drimmR-package

Description

This package performs the estimation, simulation and the exact computation of associated reliability measures of drifting Markov models (DMMs). These are particular non-homogeneous Markov chains for which the Markov transition matrix is a linear (polynomial) function of two (several) Markov transition matrices. Several statistical frameworks are taken into account (one or several samples, complete or incomplete samples, models of the same length). Computation of probabilities of appearance of a word along a sequence under a given model are also considered.

Author(s)

Maintainer: Nicolas Vergne nicolas.vergne@univ-rouen.fr

Authors:

• Vlad Stefan Barbu vladstefan.barbu@univ-rouen.fr

• Corentin Lothode corentin.lothode@univ-rouen.fr

• Alexandre Seiller

Other contributors:

• Geoffray Brelurut [contributor]

• Annthomy Gilles [contributor]

• Arnaud Lefebvre [contributor]

• Victor Mataigne [contributor]

Akaike Information Criterion (AIC)

Description

Generic function computing the Akaike Information Criterion of the model x, with the list of sequences sequences.

Usage

aic(x, sequences, ncpu = 2)

Arguments

Value

A list of AIC (numeric)

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Evaluate the AIC of a drifting Markov Model

Description

Computation of the Akaike Information Criterion.

Usage

## S3 method for class 'dmm'
aic(x, sequences, ncpu = 2)

Arguments

Value

A list of AIC (numeric)

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, loglik, aic

Examples

data(lambda, package = "drimmR")
sequence <- c("a","g","g","t","c","g","a","t","a","a","a")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
aic(dmm,sequence)

Availability function

Description

The pointwise (or instantaneous) availability of a system S_{ystem} at time k \in N is the probability that the system is operational at time k (independently of the fact that the system has failed or not in [0; k)).

Usage

availability(
  x,
  k1 = 0L,
  k2,
  upstates,
  output_file = NULL,
  plot = FALSE,
  ncpu = 2
)

Arguments

Details

Consider a system (or a component) System whose possible states during its evolution in time are E = \{1 \ldots s \}. Denote by U = \{1 \ldots s_1 \} the subset of operational states of the system (the upstates) and by D =\{s_{1}+1 \ldots s \} the subset of failure states (the down states), with 0 < s1 < s(obviously, E = U \cup D and U \cap D = \emptyset, U \neq \emptyset, D \neq \emptyset). One can think of the states of U as different operating modes or performance levels of the system, whereas the states of D can be seen as failures of the systems with different modes.

Value

A vector of length k+1 giving the values of the availability for the period [0 \ldots k]

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8.

See Also

fitdmm, getTransitionMatrix, reliability, maintainability

Examples

data(lambda, package = "drimmR")
length(lambda) <- 1000
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq",
 fit.method="sum")
k1 <- 1
k2 <- 200
upstates <- c("c","t")  # vector of working states
getA <- availability(dmm,k1,k2,upstates,plot=TRUE)

Bayesian Information Criterion (BIC)

Description

Generic function computing the Bayesian Information Criterion of the model x, with the list of sequences sequences.

Usage

bic(x, sequences, ncpu = 2)

Arguments

Value

A list of BIC (numeric)

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Evaluate the BIC of a drifting Markov Model

Description

Computation of the Bayesian Information Criterion.

Usage

## S3 method for class 'dmm'
bic(x, sequences, ncpu = 2)

Arguments

Value

A list of BIC (numeric).

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, loglik, bic

Examples

data(lambda, package = "drimmR")
sequence <- c("a","g","g","t","c","g","a","t","a","a","a")
dmm<- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
bic(dmm,sequence)

Distributions for a range of positions between <start> and <end>

Description

Distributions for a range of positions between <start> and <end>

Usage

distributions(
  x,
  start = 1,
  end = NULL,
  step = NULL,
  output_file = NULL,
  plot = FALSE,
  ncpu = 2
)

Arguments

Value

A matrix with positions and distributions of states

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getDistribution, getStationaryLaw

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
distributions(dmm,start=1,end=1000,step=100, plot=TRUE)

Failure rates function

Description

Computation of two different definition of the failure rate : the BMP-failure rate and RG-failure rate.

As for BMP-failure rate, consider a system S starting to work at time k = 0. The BMP-failure rate at time k \in N is the conditional probability that the failure of the system occurs at time k, given that the system has worked until time k - 1. The BMP-failure rate denoted by \lambda(k), k \in N is usually considered for continuous time systems.

The RG-failure rate is a discrete-time adapted failure-rate proposed by D. Roy and R. Gupta. Classification of discrete lives. Microelectronics Reliability, 32(10):1459–1473, 1992. The RG-failure rate is denoted by r(k), k \in N.

