sampler.mrf: MCMC samplers for Gibbs Random Fields
In GiRaF: Gibbs Random Fields Analysis
Description
Usage
Arguments
References
See Also
Examples
Description
sampler.mrf gives approximate sample from the likelihood of a general Potts model defined on a rectangular h x w lattice (h ≤ w) with either a first order or a second order dependency structure. Available options are the Gibbs sampler (Geman and Geman (1984)) and the Swendsen-Wang algorithm (Swendsen and Wang (1987)).
Usage
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Arguments
iter
Number of iterations of the algorithm.
sampler
The method to be used. The latter must be one of "Gibbs" or "SW" corresponding respectively to the Gibbs sampler and the Swendsen-Wang algorithm.
h
the number of rows of the rectangular lattice.
w
the number of columns of the rectangular lattice.
param
numeric entry setting the interaction parameter (edges parameter)
ncolors
the number of states for the discrete random variables. By default, ncolors = 2.
nei
the number of neighbors. The latter must be one of nei = 4 or nei = 8, which respectively correspond to a first order and a second order dependency structure. By default, nei = 4.
pot
numeric entry setting homogeneous potential on singletons (vertices parameter). By default, pot = NULL
top, left, bottom, right, corner
numeric entry setting constant borders for the lattice. By default, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL.
initialise
Logical value indicating whether initial guess should be randomly drawn.
random
Logical value indicating whether the sites should be updated sequentially or randomdly. Used only with the "Gibbs" option.
view
Logical value indicating whether the draw should be printed. Do not display the optional borders.
References
Geman, S. and Geman, D. (1984). Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Machine Intellignence, 6(6):721-741.
Swendsen, R. H. and Wang, J.-S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Pysical Review Letters, 58(2):86-88.
See Also
The “GiRaF-introduction” vignette
Examples
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# Algorithm settings
n <- 200
method <- "Gibbs"
# Dimension of the lattice
height <- width <- 100
# Interaction parameter
Beta <- 0.6 # Isotropic configuration
# Beta <- c(0.6, 0.6) # Anisotropic configuration when nei = 4
# Beta <- c(0.6, 0.6, 0.6, 0.6) # Anisotropic configuration when nei = 8
# Number of colors
K <- 2
# Number of neighbors
G <- 4
# Optional potential on sites
potential <- runif(K,-1,1)
# Optional borders.
Top <- Bottom <- sample(0:(K-1), width, replace = TRUE)
Left <- Right <- sample(0:(K-1), height, replace = TRUE)
Corner <- sample(0:(K-1), 4, replace = TRUE)
# Sampling method for the default setting
sampler.mrf(iter = n, sampler = method, h = height, w = width,
param = Beta, view = TRUE)
# Sampling using an existing configuration as starting point
sampler.mrf(iter = n, sampler = method, h = height, w = width,
ncolors = K, nei = G, param = Beta,
initialise = FALSE, view = TRUE)
# Specifying optional arguments. The users may omit to mention all
# the non-existing borders
sampler.mrf(iter = n, sampler = method, h = height, w = width,
ncolors = K, nei = G, param = Beta,
pot = potential, top = Top, left = Left, bottom = Bottom,
right = Right, corner = Corner, view = TRUE)
# Gibbs sampler with sequential updates of the sites.
sampler.mrf(iter = n, sampler = "Gibbs", h = height, w = width,
ncolors = K, nei = G, param = Beta,
random = FALSE, view = TRUE)
Example output
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
[1,] 1 1 1 1 0 0 0 0 0 0 1 1 1
[2,] 1 0 1 0 0 0 1 0 0 1 1 0 0
[3,] 0 0 0 0 0 0 0 0 0 0 1 0 0
[4,] 1 0 0 0 1 1 0 0 0 0 0 0 0
[5,] 0 0 0 0 0 0 0 0 0 0 1 1 0
[6,] 0 0 0 0 0 0 0 0 0 0 1 0 0
[7,] 0 1 0 1 0 1 0 0 0 0 1 0 0
[8,] 0 0 0 0 0 1 0 0 1 1 0 0 0
[9,] 0 0 0 0 1 0 0 0 1 1 1 0 0
[10,] 1 0 0 0 0 1 1 0 0 0 0 0 0
[11,] 0 0 1 0 0 0 1 0 0 0 0 0 0
[12,] 0 0 0 0 0 0 1 1 0 0 0 0 0
[13,] 1 1 0 0 0 0 0 0 0 0 0 0 0
[14,] 1 1 1 1 0 0 0 0 0 0 1 1 1
[15,] 1 1 1 1 0 0 1 1 0 0 0 1 1
[16,] 0 0 0 1 0 0 1 1 0 0 0 1 1
[17,] 0 0 0 0 1 0 1 1 0 0 0 0 0
[18,] 0 0 0 0 1 1 1 0 0 0 0 0 1
[19,] 1 1 0 0 0 0 1 0 1 1 0 0 1
[20,] 1 1 0 0 0 0 0 0 0 0 1 1 1
[21,] 1 1 0 0 0 0 0 0 0 0 0 1 1
[22,] 0 1 0 0 0 0 0 0 0 0 0 0 1
[23,] 1 0 1 0 1 1 0 0 1 0 0 0 1
[24,] 1 0 1 1 1 1 0 0 1 1 1 0 0
[25,] 1 1 1 1 1 0 0 1 1 1 1 0 0
[26,] 1 1 1 0 0 0 0 1 1 0 0 0 0
[27,] 1 0 1 1 1 1 0 1 1 1 0 0 0
[28,] 1 1 1 1 1 1 1 1 1 1 1 0 0
[29,] 1 0 0 1 0 1 1 0 0 0 0 0 0
[30,] 0 0 1 1 0 1 1 1 0 0 0 0 0
[31,] 0 0 0 1 1 1 1 0 0 0 0 0 0
