Package 'drimmR'

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 [email protected]

Authors:

• Vlad Stefan Barbu [email protected]

• Corentin Lothode [email protected]

• 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

x	An object for which the log-likelihood of the DMM can be computed.
sequences	A vector of characters or a list of vector of characters representing the sequences for which the AIC will be computed based on x.
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

---

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

x	An object of class dmm
sequences	A character vector or a list of character vector representing the sequences for which the AIC will be computed based on x.
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object of class dmm
k1	Start position (default value=0): a positive integer giving the start position along the sequence from which the availabilities of the DMM should be computed, such that k1<k2
k2	End position : a positive integer giving the end position along the sequence until which the availabilities of the DMM should be computed, such that k2>k1
upstates	Character vector giving the subset of operational states U.
output_file	(Optional) A file containing matrix of availability probabilities (e.g, "C:/.../AVAL.txt")
plot	FALSE (default); TRUE (display a figure plot of availability probabilities by position)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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.

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

x	An object for which the log-likelihood of the DMM can be computed.
sequences	A list of vectors representing the sequences for which the BIC will be computed based on x.
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

---

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

x	An object of class dmm
sequences	A character vector or a list of character vector representing the sequences for which the BIC will be computed based on x.
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object of class dmm
start	Start position : a positive integer giving the start position along the sequence from which the distributions of the DMM should be computed
end	End position : a positive integer giving the end position along the sequence until which the distributions of the DMM should be computed
step	A step (integer)
output_file	(Optional) A file containing matrix of distributions (e.g, "C:/.../DIST.txt")
plot	FALSE (default); TRUE (display a figure plot of distribution probabilities by position)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object of class dmm
k1	Start position (default value=0) : a positive integer giving the start position along the sequence from which the failure rates of the DMM should be computed, such that k1<k2
k2	End position : a positive integer giving the end position along the sequence until which the failure rates of the DMM should be computed, such that k2>k1
upstates	Character vector of the subspace working states among the state space vector such that upstates < s
failure.rate	Default="BMP", then BMP-failure-rate is the method used to compute the failure rate. If failure.rate= "RG", then RG-failure rate is the method used to compute the failure rate.
output_file	(Optional) A file containing matrix of failure rates at each position (e.g, "C:/.../ER.txt")
plot	FALSE (default); TRUE (display a figure plot of failure rates by position)

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. Roy D, Gupta R (1992). “Classification of discrete lives. Microelectronics Reliability.” Microelectronics Reliability, 1459–1473. doi: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

sequences	A list of character vector(s) representing one (several) sequence(s)
order	Order of the Markov chain
degree	Degree of the polynomials (e.g., linear drifting if degree=1, etc.)
states	Vector of states space of length s > 1
init.estim	Default="mle". Method used to estimate the initial law. If init.estim = "mle", then the classical Maximum Likelihood Estimator is used, if init.estim = "freq", then, the initial distribution init.estim is estimated by taking the frequences of the words of length k for all sequences. If init.estim = "prod", then, init.estim is estimated by using the product of the frequences of each letter (for all the sequences) in the word of length k. If init.estim = "stationary", then init.estim is estimated by using the stationary law of the point of support transition matrices of each letter. If init.estim = "unif", then, init.estim of each letter is estimated by using $\frac{1}{s}$. Or 'init.estim'= customisable vector of length $|E|^k$. See Details for the formulas.
fit.method	If sequences is a list of several character vectors of the same length, the usual LSE over the sample paths is proposed when fit.method="sum" (a list of a single character vector is its special case).
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object for which the distributions of the DMM can be computed.
pos	A positive integer giving the position along the sequence on which the distribution of the DMM should be computed
all.pos	'FALSE' (evaluation at position index) ; 'TRUE' (evaluation for all position indices)
internal	'FALSE' (default) ; 'TRUE' (for internal use of the distributions function)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

---

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

x	An object of class dmm
pos	A positive integer giving the position along the sequence on which the distribution of the DMM should be computed
all.pos	'FALSE' (default, evaluation at position index) ; 'TRUE' (evaluation for all position indices)
internal	'FALSE' (default) ; 'TRUE' (for internal use of distributions function)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object for which the stationary laws of the DMM can be computed.
pos	A positive integer giving the position along the sequence on which the stationary law of the DMM should be computed
all.pos	'FALSE' (default, evaluation at position index) ; 'TRUE' (evaluation for all position indices)
internal	'FALSE' (default) ; 'TRUE' (for internal use of the initial law computation)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