Usage

failureRate(
  x,
  k1 = 0L,
  k2,
  upstates,
  failure.rate = c("BMP", "RG"),
  output_file = NULL,
  plot = FALSE
)

Arguments

Details

Consider a system (or a component) System whose possible states during its evolution in time are E = \{1 \ldots s \}. Denote by U = \{1 \ldots s_1 \} the subset of operational states of the system (the upstates) and by D =\{s_{1}+1 \ldots s \} the subset of failure states (the down states), with 0 < s1 < s(obviously, E = U \cup D and U \cap D = \emptyset, U \neq \emptyset, D \neq \emptyset). One can think of the states of U as different operating modes or performance levels of the system, whereas the states of D can be seen as failures of the systems with different modes.

Value

A vector of length k + 1 giving the values of the BMP (or RG) -failure rate for the period [0 \ldots k]

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Roy D, Gupta R (1992). “Classification of discrete lives. Microelectronics Reliability.” Microelectronics Reliability, 1459–1473. doi: 10.1016/0026-2714(92)90015-D, https://doi.org/10.1016/0026-2714(92)90015-D.

See Also

fitdmm, getTransitionMatrix, reliability

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq",
 fit.method="sum")
k1 <- 1
k2 <- 200
upstates <- c("c","t")  # vector of working states
failureRate(dmm,k1,k2,upstates,failure.rate="BMP",plot=TRUE)

Point by point estimates of a k-th order drifting Markov Model

Description

Estimation of d+1 points of support transition matrices and |E|^{k} initial law of a k-th order drifting Markov Model starting from one or several sequences.

Usage

fitdmm(
  sequences,
  order,
  degree,
  states,
  init.estim = c("mle", "freq", "prod", "stationary", "unif"),
  fit.method = c("sum"),
  ncpu = 2
)

Arguments

Details

The fitdmm function creates a drifting Markov model object dmm.

Let E={1,\ldots, s}, s < \infty be random system with finite state space, with a time evolution governed by discrete-time stochastic process of values in E. A sequence X_0, X_1, \ldots, X_n with state space E= {1, 2, \ldots, s} is said to be a linear drifting Markov chain (of order 1) of length n between the Markov transition matrices \Pi_0 and \Pi_1 if the distribution of X_t, t = 1, \ldots, n, is defined by P(X_t=v \mid X_{t-1} = u, X_{t-2}, \ldots ) = \Pi_{\frac{t}{n}}(u, v), ; u, v \in E, where \Pi_{\frac{t}{n}}(u, v) = ( 1 - \frac{t}{n}) \Pi_0(u, v) + \frac{t}{n} \Pi_1(u, v), \; u, v \in E. The linear drifting Markov model of order 1 can be generalized to polynomial drifting Markov model of order k and degree d.Let \Pi_{\frac{i}{d}} = (\Pi_{\frac{i}{d}}(u_1, \dots, u_k, v))_{u_1, \dots, u_k,v \in E} be d Markov transition matrices (of order k) over a state space E.

The estimation of DMMs is carried out for 4 different types of data :

One can observe one sample path :

It is denoted by H(m,n):= (X_0,X_1, \ldots,X_{m}), where m denotes the length of the sample path and n the length of the drifting Markov chain. Two cases can be considered:

1. m=n (a complete sample path),

2. m < n (an incomplete sample path).

One can also observe H i.i.d. sample paths :

It is denoted by H_i(m_i,n_i), i=1, \ldots, H. Two cases cases are considered :

1. m_i=n_i=n \forall i=1, \ldots, H (complete sample paths of drifting Markov chains of the same length),

2. n_i=n \forall i=1, \ldots, H (incomplete sample paths of drifting Markov chains of the same length). In this case, an usual LSE over the sample paths is used.

The initial distribution of a k-th order drifting Markov Model is defined as \mu_i = P(X_1 = i). The initial distribution of the k first letters is freely customisable by the user, but five methods are proposed for the estimation of the latter :

Estimation based on the Maximum Likelihood Estimator:

The Maximum Likelihood Estimator for the initial distribution. The formula is: \widehat{\mu_i} = \frac{Nstart_i}{L}, where Nstart_i is the number of occurences of the word i (of length k) at the beginning of each sequence and L is the number of sequences. This estimator is reliable when the number of sequences L is high.

Estimation based on the frequency:

The initial distribution is estimated by taking the frequences of the words of length k for all sequences. The formula is \widehat{\mu_i} = \frac{N_i}{N}, where N_i is the number of occurences of the word i (of length k) in the sequences and N is the sum of the lengths of the sequences.

Estimation based on the product of the frequences of each state:

The initial distribution is estimated by using the product of the frequences of each state (for all the sequences) in the word of length k.