[32,] 0 0 1 0 0 1 1 0 0 0 0 0 0
[33,] 1 1 1 0 0 0 0 1 0 0 0 1 1
[34,] 1 1 1 0 0 0 1 1 1 1 0 0 0
[35,] 0 1 0 1 1 1 0 0 0 0 0 1 1
[36,] 0 1 0 1 1 1 1 1 0 0 1 1 1
[37,] 1 1 1 0 1 1 1 1 1 1 1 0 0
[38,] 1 1 0 0 1 1 1 0 1 1 1 1 1
[39,] 1 1 0 1 1 1 1 1 1 1 1 1 1
[40,] 1 1 1 1 0 0 0 0 0 0 0 1 1
[41,] 1 1 1 1 1 0 1 1 0 1 0 0 1
[42,] 1 1 0 1 1 0 0 0 1 1 0 1 1
[43,] 0 0 0 1 1 1 0 1 1 1 0 0 1
[44,] 1 0 0 1 1 1 1 1 1 0 0 0 0
[45,] 1 0 0 0 0 0 0 1 1 1 1 1 1
[46,] 0 0 0 0 1 1 1 1 1 1 1 0 0
[47,] 0 0 0 0 0 1 0 1 1 1 1 1 1
[48,] 1 1 0 0 0 0 0 1 1 1 1 1 1
[49,] 1 1 0 1 1 1 1 1 1 0 1 1 0
[50,] 0 1 0 0 0 0 1 1 0 0 0 0 0
[51,] 0 0 0 0 1 1 1 0 0 0 0 0 0
[52,] 1 1 1 1 1 1 0 1 1 0 0 0 0
[53,] 0 0 1 1 0 0 0 0 0 0 0 0 0
[54,] 0 0 1 1 1 1 0 0 0 1 0 0 0
[55,] 1 0 1 1 0 1 0 1 0 0 0 0 1
[56,] 0 0 0 1 1 1 0 0 1 0 1 0 0
[57,] 0 0 0 0 1 1 1 0 1 1 1 0 1
[58,] 0 0 0 0 1 1 1 0 0 1 1 1 1
[59,] 0 0 0 0 1 1 1 0 0 1 1 0 0
[60,] 0 0 0 0 1 1 1 0 1 1 1 0 0
[61,] 0 0 0 0 0 0 0 0 1 1 0 1 1
[62,] 0 0 1 0 0 0 0 0 0 1 0 0 0
[63,] 0 1 1 0 1 0 0 0 0 1 1 0 0
[64,] 1 1 1 1 1 1 0 1 1 1 1 0 1
[65,] 1 1 1 1 1 1 1 1 1 1 1 1 1
[66,] 1 0 1 1 1 1 0 0 0 1 1 1 1
[67,] 0 1 1 1 1 1 0 0 0 1 1 1 1
[68,] 0 0 1 1 0 0 0 0 1 1 1 1 0
[69,] 1 0 1 1 0 0 0 0 0 1 0 0 0
[70,] 1 1 0 0 0 0 0 1 0 1 1 1 0
[71,] 1 1 0 0 0 1 0 0 0 0 1 1 1
[72,] 1 1 0 0 0 0 1 0 0 0 1 0 1
[73,] 0 0 0 0 0 0 1 1 0 1 0 0 0
[74,] 1 0 0 1 0 0 1 0 0 1 1 0 0
[75,] 1 0 0 1 0 1 1 0 0 1 1 1 1
[76,] 1 0 1 0 1 1 0 0 1 1 1 1 0
[77,] 1 1 1 1 1 1 1 1 1 1 1 1 1
[78,] 0 0 1 1 1 0 1 1 1 1 1 1 1
[79,] 1 1 1 1 1 1 1 0 1 1 1 1 1
[80,] 0 1 1 1 1 1 1 1 1 0 0 1 1
[81,] 1 0 1 1 1 0 0 0 1 0 0 1 1
[82,] 0 1 1 1 1 0 0 0 0 0 0 0 0
[83,] 0 1 1 1 0 0 0 0 0 0 0 0 0
[84,] 1 1 1 1 1 0 0 0 0 0 0 0 0
[85,] 1 1 0 1 1 0 0 1 1 0 0 1 0
[86,] 1 0 0 0 1 1 0 1 0 0 0 0 0
[87,] 0 0 0 0 0 0 0 0 0 0 0 0 0
[88,] 1 0 0 1 0 0 0 1 0 0 0 0 0
[89,] 0 0 0 1 0 1 1 1 0 0 0 0 0
[90,] 1 1 0 0 0 1 1 1 0 0 0 0 0
[91,] 1 0 0 0 0 1 1 1 0 1 0 1 0
[92,] 1 0 1 0 0 0 0 0 0 1 1 1 1
[93,] 1 1 1 1 1 1 1 0 1 1 0 1 1
[94,] 1 1 0 1 1 1 1 0 0 0 1 1 0
[95,] 1 1 1 1 1 1 0 0 0 1 1 1 1
[96,] 1 1 1 1 1 1 0 0 1 1 1 1 1
[97,] 0 1 1 1 0 0 0 0 0 1 0 0 0
[98,] 0 1 1 1 1 1 1 0 0 1 0 0 1
[99,] 1 1 1 1 0 0 1 0 0 1 0 0 0
[100,] 1 1 0 1 1 0 1 0 0 0 1 0 0
[,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
[1,] 0 0 0 0 0 0 0 1 0 0 0 1
[2,] 0 0 0 0 0 0 0 1 1 0 0 1
[3,] 0 0 1 0 0 0 0 0 1 1 0 0
[4,] 0 0 1 1 0 1 0 0 0 1 1 0
[5,] 0 1 1 1 0 0 0 1 1 1 1 1
[6,] 1 1 1 0 0 0 1 1 1 1 0 1
[7,] 0 1 1 1 0 0 0 0 0 0 1 0
[8,] 1 0 0 1 0 0 0 1 0 0 0 0
[9,] 0 1 0 0 0 0 0 0 0 0 0 0
[10,] 0 0 0 0 0 0 1 1 1 1 0 1
[11,] 0 0 0 0 1 0 1 1 1 1 0 1
[12,] 0 0 0 0 1 0 1 1 1 0 0 1
[13,] 0 0 0 1 1 0 0 1 0 0 0 0
[14,] 1 1 0 1 1 1 0 1 1 1 0 0
[15,] 1 1 1 1 1 1 0 1 1 1 0 1
[16,] 0 1 1 1 1 1 0 0 1 1 1 0
[17,] 0 1 0 1 1 1 0 0 0 1 1 1
[18,] 1 1 1 0 0 0 0 0 1 1 0 1
[19,] 0 1 1 1 1 0 0 0 1 1 0 1
[20,] 0 1 1 0 1 1 1 0 1 1 1 1
[21,] 1 0 0 1 1 1 1 0 1 1 1 0
[22,] 1 1 1 1 1 0 1 0 0 1 0 0
[23,] 1 0 1 1 0 0 0 1 1 1 1 0
[24,] 1 1 0 0 0 1 1 1 1 1 1 0
[25,] 1 0 1 0 0 1 1 1 1 1 1 0
[26,] 1 0 0 0 0 1 1 1 1 1 1 0
[27,] 0 0 0 1 1 1 1 1 0 0 1 1
[28,] 1 1 1 1 1 1 1 0 0 1 1 1
[29,] 0 0 0 0 1 0 1 1 1 1 1 1
[30,] 1 1 0 1 0 1 0 1 1 1 1 1
[31,] 0 0 0 0 1 1 1 1 1 0 0 0
[32,] 1 0 0 0 0 0 1 1 0 0 0 0
[33,] 0 0 0 0 1 1 1 0 0 0 0 0
[34,] 0 1 1 0 1 1 1 0 0 0 0 1
[35,] 1 1 0 1 1 0 0 0 0 1 0 0
[36,] 0 0 0 1 1 0 0 0 0 0 1 0
[37,] 0 0 0 1 1 1 1 1 0 0 1 1
[38,] 1 0 1 0 0 1 1 1 0 0 1 1
[39,] 0 0 0 0 0 1 1 1 1 1 1 1
[40,] 1 1 1 1 1 0 0 1 1 0 1 1
[41,] 1 1 0 1 0 1 0 0 1 1 1 0
[42,] 1 1 1 1 1 1 1 1 0 1 1 1
[43,] 1 1 1 1 1 1 0 0 0 0 1 0
[44,] 0 1 0 0 1 0 0 0 0 0 0 0
[45,] 1 1 0 0 0 0 1 0 0 0 0 0
[46,] 0 1 0 0 1 0 0 0 0 0 0 1
[47,] 1 1 0 0 0 1 1 1 1 1 1 1
[48,] 1 0 1 1 1 1 0 1 1 1 1 1
[49,] 1 1 1 1 1 0 0 1 0 0 1 1
[50,] 1 1 1 1 1 0 0 1 1 0 1 0
[51,] 0 0 0 1 1 1 1 1 0 0 1 1
[52,] 1 0 0 0 1 1 0 0 0 0 0 1
[53,] 0 0 0 0 0 1 1 0 0 0 0 1
[54,] 0 1 0 0 0 1 1 1 1 0 0 0
[55,] 0 0 0 0 0 1 1 1 1 1 0 0
[56,] 0 1 0 0 1 1 1 1 1 1 0 1
[57,] 1 0 0 0 0 0 1 1 0 0 0 0
[58,] 0 0 0 0 0 0 1 1 1 0 1 0
[59,] 0 0 0 0 0 0 0 0 1 0 1 1
[60,] 0 0 1 1 0 0 0 1 1 0 1 1
[61,] 1 1 0 1 0 0 1 1 1 0 0 1
[62,] 1 0 0 0 0 0 1 1 1 1 1 1
[63,] 1 1 0 1 1 0 0 1 1 1 1 1
[64,] 0 1 1 1 1 1 1 1 1 1 1 1
[65,] 0 1 1 1 1 1 1 1 1 1 1 1
[66,] 0 0 0 1 1 1 1 1 1 1 1 1
[67,] 0 0 0 0 0 0 0 0 1 0 1 1
[68,] 0 0 0 1 1 1 1 0 0 1 1 1
[69,] 0 0 0 1 1 1 1 0 0 0 0 1
[70,] 0 0 0 1 1 0 0 0 1 1 1 1
[71,] 0 1 1 0 0 0 0 0 1 0 0 0
[72,] 0 1 1 0 0 1 1 1 1 1 0 0
[73,] 0 0 0 0 0 1 1 1 1 1 0 1
[74,] 0 0 0 1 0 0 1 1 1 1 1 1
[75,] 0 0 0 0 0 1 1 1 1 1 1 1
[76,] 0 0 0 0 1 0 0 0 1 1 1 0
[77,] 0 0 0 1 1 1 1 0 1 1 1 0
