SMM-package: SMM : Semi-Markov and Markov Models

Description Details Author(s) See Also Examples

Description

This package 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. Estimation and simulation of discrete-time k-th order Markov chains are also considered.

Details

This R package provides the following functions: estimSM, simulSM, LoglikelihoodSM, AIC_SM, BIC_SM, InitialLawSM, estimMk, simulMk, LoglikelihoodMK, AIC_Mk, BIC_Mk and InitialLawMk.

Author(s)

Vlad Stefan Barbu, barbu@univ-rouen.fr
Caroline Berard, caroline.berard@univ-rouen.fr
Dominique Cellier, dominique.cellier@laposte.net
Mathilde Sautreuil, mathilde.sautreuil@etu.univ-rouen.fr
Nicolas Vergne, nicolas.vergne@univ-rouen.fr

See Also

simulMk estimMk simulSM estimSM

Examples

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##--------------------------------------##
#  An example for the simulMk function   #
#  see more examples >help(simulMk)      #
##--------------------------------------##
### Example 1 ###
# Second order model with the finite state space {a,c,g,t}
E <- c("a","c","g","t")
S = length(E)
init.distribution <- c(1/6,1/6,5/12,3/12)
k<-2
p <- matrix(0.25, nrow = S^k, ncol = S)

# We simulate 3 sequences of size 1000, 10000 and 2000 respectively
simulMk(E = E, nbSeq = 3, lengthSeq = c(1000, 10000, 2000), Ptrans = p, 
init = init.distribution, k = k)

##--------------------------------------##
#  An example for the estimMk function   #
#  see more examples >help(estimMk)      #
##--------------------------------------##
### Example 1 ###
# Second order model with the finite state space {a,c,g,t}
E <- c("a","c","g","t")
S = length(E)
init.distribution <- c(1/6,1/6,5/12,3/12)
k<-2
p <- matrix(0.25, nrow = S^k, ncol = S)

## simulation of 3 sequences with the simulMk function
seq3 = simulMk(E = E, nbSeq = 3, lengthSeq = c(1000, 10000, 2000), Ptrans = p, 
init = init.distribution, k = 2)

## estimation of simulated sequences 
res3 = estimMk(seq = seq3, E = E, k = 2)

##--------------------------------------##
#  An example for the simulSM function   #
#  see more examples >help(simulSM)      #
##--------------------------------------##
alphabet = c("a","c","g","t")
S = length(alphabet)
## creation of the initial distribution
vect.init = c(1/4,1/4,1/4,1/4)
# creation of the transition matrix
Pij = matrix(c(0,0.2,0.3,0.5,0.4,0,0.2,0.4,0.1,0.2,0,0.7,0.8,0.1,0.1,0), 
nrow = S, ncol = S, byrow = TRUE)

################################
## Parametric simulation of a trajectory (of length equal to 50) 
## where the sojourn times do not depend on the present state and on the next state
################################
## Simulation of a sequence of length 50
seq50 = simulSM(E = alphabet, NbSeq = 1, lengthSeq = 50, TypeSojournTime = "f", 
                init = vect.init, Ptrans = Pij, distr = "pois", param = 2)
                
##--------------------------------------##
#  An example for the simulSM function   #
#  see more examples >help(simulSM)      #
##--------------------------------------##
alphabet = c("a","c","g","t")
S = length(alphabet)
# creation of the transition matrix
Pij = matrix(c(0,0.2,0.3,0.5,0.4,0,0.2,0.4,0.1,0.2,0,0.7,0.8,0.1,0.1,0), 
nrow = S, ncol = S, byrow = TRUE)

Pij
#     [,1] [,2] [,3] [,4]
#[1,]  0.0  0.2  0.3  0.5
#[2,]  0.4  0.0  0.2  0.4
#[3,]  0.1  0.2  0.0  0.7
#[4,]  0.8  0.1  0.1  0.0

################################
## Parametric estimation of a trajectory (of length equal to 5000) 
## where the sojourn times do not depend on the present state and on the next state
################################
## Simulation of a sequence of length 5000
seq5000 = simulSM(E = alphabet, NbSeq = 1, lengthSeq = 5000, TypeSojournTime = "f", 
                init = c(1/4,1/4,1/4,1/4), Ptrans = Pij, distr = "pois", param = 2)
                
                
## Estimation of the simulated sequence                
estSeq5000 = estimSM(seq = seq5000, E = alphabet, TypeSojournTime = "f", 
                   distr = "pois", cens.end = 0, cens.beg = 0)
                   

