BlocksGibbs {PottsUtils}

Generate Random Samples from a Potts Model Using the Checkerboard Idea

Description

Generate random samples from a Potts model by Gibbs Sampling that takes advantage of conditional independence.

Usage

  BlocksGibbs(n, nvertex, ncolor, neighbors, blocks,
              weights=1, spatialMat=NULL, beta)

Arguments

n

number of samples.

nvertex

number of vertices in a graph.

ncolor

number of colors each vertex can take.

neighbors

a matrix of all neighbors in a graph, one row per vertex.

blocks

a list of blocks of vertices in a graph.

weights

weights between neighbors. One for each corresponding neighbor in neighbors. The default values are 1s for all.

spatialMat

the matrix that describes the relationship among vertices in neighbor. The default value is NULL corresponding to the simple or compound Potts model.

beta

the parameter inverse temperature of the Potts model.

Details

We use the Gibbs algorithm that takes advantage of conditional independence to speed up the generation of random samples from a Potts model. The idea is that if we can divide variables that need to be updated into different blocks and given the variables in other blocks, all the variables within the same block are conditionally independent, then we can update all blocks iteratively with the variables within the same block being updated simultaneously.

The spatialMat is the argument used to specify the relationship among vertices in neighbor. See rPotts1 for more information on the Potts model and spatialMat.

Value

The output is a nvertex by n matrix with the kth column being the kth sample.

References

Dai Feng (2008) Bayesian Hidden Markov Normal Mixture Models with Application to MRI Tissue Classification Ph. D. Dissertation, The University of Iowa

See Also

Wolff, SW

Examples

  #Example 1: Generate 100 samples from a repulsion Potts model with the
  #           neighborhood structure corresponding to a first-order
  #           Markov random field defined on a 3*3 2D graph.
  #           The number of colors is 3 and beta=0.1,a_1=2,a_2=1,a_3=0.
  #           All weights are equal to 1.
 
  neighbors <- getNeighbors(mask=matrix(1, 3, 3), neiStruc=c(2,2,0,0))
  blocks <- getBlocks(mask=matrix(1, 3, 3), nblock=2)
  spatialMat <- matrix(c(2,1,0, 1,2,1,0,1,2), ncol=3)
  BlocksGibbs(n=100, nvertex=9, ncolor=3, neighbors=neighbors, blocks=blocks,
              spatialMat=spatialMat, beta=0.1)