[78,] 1 1 1 1 1 1 0 0 1 0 0 0
[79,] 1 1 0 1 0 0 0 0 1 0 0 0
[80,] 1 1 0 1 0 0 0 0 1 0 0 0
[81,] 1 1 1 1 1 1 0 0 0 0 0 1
[82,] 1 1 1 0 0 1 0 0 1 1 0 1
[83,] 1 1 1 1 0 1 1 0 0 1 0 0
[84,] 0 0 0 1 1 1 0 1 1 1 0 0
[85,] 0 0 0 0 1 1 1 1 0 1 1 1
[86,] 0 1 1 1 1 1 1 1 1 0 0 0
[87,] 0 0 1 1 0 0 0 1 1 1 1 1
[88,] 1 0 1 1 1 0 0 1 1 1 1 1
[89,] 1 1 1 0 0 0 0 1 1 1 0 0
[90,] 1 0 1 1 1 0 1 1 1 0 0 0
[91,] 1 0 1 1 1 0 1 1 1 0 0 1
[92,] 1 1 1 1 1 1 1 1 0 0 0 1
[93,] 1 1 1 1 1 1 1 0 0 0 1 1
[94,] 1 0 1 1 1 1 0 0 0 1 1 1
[95,] 0 1 1 1 0 0 0 1 1 1 0 1
[96,] 1 0 1 1 1 1 0 0 1 1 1 1
[97,] 1 1 1 0 0 0 0 0 0 1 1 1
[98,] 1 0 0 0 1 0 1 1 1 1 1 1
[99,] 1 0 0 0 0 0 0 1 1 1 1 1
[100,] 1 0 0 0 0 1 1 1 0 0 0 0
[,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
[1,] 0 0 0 0 0 0 0 1 1 1 0 0
[2,] 0 0 1 1 0 0 0 1 1 1 0 1
[3,] 0 1 1 1 0 1 0 0 1 1 1 1
[4,] 1 1 0 1 1 0 0 1 1 1 1 1
[5,] 1 1 0 0 0 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 0 1 1 1 1 1
[7,] 0 1 1 1 1 1 0 1 1 1 0 1
[8,] 1 0 1 1 1 0 0 0 1 0 0 1
[9,] 0 0 1 1 1 0 0 1 0 0 0 1
[10,] 0 0 0 0 1 0 0 0 0 1 1 0
[11,] 0 0 0 0 1 1 1 0 0 1 1 1
[12,] 0 0 0 1 1 1 0 0 0 1 1 0
[13,] 1 0 0 1 1 1 1 0 1 1 1 1
[14,] 1 0 0 0 0 1 1 1 0 1 1 1
[15,] 1 1 0 0 0 1 0 1 1 1 1 1
[16,] 0 1 0 0 0 1 1 1 1 0 1 1
[17,] 1 0 0 0 0 0 1 1 1 1 1 1
[18,] 1 1 1 0 0 0 1 0 1 1 1 1
[19,] 1 1 0 0 0 0 0 0 0 0 1 1
[20,] 1 1 1 1 0 1 0 1 0 0 0 0
[21,] 1 0 1 0 0 0 0 0 0 0 0 0
[22,] 0 0 1 0 0 0 0 0 0 1 1 1
[23,] 0 0 1 0 0 0 1 1 1 1 1 1
[24,] 0 0 1 0 0 0 1 1 1 1 1 1
[25,] 0 0 0 0 0 1 1 0 0 0 1 0
[26,] 0 1 0 0 0 0 0 0 0 0 1 0
[27,] 0 1 1 0 0 0 0 0 0 0 0 0
[28,] 1 0 0 0 0 0 0 0 0 0 0 0
[29,] 1 0 0 0 1 1 0 0 1 0 1 1
[30,] 1 0 0 0 1 1 1 1 1 1 1 1
[31,] 1 0 0 0 0 1 0 1 1 0 0 1
[32,] 0 1 0 1 0 1 0 0 0 0 0 0
[33,] 0 0 1 1 1 0 0 0 0 0 1 0
[34,] 1 1 1 1 1 0 0 0 0 1 0 1
[35,] 0 0 1 1 1 1 1 1 0 1 1 1
[36,] 0 0 1 0 1 1 1 1 1 1 0 0
[37,] 1 1 0 0 0 1 1 1 1 1 1 1
[38,] 1 1 0 0 0 1 1 1 1 1 1 1
[39,] 1 1 0 1 1 1 1 1 1 0 1 1
[40,] 0 0 1 1 1 0 0 0 1 0 1 1
[41,] 1 1 1 1 1 1 0 0 0 0 1 1
[42,] 1 1 1 0 1 1 0 0 0 0 0 1
[43,] 1 0 0 0 0 0 1 1 0 0 0 1
[44,] 0 0 0 0 0 1 1 0 0 1 0 1
[45,] 0 0 0 1 0 0 0 0 1 0 0 0
[46,] 0 1 1 1 0 0 1 1 1 0 0 0
[47,] 1 1 0 1 1 0 0 1 1 0 0 0
[48,] 0 1 1 1 0 0 0 0 1 1 0 0
[49,] 0 0 0 0 0 0 0 1 1 1 1 0
[50,] 1 0 0 0 0 0 0 0 0 0 1 0
[51,] 0 0 0 0 0 0 1 1 0 0 1 1
[52,] 1 0 0 0 0 0 1 1 0 0 0 0
[53,] 1 1 0 1 1 0 1 1 0 0 0 0
[54,] 1 1 1 1 0 1 1 1 1 0 0 0
[55,] 0 0 1 1 1 1 1 1 1 0 0 0
[56,] 1 0 1 1 1 1 1 1 0 1 1 0
[57,] 1 0 1 1 1 1 0 1 1 1 1 0
[58,] 0 0 1 0 0 0 1 1 1 1 0 1
[59,] 0 0 0 0 0 1 1 1 1 0 0 0
[60,] 0 1 0 1 0 1 0 0 0 0 1 0
[61,] 1 1 0 0 0 1 1 0 0 1 1 1
[62,] 1 1 0 1 0 0 0 0 0 1 1 1
[63,] 1 0 0 0 0 0 0 0 0 0 1 1
[64,] 0 0 1 1 1 1 1 1 0 0 1 1
[65,] 0 0 0 1 1 1 0 0 1 1 0 1
[66,] 1 1 1 1 1 1 0 0 1 1 0 0
[67,] 1 1 1 1 1 1 1 1 0 1 0 0
[68,] 1 1 1 1 1 0 0 0 0 0 0 0
[69,] 1 1 1 1 0 0 0 1 0 0 0 1
[70,] 1 0 0 0 0 1 0 0 0 0 1 1
[71,] 0 0 0 1 1 1 1 0 0 0 1 0
[72,] 0 1 1 1 1 1 1 0 0 0 1 1
[73,] 1 0 0 1 1 1 1 1 1 0 0 0
[74,] 1 1 0 1 1 1 1 1 0 0 0 0
[75,] 1 1 0 1 1 1 1 0 0 0 0 0
[76,] 1 1 1 1 1 1 1 1 1 1 0 1
[77,] 1 1 1 1 1 1 0 0 0 0 1 1
[78,] 1 1 0 0 0 1 1 1 0 0 1 1
[79,] 0 0 0 0 0 0 0 1 0 0 0 1
[80,] 0 0 1 0 0 0 1 1 0 0 0 0
[81,] 0 1 1 0 0 0 0 0 1 1 0 0
[82,] 0 0 0 1 0 0 1 1 1 1 0 0
[83,] 0 0 0 0 0 1 1 0 1 1 0 0
[84,] 1 0 0 1 1 1 1 0 0 1 1 1
[85,] 0 0 0 1 0 0 0 0 0 1 1 1
[86,] 0 0 0 0 0 0 0 0 0 1 1 1
[87,] 1 0 0 0 0 0 0 1 1 1 1 1
[88,] 0 0 0 0 0 0 1 1 1 0 0 0
[89,] 0 1 1 1 1 0 0 0 0 0 0 0
[90,] 1 1 0 1 1 1 0 0 0 0 0 0
[91,] 1 1 1 1 1 1 1 1 1 0 1 0
[92,] 1 1 1 0 1 1 0 0 0 0 1 1
[93,] 1 1 1 0 0 1 0 0 0 0 1 1
[94,] 1 1 1 1 0 1 0 1 0 1 1 1
[95,] 1 1 0 1 1 1 0 1 1 0 0 1
[96,] 1 0 0 1 1 1 1 1 1 1 0 1
[97,] 1 1 1 1 1 1 1 0 1 1 0 1
[98,] 1 1 1 1 1 1 1 0 1 1 1 1
[99,] 0 1 0 0 0 0 1 1 1 1 1 0
[100,] 1 1 0 0 0 0 0 1 1 0 0 0
[,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49]
[1,] 0 0 1 1 0 0 0 0 0 0 0 0
[2,] 1 0 0 0 0 0 0 0 0 1 0 0
[3,] 1 0 0 0 0 1 0 0 0 0 0 0
[4,] 1 0 0 0 0 0 0 0 0 1 1 0
[5,] 1 0 0 0 0 0 0 0 0 0 0 1
[6,] 0 1 1 0 0 0 0 0 0 0 1 1
[7,] 1 1 0 0 0 0 1 0 0 0 0 0
[8,] 1 0 0 0 0 0 1 1 0 0 0 0
[9,] 0 1 1 0 1 1 1 1 0 0 1 0
[10,] 0 1 0 1 1 1 0 0 0 0 1 1
[11,] 1 1 1 1 1 1 1 1 0 0 1 0
[12,] 1 0 0 0 0 1 1 1 0 0 0 0
[13,] 1 0 0 1 0 0 0 0 0 0 0 0
[14,] 1 0 1 1 0 0 0 0 0 0 1 0
[15,] 1 1 1 1 1 0 1 0 0 0 1 0
[16,] 1 1 0 1 1 0 0 1 1 0 1 0
[17,] 0 0 1 1 1 1 0 0 0 1 1 1
[18,] 0 0 1 1 0 1 0 0 1 0 1 1
[19,] 0 1 1 1 1 1 0 0 1 1 1 1
[20,] 1 1 1 1 0 1 1 0 1 0 0 1
[21,] 1 1 1 0 0 1 1 0 1 1 0 1
[22,] 1 1 1 1 0 0 0 0 1 1 1 1
[23,] 1 0 1 1 0 0 0 0 0 1 0 0
[24,] 1 1 1 1 0 0 0 0 0 1 1 1
[25,] 0 1 1 0 0 1 1 0 1 0 1 1
