BlocksGibbs: Generate Random Samples from a Potts Model Using the...
In PottsUtils: Utility Functions of the Potts Models

Description Usage Arguments Details Value References See Also Examples

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

1 2	BlocksGibbs(n, nvertex, ncolor, neighbors, blocks, weights=1, spatialMat=NULL, beta)

`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.

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.

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

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

Wolff, SW

  #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)