Example output

Loading required package: seqinr
Loading required package: DiscreteWeibull
Loading required package: Rsolnp
[[1]]
   [1] "c" "t" "t" "g" "g" "c" "g" "c" "t" "g" "a" "g" "t" "c" "a" "c" "t" "c"
  [19] "a" "t" "t" "a" "g" "g" "a" "t" "t" "c" "a" "a" "g" "c" "c" "a" "a" "g"
  [37] "a" "a" "g" "c" "a" "c" "g" "t" "a" "t" "c" "t" "c" "a" "t" "c" "g" "g"
  [55] "g" "a" "c" "a" "c" "c" "t" "c" "t" "a" "t" "g" "a" "a" "g" "t" "a" "a"
  [73] "g" "a" "a" "c" "t" "c" "t" "c" "c" "c" "c" "t" "g" "c" "t" "c" "c" "a"
  [91] "t" "g" "a" "t" "a" "a" "t" "a" "a" "t" "t" "c" "a" "t" "t" "c" "c" "a"
 [109] "c" "a" "c" "g" "g" "a" "g" "c" "c" "c" "a" "c" "t" "c" "g" "t" "a" "c"
 [127] "t" "t" "a" "t" "t" "g" "t" "g" "c" "t" "c" "a" "c" "g" "g" "g" "t" "t"
 [145] "a" "a" "c" "a" "a" "g" "t" "a" "g" "g" "g" "t" "t" "a" "g" "a" "g" "a"
 [163] "g" "c" "a" "g" "a" "g" "t" "a" "a" "c" "a" "g" "t" "a" "a" "c" "t" "t"
 [181] "g" "c" "a" "c" "c" "c" "t" "t" "g" "a" "a" "g" "g" "t" "t" "g" "g" "c"
 [199] "a" "t" "g" "g" "a" "g" "c" "c" "a" "t" "a" "t" "g" "a" "t" "a" "t" "a"
 [217] "c" "c" "c" "t" "g" "g" "c" "a" "a" "t" "t" "c" "c" "a" "c" "a" "c" "t"
 [235] "a" "g" "t" "a" "g" "c" "t" "c" "a" "a" "g" "g" "a" "a" "c" "g" "t" "c"
 [253] "a" "g" "g" "g" "t" "g" "t" "g" "a" "g" "c" "c" "t" "g" "a" "a" "a" "c"
 [271] "t" "t" "a" "g" "c" "t" "g" "a" "g" "c" "g" "a" "g" "a" "g" "a" "c" "g"
 [289] "a" "t" "g" "t" "a" "c" "a" "a" "g" "t" "g" "g" "g" "a" "a" "t" "c" "g"
 [307] "a" "a" "t" "g" "g" "c" "a" "c" "t" "c" "g" "a" "c" "t" "g" "c" "a" "a"
 [325] "c" "a" "a" "a" "c" "a" "c" "a" "c" "g" "a" "g" "t" "c" "c" "g" "g" "c"
 [343] "c" "a" "g" "c" "g" "a" "c" "g" "g" "g" "a" "g" "c" "g" "t" "t" "t" "t"
 [361] "t" "g" "c" "c" "g" "t" "t" "a" "g" "a" "g" "c" "c" "a" "c" "g" "t" "a"
 [379] "c" "t" "c" "g" "a" "t" "c" "t" "c" "c" "g" "c" "a" "t" "a" "g" "t" "g"
 [397] "a" "c" "g" "g" "g" "c" "a" "a" "t" "t" "a" "g" "c" "g" "t" "a" "t" "a"
 [415] "a" "t" "c" "c" "c" "t" "a" "g" "a" "g" "t" "g" "a" "c" "a" "c" "t" "a"
 [433] "g" "t" "a" "t" "c" "g" "t" "c" "t" "c" "c" "c" "c" "a" "g" "c" "c" "t"
 [451] "g" "a" "c" "g" "a" "a" "a" "c" "c" "c" "c" "t" "g" "t" "c" "g" "a" "a"
 [469] "c" "t" "c" "c" "t" "c" "t" "g" "g" "c" "a" "c" "a" "t" "c" "c" "c" "t"
