Package 'potts'

Cache calculated Canonical Statistics for Potts Models.

Description

Variety of functions to support caching of calculated canonical statistics for Potts Models. There is some attempt at being 'smart' with when to regenerate the statistics.

Usage

generate_t_cache(x, ncolor, t_stat, sizeA, npixel, f,
                 fapply=lapply, gridcache=NULL)

gengridcache(ncolor, sizeCA, ncol)

gensingleton(ncolor)

singleton(x, ncolor, a, idx, gridcache=NULL)

gentwopixel(ncolor)

twopixel(x, ncolor, a, idx, gridcache=NULL)

twopixel.nonoverlap(x, ncolor, a, idx, gridcache=NULL)

genfourpixel(ncolor)

fourpixel(x, ncolor, a, idx, gridcache=NULL)

fourpixel.nonoverlap(x, ncolor, a, idx, gridcache=NULL)

genthreebythree(ncolor)

ninepixel.nonoverlap(x, ncolor, a, idx, gridcache=NULL)

genfourbyfour(ncolor)

sixteenpixel.nonoverlap(x, ncolor, a, idx, gridcache=NULL)

Arguments

t_stat	numerical vector of length ncolor. Contains the canonical statistic for the whole image.
sizeA	numerical. The number of elements in $\mathcal{A}$.
sizeCA	numerical. The number of elements in $C^A$.
npixel	numerical. The number of pixels in one element of $\mathcal{A}$.
f	function. Takes arguments x, ncolor, a, idx and ncolor. Returns value of t_calc_innergrid with window $A_a$ replaced by the $\texttt{idx}^{th}$ element of $C^A$.
fapply	function. It should behave exactly as lapply does. You can use this argument to enable parallel computing.
gridcache	list. Optional. If non-null, it is a list of the elements of $C^A$.
x	numeric, 2 dimensional matrix, elements in 1, ..., ncolor, representing a Potts model.
ncolor	numeric. Number of colors in this Potts Model.
ncol	numeric. Gives the number of columns in a rectangular window.
a	numeric. Indicates which member of $\mathcal{A}$ is being referenced.
idx	numeric. Indicates which element of $C^A$ is being referenced.

Details

For a description of notation and terminology, see composite.ll.

This set of functions is used to generate cached calculations of the canonical statistic of a Potts model suitable for passing into composite.ll or gr.composite.ll.

All of the calculations using composite.ll and these caching functions need one of the color components to be dropped for the model to be identifiable. For simplicity, the first color is dropped by generate_t_cache. In computing the composite log likelihood for a Potts model with ncolor colors, we are interested in many calculations across $C^A$, the set of all permutations of colors across a window. These functions facilitate those calculations. gridcache is a list of $C^A$.

generate_t_cache is the main function, and the others are intended to be used in conjunction with it. generate_t_cache creates a list of arrays. Each array represents one window in the image, and each row of the array contains the value of $t(x)$ (with one component dropped) found by replacing the pixels in that window with one of the elements of $C^A$.

gengridcache can generate the gridcache for any rectangular window, give the number of colors, size of $C^A$, and number of columns in the window. gensingleton, gentwopixel, genfourpixel, genthreebythree and genfourbyfour are all just simple wrappers for gengridcache.

singleton, twopixel, twopixel.nonoverlap, fourpixel, fourpixel.nonoverlap, ninepixel.nonoverlap and sixteenpixel.nonoverlap are intended to be passed to generate_t_cache in the argument f. They are used to calculate $t(ca_{idx} \cup X \setminus A_a)$ for the $\texttt{idx}^{th}$ element of $C^{A_a}$.

Functions that have overlap and nonoverlap versions generate a overlapping and nonoverlapping set of windows respectively.

singleton is for a single pixel window (Besag or MPLE).

twopixel does a two horizontal pixel window.

fourpixel does a two by two pixel window.

ninepixel does a three by three pixel window.

sixteenpixel does a four by four pixel window.

Value

Functions that start with gen return a list of the elements of $C^A$.