Estimation based on the stationary law of point of support transition matrix for a word of length k :

The initial distribution is estimated using \mu(\Pi_{\frac{k-1}{n}})

Estimation based on the uniform law :

\frac{1}{s}

Value

An object of class dmm

Author(s)

Geoffray Brelurut, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Examples

data(lambda, package = "drimmR")
states <- c("a","c","g","t")
order <- 1
degree <- 1
fitdmm(lambda,order,degree,states, init.estim = "freq",fit.method="sum")

Distributions of the drifting Markov Model

Description

Generic function evaluating the distribution of a model x at a given position pos or at every position all.pos

Usage

getDistribution(x, pos, all.pos = FALSE, internal = FALSE, ncpu = 2)

Arguments

Details

Distribution at position l is evaluated by \mu_{l} =\mu_0 \prod_{t=k}^{l} \ \pi_{\frac{t}{n}}, \forall l \ge k, k \in N^* order of the DMM

Value

A vector or matrix of distribution probabilities

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Get the distributions of the DMM

Description

Evaluate the distribution of the DMM at a given position or at every position

Usage

## S3 method for class 'dmm'
getDistribution(x, pos, all.pos = FALSE, internal = FALSE, ncpu = 2)

Arguments

Details

Distribution at position l is evaluated by \mu_{l} =\mu_0 \prod_{t=k}^{l} \ \pi_{\frac{t}{n}}, \forall l \ge k, k \in N^* order of the DMM

Value

A vector or matrix of distribution probabilities

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, distributions, getStationaryLaw

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
t <- 10
getDistribution(dmm,pos=t)

Stationary laws of the drifting Markov Model

Description

Generic function evaluating the stationary law of a model x at a given position pos or at every position all.pos

Usage

getStationaryLaw(x, pos, all.pos = FALSE, internal = FALSE, ncpu = 2)

Arguments

Details

Stationary law at position t is evaluated by solving \mu_t \ \pi_{\frac{t}{n}} = \mu

Value

A vector or matrix of stationary law probabilities

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Get the stationary laws of the DMM

Description

Evaluate the stationary law of the DMM at a given position or at every position

Usage

## S3 method for class 'dmm'
getStationaryLaw(x, pos, all.pos = FALSE, internal = FALSE, ncpu = 2)

Arguments

Details

Stationary law at position t is evaluated by solving \mu_t \ \pi_{\frac{t}{n}} = \mu

Value

A vector or matrix of stationary law probabilities

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, stationary_distributions, getDistribution

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
t <- 10
getStationaryLaw(dmm,pos=t)

Transition matrix of the drifting Markov Model

Description

Generic function evaluating the transition matrix of a model x at a given position pos

Usage

getTransitionMatrix(x, pos)

Arguments

Value

A transition matrix at a given position

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Get transition matrix of the drifting Markov Model

Description

Evaluate the transition matrix of the DMM at a given position

Usage

## S3 method for class 'dmm'
getTransitionMatrix(x, pos)

Arguments

Value

A transition matrix at a given position

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'),init.estim = "freq", fit.method="sum")
t <- 10
getTransitionMatrix(dmm,pos=t)

lambda genome

Description

Complete data from phage genome [WT71] of length 48502

Usage

data(lambda)

Format

A vector object of class "Rdata".

Examples

data(lambda)

Probability of occurrence of the observed word of size m in a sequence at several positions

Description

Probability of occurrence of the observed word of size m in a sequence at several positions

Usage

lengthWord_probabilities(m, sequence, pos, x, output_file = NULL, plot = FALSE)

Arguments

Value

A dataframe of probability by position

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, word_probability

Examples

data(lambda, package = "drimmR")
length(lambda) <- 1000
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
m <- 2
lengthWord_probabilities(m, lambda, c(1,length(lambda)-m), dmm, plot=TRUE)

Log-likelihood of the drifting Markov Model

Description

Generic function computing the log-likelihood of the model x, with the list of sequences sequences.

Usage

loglik(x, sequences, ncpu = 2)

Arguments

Value

A list of loglikelihood (numeric)

Author(s)

Annthomy Gilles, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Evaluate the log-likelihood of a drifting Markov Model

Description

Evaluate the log-likelihood of a drifting Markov Model

Usage

## S3 method for class 'dmm'
loglik(x, sequences, ncpu = 2)

Arguments

Value

A list of log-likelihood (numeric)

Author(s)

Annthomy Gilles, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix

Examples

data(lambda, package = "drimmR")
sequence <- c("a","g","g","t","c","g","a","t","a","a","a")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
loglik(dmm,sequence)

Maintainability function

Description

Maintainability of a system at time k \in N is the probability that the system is repaired up to time k, given that is has failed at time k=0.