 [487] "g" "a" "c" "c" "g" "a" "a" "g" "c" "a" "t" "a" "g" "c" "c" "a" "c" "g"
 [505] "t" "c" "c" "c" "g" "a" "a" "t" "a" "t" "c" "g" "c" "t" "c" "t" "c" "g"
 [523] "a" "t" "a" "t" "c" "t" "t" "t" "t" "g" "a" "t" "g" "c" "c" "c" "t" "a"
 [541] "c" "t" "t" "c" "a" "t" "a" "t" "c" "a" "a" "c" "g" "t" "t" "g" "g" "c"
 [559] "t" "c" "a" "t" "g" "c" "c" "a" "a" "a" "c" "g" "c" "t" "t" "g" "g" "t"
 [577] "c" "t" "g" "g" "g" "c" "t" "t" "c" "a" "c" "t" "c" "g" "g" "c" "c" "c"
 [595] "c" "g" "c" "g" "c" "c" "t" "g" "a" "g" "g" "g" "g" "c" "g" "c" "t" "t"
 [613] "a" "c" "c" "g" "c" "c" "c" "c" "t" "g" "t" "a" "t" "c" "c" "g" "a" "c"
 [631] "g" "g" "c" "t" "g" "t" "t" "a" "t" "t" "g" "a" "t" "c" "g" "g" "g" "c"
 [649] "t" "t" "g" "a" "c" "c" "c" "t" "a" "g" "c" "c" "a" "c" "g" "a" "a" "a"
 [667] "g" "a" "t" "g" "a" "a" "t" "g" "c" "t" "g" "g" "t" "t" "a" "t" "a" "c"
 [685] "c" "a" "t" "a" "g" "c" "t" "t" "t" "a" "c" "t" "g" "g" "c" "g" "c" "t"
 [703] "a" "c" "a" "g" "g" "a" "c" "a" "c" "t" "g" "t" "t" "c" "t" "t" "t" "a"
 [721] "a" "g" "g" "c" "t" "g" "t" "a" "a" "c" "t" "a" "a" "t" "t" "g" "a" "t"
 [739] "t" "c" "a" "t" "a" "a" "a" "t" "a" "c" "a" "g" "a" "a" "a" "g" "t" "c"
 [757] "c" "a" "a" "g" "t" "g" "g" "a" "g" "a" "c" "t" "a" "c" "g" "g" "a" "g"
 [775] "g" "a" "g" "c" "g" "t" "g" "g" "g" "a" "a" "t" "c" "a" "g" "a" "a" "a"
 [793] "g" "t" "t" "g" "a" "a" "c" "a" "t" "g" "c" "a" "g" "t" "a" "t" "g" "g"
 [811] "a" "g" "a" "g" "c" "a" "a" "t" "a" "a" "t" "g" "c" "g" "a" "a" "c" "c"
 [829] "g" "g" "c" "c" "a" "c" "t" "c" "a" "a" "g" "c" "a" "t" "c" "t" "a" "g"
 [847] "a" "a" "t" "a" "t" "t" "a" "c" "g" "g" "a" "g" "g" "a" "g" "g" "a" "t"
 [865] "a" "a" "g" "c" "a" "a" "g" "a" "t" "t" "c" "g" "g" "c" "c" "a" "a" "t"
 [883] "t" "c" "c" "a" "t" "t" "a" "g" "g" "a" "t" "g" "g" "t" "c" "a" "c" "a"
 [901] "t" "a" "c" "t" "c" "g" "t" "t" "g" "t" "t" "c" "c" "t" "g" "t" "c" "a"
 [919] "t" "a" "c" "a" "g" "t" "g" "t" "a" "a" "c" "a" "t" "c" "a" "c" "a" "t"
 [937] "c" "t" "t" "t" "a" "g" "c" "a" "g" "a" "g" "t" "g" "g" "a" "t" "a" "a"
 [955] "c" "c" "a" "a" "t" "t" "g" "c" "g" "t" "t" "g" "g" "c" "a" "a" "c" "a"
 [973] "g" "t" "a" "g" "c" "a" "c" "t" "a" "g" "t" "a" "g" "c" "t" "g" "t" "g"
 [991] "t" "a" "c" "a" "c" "g" "g" "g" "t" "c"

