Jackknife and bootstrap stability of a SMACOF solution

MdsJackknife: leave-one-out refits rotated to a common comparison configuration, with stability, cross-validity and dispersion (De Leeuw and Meulman 1986), as smacof::jackmds. MdsBootstrap: row resampling of an \(N \times n\) data matrix whose columns are the MDS objects, dissimilarities \(\sqrt{1 - r}\) (Pearson or Spearman) or Euclidean, each refit Procrustes-fitted onto the original; per-object covariance matrices, stress percentile interval and stability (Jacoby and Armstrong 2014), as smacof::bootmds. method = "standard" uses the full sum in the jackknife Procrustes update and Frobenius norms; method = "smacof" reproduces smacof 2.1 (update over later objects only, base::norm's default one-norm). Identical to the Python arm morie.fn.mdsext; resamples are 1-based here.

Usage

MdsJackknife(
  delta,
  ndim = 2,
  type = "ratio",
  method = "standard",
  eps = 1e-06,
  itmax = 100L
)

MdsBootstrap(
  data,
  ndim = 2,
  method_dat = "pearson",
  nrep = 100L,
  alpha = 0.05,
  type = "ratio",
  method = "standard",
  seed = 1,
  resamples = NULL
)

Arguments

delta: Dissimilarity matrix.
ndim: Dimensions.
type: SMACOF type (see SmacofMds).
method: "standard" or "smacof".
eps, itmax: Jackknife convergence tolerance and iteration limit.
data: Data matrix, columns are objects.
method_dat: "pearson", "spearman" or "euclidean".
nrep: Bootstrap replicates.
alpha: Stress interval level.
seed: Philox seed.
resamples: Optional list of row-index vectors.

Value

List.

References

De Leeuw, J. and Meulman, J. (1986). A special jackknife for multidimensional scaling. Journal of Classification 3, 97-112.

Jacoby, W. G. and Armstrong, D. A. (2014). Bootstrap confidence regions for multidimensional scaling solutions. American Journal of Political Science 58, 264-278.

Examples

D <- rbind(c(0, 1, 2, 1), c(1, 0, 1, 2), c(2, 1, 0, 1), c(1, 2, 1, 0))
MdsJackknife(D)$stab
#> [1] 0.8181818
X <- cbind(1:6, c(2, 1, 5, 3, 6, 4), c(3, 4, 2, 6, 5, 8), c(1, 0, 2, 1, 3, 2))
MdsBootstrap(X, 2, method_dat = "euclidean", nrep = 5)$bootci
#> [1] 0.001488274 0.021511841