potts {potts}

Potts Models

Description

Simulate Potts model using Swendsen-Wang algorithm.

Usage

potts(obj, param, nbatch, blen = 1, nspac = 1,
    boundary = c("torus", "free", "condition"), debug = FALSE,
    outfun = NULL, ...)

Arguments

obj

an R vector of class "raw" that encodes a realization of a Potts model, typically the output of packPotts. Alternatively, an object of class "potts" from a previous invocation of this function can be supplied, in which case any missing arguments are taken from this object.

param

numeric, canonical parameter vector. Last component must nonnegative (see Details below).

nbatch

the number of batches.

blen

the length of batches.

nspac

the spacing of iterations that contribute to batches.

boundary

type of boundary conditions. The value of this argument can be abbreviated.

debug

return additional debugging information.

outfun

controls the output. If a function, then the batch means of outfun(tt, ...) are returned. The argument tt is the canonical statistic of the Potts model having the same length as the argument param of this function. If NULL, the batch means of the canonical statistic are returned.

...

additional arguments for outfun.

Details

Runs a Swendsen-Wang algorithm producing a Markov chain with equilibrium distribution having the specified Potts model. The state of a Potts model is a collection of random variables taking values in a finite set. Here the finite set is 1, ..., ncolor and the elements are called “colors”. The random variables are associated with the nodes of a rectangular lattice, represented by unpackPotts as a matrix. In keeping with calling the values “colors”, the random variables themselves are often called “pixels”. The probability model is an exponential family with canonical statistic vector of length ncolor + 1. The first ncolor components are the counts of the number of pixels of each color. The last component is the number of pairs of neighboring pixels colored the same. The corresponding canonical parameter, last component of the canonical parameter vector (argument param) must be nonnegative for the Swendsen-Wang algorithm to work (Potts models are defined for negative dependence parameter, but can't be simulated using this algorithm).

In the default boundary specification ("torus"), also called toroidal or periodic boundary conditions, the vertical edges of the pixel matrix are considered glued together, as are the horizontal edges. Thus corresponding pixels in the first and last rows are considered neighbors, as are corresponding pixels in the first and last columns. In the other boundary specifications there is no such gluing: pixels in the the relative interiors of the first and last rows and first and last columns have only three neighbors, and the four corner pixels have only two neighbors.

In the "torus" and "free" boundary specifications, all pixels are counted in determining the color count canonical statistics, which thus range from zero to nrow * ncol, where nrow and ncol are the number of rows and columns of the pixel matrix. In the "condition" boundary specification, all pixels in the first and last rows and first and last columns are fixed (conditioned on), and only the random pixels are counted in determining the color count canonical statistics, which thus range from zero to (nrow - 2) * (ncol - 2).

In the "torus" boundary specification, all pixels have four neighbors, so the neighbor pair canonical statistic ranges from zero to 2 * nrow * ncol. In the "free" boundary specification, pixels in the interior have four neighbors, those in the relative interior of edges have three, and those in the corners have two, so the neighbor pair canonical statistic ranges from zero to nrow * (ncol - 1) + (nrow - 1) * ncol. In the "condition" boundary specification, only neighbor pairs in which at least one pixel is random are counted, so the neighbor pair canonical statistic ranges from zero to (nrow - 2) * (ncol - 1) + (nrow - 1) * (ncol - 2).

Value

an object of class "potts", which is a list containing at least the following components:

initial

initial state of Markov chain in the format output by packPotts.

final

final state of Markov chain in the same format.

initial.seed

value of .Random.seed before the run.

final.seed

value of .Random.seed after the run.

time

running time of Markov chain from system.time.

param

canonical parameter vector.

nbatch

the number of batches.

blen

the length of batches.

nspac

the spacing of iterations that contribute to batches.

boundary

the argument boundary.

batch

an nbatch by nout matrix, where nout is the length of the result returned by outfun or length(param) if outfun == NULL; each row is the batch means for the result of outfun or the canonical statistic vector for one batch of Markov chain iterations.

Examples

ncolor <- as.integer(4)
beta <- log(1 + sqrt(ncolor))
theta <- c(rep(0, ncolor), beta)

nrow <- 100
ncol <- 100
x <- matrix(1, nrow = nrow, ncol = ncol)
foo <- packPotts(x, ncolor)

out <- potts(foo, theta, nbatch = 10)
out$batch
## Not run: image(out$final)