[26,] 0 0 0 0 0 0 0 0 0 0 1 1
[27,] 0 0 0 0 0 1 1 0 0 0 1 1
[28,] 0 0 0 0 1 1 1 0 1 1 1 1
[29,] 1 0 0 0 1 1 0 0 0 0 0 1
[30,] 0 0 0 1 1 0 0 0 0 1 1 1
[31,] 0 1 0 0 0 0 0 0 0 1 0 1
[32,] 0 0 1 1 1 1 0 0 0 1 1 0
[33,] 0 0 1 0 0 1 0 0 0 0 0 0
[34,] 1 1 1 0 0 0 0 0 0 0 0 0
[35,] 1 1 1 1 1 0 0 0 0 0 0 0
[36,] 0 1 1 1 1 0 0 0 0 0 0 0
[37,] 1 0 1 1 0 1 0 0 0 1 1 0
[38,] 1 1 1 0 0 1 1 0 0 1 1 0
[39,] 1 1 1 0 0 0 0 0 0 1 1 1
[40,] 1 1 1 0 0 0 0 0 0 0 0 0
[41,] 1 1 0 0 0 0 0 0 0 1 0 1
[42,] 1 1 0 0 0 0 0 0 1 1 1 0
[43,] 0 0 0 1 0 0 0 1 1 0 0 0
[44,] 0 0 0 0 0 0 0 0 0 0 1 0
[45,] 0 0 0 1 0 0 0 0 0 0 0 0
[46,] 0 0 1 1 0 0 0 0 0 0 0 0
[47,] 0 0 0 0 0 0 0 0 0 0 0 0
[48,] 1 0 0 0 1 0 0 0 1 0 0 0
[49,] 0 0 0 0 0 1 0 1 0 0 1 1
[50,] 0 1 0 0 1 1 1 0 1 0 1 1
[51,] 0 0 0 1 0 1 0 1 1 0 1 0
[52,] 0 0 0 0 1 1 1 1 0 0 1 1
[53,] 0 0 0 0 1 0 0 0 0 0 0 1
[54,] 0 0 0 0 1 0 0 0 1 1 0 0
[55,] 1 0 0 0 1 1 1 0 0 0 0 0
[56,] 0 0 0 0 1 1 0 0 0 0 0 0
[57,] 0 0 0 1 1 1 0 0 0 0 1 0
[58,] 0 0 0 0 1 1 0 1 0 0 1 1
[59,] 0 0 0 1 1 0 1 1 0 0 1 1
[60,] 0 0 0 0 0 1 1 1 1 0 1 1
[61,] 0 0 0 0 0 0 1 1 1 1 1 0
[62,] 0 0 0 0 0 0 1 1 1 1 1 1
[63,] 0 0 0 0 0 0 1 1 0 0 1 1
[64,] 0 0 0 1 1 1 1 1 0 0 1 1
[65,] 1 1 1 1 0 1 1 0 0 0 1 1
[66,] 0 0 1 1 0 0 0 0 0 0 1 1
[67,] 0 1 1 1 1 1 1 1 0 1 1 1
[68,] 0 1 1 1 0 0 1 1 0 0 0 1
[69,] 1 1 1 1 0 0 0 0 0 0 1 1
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[,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
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[66,] 1 1 1 1 1 1 1 1 1 1 1 1
[67,] 1 1 1 1 1 1 1 1 1 1 1 0
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[,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
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[24,] 1 1 1 1 1 1 1 1 0 1 1 1
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[60,] 1 0 1 1 0 0 1 1 0 1 1 1
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[63,] 0 1 0 1 1 1 1 1 0 0 0 0
[64,] 0 1 1 1 1 1 0 0 0 1 1 0
[65,] 1 1 1 0 0 0 0 0 0 1 0 1
[66,] 1 1 1 1 1 1 0 0 0 1 1 1
[67,] 1 1 1 1 1 1 1 0 0 0 1 0
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[69,] 1 1 1 1 1 1 1 1 0 1 1 1
[70,] 1 1 1 1 1 1 0 1 1 1 1 1
[71,] 1 1 1 1 1 1 1 1 1 1 1 1
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[100,] 1 0 0 0 1 1 0 1 1 0 1 1
[,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49]
[1,] 0 0 1 1 1 1 1 1 0 0 0 1
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[,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61]
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GiRaF documentation built on Oct. 23, 2020, 6:07 p.m.
R Package Documentation
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Description Usage Arguments References See Also Examples
Description
sampler.mrf gives approximate sample from the likelihood of a general Potts model defined on a rectangular h x w lattice (h ≤ w) with either a first order or a second order dependency structure. Available options are the Gibbs sampler (Geman and Geman (1984)) and the Swendsen-Wang algorithm (Swendsen and Wang (1987)).
Usage
1 2 3 4 |
Arguments
iter |
Number of iterations of the algorithm. |
sampler |
The method to be used. The latter must be one of "Gibbs" or "SW" corresponding respectively to the Gibbs sampler and the Swendsen-Wang algorithm. |
h |
the number of rows of the rectangular lattice. |
w |
the number of columns of the rectangular lattice. |
param |
numeric entry setting the interaction parameter (edges parameter) |
ncolors |
the number of states for the discrete random variables. By default, ncolors = 2. |
nei |
the number of neighbors. The latter must be one of nei = 4 or nei = 8, which respectively correspond to a first order and a second order dependency structure. By default, nei = 4. |
pot |
numeric entry setting homogeneous potential on singletons (vertices parameter). By default, pot = NULL |
top, left, bottom, right, corner |
numeric entry setting constant borders for the lattice. By default, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL. |
initialise |
Logical value indicating whether initial guess should be randomly drawn. |
random |
Logical value indicating whether the sites should be updated sequentially or randomdly. Used only with the "Gibbs" option. |
view |
Logical value indicating whether the draw should be printed. Do not display the optional borders. |
References
Geman, S. and Geman, D. (1984). Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Machine Intellignence, 6(6):721-741.
Swendsen, R. H. and Wang, J.-S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Pysical Review Letters, 58(2):86-88.
