Three-way SMACOF (INDSCAL, IDIOSCAL, replicated MDS)

Ratio three-way SMACOF as smacof::smacofIndDiff: each source gets \(X_j = Z C_j\) with diagonal \(C_j\) ("indscal", subject weights), general \(C_j\) ("idioscal") or \(C_j = I\) ("identity", replicated MDS). Unconstrained Guttman transforms are projected on the constraint each iteration; dissimilarities are normalised per source, missing ones get weight 0; the start is the Torgerson solution of the summed dissimilarities (Carroll and Chang 1970; De Leeuw and Mair 2009). Identical to the Python arm morie.fn.mdsext.smacof_indiff.

Usage

SmacofIndDiff(
  deltas,
  ndim = 2,
  constraint = "indscal",
  itmax = 1000L,
  eps = 1e-06
)

Arguments

deltas

List of dissimilarity matrices.

ndim

Dimensions.

constraint

"indscal", "idioscal" or "identity".

itmax

Maximum iterations.

eps

Convergence tolerance on raw stress.

Value

List with conf, gspace, cweights, stress, sps, spp, confdist, niter.

References

Carroll, J. D. and Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an N-way generalization of Eckart-Young decomposition. Psychometrika 35, 283-319.

De Leeuw, J. and Mair, P. (2009). Multidimensional scaling using majorization: SMACOF in R. Journal of Statistical Software 31(3), 1-30.

Examples

D1 <- as.matrix(dist(1:4))
SmacofIndDiff(list(D1, D1), 1, constraint = "identity")$stress
#> [1] 1.570092e-16