The other functions (e.g. twopixel, fourpixel, ...) return the result of replacing the a-th window of x with the idx-th element of $C^A$ and calculating calc_t_innergrid for that window.

See Also

composite.ll, calc_t.

Examples

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

nrow <- 32
ncol <- 32

x <- matrix(sample(ncolor, nrow*ncol, replace=TRUE), nrow=nrow, ncol=ncol)
foo <- packPotts(x, ncolor)
out <- potts(foo, theta, nbatch=10)
x <- unpackPotts(out$final)

t_stat <- calc_t(x, ncolor)
t_cache_mple <- generate_t_cache(x, ncolor, t_stat, nrow*ncol, 1,
                                 singleton)

## Not run: 
# use multicore to speed things up.
library(multicore)
t_cache_mple <- generate_t_cache(x, ncolor, t_stat, nrow*ncol, 1,
                                 singleton, fapply=mclapply)

## End(Not run)

---

Calculate Canonical Statistic for Potts Model

Description

Calculate the canonical statistic 't' for a realization of a Potts Model

Usage

calc_t_full(x,ncolor)
calc_t_innergrid(x, ncolor, grid, i, j)
calc_t(x, ncolor, grid=NULL, i=NULL, j=NULL)

Arguments

x	2 dimensional matrix, elements in 1, ..., ncolor, representing a Potts model
ncolor	numeric. Number of colors in this Potts Model.
grid	numeric. 2 dimensional matrix, elements in 1, ..., ncolor. If non-NULL it is placed into x at the location x[i,j].
i	numeric. Row to place the grid.
j	numeric. Column to place the grid.

Details

For a description of notation and terminology, see composite.ll.

Calculates the canonical statistics for a realized Potts Model. calc_t calls calc_t_full if grid is NULL and calc_t_innergrid otherwise.

calc_t_full calculates the canonical statistics for the full image.

calc_t_innergrid calculates the canonical statistics for the a window of the image, but with that window replaced by grid, with the upper left corner of grid located at x[i,j].

Value

For a description of notation and terminology, see composite.ll.

All functions return a vector of length ncolor+1. Elements 1,...,ncolor contain the number of pixels of each color. Element ncolor+1 contains the number of matching neighbor pairs for the image.

calc_t_full returns the values for the whole image.

calc_t_innergrid returns the value for just the selected window, but this includes the number of matching pairs from the replaced window to it's neighbors. E.g. if X is the full image, and $A_a$ is the value of some window in the image and we want to know the value of $t(y \cup X \setminus A_a)$ this would be calc_t_full(X, ncolor) + calc_t_innergrid(X, ncolor, y, i, j) - calc_t_innergrid(X, ncolor, A(a), i, j)

See Also

generate_t_cache, composite.ll.

Examples

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

nrow <- 32
ncol <- 32

x <- matrix(sample(ncolor, nrow*ncol, replace=TRUE), nrow=nrow, ncol=ncol)
foo <- packPotts(x, ncolor)
out <- potts(foo, theta, nbatch=10)
x <- unpackPotts(out$final)

t_stat <- calc_t(x, ncolor)
t_stat_inner <- calc_t(x, ncolor, matrix(1, nrow=2, ncol=2), 1, 1)

---

Composite Log Likelihood for Potts Models

Description

Calculate Composite Log Likelihood (CLL) and the gradient of the CLL for Potts models.

Usage

composite.ll(theta, t_stat, t_cache=NULL, fapply=lapply)
gr.composite.ll(theta, t_stat, t_cache=NULL, fapply=lapply)

Arguments

theta	numeric canonical parameter vector. The CLL will be evaluated at this point. It is assumed that the component corresponding to the first color has been dropped.
t_stat	numeric, canonical statistic vector. The value of the canonical statistic for the full image.
t_cache	list of arrays. t_cache[[i]][j,] = the value of $t$ with window $A_i$ replaced by the $j^{th}$ element of $C^A$.
fapply	function. Expected to function as lapply does. Useful for enabling parallel processing. E.g. use the mclapply function from the multicore package.

Details

For the given value of theta composite.ll and gr.composite.ll calculate the CLL and the gradient of the CLL respectively for a realized Potts model represented by t_stat and t_cache.