See Also
The “GiRaF-introduction” vignette
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | # Algorithm settings
n <- 200
method <- "Gibbs"
# Dimension of the lattice
height <- width <- 100
# Interaction parameter
Beta <- 0.6 # Isotropic configuration
# Beta <- c(0.6, 0.6) # Anisotropic configuration when nei = 4
# Beta <- c(0.6, 0.6, 0.6, 0.6) # Anisotropic configuration when nei = 8
# Number of colors
K <- 2
# Number of neighbors
G <- 4
# Optional potential on sites
potential <- runif(K,-1,1)
# Optional borders.
Top <- Bottom <- sample(0:(K-1), width, replace = TRUE)
Left <- Right <- sample(0:(K-1), height, replace = TRUE)
Corner <- sample(0:(K-1), 4, replace = TRUE)
# Sampling method for the default setting
sampler.mrf(iter = n, sampler = method, h = height, w = width,
param = Beta, view = TRUE)
# Sampling using an existing configuration as starting point
sampler.mrf(iter = n, sampler = method, h = height, w = width,
ncolors = K, nei = G, param = Beta,
initialise = FALSE, view = TRUE)
# Specifying optional arguments. The users may omit to mention all
# the non-existing borders
sampler.mrf(iter = n, sampler = method, h = height, w = width,
ncolors = K, nei = G, param = Beta,
pot = potential, top = Top, left = Left, bottom = Bottom,
right = Right, corner = Corner, view = TRUE)
# Gibbs sampler with sequential updates of the sites.
sampler.mrf(iter = n, sampler = "Gibbs", h = height, w = width,
ncolors = K, nei = G, param = Beta,
random = FALSE, view = TRUE)
|
Example output
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
[1,] 1 1 1 1 0 0 0 0 0 0 1 1 1
[2,] 1 0 1 0 0 0 1 0 0 1 1 0 0
[3,] 0 0 0 0 0 0 0 0 0 0 1 0 0
[4,] 1 0 0 0 1 1 0 0 0 0 0 0 0
[5,] 0 0 0 0 0 0 0 0 0 0 1 1 0
[6,] 0 0 0 0 0 0 0 0 0 0 1 0 0
[7,] 0 1 0 1 0 1 0 0 0 0 1 0 0
[8,] 0 0 0 0 0 1 0 0 1 1 0 0 0
[9,] 0 0 0 0 1 0 0 0 1 1 1 0 0
[10,] 1 0 0 0 0 1 1 0 0 0 0 0 0
[11,] 0 0 1 0 0 0 1 0 0 0 0 0 0
[12,] 0 0 0 0 0 0 1 1 0 0 0 0 0
[13,] 1 1 0 0 0 0 0 0 0 0 0 0 0
[14,] 1 1 1 1 0 0 0 0 0 0 1 1 1
[15,] 1 1 1 1 0 0 1 1 0 0 0 1 1
[16,] 0 0 0 1 0 0 1 1 0 0 0 1 1
[17,] 0 0 0 0 1 0 1 1 0 0 0 0 0
[18,] 0 0 0 0 1 1 1 0 0 0 0 0 1
[19,] 1 1 0 0 0 0 1 0 1 1 0 0 1
[20,] 1 1 0 0 0 0 0 0 0 0 1 1 1
[21,] 1 1 0 0 0 0 0 0 0 0 0 1 1
[22,] 0 1 0 0 0 0 0 0 0 0 0 0 1
[23,] 1 0 1 0 1 1 0 0 1 0 0 0 1
[24,] 1 0 1 1 1 1 0 0 1 1 1 0 0
[25,] 1 1 1 1 1 0 0 1 1 1 1 0 0
[26,] 1 1 1 0 0 0 0 1 1 0 0 0 0
[27,] 1 0 1 1 1 1 0 1 1 1 0 0 0
[28,] 1 1 1 1 1 1 1 1 1 1 1 0 0
[29,] 1 0 0 1 0 1 1 0 0 0 0 0 0
[30,] 0 0 1 1 0 1 1 1 0 0 0 0 0
[31,] 0 0 0 1 1 1 1 0 0 0 0 0 0
[32,] 0 0 1 0 0 1 1 0 0 0 0 0 0
[33,] 1 1 1 0 0 0 0 1 0 0 0 1 1
[34,] 1 1 1 0 0 0 1 1 1 1 0 0 0
[35,] 0 1 0 1 1 1 0 0 0 0 0 1 1
[36,] 0 1 0 1 1 1 1 1 0 0 1 1 1
[37,] 1 1 1 0 1 1 1 1 1 1 1 0 0
[38,] 1 1 0 0 1 1 1 0 1 1 1 1 1
[39,] 1 1 0 1 1 1 1 1 1 1 1 1 1
[40,] 1 1 1 1 0 0 0 0 0 0 0 1 1
[41,] 1 1 1 1 1 0 1 1 0 1 0 0 1
[42,] 1 1 0 1 1 0 0 0 1 1 0 1 1
[43,] 0 0 0 1 1 1 0 1 1 1 0 0 1
[44,] 1 0 0 1 1 1 1 1 1 0 0 0 0
[45,] 1 0 0 0 0 0 0 1 1 1 1 1 1
[46,] 0 0 0 0 1 1 1 1 1 1 1 0 0
[47,] 0 0 0 0 0 1 0 1 1 1 1 1 1
[48,] 1 1 0 0 0 0 0 1 1 1 1 1 1
[49,] 1 1 0 1 1 1 1 1 1 0 1 1 0
[50,] 0 1 0 0 0 0 1 1 0 0 0 0 0
[51,] 0 0 0 0 1 1 1 0 0 0 0 0 0
[52,] 1 1 1 1 1 1 0 1 1 0 0 0 0
[53,] 0 0 1 1 0 0 0 0 0 0 0 0 0
[54,] 0 0 1 1 1 1 0 0 0 1 0 0 0
[55,] 1 0 1 1 0 1 0 1 0 0 0 0 1
[56,] 0 0 0 1 1 1 0 0 1 0 1 0 0
[57,] 0 0 0 0 1 1 1 0 1 1 1 0 1
[58,] 0 0 0 0 1 1 1 0 0 1 1 1 1
[59,] 0 0 0 0 1 1 1 0 0 1 1 0 0
[60,] 0 0 0 0 1 1 1 0 1 1 1 0 0
[61,] 0 0 0 0 0 0 0 0 1 1 0 1 1
[62,] 0 0 1 0 0 0 0 0 0 1 0 0 0
[63,] 0 1 1 0 1 0 0 0 0 1 1 0 0
[64,] 1 1 1 1 1 1 0 1 1 1 1 0 1
[65,] 1 1 1 1 1 1 1 1 1 1 1 1 1
[66,] 1 0 1 1 1 1 0 0 0 1 1 1 1
[67,] 0 1 1 1 1 1 0 0 0 1 1 1 1
[68,] 0 0 1 1 0 0 0 0 1 1 1 1 0
[69,] 1 0 1 1 0 0 0 0 0 1 0 0 0
[70,] 1 1 0 0 0 0 0 1 0 1 1 1 0
[71,] 1 1 0 0 0 1 0 0 0 0 1 1 1
[72,] 1 1 0 0 0 0 1 0 0 0 1 0 1
[73,] 0 0 0 0 0 0 1 1 0 1 0 0 0
[74,] 1 0 0 1 0 0 1 0 0 1 1 0 0
[75,] 1 0 0 1 0 1 1 0 0 1 1 1 1
[76,] 1 0 1 0 1 1 0 0 1 1 1 1 0
[77,] 1 1 1 1 1 1 1 1 1 1 1 1 1
[78,] 0 0 1 1 1 0 1 1 1 1 1 1 1
[79,] 1 1 1 1 1 1 1 0 1 1 1 1 1