[[2]]
    [1] "g" "t" "c" "t" "c" "g" "c" "g" "c" "g" "a" "g" "g" "g" "t" "a" "c" "a"
   [19] "c" "a" "a" "t" "g" "a" "t" "a" "g" "g" "c" "c" "t" "g" "t" "g" "a" "a"
   [37] "t" "a" "g" "c" "t" "g" "t" "g" "g" "g" "a" "t" "c" "c" "t" "a" "t" "c"
   [55] "g" "c" "g" "g" "g" "g" "g" "g" "c" "c" "c" "c" "a" "g" "t" "t" "g" "a"
   [73] "c" "t" "t" "a" "a" "g" "c" "c" "c" "g" "t" "c" "a" "a" "c" "a" "t" "c"
   [91] "t" "g" "t" "c" "a" "a" "c" "c" "t" "c" "g" "t" "t" "a" "a" "a" "a" "a"
  [109] "c" "c" "t" "t" "g" "g" "c" "c" "a" "a" "a" "a" "a" "c" "g" "t" "t" "g"
  [127] "t" "a" "t" "a" "g" "a" "t" "g" "a" "a" "g" "a" "c" "t" "t" "t" "a" "c"
  [145] "c" "t" "c" "g" "a" "t" "c" "g" "g" "t" "t" "t" "a" "c" "a" "a" "a" "c"
  [163] "t" "c" "a" "c" "a" "a" "g" "t" "a" "c" "c" "g" "a" "a" "a" "c" "g" "g"
  [181] "c" "t" "g" "g" "g" "c" "c" "a" "c" "a" "c" "c" "g" "a" "a" "a" "a" "g"
  [199] "a" "g" "c" "g" "g" "c" "g" "t" "g" "t" "a" "g" "t" "t" "a" "a" "c" "t"
  [217] "c" "t" "t" "a" "a" "g" "a" "t" "c" "g" "t" "c" "a" "g" "g" "c" "g" "a"
  [235] "c" "c" "g" "g" "c" "t" "g" "g" "g" "c" "g" "a" "a" "c" "a" "a" "g" "a"
  [253] "g" "a" "g" "g" "a" "t" "t" "t" "a" "a" "c" "g" "a" "c" "c" "c" "g" "t"
  [271] "g" "g" "g" "c" "t" "a" "t" "t" "g" "c" "c" "a" "g" "g" "a" "t" "a" "c"
  [289] "t" "a" "c" "c" "a" "t" "a" "c" "t" "a" "a" "c" "c" "a" "a" "c" "t" "t"
  [307] "c" "c" "c" "g" "g" "c" "g" "g" "g" "a" "c" "g" "t" "a" "t" "g" "a" "g"
  [325] "a" "a" "a" "g" "t" "g" "t" "a" "g" "t" "t" "a" "a" "t" "a" "t" "t" "a"
  [343] "c" "g" "a" "c" "g" "c" "g" "g" "a" "g" "g" "c" "c" "c" "c" "g" "t" "g"
  [361] "t" "t" "c" "a" "t" "t" "c" "a" "t" "t" "a" "c" "a" "t" "c" "g" "c" "t"
  [379] "c" "g" "g" "g" "c" "g" "g" "a" "a" "t" "t" "g" "a" "g" "g" "a" "g" "a"
  [397] "c" "g" "c" "a" "t" "c" "g" "t" "t" "g" "c" "a" "t" "t" "c" "g" "c" "a"
  [415] "g" "a" "c" "g" "g" "g" "a" "a" "a" "a" "g" "a" "t" "a" "c" "g" "g" "c"
  [433] "g" "c" "t" "g" "t" "c" "a" "g" "g" "t" "g" "g" "a" "a" "c" "g" "a" "t"
  [451] "t" "g" "a" "t" "a" "c" "g" "c" "c" "g" "g" "a" "c" "t" "a" "a" "g" "c"
  [469] "t" "a" "a" "g" "g" "t" "a" "a" "c" "a" "g" "t" "g" "c" "c" "t" "a" "t"
  [487] "c" "c" "a" "g" "c" "g" "c" "g" "c" "c" "t" "a" "c" "c" "g" "g" "a" "t"
  [505] "g" "a" "c" "g" "a" "a" "g" "c" "a" "a" "g" "t" "c" "t" "c" "a" "c" "t"
  [523] "c" "t" "c" "c" "a" "c" "g" "t" "t" "a" "c" "g" "g" "t" "t" "g" "t" "g"
  [541] "g" "c" "c" "t" "c" "g" "c" "g" "a" "a" "t" "c" "g" "a" "c" "a" "g" "t"
  [559] "g" "t" "t" "t" "a" "c" "c" "a" "g" "g" "c" "a" "g" "g" "c" "g" "c" "t"
  [577] "a" "a" "c" "g" "c" "g" "c" "t" "c" "t" "c" "c" "t" "g" "t" "a" "c" "g"
  [595] "c" "a" "g" "t" "c" "a" "c" "g" "c" "g" "t" "c" "a" "c" "c" "a" "c" "t"
  [613] "c" "c" "a" "a" "c" "t" "c" "c" "g" "c" "a" "a" "c" "g" "c" "a" "t" "g"
  [631] "a" "t" "c" "c" "g" "a" "c" "g" "c" "g" "t" "a" "g" "g" "g" "t" "t" "c"
  [649] "c" "a" "t" "c" "g" "t" "a" "a" "t" "a" "t" "a" "c" "g" "a" "c" "a" "c"
  [667] "t" "a" "c" "t" "c" "a" "a" "g" "a" "g" "a" "g" "c" "t" "g" "c" "t" "t"
