Bayesian Multidimensional Scaling

A Bayesian formulation of classical Multidimensional Scaling is presented. Even though this method is based on MCMC sampling, we only return maximum a posterior (MAP) estimate that maximizes the posterior distribution. Due to its nature without any special tuning, increasing mc.iter requires much computation.

bmds(
  data,
  ndim = 2,
  par.a = 5,
  par.alpha = 0.5,
  par.step = 1,
  mc.iter = 8128,
  verbose = TRUE
)

Arguments

data

an \((n\times p)\) matrix whose rows are observations.

ndim

an integer-valued target dimension.

par.a

hyperparameter for conjugate prior on variance term, i.e., \(\sigma^2 \sim IG(a,b)\). Note that \(b\) is chosen appropriately as in paper.

par.alpha

hyperparameter for conjugate prior on diagonal term, i.e., \(\lambda_j \sim IG(\alpha, \beta_j)\). Note that \(\beta_j\) is chosen appropriately as in paper.

par.step

stepsize for random-walk, which is standard deviation of Gaussian proposal.

mc.iter

the number of MCMC iterations.

verbose

a logical; TRUE to show iterations, FALSE otherwise.

Value

a named list containing

embed

an \((n\times ndim)\) matrix whose rows are embedded observations.

stress

discrepancy between embedded and origianl data as a measure of error.

References

Oh M, Raftery AE (2001). “Bayesian Multidimensional Scaling and Choice of Dimension.” Journal of the American Statistical Association, 96(455), 1031--1044.

Examples

# \donttest{
## use simple example of iris dataset
data(iris) 
idata = as.matrix(iris[,1:4])

## run Bayesian MDS
#  let's run 10 iterations only.
iris.cmds = cmds(idata, ndim=2)
iris.bmds = bmds(idata, ndim=2, mc.iter=5, par.step=(2.38^2)) 
#> ** bmds : iteration 1/5 complete.
#> ** bmds : iteration 2/5 complete.
#> ** bmds : iteration 3/5 complete.
#> ** bmds : iteration 4/5 complete.
#> ** bmds : iteration 5/5 complete.

## extract coordinates and class information
cx = iris.cmds$embed # embedded coordinates of CMDS
bx = iris.bmds$embed #                         BMDS
icol = iris[,5]      # class information

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,1))
mc = paste0("CMDS with STRESS=",round(iris.cmds$stress,4))
mb = paste0("BMDS with STRESS=",round(iris.bmds$stress,4))
plot(cx, col=icol,pch=19,main=mc)
plot(bx, col=icol,pch=19,main=mb)

par(opar)
# }