Simulate Potts model using Swendsen-Wang algorithm.
potts(obj, param, nbatch, blen = 1, nspac = 1,
boundary = c("torus", "free", "condition"), debug = FALSE,
outfun = NULL, ...)an object of class "potts",
which is a list containing at least the following components:
initial state of Markov chain in the format output
by packPotts.
final state of Markov chain in the same format.
value of .Random.seed before the run.
value of .Random.seed after the run.
running time of Markov chain from system.time.
canonical parameter vector.
the number of batches.
the length of batches.
the spacing of iterations that contribute to batches.
the argument boundary.
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.
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.
numeric, canonical parameter vector. Last component must nonnegative (see Details below).
the number of batches.
the length of batches.
the spacing of iterations that contribute to batches.
type of boundary conditions. The value of this argument can be abbreviated.
return additional debugging information.
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.
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).
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$batch
if (FALSE) image(out$final)
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