  [685] "g" "t" "g" "g" "a" "c" "c" "c" "t" "g" "t" "t" "a" "t" "c" "a" "t" "c"
  [703] "t" "g" "t" "t" "c" "t" "g" "g" "g" "t" "t" "t" "a" "c" "g" "t" "t" "t"
  [721] "g" "a" "t" "t" "g" "a" "a" "g" "c" "g" "c" "c" "a" "t" "a" "g" "a" "c"
  [739] "t" "g" "c" "a" "t" "c" "c" "c" "g" "g" "g" "a" "c" "c" "a" "g" "g" "a"
  [757] "c" "c" "a" "a" "c" "c" "t" "c" "c" "a" "g" "c" "t" "c" "g" "t" "c" "c"
  [775] "c" "c" "t" "c" "t" "c" "a" "a" "g" "c" "g" "g" "t" "a" "t" "g" "a" "c"
  [793] "g" "t" "g" "a" "a" "t" "g" "c" "t" "a" "g" "g" "c" "t" "t" "t" "t" "t"
  [811] "g" "t" "a" "a" "g" "c" "g" "g" "g" "t" "g" "t" "g" "g" "g" "a" "a" "c"
  [829] "t" "c" "a" "c" "c" "a" "c" "a" "a" "t" "a" "g" "a" "t" "g" "c" "t" "t"
  [847] "g" "g" "t" "t" "t" "g" "g" "c" "g" "t" "g" "g" "g" "c" "t" "t" "a" "c"
  [865] "t" "c" "c" "a" "a" "a" "c" "g" "c" "c" "a" "g" "c" "a" "c" "a" "t" "t"
  [883] "a" "g" "t" "t" "a" "c" "g" "c" "a" "a" "c" "g" "a" "c" "c" "a" "t" "c"
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 [1891] "t" "t" "c" "a" "g" "g" "g" "c" "a" "g" "g" "c" "a" "a" "t" "c" "t" "t"
 [1909] "g" "c" "g" "g" "c" "t" "g" "g" "c" "c" "t" "g" "c" "c" "t" "t" "t" "c"
 [1927] "g" "c" "g" "g" "t" "t" "g" "c" "g" "g" "g" "t" "t" "t" "c" "g" "g" "c"
 [1945] "t" "a" "t" "g" "g" "t" "c" "c" "g" "t" "g" "g" "c" "g" "g" "a" "c" "c"
 [1963] "c" "t" "t" "a" "t" "g" "g" "t" "g" "g" "t" "a" "a" "g" "t" "t" "a" "a"
 [1981] "c" "t" "c" "c" "c" "t" "c" "c" "a" "t" "g" "g" "g" "c" "g" "a" "a" "t"
 [1999] "g" "a" "t" "t" "a" "t" "g" "t" "g" "g" "t" "g" "g" "t" "t" "c" "t" "t"
 [2017] "c" "c" "a" "c" "t" "t" "t" "t" "c" "t" "c" "a" "c" "a" "a" "g" "g" "c"
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 [1045] "t" "c" "t" "c" "g" "a" "c" "t" "t" "c" "g" "t" "t" "g" "g" "t" "a" "t"
 [1063] "t" "a" "c" "g" "a" "a" "t" "g" "a" "g" "a" "t" "t" "g" "c" "c" "t" "c"
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 [1873] "a" "a" "c" "a" "c" "c" "a" "a" "g" "g" "t" "t" "g" "c" "c" "t" "c" "g"
 [1891] "a" "c" "c" "g" "t" "c" "g" "c" "g" "c" "a" "c" "t" "t" "t" "c" "a" "g"
 [1909] "t" "a" "t" "g" "c" "g" "a" "a" "a" "g" "c" "t" "t" "c" "a" "t" "g" "t"
 [1927] "c" "a" "a" "a" "c" "c" "t" "a" "a" "c" "c" "t" "t" "t" "c" "a" "a" "c"
 [1945] "a" "a" "c" "c" "g" "g" "t" "t" "c" "a" "t" "a" "g" "g" "a" "g" "c" "a"
 [1963] "c" "c" "t" "c" "t" "t" "g" "c" "g" "c" "a" "t" "c" "a" "t" "c" "a" "a"
 [1981] "c" "c" "t" "g" "c" "a" "c" "g" "a" "t" "t" "t" "a" "a" "g" "t" "c" "t"
 [1999] "g" "t" "t" "t" "a" "t" "g" "t" "g" "g" "t" "g" "g" "t" "t" "c" "t" "t"
 [2017] "c" "c" "a" "c" "t" "t" "t" "t" "c" "t" "c" "a" "c" "a" "a" "g" "g" "c"
 [2035] "c" "t" "a" "t" "t" "g" "t" "c" "c" "t" "a" "t" "g" "g" "c" "t" "t" "g"
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 [2071] "a" "t" "g" "a" "t" "a" "c" "g" "a" "a" "g" "t" "a" "a" "t" "g" "g" "t"
 [2089] "t" "t" "c" "t" "c" "a" "a" "g" "a" "c" "t" "c" "a" "c" "a" "a" "a" "t"
 [2107] "g" "c" "g" "a" "t" "c" "t" "a" "c" "t" "c" "t" "t" "a" "a" "t" "a" "a"
