DISTATIS for Multiple Distance Matrices

Fits a consensus object map, an assessor map based on RV similarities, and partial object maps for distances measured on the same objects.

Usage

distatis(
  data,
  ncomp = 2,
  squared = FALSE,
  norm_method = c("mfa", "frobenius", "none"),
  labels = NULL,
  tol = 1e-08,
  ...
)

# Default S3 method
distatis(
  data,
  ncomp = 2,
  squared = FALSE,
  norm_method = c("mfa", "frobenius", "none"),
  labels = NULL,
  tol = 1e-08,
  ...
)

Arguments

data: A list of numeric square distance matrices or stats::dist() objects, an object-by-object-by-assessor numeric array, or a single distance matrix or dist object. Object names and order must agree across tables.
ncomp: Positive integer; maximum number of axes retained in each map. Object and assessor ranks are determined independently.
squared: Logical; are the supplied distances already squared? Defaults to FALSE, including for dist objects.
norm_method: Table normalization: "mfa" (default) divides each centered Gram matrix by its largest eigenvalue; "frobenius" gives unit Frobenius norm; "none" preserves scale. RV similarities are used in all three cases.
labels: Optional unique object labels. These override display labels, but do not bypass checks of input object names and order.
tol: Relative numerical tolerance, strictly between zero and one, for input symmetry, Euclidean geometry, spectral rank, and leading-root ties.
...: Reserved for methods. Unused arguments are rejected.

Value

A distatis object inheriting from covstatis and projector. object_scores (also roi_scores) and subject_scores contain the two maps; partial_scores contains one object map per assessor; alpha contains weights; C is the RV matrix; compromise_full is the normalized compromise Gram matrix. eigenvalues and interstructure_eigenvalues retain the full spectra, while inertia and subject_inertia use their complete positive inertia as denominators. normalization_constants, input_labels, squared, norm_method, tol, and diagnostics record preprocessing.

Details

For ordinary distance matrices \(D_k\), DISTATIS computes \(B_k=-HD_k^2H/2\), with elementwise squaring and \(H=I-11^T/n\). The normalized Gram matrices \(S_k\) are compared using RV coefficients. The leading eigenvector of the RV matrix, oriented nonnegatively and rescaled to sum to one, supplies the compromise weights. The weighted compromise has object coordinates \(V\Lambda^{1/2}\); each partial map is \(S_kV\Lambda^{-1/2}\). Their weighted mean equals the object map. Assessor coordinates use the RV matrix's own spectrum.

Non-Euclidean distances, zero Gram matrices, missing values, and a tied leading RV eigenvalue are rejected. Small negative eigenvalues within tol are treated as numerical zero for rank and inertia calculations. No distance correction or imputation is performed. Weights describe agreement with the fitted panel, rather than assessor accuracy. Inference is not currently implemented.

References

Abdi, H., Valentin, D., Chollet, S., & Chrea, C. (2007). Analyzing assessors and products in sorting tasks: DISTATIS, theory and applications. Food Quality and Preference, 18, 627-640. doi:10.1016/j.foodqual.2006.09.003 .

Examples

partitions <- data.frame(A = c(1, 1, 2, 3), B = c(1, 1, 2, 2),
                         C = c(1, 2, 3, 3))
distances <- sorting_distances(partitions)
fit <- distatis(distances)
fit$alpha
#> [1] 0.3269993 0.3460015 0.3269993
multivarious::scores(fit)
#>   Dim1          Dim2
#> 1 -0.5  0.000000e+00
#> 2 -0.5  3.233164e-17
#> 3  0.5 -3.301511e-01
#> 4  0.5  3.301511e-01
project_distance(fit, distances[[1]])
#>   Dim1          Dim2
#> 1 -0.5 -1.347650e-17
#> 2 -0.5 -1.347650e-17
#> 3  0.5 -1.009639e+00
#> 4  0.5  1.009639e+00