[80,] 0 1 1 1 1 1 1 1 1 0 0 1 1
[81,] 1 0 1 1 1 0 0 0 1 0 0 1 1
[82,] 0 1 1 1 1 0 0 0 0 0 0 0 0
[83,] 0 1 1 1 0 0 0 0 0 0 0 0 0
[84,] 1 1 1 1 1 0 0 0 0 0 0 0 0
[85,] 1 1 0 1 1 0 0 1 1 0 0 1 0
[86,] 1 0 0 0 1 1 0 1 0 0 0 0 0
[87,] 0 0 0 0 0 0 0 0 0 0 0 0 0
[88,] 1 0 0 1 0 0 0 1 0 0 0 0 0
[89,] 0 0 0 1 0 1 1 1 0 0 0 0 0
[90,] 1 1 0 0 0 1 1 1 0 0 0 0 0
[91,] 1 0 0 0 0 1 1 1 0 1 0 1 0
[92,] 1 0 1 0 0 0 0 0 0 1 1 1 1
[93,] 1 1 1 1 1 1 1 0 1 1 0 1 1
[94,] 1 1 0 1 1 1 1 0 0 0 1 1 0
[95,] 1 1 1 1 1 1 0 0 0 1 1 1 1
[96,] 1 1 1 1 1 1 0 0 1 1 1 1 1
[97,] 0 1 1 1 0 0 0 0 0 1 0 0 0
[98,] 0 1 1 1 1 1 1 0 0 1 0 0 1
[99,] 1 1 1 1 0 0 1 0 0 1 0 0 0
[100,] 1 1 0 1 1 0 1 0 0 0 1 0 0
[,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
[1,] 0 0 0 0 0 0 0 1 0 0 0 1
[2,] 0 0 0 0 0 0 0 1 1 0 0 1
[3,] 0 0 1 0 0 0 0 0 1 1 0 0
[4,] 0 0 1 1 0 1 0 0 0 1 1 0
[5,] 0 1 1 1 0 0 0 1 1 1 1 1
[6,] 1 1 1 0 0 0 1 1 1 1 0 1
[7,] 0 1 1 1 0 0 0 0 0 0 1 0
[8,] 1 0 0 1 0 0 0 1 0 0 0 0
[9,] 0 1 0 0 0 0 0 0 0 0 0 0
[10,] 0 0 0 0 0 0 1 1 1 1 0 1
[11,] 0 0 0 0 1 0 1 1 1 1 0 1
[12,] 0 0 0 0 1 0 1 1 1 0 0 1
[13,] 0 0 0 1 1 0 0 1 0 0 0 0
[14,] 1 1 0 1 1 1 0 1 1 1 0 0
[15,] 1 1 1 1 1 1 0 1 1 1 0 1
[16,] 0 1 1 1 1 1 0 0 1 1 1 0
[17,] 0 1 0 1 1 1 0 0 0 1 1 1
[18,] 1 1 1 0 0 0 0 0 1 1 0 1
[19,] 0 1 1 1 1 0 0 0 1 1 0 1
[20,] 0 1 1 0 1 1 1 0 1 1 1 1
[21,] 1 0 0 1 1 1 1 0 1 1 1 0
[22,] 1 1 1 1 1 0 1 0 0 1 0 0
[23,] 1 0 1 1 0 0 0 1 1 1 1 0
[24,] 1 1 0 0 0 1 1 1 1 1 1 0
[25,] 1 0 1 0 0 1 1 1 1 1 1 0
[26,] 1 0 0 0 0 1 1 1 1 1 1 0
[27,] 0 0 0 1 1 1 1 1 0 0 1 1
[28,] 1 1 1 1 1 1 1 0 0 1 1 1
[29,] 0 0 0 0 1 0 1 1 1 1 1 1
[30,] 1 1 0 1 0 1 0 1 1 1 1 1
[31,] 0 0 0 0 1 1 1 1 1 0 0 0
[32,] 1 0 0 0 0 0 1 1 0 0 0 0
[33,] 0 0 0 0 1 1 1 0 0 0 0 0
[34,] 0 1 1 0 1 1 1 0 0 0 0 1
[35,] 1 1 0 1 1 0 0 0 0 1 0 0
[36,] 0 0 0 1 1 0 0 0 0 0 1 0
[37,] 0 0 0 1 1 1 1 1 0 0 1 1
[38,] 1 0 1 0 0 1 1 1 0 0 1 1
[39,] 0 0 0 0 0 1 1 1 1 1 1 1
[40,] 1 1 1 1 1 0 0 1 1 0 1 1
[41,] 1 1 0 1 0 1 0 0 1 1 1 0
[42,] 1 1 1 1 1 1 1 1 0 1 1 1
[43,] 1 1 1 1 1 1 0 0 0 0 1 0
[44,] 0 1 0 0 1 0 0 0 0 0 0 0
[45,] 1 1 0 0 0 0 1 0 0 0 0 0
[46,] 0 1 0 0 1 0 0 0 0 0 0 1
[47,] 1 1 0 0 0 1 1 1 1 1 1 1
[48,] 1 0 1 1 1 1 0 1 1 1 1 1
[49,] 1 1 1 1 1 0 0 1 0 0 1 1
[50,] 1 1 1 1 1 0 0 1 1 0 1 0
[51,] 0 0 0 1 1 1 1 1 0 0 1 1
[52,] 1 0 0 0 1 1 0 0 0 0 0 1
[53,] 0 0 0 0 0 1 1 0 0 0 0 1
[54,] 0 1 0 0 0 1 1 1 1 0 0 0
[55,] 0 0 0 0 0 1 1 1 1 1 0 0
[56,] 0 1 0 0 1 1 1 1 1 1 0 1
[57,] 1 0 0 0 0 0 1 1 0 0 0 0
[58,] 0 0 0 0 0 0 1 1 1 0 1 0
[59,] 0 0 0 0 0 0 0 0 1 0 1 1
[60,] 0 0 1 1 0 0 0 1 1 0 1 1
[61,] 1 1 0 1 0 0 1 1 1 0 0 1
[62,] 1 0 0 0 0 0 1 1 1 1 1 1
[63,] 1 1 0 1 1 0 0 1 1 1 1 1
[64,] 0 1 1 1 1 1 1 1 1 1 1 1
[65,] 0 1 1 1 1 1 1 1 1 1 1 1
[66,] 0 0 0 1 1 1 1 1 1 1 1 1
[67,] 0 0 0 0 0 0 0 0 1 0 1 1
[68,] 0 0 0 1 1 1 1 0 0 1 1 1
[69,] 0 0 0 1 1 1 1 0 0 0 0 1
[70,] 0 0 0 1 1 0 0 0 1 1 1 1
[71,] 0 1 1 0 0 0 0 0 1 0 0 0
[72,] 0 1 1 0 0 1 1 1 1 1 0 0
[73,] 0 0 0 0 0 1 1 1 1 1 0 1
[74,] 0 0 0 1 0 0 1 1 1 1 1 1
[75,] 0 0 0 0 0 1 1 1 1 1 1 1
[76,] 0 0 0 0 1 0 0 0 1 1 1 0
[77,] 0 0 0 1 1 1 1 0 1 1 1 0
[78,] 1 1 1 1 1 1 0 0 1 0 0 0
[79,] 1 1 0 1 0 0 0 0 1 0 0 0
[80,] 1 1 0 1 0 0 0 0 1 0 0 0
[81,] 1 1 1 1 1 1 0 0 0 0 0 1
[82,] 1 1 1 0 0 1 0 0 1 1 0 1
[83,] 1 1 1 1 0 1 1 0 0 1 0 0
[84,] 0 0 0 1 1 1 0 1 1 1 0 0
[85,] 0 0 0 0 1 1 1 1 0 1 1 1
[86,] 0 1 1 1 1 1 1 1 1 0 0 0
[87,] 0 0 1 1 0 0 0 1 1 1 1 1
[88,] 1 0 1 1 1 0 0 1 1 1 1 1
[89,] 1 1 1 0 0 0 0 1 1 1 0 0
[90,] 1 0 1 1 1 0 1 1 1 0 0 0
[91,] 1 0 1 1 1 0 1 1 1 0 0 1
[92,] 1 1 1 1 1 1 1 1 0 0 0 1
[93,] 1 1 1 1 1 1 1 0 0 0 1 1
[94,] 1 0 1 1 1 1 0 0 0 1 1 1
[95,] 0 1 1 1 0 0 0 1 1 1 0 1
[96,] 1 0 1 1 1 1 0 0 1 1 1 1
[97,] 1 1 1 0 0 0 0 0 0 1 1 1
[98,] 1 0 0 0 1 0 1 1 1 1 1 1
[99,] 1 0 0 0 0 0 0 1 1 1 1 1
[100,] 1 0 0 0 0 1 1 1 0 0 0 0
[,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
[1,] 0 0 0 0 0 0 0 1 1 1 0 0
[2,] 0 0 1 1 0 0 0 1 1 1 0 1
[3,] 0 1 1 1 0 1 0 0 1 1 1 1
[4,] 1 1 0 1 1 0 0 1 1 1 1 1
[5,] 1 1 0 0 0 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 0 1 1 1 1 1
[7,] 0 1 1 1 1 1 0 1 1 1 0 1
[8,] 1 0 1 1 1 0 0 0 1 0 0 1
[9,] 0 0 1 1 1 0 0 1 0 0 0 1