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 [2809] "t" "t" "a" "g" "a" "a" "c" "a" "g" "g" "a" "g" "a" "a" "c" "a" "c" "a"
 [2827] "g" "g" "g" "a" "t" "c" "a" "t" "c" "g" "a" "a" "c" "c" "c" "g" "c" "t"
 [2845] "c" "a" "g" "a" "c" "g" "g" "g" "t" "g" "g" "a" "c" "t" "a" "t" "c" "t"
 [2863] "t" "c" "a" "t" "g" "t" "a" "a" "a" "g" "g" "t" "g" "g" "g" "g" "g" "g"
 [2881] "t" "c" "a" "g" "a" "t" "t" "a" "g" "c" "c" "c" "t" "c" "t" "t" "a" "t"
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 [7795] "t" "c" "g" "a" "a" "t" "a" "g" "a" "g" "g" "c" "t" "a" "a" "g" "c" "a"
 [7813] "t" "t" "c" "a" "t" "g" "c" "g" "c" "g" "t" "t" "t" "g" "a" "a" "a" "t"
 [7831] "t" "t" "a" "g" "g" "t" "t" "c" "g" "a" "t" "a" "a" "c" "t" "a" "a" "t"
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 [7939] "t" "c" "g" "c" "c" "a" "c" "t" "g" "a" "c" "t" "t" "c" "a" "c" "a" "a"
 [7957] "g" "c" "a" "t" "c" "c" "t" "c" "c" "t" "t" "c" "t" "a" "a" "c" "t" "t"
 [7975] "g" "g" "t" "c" "t" "c" "c" "t" "t" "c" "t" "c" "c" "a" "g" "g" "t" "c"
 [7993] "t" "c" "t" "a" "g" "c" "g" "g" "a" "c" "g" "t" "t" "g" "t" "t" "g" "t"
 [8011] "a" "a" "g" "c" "g" "c" "g" "a" "t" "a" "c" "a" "g" "g" "c" "c" "t" "t"
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 [8047] "t" "a" "t" "t" "t" "c" "a" "g" "c" "a" "g" "t" "c" "a" "a" "c" "a" "a"
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 [8119] "g" "t" "c" "c" "g" "t" "c" "g" "g" "t" "c" "t" "t" "c" "a" "a" "c" "c"
 [8137] "g" "t" "g" "a" "c" "c" "c" "a" "t" "a" "t" "c" "c" "g" "g" "c" "g" "t"
 [8155] "t" "a" "g" "c" "g" "c" "a" "c" "a" "a" "a" "g" "g" "a" "t" "c" "c" "c"
 [8173] "t" "g" "a" "g" "a" "c" "t" "c" "a" "t" "a" "g" "a" "a" "t" "t" "a" "g"
 [8191] "t" "g" "g" "a" "c" "c" "g" "c" "a" "t" "t" "a" "t" "a" "a" "c" "t" "t"
 [8209] "t" "t" "a" "t" "c" "g" "t" "g" "t" "t" "c" "c" "a" "a" "g" "c" "c" "t"
 [8227] "t" "a" "g" "a" "a" "c" "g" "t" "t" "t" "a" "a" "g" "c" "t" "c" "a" "a"
 [8245] "g" "t" "c" "a" "g" "c" "t" "t" "a" "t" "c" "c" "a" "a" "t" "c" "a" "a"
 [8263] "t" "t" "a" "a" "c" "g" "g" "t" "g" "t" "g" "c" "c" "c" "a" "g" "a" "t"
 [8281] "c" "t" "t" "a" "c" "g" "t" "a" "t" "a" "a" "c" "c" "c" "a" "g" "t" "t"
 [8299] "g" "a" "c" "c" "c" "g" "c" "c" "g" "g" "t" "c" "g" "c" "a" "g" "a" "a"
 [8317] "a" "g" "t" "a" "t" "a" "c" "c" "c" "g" "g" "t" "g" "a" "g" "g" "t" "g"
 [8335] "c" "g" "a" "a" "t" "c" "c" "t" "c" "c" "g" "a" "c" "c" "c" "g" "g" "t"
 [8353] "g" "g" "t" "g" "a" "c" "g" "a" "g" "t" "a" "a" "c" "a" "t" "g" "g" "c"
 [8371] "g" "a" "g" "t" "a" "g" "c" "t" "a" "a" "a" "c" "t" "a" "c" "g" "t" "c"
 [8389] "a" "g" "g" "g" "g" "c" "a" "g" "a" "t" "c" "t" "a" "c" "c" "c" "g" "a"
 [8407] "c" "c" "g" "a" "a" "g" "g" "g" "g" "a" "c" "g" "a" "t" "c" "g" "g" "g"
 [8425] "t" "c" "c" "a" "t" "a" "t" "c" "a" "t" "t" "a" "g" "c" "t" "c" "g" "c"
 [8443] "t" "t" "g" "g" "t" "c" "t" "a" "t" "g" "c" "g" "t" "g" "c" "t" "a" "g"
 [8461] "t" "t" "a" "t" "c" "c" "t" "c" "c" "a" "a" "c" "t" "t" "g" "g" "c" "c"