$\mathcal{A}$ is the set of all windows to be used in calculating the Composite Log Likelihood (CLL) for a Potts model. A window is a collection of adjacent pixels on the lattice of the Potts model. $A$ is used to represent a generic window in $\mathcal{A}$, the code in this package expects that all the windows in $\mathcal{A}$ have the same size and shape. $|A|$ is used to denote the size, or number of pixels in a window. Each pixel in a Potts takes on a value in $C$, the set of possible colors. For simplicity, this implementation takes $C = \{1,\dots,\texttt{ncolor}\}$. Elements of $C$ will be referenced using $c_j$ with $j \in \{1,\dots,\texttt{ncolor}\}$. $C^A$ is used to denote all the permutations of $C$ across the window $A$, and $|C|^{|A|}$ is used to denote the size of $C^A$. In an abuse of notation, we use $A_a$ to refer to the $a^{th}$ element of $\mathcal{A}$. No ordinal or numerical properties of $\mathcal{A}$, $C$ or $C^A$ are used, only that each element in the sets are referenced by one and only one indexing value.

Value

composite.ll returns CLL evaluated at theta.

gr.composite.ll returns a numeric vector of length length(theta) containing the gradient of the CLL at theta.

See Also

generate_t_cache, calc_t.

Examples

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

nrow <- 32
ncol <- 32

x <- matrix(sample(ncolor, nrow*ncol, replace=TRUE), nrow=nrow, ncol=ncol)
foo <- packPotts(x, ncolor)
out <- potts(foo, theta, nbatch=10)
x <- unpackPotts(out$final)

t_stat <- calc_t(x, ncolor)
t_cache_mple <- generate_t_cache(x, ncolor, t_stat, nrow*ncol, 1,
                                 singleton)
t_cache_two <- generate_t_cache(x, ncolor, t_stat, nrow*ncol/2, 2,
                                twopixel.nonoverlap)

composite.ll(theta[-1], t_stat, t_cache_mple)
gr.composite.ll(theta[-1], t_stat, t_cache_mple)

## Not run: 
optim.mple <- optim(theta.initial, composite.ll, gr=gr.composite.ll,
                    t_stat, t_cache_mple, method="BFGS",
                    control=list(fnscale=-1))
optim.mple$par

optim.two <- optim(theta.initial, composite.ll, gr=gr.composite.ll,
                   t_stat, t_cache_two, method="BFGS",
                   control=list(fnscale=-1))
optim.two$par

## End(Not run)

## Not run: 
# or use mclapply to speed things up.
library(multicore)
optim.two <- optim(theta.initial, composite.ll, gr=gr.composite.ll,
                   t_stat, t_cache_two, mclapply, method="BFGS",
                   control=list(fnscale=-1))
optim.two$par


## End(Not run)

---

Plot Potts Model Data

Description

plot Potts model data.

Usage

## S3 method for class 'raw'
image(x, col = c("white", "red", "blue", "green",
    "black", "cyan", "yellow", "magenta"), ...)

Arguments

x	an R vector of class "raw" that encodes a realization of a Potts model, typically the output of packPotts or of potts.
col	a vector of colors. Must be as many as number of colors of Potts model.
...	other arguments passed to image.default.

Bugs

Too slow for large images. Needs to be rewritten for efficient plotting.

See Also

potts

---

Transform Potts Model Data

Description

transform Potts model data from integer matrix to raw vector and vice versa.

Usage

packPotts(x, ncolor)
inspectPotts(raw)
unpackPotts(raw)

Arguments

x	integer matrix containing Potts model data. Colors are coded from one to ncolor.
ncolor	integer scalar, number of colors.
raw	vector of type "raw".

Value

for packPotts a vector of type "raw". for inspectPotts a list containing components ncolor, nrow, and ncol. for unpackPotts an integer matrix.

Examples

x <- matrix(sample(4, 2 * 3, replace = TRUE), nrow = 2)
x
foo <- packPotts(x, ncolor = 4)
foo
inspectPotts(foo)
unpackPotts(foo)

---

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)

---