[10,] 0 0 0 0 1 0 0 0 0 1 1 0
[11,] 0 0 0 0 1 1 1 0 0 1 1 1
[12,] 0 0 0 1 1 1 0 0 0 1 1 0
[13,] 1 0 0 1 1 1 1 0 1 1 1 1
[14,] 1 0 0 0 0 1 1 1 0 1 1 1
[15,] 1 1 0 0 0 1 0 1 1 1 1 1
[16,] 0 1 0 0 0 1 1 1 1 0 1 1
[17,] 1 0 0 0 0 0 1 1 1 1 1 1
[18,] 1 1 1 0 0 0 1 0 1 1 1 1
[19,] 1 1 0 0 0 0 0 0 0 0 1 1
[20,] 1 1 1 1 0 1 0 1 0 0 0 0
[21,] 1 0 1 0 0 0 0 0 0 0 0 0
[22,] 0 0 1 0 0 0 0 0 0 1 1 1
[23,] 0 0 1 0 0 0 1 1 1 1 1 1
[24,] 0 0 1 0 0 0 1 1 1 1 1 1
[25,] 0 0 0 0 0 1 1 0 0 0 1 0
[26,] 0 1 0 0 0 0 0 0 0 0 1 0
[27,] 0 1 1 0 0 0 0 0 0 0 0 0
[28,] 1 0 0 0 0 0 0 0 0 0 0 0
[29,] 1 0 0 0 1 1 0 0 1 0 1 1
[30,] 1 0 0 0 1 1 1 1 1 1 1 1
[31,] 1 0 0 0 0 1 0 1 1 0 0 1
[32,] 0 1 0 1 0 1 0 0 0 0 0 0
[33,] 0 0 1 1 1 0 0 0 0 0 1 0
[34,] 1 1 1 1 1 0 0 0 0 1 0 1
[35,] 0 0 1 1 1 1 1 1 0 1 1 1
[36,] 0 0 1 0 1 1 1 1 1 1 0 0
[37,] 1 1 0 0 0 1 1 1 1 1 1 1
[38,] 1 1 0 0 0 1 1 1 1 1 1 1
[39,] 1 1 0 1 1 1 1 1 1 0 1 1
[40,] 0 0 1 1 1 0 0 0 1 0 1 1
[41,] 1 1 1 1 1 1 0 0 0 0 1 1
[42,] 1 1 1 0 1 1 0 0 0 0 0 1
[43,] 1 0 0 0 0 0 1 1 0 0 0 1
[44,] 0 0 0 0 0 1 1 0 0 1 0 1
[45,] 0 0 0 1 0 0 0 0 1 0 0 0
[46,] 0 1 1 1 0 0 1 1 1 0 0 0
[47,] 1 1 0 1 1 0 0 1 1 0 0 0
[48,] 0 1 1 1 0 0 0 0 1 1 0 0
[49,] 0 0 0 0 0 0 0 1 1 1 1 0
[50,] 1 0 0 0 0 0 0 0 0 0 1 0
[51,] 0 0 0 0 0 0 1 1 0 0 1 1
[52,] 1 0 0 0 0 0 1 1 0 0 0 0
[53,] 1 1 0 1 1 0 1 1 0 0 0 0
[54,] 1 1 1 1 0 1 1 1 1 0 0 0
[55,] 0 0 1 1 1 1 1 1 1 0 0 0
[56,] 1 0 1 1 1 1 1 1 0 1 1 0
[57,] 1 0 1 1 1 1 0 1 1 1 1 0
[58,] 0 0 1 0 0 0 1 1 1 1 0 1
[59,] 0 0 0 0 0 1 1 1 1 0 0 0
[60,] 0 1 0 1 0 1 0 0 0 0 1 0
[61,] 1 1 0 0 0 1 1 0 0 1 1 1
[62,] 1 1 0 1 0 0 0 0 0 1 1 1
[63,] 1 0 0 0 0 0 0 0 0 0 1 1
[64,] 0 0 1 1 1 1 1 1 0 0 1 1
[65,] 0 0 0 1 1 1 0 0 1 1 0 1
[66,] 1 1 1 1 1 1 0 0 1 1 0 0
[67,] 1 1 1 1 1 1 1 1 0 1 0 0
[68,] 1 1 1 1 1 0 0 0 0 0 0 0
[69,] 1 1 1 1 0 0 0 1 0 0 0 1
[70,] 1 0 0 0 0 1 0 0 0 0 1 1
[71,] 0 0 0 1 1 1 1 0 0 0 1 0
[72,] 0 1 1 1 1 1 1 0 0 0 1 1
[73,] 1 0 0 1 1 1 1 1 1 0 0 0
[74,] 1 1 0 1 1 1 1 1 0 0 0 0
[75,] 1 1 0 1 1 1 1 0 0 0 0 0
[76,] 1 1 1 1 1 1 1 1 1 1 0 1
[77,] 1 1 1 1 1 1 0 0 0 0 1 1
[78,] 1 1 0 0 0 1 1 1 0 0 1 1
[79,] 0 0 0 0 0 0 0 1 0 0 0 1
[80,] 0 0 1 0 0 0 1 1 0 0 0 0
[81,] 0 1 1 0 0 0 0 0 1 1 0 0
[82,] 0 0 0 1 0 0 1 1 1 1 0 0
[83,] 0 0 0 0 0 1 1 0 1 1 0 0
[84,] 1 0 0 1 1 1 1 0 0 1 1 1
[85,] 0 0 0 1 0 0 0 0 0 1 1 1
[86,] 0 0 0 0 0 0 0 0 0 1 1 1
[87,] 1 0 0 0 0 0 0 1 1 1 1 1
[88,] 0 0 0 0 0 0 1 1 1 0 0 0
[89,] 0 1 1 1 1 0 0 0 0 0 0 0
[90,] 1 1 0 1 1 1 0 0 0 0 0 0
[91,] 1 1 1 1 1 1 1 1 1 0 1 0
[92,] 1 1 1 0 1 1 0 0 0 0 1 1
[93,] 1 1 1 0 0 1 0 0 0 0 1 1
[94,] 1 1 1 1 0 1 0 1 0 1 1 1
[95,] 1 1 0 1 1 1 0 1 1 0 0 1
[96,] 1 0 0 1 1 1 1 1 1 1 0 1
[97,] 1 1 1 1 1 1 1 0 1 1 0 1
[98,] 1 1 1 1 1 1 1 0 1 1 1 1
[99,] 0 1 0 0 0 0 1 1 1 1 1 0
[100,] 1 1 0 0 0 0 0 1 1 0 0 0
[,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49]
[1,] 0 0 1 1 0 0 0 0 0 0 0 0
[2,] 1 0 0 0 0 0 0 0 0 1 0 0
[3,] 1 0 0 0 0 1 0 0 0 0 0 0
[4,] 1 0 0 0 0 0 0 0 0 1 1 0
[5,] 1 0 0 0 0 0 0 0 0 0 0 1
[6,] 0 1 1 0 0 0 0 0 0 0 1 1
[7,] 1 1 0 0 0 0 1 0 0 0 0 0
[8,] 1 0 0 0 0 0 1 1 0 0 0 0
[9,] 0 1 1 0 1 1 1 1 0 0 1 0
[10,] 0 1 0 1 1 1 0 0 0 0 1 1
[11,] 1 1 1 1 1 1 1 1 0 0 1 0
[12,] 1 0 0 0 0 1 1 1 0 0 0 0
[13,] 1 0 0 1 0 0 0 0 0 0 0 0
[14,] 1 0 1 1 0 0 0 0 0 0 1 0
[15,] 1 1 1 1 1 0 1 0 0 0 1 0
[16,] 1 1 0 1 1 0 0 1 1 0 1 0
[17,] 0 0 1 1 1 1 0 0 0 1 1 1
[18,] 0 0 1 1 0 1 0 0 1 0 1 1
[19,] 0 1 1 1 1 1 0 0 1 1 1 1
[20,] 1 1 1 1 0 1 1 0 1 0 0 1
[21,] 1 1 1 0 0 1 1 0 1 1 0 1
[22,] 1 1 1 1 0 0 0 0 1 1 1 1
[23,] 1 0 1 1 0 0 0 0 0 1 0 0
[24,] 1 1 1 1 0 0 0 0 0 1 1 1
[25,] 0 1 1 0 0 1 1 0 1 0 1 1
[26,] 0 0 0 0 0 0 0 0 0 0 1 1
[27,] 0 0 0 0 0 1 1 0 0 0 1 1
[28,] 0 0 0 0 1 1 1 0 1 1 1 1
[29,] 1 0 0 0 1 1 0 0 0 0 0 1
[30,] 0 0 0 1 1 0 0 0 0 1 1 1
[31,] 0 1 0 0 0 0 0 0 0 1 0 1
[32,] 0 0 1 1 1 1 0 0 0 1 1 0
[33,] 0 0 1 0 0 1 0 0 0 0 0 0
[34,] 1 1 1 0 0 0 0 0 0 0 0 0
[35,] 1 1 1 1 1 0 0 0 0 0 0 0
[36,] 0 1 1 1 1 0 0 0 0 0 0 0
[37,] 1 0 1 1 0 1 0 0 0 1 1 0
[38,] 1 1 1 0 0 1 1 0 0 1 1 0
[39,] 1 1 1 0 0 0 0 0 0 1 1 1
[40,] 1 1 1 0 0 0 0 0 0 0 0 0
[41,] 1 1 0 0 0 0 0 0 0 1 0 1
[42,] 1 1 0 0 0 0 0 0 1 1 1 0