 [8479] "g" "g" "a" "c" "c" "t" "g" "t" "g" "t" "t" "a" "c" "g" "t" "t" "a" "t"
 [8497] "t" "g" "c" "g" "g" "a" "a" "a" "c" "a" "t" "a" "t" "t" "a" "g" "a" "g"
 [8515] "a" "g" "c" "t" "c" "g" "c" "t" "c" "t" "t" "a" "t" "t" "a" "t" "c" "c"
 [8533] "t" "g" "g" "g" "t" "c" "g" "g" "a" "a" "a" "t" "t" "a" "t" "t" "t" "c"
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 [8605] "g" "g" "c" "t" "t" "g" "c" "t" "c" "c" "g" "t" "a" "a" "t" "c" "c" "t"
 [8623] "c" "g" "c" "c" "c" "a" "g" "c" "c" "c" "t" "g" "t" "t" "t" "a" "c" "g"
 [8641] "g" "g" "g" "a" "c" "g" "a" "c" "c" "t" "g" "t" "g" "t" "g" "a" "c" "t"
 [8659] "t" "t" "g" "t" "a" "g" "t" "c" "c" "t" "c" "g" "g" "t" "t" "t" "c" "t"
 [8677] "a" "a" "t" "c" "c" "a" "g" "a" "g" "t" "t" "g" "t" "g" "a" "c" "g" "a"
 [8695] "c" "a" "a" "c" "a" "c" "t" "c" "t" "a" "g" "g" "c" "a" "c" "g" "t" "t"
 [8713] "c" "t" "g" "c" "c" "t" "a" "a" "a" "c" "t" "a" "t" "g" "t" "c" "t" "c"
 [8731] "c" "t" "g" "t" "g" "c" "a" "c" "c" "g" "t" "g" "a" "a" "a" "t" "a" "t"
 [8749] "t" "c" "g" "t" "t" "a" "a" "c" "a" "c" "t" "t" "c" "g" "g" "a" "g" "t"
 [8767] "a" "t" "c" "a" "a" "t" "a" "c" "g" "a" "c" "g" "c" "c" "t" "c" "t" "a"
 [8785] "a" "g" "t" "a" "g" "a" "c" "t" "t" "g" "g" "t" "a" "g" "a" "c" "t" "g"
 [8803] "g" "g" "a" "t" "g" "g" "g" "t" "t" "a" "a" "a" "a" "c" "a" "c" "t" "t"
 [8821] "g" "t" "c" "c" "a" "t" "t" "c" "c" "t" "c" "g" "g" "c" "t" "t" "a" "t"
 [8839] "t" "a" "c" "t" "t" "a" "g" "t" "c" "a" "a" "g" "t" "a" "c" "g" "c" "g"
 [8857] "t" "t" "t" "a" "a" "t" "g" "a" "g" "c" "a" "g" "a" "g" "a" "t" "a" "c"
 [8875] "c" "a" "c" "c" "g" "g" "t" "a" "a" "c" "c" "a" "g" "t" "g" "c" "a" "c"
 [8893] "a" "c" "a" "g" "c" "a" "c" "g" "a" "a" "t" "t" "g" "g" "g" "t" "g" "g"
 [8911] "a" "c" "g" "g" "g" "g" "t" "c" "t" "c" "t" "a" "g" "g" "g" "g" "g" "a"
 [8929] "a" "a" "a" "a" "c" "c" "a" "t" "t" "g" "g" "a" "a" "c" "a" "a" "t" "c"
 [8947] "g" "c" "g" "a" "a" "c" "g" "t" "g" "a" "a" "c" "g" "g" "c" "c" "t" "t"
 [8965] "g" "t" "t" "t" "a" "a" "c" "a" "a" "a" "c" "t" "a" "c" "t" "c" "a" "a"
 [8983] "g" "a" "t" "t" "t" "g" "a" "g" "a" "g" "t" "t" "a" "c" "g" "c" "c" "t"
 [9001] "g" "t" "g" "c" "c" "g" "g" "t" "t" "a" "g" "a" "t" "g" "t" "c" "g" "c"
 [9019] "c" "c" "a" "g" "t" "c" "a" "a" "g" "c" "g" "c" "c" "c" "t" "a" "a" "a"
 [9037] "c" "a" "a" "t" "a" "t" "t" "t" "t" "t" "c" "a" "t" "a" "a" "a" "c" "a"
 [9055] "t" "a" "g" "c" "t" "t" "c" "g" "g" "c" "c" "c" "t" "t" "g" "a" "t" "g"
 [9073] "t" "g" "c" "t" "g" "t" "t" "t" "c" "a" "g" "t" "a" "c" "a" "a" "t" "a"
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 [9109] "t" "c" "t" "g" "c" "a" "c" "t" "g" "c" "a" "g" "t" "a" "a" "c" "g" "g"
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 [9289] "a" "t" "a" "a" "a" "t" "g" "a" "a" "a" "g" "t" "t" "t" "c" "c" "t" "t"
 [9307] "t" "t" "t" "c" "g" "c" "t" "a" "c" "g" "g" "c" "t" "t" "c" "c" "a" "c"
 [9325] "c" "t" "g" "g" "a" "t" "a" "g" "c" "a" "g" "c" "g" "g" "g" "c" "t" "c"
 [9343] "a" "t" "g" "g" "a" "c" "c" "g" "g" "g" "g" "c" "t" "c" "c" "c" "g" "a"
