| Title: | Gibbs Random Fields Analysis |
|---|---|
| Description: | Allows calculation on, and sampling from Gibbs Random Fields, and more precisely general homogeneous Potts model. The primary tool is the exact computation of the intractable normalising constant for small rectangular lattices. Beside the latter function, it contains method that give exact sample from the likelihood for small enough rectangular lattices or approximate sample from the likelihood using MCMC samplers for large lattices. |
| Authors: | Julien Stoehr, Pierre Pudlo and Nial Friel |
| Maintainer: | Julien Stoehr <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.0.1 |
| Built: | 2024-12-03 06:49:31 UTC |
| Source: | CRAN |
GiRaF is a package for calculations on, and sampling from Gibbs (or discrete Markov) random fields.
GiRaF offers various tools for the analysis of Gibbs random fields and more precisely general homogeneous Potts model with possible anisotropy and potential on singletons (cliques composed of single vertex). GiRaF substantially lowers the barrier for practitioners aiming at analysing such Gibbs random fields. GiRaF contains exact methods for small lattices and several approximate methods for larger lattices that make the analysis easier for practitioners.
The “GiRaF-introduction” vignette gives a detailled introduction on the package.
For a complete list of functions, use library(help = "GiRaF").
Julien Stoehr, Pierre Pudlo and Nial Friel.
Maintainer: Julien Stoehr <[email protected]>
Friel, N. and Rue, H. (2007). Recursive computing and simulation-free inference for general factorizable models. Biometrika, 94(3):661–672.
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.
Reeves, R. and Pettitt, A. N. (2004). Efficient recursions for general factorisable models. Biometrika, 91(3):751–757.
Swendsen, R. H. and Wang, J.-S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Pysical Review Letters, 58(2):86-88.
The “GiRaF-introduction” vignette
# Dimension of the lattice height <- 8 width <- 10 # 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) # Partition function for the default setting NC.mrf(h = height, w = width, param = Beta) # When specifying the number of colors and neighbors NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta) # When specifying an optional potential on sites NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential) # When specifying possible borders. The users will omit to mention all # the non-existing borders NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner) # Exact sampling for the default setting exact.mrf(h = height, w = width, param = Beta, view = TRUE) # When specifying the number of colors and neighbors exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, view = TRUE) # When specifying an optional potential on sites exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential, view = TRUE) # When specifying possible borders. The users will omit to mention all # the non-existing borders exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner, view = TRUE) # Algorithm settings n <- 200 method <- "Gibbs" # 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)# Dimension of the lattice height <- 8 width <- 10 # 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) # Partition function for the default setting NC.mrf(h = height, w = width, param = Beta) # When specifying the number of colors and neighbors NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta) # When specifying an optional potential on sites NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential) # When specifying possible borders. The users will omit to mention all # the non-existing borders NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner) # Exact sampling for the default setting exact.mrf(h = height, w = width, param = Beta, view = TRUE) # When specifying the number of colors and neighbors exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, view = TRUE) # When specifying an optional potential on sites exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential, view = TRUE) # When specifying possible borders. The users will omit to mention all # the non-existing borders exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner, view = TRUE) # Algorithm settings n <- 200 method <- "Gibbs" # 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)
exact.mrf gives exact sample from the likelihood of a general Potts model defined on a rectangular lattice () with either a first order or a second order dependency structure and a small number of rows (up to 19 for 2-state models).
exact.mrf(h, w, param, ncolors = 2, nei = 4, pot = NULL, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL, view = FALSE)exact.mrf(h, w, param, ncolors = 2, nei = 4, pot = NULL, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL, view = FALSE)
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. |
view |
Logical value indicating whether the draw should be printed. Do not display the optional borders. |
Friel, N. and Rue, H. (2007). Recursive computing and simulation-free inference for general factorizable models. Biometrika, 94(3):661–672.
The “GiRaF-introduction” vignette
# Dimension of the lattice height <- 8 width <- 10 # 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) # Exact sampling for the default setting exact.mrf(h = height, w = width, param = Beta, view = TRUE) # When specifying the number of colors and neighbors exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, view = TRUE) # When specifying an optional potential on sites exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential, view = TRUE) # When specifying possible borders. The users will omit to mention all # the non-existing borders exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner, view = TRUE)# Dimension of the lattice height <- 8 width <- 10 # 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) # Exact sampling for the default setting exact.mrf(h = height, w = width, param = Beta, view = TRUE) # When specifying the number of colors and neighbors exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, view = TRUE) # When specifying an optional potential on sites exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential, view = TRUE) # When specifying possible borders. The users will omit to mention all # the non-existing borders exact.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner, view = TRUE)
Partition function of a general Potts model defined on a rectangular lattice () with either a first order or a second order dependency structure and a small number of rows (up to 25 for 2-state models).
NC.mrf(h, w, param, ncolors = 2, nei = 4, pot = NULL, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL)NC.mrf(h, w, param, ncolors = 2, nei = 4, pot = NULL, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL)
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. |
Friel, N. and Rue, H. (2007). Recursive computing and simulation-free inference for general factorizable models. Biometrika, 94(3):661–672.
Reeves, R. and Pettitt, A. N. (2004). Efficient recursions for general factorisable models. Biometrika, 91(3):751–757.
The “GiRaF-introduction” vignette
# Dimension of the lattice height <- 8 width <- 10 # 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) # Partition function for the default setting NC.mrf(h = height, w = width, param = Beta) # When specifying the number of colors and neighbors NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta) # When specifying an optional potential on sites NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential) # When specifying possible borders. The users will omit to mention all # the non-existing borders NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner)# Dimension of the lattice height <- 8 width <- 10 # 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) # Partition function for the default setting NC.mrf(h = height, w = width, param = Beta) # When specifying the number of colors and neighbors NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta) # When specifying an optional potential on sites NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, pot = potential) # When specifying possible borders. The users will omit to mention all # the non-existing borders NC.mrf(h = height, w = width, ncolors = K, nei = G, param = Beta, top = Top, left = Left, bottom = Bottom, right = Right, corner = Corner)
sampler.mrf gives approximate sample from the likelihood of a general Potts model defined on a rectangular lattice () 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)).
sampler.mrf(iter, sampler = "Gibbs" , h, w, param, ncolors = 2, nei = 4, pot = NULL, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL, initialise = TRUE, random = TRUE, view = FALSE)sampler.mrf(iter, sampler = "Gibbs" , h, w, param, ncolors = 2, nei = 4, pot = NULL, top = NULL, left = NULL, bottom = NULL, right = NULL, corner = NULL, initialise = TRUE, random = TRUE, view = FALSE)
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. |
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.
The “GiRaF-introduction” vignette
# 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)# 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)