[43,] 0 0 0 1 0 0 0 1 1 0 0 0
[44,] 0 0 0 0 0 0 0 0 0 0 1 0
[45,] 0 0 0 1 0 0 0 0 0 0 0 0
[46,] 0 0 1 1 0 0 0 0 0 0 0 0
[47,] 0 0 0 0 0 0 0 0 0 0 0 0
[48,] 1 0 0 0 1 0 0 0 1 0 0 0
[49,] 0 0 0 0 0 1 0 1 0 0 1 1
[50,] 0 1 0 0 1 1 1 0 1 0 1 1
[51,] 0 0 0 1 0 1 0 1 1 0 1 0
[52,] 0 0 0 0 1 1 1 1 0 0 1 1
[53,] 0 0 0 0 1 0 0 0 0 0 0 1
[54,] 0 0 0 0 1 0 0 0 1 1 0 0
[55,] 1 0 0 0 1 1 1 0 0 0 0 0
[56,] 0 0 0 0 1 1 0 0 0 0 0 0
[57,] 0 0 0 1 1 1 0 0 0 0 1 0
[58,] 0 0 0 0 1 1 0 1 0 0 1 1
[59,] 0 0 0 1 1 0 1 1 0 0 1 1
[60,] 0 0 0 0 0 1 1 1 1 0 1 1
[61,] 0 0 0 0 0 0 1 1 1 1 1 0
[62,] 0 0 0 0 0 0 1 1 1 1 1 1
[63,] 0 0 0 0 0 0 1 1 0 0 1 1
[64,] 0 0 0 1 1 1 1 1 0 0 1 1
[65,] 1 1 1 1 0 1 1 0 0 0 1 1
[66,] 0 0 1 1 0 0 0 0 0 0 1 1
[67,] 0 1 1 1 1 1 1 1 0 1 1 1
[68,] 0 1 1 1 0 0 1 1 0 0 0 1
[69,] 1 1 1 1 0 0 0 0 0 0 1 1
[70,] 0 1 0 1 0 0 0 0 1 1 1 1
[71,] 0 1 0 0 0 0 0 0 0 1 1 1
[72,] 1 1 1 1 0 1 0 0 1 0 1 1
[73,] 0 0 0 1 1 1 1 1 0 1 1 1
[74,] 0 0 1 0 1 1 1 1 1 1 1 1
[75,] 0 0 1 0 1 1 1 0 0 1 0 0
[76,] 1 1 0 0 0 0 1 0 1 1 1 0
[77,] 0 0 0 0 0 0 0 0 0 0 1 0
[78,] 1 0 0 0 0 0 1 1 0 0 1 1
[79,] 0 0 0 0 0 0 1 0 1 1 1 1
[80,] 0 0 0 1 1 0 0 0 0 1 1 1
[81,] 0 0 0 1 1 1 0 0 0 0 0 1
[82,] 1 1 1 1 1 1 0 0 1 0 0 1
[83,] 1 1 1 1 1 1 1 0 0 0 0 0
[84,] 1 1 1 1 1 1 1 1 0 0 0 0
[85,] 1 1 0 1 1 1 1 1 0 0 1 1
[86,] 1 1 1 1 1 0 0 1 1 0 1 1
[87,] 1 1 1 1 1 1 0 1 1 1 1 1
[88,] 1 1 0 1 1 1 1 1 1 1 0 1
[89,] 0 0 0 1 1 1 0 1 0 0 0 0
[90,] 1 0 0 0 1 1 0 1 1 1 0 1
[91,] 1 0 0 0 1 0 0 1 1 1 0 0
[92,] 0 0 0 0 1 0 0 0 1 1 1 0
[93,] 0 0 1 0 0 0 0 0 0 1 1 1
[94,] 0 0 0 1 0 0 1 1 1 1 1 0
[95,] 0 0 0 0 0 0 1 1 1 1 1 1
[96,] 0 0 1 1 1 0 1 1 1 1 1 1
[97,] 1 1 1 1 0 0 1 1 1 1 1 1
[98,] 0 1 1 1 0 0 0 0 0 0 1 1
[99,] 0 0 0 0 0 0 0 0 0 0 0 1
[100,] 0 1 0 0 0 1 0 1 0 0 0 0
[,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61]
[1,] 0 0 1 1 0 0 0 1 1 0 0 0
[2,] 0 0 0 1 0 1 0 0 0 0 0 0
[3,] 0 1 0 0 0 1 0 1 1 1 1 1
[4,] 0 1 0 1 0 1 1 1 1 1 1 0
[5,] 0 1 1 0 0 0 0 0 1 1 0 0
[6,] 0 0 0 0 1 0 0 0 1 1 1 0
[7,] 0 0 1 1 1 1 0 0 1 0 0 1
[8,] 0 0 0 0 1 1 0 0 0 1 1 1
[9,] 0 0 0 1 0 0 0 0 1 1 1 0
[10,] 1 0 0 0 0 0 0 0 1 1 1 1
[11,] 0 0 0 1 0 0 1 1 1 1 1 1
[12,] 1 1 1 0 0 0 1 1 0 0 1 1
[13,] 0 0 0 0 0 0 1 1 1 0 0 1
[14,] 0 0 0 0 0 0 0 1 1 0 1 1
[15,] 1 0 1 0 0 0 0 0 0 1 0 0
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[,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
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[63,] 0 1 1 0 0 0 1 1 1 0 0 1
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[100,] 1 1 1 1 1 0 0 0 1 1 1 1
[,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49]
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[,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61]
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[,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
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[,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61]
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[,86] [,87] [,88] [,89] [,90] [,91] [,92] [,93] [,94] [,95] [,96] [,97]
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[25,] 1 1 1
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[28,] 0 1 1
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[34,] 0 1 1
[35,] 1 1 1
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[53,] 1 1 1
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[55,] 0 1 1
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[59,] 1 1 1
[60,] 0 1 1
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[65,] 1 1 1
[66,] 1 1 1
[67,] 0 0 0
[68,] 1 1 0
[69,] 1 0 1
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[71,] 1 1 1
[72,] 1 1 0
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[74,] 0 0 0
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[81,] 0 1 0
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[84,] 1 0 1
[85,] 1 1 0
[86,] 1 0 0
[87,] 1 1 1
[88,] 1 1 1
[89,] 1 1 1
[90,] 1 1 1
[91,] 0 1 1
[92,] 1 1 1
[93,] 0 1 1
[94,] 1 1 0
[95,] 1 1 1
[96,] 1 1 1
[97,] 1 1 1
[98,] 1 0 0
[99,] 1 0 0
[100,] 0 0 0
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