 [9361] "c" "g" "t" "c" "c" "t" "t" "c" "g" "g" "c" "c" "g" "t" "t" "g" "t" "g"
 [9379] "t" "c" "c" "c" "c" "t" "a" "c" "c" "t" "a" "t" "c" "c" "c" "c" "a" "t"
 [9397] "a" "t" "a" "c" "g" "g" "g" "t" "a" "c" "t" "g" "g" "c" "c" "g" "g" "t"
 [9415] "t" "g" "t" "t" "c" "c" "g" "c" "t" "c" "a" "g" "c" "t" "t" "g" "t" "c"
 [9433] "a" "a" "g" "c" "t" "g" "g" "c" "g" "a" "c" "a" "g" "g" "a" "c" "c" "t"
 [9451] "g" "c" "g" "a" "t" "t" "g" "c" "t" "a" "a" "g" "t" "g" "a" "a" "a" "g"
 [9469] "t" "t" "t" "a" "a" "a" "g" "t" "t" "t" "g" "a" "t" "t" "t" "a" "c" "t"
 [9487] "g" "g" "a" "a" "g" "c" "g" "t" "t" "c" "c" "c" "t" "t" "g" "t" "c" "a"
 [9505] "a" "a" "g" "t" "a" "g" "c" "c" "g" "c" "c" "c" "t" "c" "t" "t" "a" "g"
 [9523] "c" "c" "g" "a" "a" "a" "c" "c" "c" "t" "t" "c" "a" "t" "t" "a" "g" "g"
 [9541] "c" "g" "c" "t" "c" "t" "g" "t" "a" "a" "a" "c" "a" "t" "a" "g" "a" "g"
 [9559] "g" "a" "t" "t" "t" "g" "g" "a" "c" "g" "t" "a" "g" "g" "a" "c" "c" "c"
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 [9595] "t" "g" "c" "g" "t" "a" "g" "t" "a" "c" "t" "t" "g" "a" "c" "a" "g" "g"
 [9613] "g" "g" "g" "t" "c" "t" "g" "t" "a" "a" "c" "a" "c" "g" "c" "t" "g" "a"
 [9631] "c" "g" "t" "t" "c" "c" "t" "a" "c" "c" "a" "t" "g" "c" "a" "c" "t" "g"
 [9649] "g" "t" "g" "g" "t" "g" "g" "g" "c" "g" "t" "a" "c" "a" "c" "g" "g" "c"
 [9667] "t" "c" "a" "g" "g" "t" "g" "g" "g" "a" "c" "g" "c" "g" "a" "c" "c" "a"
 [9685] "g" "c" "c" "g" "c" "t" "c" "g" "c" "g" "c" "g" "t" "g" "c" "g" "c" "g"
 [9703] "c" "t" "a" "c" "t" "g" "g" "a" "t" "t" "t" "c" "c" "a" "t" "a" "t" "c"
 [9721] "a" "c" "a" "c" "g" "a" "c" "c" "a" "a" "c" "c" "g" "c" "g" "g" "g" "t"
 [9739] "a" "g" "a" "c" "g" "g" "a" "t" "c" "a" "c" "a" "g" "g" "g" "a" "g" "t"
 [9757] "t" "c" "c" "c" "a" "c" "c" "c" "t" "a" "t" "c" "a" "c" "c" "t" "c" "g"
 [9775] "a" "c" "t" "g" "a" "g" "c" "t" "c" "a" "c" "c" "g" "t" "a" "c" "c" "t"
 [9793] "c" "a" "t" "t" "t" "t" "c" "g" "c" "c" "c" "g" "g" "t" "a" "g" "a" "c"
 [9811] "t" "g" "a" "t" "t" "c" "t" "t" "t" "g" "g" "a" "t" "a" "a" "a" "c" "t"
 [9829] "c" "a" "t" "a" "t" "g" "g" "g" "a" "g" "c" "t" "a" "a" "g" "g" "t" "a"
 [9847] "g" "g" "a" "t" "t" "g" "c" "a" "g" "c" "t" "g" "a" "t" "t" "c" "t" "g"
 [9865] "g" "g" "t" "c" "c" "g" "t" "c" "t" "g" "g" "t" "c" "c" "c" "a" "t" "t"
 [9883] "a" "c" "a" "a" "t" "g" "g" "c" "t" "c" "c" "a" "c" "a" "a" "g" "g" "c"
 [9901] "t" "c" "a" "g" "c" "c" "t" "c" "a" "c" "a" "a" "t" "t" "c" "c" "a" "c"
 [9919] "c" "c" "g" "t" "t" "t" "c" "c" "c" "c" "a" "a" "a" "g" "t" "t" "c" "c"
 [9937] "g" "c" "a" "t" "a" "t" "g" "g" "c" "g" "c" "g" "c" "t" "g" "c" "t" "c"
 [9955] "c" "c" "g" "c" "c" "t" "t" "c" "a" "g" "g" "g" "a" "g" "c" "t" "a" "c"
 [9973] "t" "c" "t" "c" "t" "g" "g" "c" "g" "a" "c" "a" "g" "g" "c" "c" "a" "c"
 [9991] "t" "g" "a" "c" "g" "c" "a" "a" "g" "a"

     [,1] [,2] [,3] [,4]
[1,]  0.0  0.2  0.3  0.5
[2,]  0.4  0.0  0.2  0.4
[3,]  0.1  0.2  0.0  0.7
[4,]  0.8  0.1  0.1  0.0

SMM documentation built on Jan. 31, 2020, 5:07 p.m.