bmds {bayMDS}

run bmdsMCMC for various number of dimensions

Description

Provide object configuration and estimates of parameters, for number of dimensions from min_p to max_p

Usage

bmds(DIST,min_p=1, max_p=6,nwarm = 1000,niter = 5000,...)

Arguments

DIST

symmetric data matrix of dissimilarity measures for pairs of objects

min_p

minimum number of dimensions for object configuration (default=1)

max_p

maximum number of dimensions for object configuration (default=6)

nwarm

number of iterations for burn-in period in MCMC (default=1000)

niter

number of MCMC iterations after burn-in period (default=5000)

...

arguments to be passed to methods.

Details

Model

The basic model for Bayesian multidimensional scaling given in Oh and Raftery (2001) is as follows. Given the number of dimensions p, we assume that an observed dissimilarity measure follows a truncated multivariate normal distribution with mean equal to Euclidean distance, i.e.,

d_{ij} \sim N ( \delta_{ij}, \sigma^2 )I( d_{ij} > 0), independently for i \ne j, i,j=1, \cdots,n,

where

• n is the number of objects, i.e, numner of rows in DIST

• d_{ij} is an observed dissimilarity measure between objects i and j

• \delta_{ij} is the distance between objects i and j in a p-dimensional Euclidean space, i.e.,

  \delta_{ij} = \sqrt{ \sum_{k=1}^p (x_{ik}-x_{jk})^2 }

• x_i=(x_{i1},...,x_{ip}) denotes the values of the attributes possessed by object i, i.e., the coordinates of object i in a p-dimensional Euclidean space.

Priors

• Prior distribution of x_i is given as a multivariate normal distribution with mean 0 and a diagonal covariance matrix \Lambda, i.e., x_i \sim N(0,\Lambda), independently for i = 1,\cdots,n. Note that the zero mean and diagonal covariance matrix is assumed because Euclidean distance is invariant under translation and rotation of X=\{x_i\}.

• Prior distribution of the error variance \sigma^2 is given as \sigma^2 \sim IG(a,b), the inverse Gamma distribution with mode b/(a+1).

• Hyperpriors for the elements of \Lambda = diag (\lambda_1,...,\lambda_p) are given as \lambda_j \sim IG(\alpha, \beta_j), independently for j=1,\cdots,p.

• We assume prior independence among X, \Lambda,\sigma^2.

Measure of fit

A measure of fit, called STRESS, is defined as

STRESS =\sqrt{{\sum_{i > j} (d_{ij}-\hat{\delta}_{ij})^2 } \over {\sum_{i > j} d_{ij}^2 }},

where \hat{\delta}_{ij} is the Euclidean distance between objects i and j, computed from the estimated object configuration. Note that the squared STRESS is proportional to the sum of squared residuals, SSR=\sum_{i > j} (d_{ij}-\hat{\delta}_{ij})^2.

Value

in bmds object

n

number of objects, i.e., number of rows in DIST

min_p

minimum number of dimensions

max_p

maximum number of dimensions

niter

number of MCMC iterations

nwarm

number of burn-in in MCMC

*

the following lists contains objects from bmdsMCMC for number of dimensions from min_p to max_p

x_bmds

a list of object configurations

minSSR.L

a list of minimum sum of squares of residuals between the observed dissimilarities and the estimated Euclidean distances between pairs of objects

minSSR_id.L

a list of the indecies of the iteration corresponding to minimum SSR

stress.L

a list of STRESS values

e_sigma.L

a list of posterior mean of \sigma^2

var_sigma.L

a list of posterior variance of \sigma^2

SSR.L

a list of posterior samples of SSR

lam.L

a list of posterior samples of elements of \Lambda

sigma.L

a list of posterior samples of \sigma^2, the error variance

del.L

a list of posterior samples of \deltas,Euclidean distances between pairs of objects)

cmds.L

a list of object configuration from the classical multidimensional scaling of Togerson(1952)

BMDSp

a list of outputs from bmdsMCMC founction for each number of dimensions

References

Oh, M-S., Raftery A.E. (2001). Bayesian Multidimensional Scaling and Choice of Dimension, Journal of the American Statistical Association, 96, 1031-1044.

Torgerson, W.S. (1952). Multidimensional Scaling: I. Theory and Methods, Psychometrika, 17, 401-419.

Examples

data(cityDIST)
out <- bmds(cityDIST)