Usage

maintainability(
  x,
  k1 = 0L,
  k2,
  upstates,
  output_file = NULL,
  plot = FALSE,
  ncpu = 2
)

Arguments

Details

Consider a system (or a component) System whose possible states during its evolution in time are E = \{1 \ldots s \}. Denote by U = \{1 \ldots s_1 \} the subset of operational states of the system (the upstates) and by D =\{s_{1}+1 \ldots s \} the subset of failure states (the down states), with 0 < s1 < s(obviously, E = U \cup D and U \cap D = \emptyset, U \neq \emptyset, D \neq \emptyset). One can think of the states of U as different operating modes or performance levels of the system, whereas the states of D can be seen as failures of the systems with different modes.

Value

A vector of length k + 1 giving the values of the maintainability for the period [0 \ldots k]

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8.

See Also

fitdmm, getTransitionMatrix

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'),
init.estim = "freq", fit.method="sum")
k1 <- 1
k2 <- 200
upstates <- c("c","t")  # vector of working states
maintainability(dmm,k1,k2,upstates,plot=TRUE)

Reliability function

Description

Reliability or the survival function of a system at time k \in N

Usage

reliability(
  x,
  k1 = 0L,
  k2,
  upstates,
  output_file = NULL,
  plot = FALSE,
  ncpu = 2
)

Arguments

Details

The reliability at time k \in N is the probability that the system has functioned without failure in the period [0, k]

Value

A vector of length k + 1 giving the values of the reliability for the period [0 \ldots k]

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8.

See Also

fitdmm, getTransitionMatrix

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq",
fit.method="sum")
k1 <- 1
k2 <- 200
upstates <- c("c","t")  # vector of working states
reliability(dmm,k1,k2,upstates,plot=TRUE)

Simulate a sequence under a drifting Markov model

Description

Generic function simulating a sequence of length model_size under a model x

Usage

simulate(x, output_file, model_size = 1e+05, ncpu = 2)

Arguments

Value

the vector of simulated sequence

Author(s)

Annthomy Gilles, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

Simulate a sequence under a drifting Markov model

Description

Simulate a sequence under a k-th order DMM.

Usage

## S3 method for class 'dmm'
simulate(x, output_file = NULL, model_size = NULL, ncpu = 2)

Arguments

Value

the vector of simulated sequence

Author(s)

Annthomy Gilles, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, getStationaryLaw

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
simulate(dmm, model_size=100)

Stationary laws for a range of positions between <start> and <end>

Description

Stationary laws for a range of positions between <start> and <end>

Usage

stationary_distributions(
  x,
  start = 1,
  end = NULL,
  step = NULL,
  output_file = NULL,
  plot = FALSE
)

Arguments

Value

A matrix with positions and stationary laws of states

Author(s)

Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getStationaryLaw

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq",fit.method="sum")
stationary_distributions(dmm,start=1,end=1000,step=100, plot=TRUE)

Probabilities of a word at several positions of a DMM

Description

Probabilities of a word at several positions of a DMM

Usage

word_probabilities(word, pos, x, output_file = NULL, plot = FALSE)

Arguments

Value

A numeric vector, probabilities of word

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, word_probability, words_probabilities

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'),
init.estim = "freq", fit.method="sum")
word_probabilities("aggctga",c(100,300),dmm, plot=TRUE)

Probability of a word at a position t of a DMM

Description

Probability of a word at a position t of a DMM

Usage

word_probability(word, pos, x, output_file = NULL, internal = FALSE, ncpu = 2)

Arguments

Value

A numeric, probability of word

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, word_probabilities, words_probabilities

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq", fit.method="sum")
word_probability("aggctga",4,dmm)

Probability of appearance of several words at several positions of a DMM

Description

Probability of appearance of several words at several positions of a DMM

Usage

words_probabilities(words, pos, x, output_file = NULL, plot = FALSE)

Arguments

Value

A dataframe of word probabilities along the positions of the sequence

Author(s)

Victor Mataigne, Alexandre Seiller

References

Barbu VS, Vergne N (2018). “Reliability and survival analysis for drifting Markov models: modelling and estimation.” Methodology and Computing in Applied Probability, 1–33. doi: 10.1007/s11009-018-9682-8, https://doi.org/10.1007/s11009-018-9682-8. Vergne N (2008). “Drifting Markov models with polynomial drift and applications to DNA sequences.” Statistical Applications in Genetics Molecular Biology , 7(1) . doi: 10.2202/1544-6115.1326, https://doi.org/10.2202/1544-6115.1326.

See Also

fitdmm, getTransitionMatrix, word_probability, word_probabilities

Examples

data(lambda, package = "drimmR")
dmm <- fitdmm(lambda, 1, 1, c('a','c','g','t'), init.estim = "freq",
fit.method="sum")
words <- c("atcgattc", "taggct", "ggatcgg")
pos <- c(100,300)
words_probabilities(words=words,pos=pos,dmm,plot=TRUE)