---

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

x	An object of class dmm
pos	A positive integer giving the position along the sequence on which the stationary law of the DMM should be computed
all.pos	'FALSE' (default, evaluation at position index) ; 'TRUE' (evaluation for all position indices)
internal	'FALSE' (default) ; 'TRUE' (for internal use of th initial law of fitdmm and word applications)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object for which the transition matrices of the DMM can be computed.
pos	A positive integer giving the position along the sequence on which the transition matrix of the DMM should be computed

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. 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.

---

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

x	An object of class dmm
pos	A positive integer giving the position along the sequence on which the transition matrix of the DMM should be computed

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. 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.

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

m	An integer, the length word
sequence	A vector of characters
pos	A vector of integer positions
x	An object of class dmm
output_file	(Optional) A file containing the vector of probabilities (e.g,"C:/.../PROB.txt")
plot	FALSE (default); TRUE (display figure plots of probabilities of occurrence of the observed word of size m by position)

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. 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.

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

x	An object for which the log-likelihood of the DMM can be computed.
sequences	A vector of character or list of vectors representing the sequences for which the
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores. log-likelihood of the model must be computed.

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. 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.

---

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

x	An object of class dmm
sequences	A character vector or a list of character vectors representing the sequence
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object of class dmm
k1	Start position (default value=0) : a positive integer giving the start position along the sequence from which the maintainabilities of the DMM should be computed, such that k1<k2
k2	End position : a positive integer giving the end position along the sequence until which the maintainabilities of the DMM should be computed, such that k2>k1
upstates	Character vector of the subspace working states among the state space vector such that upstates < s
output_file	(Optional) A file containing matrix of maintainability probabilities (e.g, "C:/.../MAIN.txt")
plot	FALSE (default); TRUE (display a figure plot of maintainability probabilities by position)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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.

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

x	An object of class dmm
k1	Start position (default value=0) : a positive integer giving the start position along the sequence from which the reliabilities of the DMM should be computed, such that k1<k2
k2	End position : a positive integer giving the end position along the sequence until which the reliabilities of the DMM should be computed, such that k2>k1
upstates	Character vector of the subspace working states among the state space vector such that upstates < s
output_file	(Optional) A file containing matrix of reliability probabilities (e.g, "C:/.../REL.txt")
plot	FALSE (default); TRUE (display a figure plot of reliability probabilities by position)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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.

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

x	An object for which simulated sequences of the DMM can be computed.
output_file	(Optional) File containing the simulated sequence (e.g, "C:/.../SIM.txt")
model_size	Size of the model
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

---

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

x	An object of class dmm
output_file	(Optional) File containing the simulated sequence (e.g, "C:/.../SIM.txt")
model_size	Size of the model
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

x	An object of class dmm
start	Start position : a positive integer giving the start position along the sequence from which the stationary laws of the DMM should be computed
end	End position : a positive integer giving the end position along the sequence until which the stationary laws of the DMM should be computed
step	A step (integer)
output_file	(Optional) A file containing matrix of stationary laws (e.g, "C:/.../SL.txt")
plot	FALSE (default); TRUE (display a figure plot of stationary distribution probabilities by position)

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. 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.

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

word	A subsequence (string of characters)
pos	A vector of integer positions
x	An object of class dmm
output_file	(Optional) A file containing the vector of probabilities (e.g,"C:/.../PROB.txt")
plot	FALSE (default); TRUE (display figure plot of word's probabilities by position)

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. 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.

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

word	A subsequence (string of characters)
pos	A position (numeric)
x	An object of class dmm
output_file	(Optional) A file containing the probability (e.g,"C:/.../PROB.txt")
internal	FALSE (default) ; TRUE (for internal use of word applications)
ncpu	Default=2. Represents the number of cores used to parallelized computation. If ncpu=-1, then it uses all available cores.

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. 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.

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

words	A vector of characters containing words
pos	A vector of integer positions
x	An object of class dmm
output_file	(Optional) A file containing the matrix of probabilities (e.g,"C:/.../PROB.txt")
plot	FALSE (default); TRUE (display figure plots of words' probabilities by position)

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. 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.

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)

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