smmnonparametric {smmR}

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

states

Vector of state space of length s.

init

Vector of initial distribution of length s.

ptrans

Matrix of transition probabilities of the embedded Markov chain J=(J_m)_{m} of dimension (s, s).

type.sojourn

Type of sojourn time (for more explanations, see Details).

distr

• Array of dimension (s, s, kmax) if type.sojourn = "fij";

• Matrix of dimension (s, kmax) if type.sojourn = "fi" or "fj";

• Vector of length kmax if the type.sojourn = "f".

kmax is the maximum length of the sojourn times.

cens.beg

Optional. A logical value indicating whether or not sequences are censored at the beginning.

cens.end

Optional. A logical value indicating whether or not sequences are censored at the end.

Details

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

The difference between the Markov model and the semi-Markov model concerns the modeling of the sojourn time. With a Markov chain, the sojourn time distribution 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 kernel q_{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 chain J = (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 argument distr 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 define kmax the maximum length of the sojourn times. If type.sojourn = "fij", distr is an array of dimension (s, s, kmax). If type.sojourn = "fi" or "fj", distr must be a matrix of dimension (s, kmax). If type.sojourn = "f", distr must be a vector of length kmax.

If the sequence is censored at the beginning and/or at the end, cens.beg must be equal to TRUE and/or cens.end must be equal to TRUE. All the sequences must be censored in the same way.

Value

Returns an object of class smm, smmnonparametric.

References

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

See Also

simulate, fitsmm, smmparametric

Examples

states <- c("a", "c", "g", "t")
s <- length(states)

# Creation of the initial distribution
vect.init <- c(1 / 4, 1 / 4, 1 / 4, 1 / 4)

# Creation of the transition matrix
pij <- 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 distributions
kmax <- 6
nparam.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