in states of the embedded Markov chainJ = (J_m)_m,p_{trans}(i,j) = P( J_{m+1} = j | J_m = i );

• the initial distribution\mu_i = P(J_1 = i) = P(Z_1 = i),i \in 1, 2, \dots, s;

• the conditional sojourn time distributions(f_{ij}(k))_{i,j} \in states,\ k \in N ,\ f_{ij}(k) = P(T_{m+1} - T_m = k | J_m = i, J_{m+1} = j ),f is specified by the argumentparam in the parametric case.

In this package we can choose different types of sojourn time.Four options are available for the sojourn times:

• depending on the present state and on the next state (f_{ij});

• depending only on the present state (f_{i});

• depending only on the next state (f_{j});

• depending neither on the current, nor on the next state (f).

Iftype.sojourn = "fij",distr is a matrix of dimension(s, s)(e.g., if the row 1 of the 2nd column is"pois", that is to say we go fromthe first state to the second state following a Poisson distribution).Iftype.sojourn = "fi" or"fj",distr must be a vector (e.g., if thefirst element of vector is"geom", that is to say we go from the firststate to any state according to a Geometric distribution).Iftype.sojourn = "f",distr must be one of"unif","geom","pois","dweibull","nbinom" (e.g., ifdistr is equal to"nbinom", that isto say that the sojourn times when going from any state to any state followsa Negative Binomial distribution).For the non-parametric case,distr is equal to"nonparametric" whatevertype of sojourn time given.

If the sequence is censored at the beginning and/or at the end,cens.begmust be equal toTRUE and/orcens.end must be equal toTRUE.All the sequences must be censored in the same way.

Value

Returns an object of classsmmparametric.

References

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

See Also

simulate,fitsmm,smmnonparametric

Examples

states <- c("a", "c", "g", "t")s <- length(states)# Creation of the initial distributionvect.init <- c(1 / 4, 1 / 4, 1 / 4, 1 / 4)# Creation of the transition matrixpij <- matrix(c(0, 0.2, 0.5, 0.3,                0.2, 0, 0.3, 0.5,                0.3, 0.5, 0, 0.2,                0.4, 0.2, 0.4, 0),              ncol = s, byrow = TRUE)# Creation of the distribution matrixdistr.matrix <- matrix(c(NA, "pois", "geom", "nbinom",                         "geom", NA, "pois", "dweibull",                         "pois", "pois", NA, "geom",                         "pois", "geom", "geom", NA),                       nrow = s, ncol = s, byrow = TRUE)# Creation of an array containing the parametersparam1.matrix <- matrix(c(NA, 2, 0.4, 4,                          0.7, NA, 5, 0.6,                          2, 3, NA, 0.6,                          4, 0.3, 0.4, NA),                        nrow = s, ncol = s, byrow = TRUE)param2.matrix <- matrix(c(NA, NA, NA, 0.6,                          NA, NA, NA, 0.8,                          NA, NA, NA, NA,                          NA, NA, NA, NA),                        nrow = s, ncol = s, byrow = TRUE)param.array <- array(c(param1.matrix, param2.matrix), c(s, s, 2))# Specify the semi-Markov modelsemimarkov <- smmparametric(states = states, init = vect.init, ptrans = pij,                             type.sojourn = "fij", distr = distr.matrix,                             param = param.array)semimarkov