distatis {DistatisR}

3-Way MDS based on the "STATIS" optimization procedure.

Description

distatis: Implements the DISTATIS method which is a 3-way generalization of metric multidimensional scaling (a.k.a. classical MDS or principal coordinate analysis).

Usage

distatis(
  LeCube2Distance,
  Norm = "MFA",
  Distance = TRUE,
  double_centering = TRUE,
  RV = TRUE,
  nfact2keep = 3,
  compact = FALSE
)

Arguments

LeCube2Distance

an "observations \times observations \times distance matrices" array of dimensions I\times I \times K. Each of the K "slices" is a I\times I square distance (or covariance) matrix describing the I observations.

Norm

Type of normalization used for each cross-product matrix derived from the distance (or covariance) matrices. Current options are NONE (do nothing), SUMPCA (normalize by the total inertia) or MFA (default) that normalizes each matrix so that its first eigenvalue is equal to one or NUCLEAR (i.e., the of the squarae root of the eigenvalues).

Distance

if TRUE (default) the matrices are distance matrices, FALSE the matrices are treated as positive semi-definite matrices (e.g., scalar products, covariance, or correlation matrices).

double_centering

if TRUE (default) the matrices are double-centered (should always be used for distances). if FALSE the matrices will not be double centered (note that these matrices should be semi positive definite matrices such as, for example, covariance matrices).

RV

if TRUE (default) we use the R_V coefficient to compute the \alpha, if FALSE we use the matrix scalar product.

nfact2keep

(default: 3) Number of factors to keep for the computation of the factor scores of the observations.

compact

if FALSE (default), distatis provides detailed output, if TRUE, distatis sends back only the \alpha weights (this option is used to make the bootstrap routine BootFromCompromise more computationally efficient).

Details

distatis takes as input a set of K distance matrices (or positive semi-definite matrices such as scalar products, covariance, or correlation matrices) describing a set of I observations. From this set of matrices distatis computes: (1) a set of factor scores that describes the similarity structure of the K distance matrices (e.g., what distance matrices describe the observations in the same way, what distance matrices differ from each other) (2) a set of factor scores (called the compromise factor scores) that best describes the similarity structure of the I observations and (3) I sets of partial factor scores that show how each individual distance matrix "sees" the compromise space.

distatis computes the compromise as an optimum linear combination of the cross-product matrices associated to each distance (or positive positive semi-definite) matrix.

distatis can also be applied to a set of scalar products, covariance, or correlation matrices.

DISTATIS is part of the STATIS family. It is often used to analyze the results of sorting tasks.

Value

distatis sends back the results via two lists: res.Cmat and res.Splus. Note that items with a * are the only ones sent back when using the compact = TRUE option.

res.Cmat

Results for the between distance matrices analysis.

• res.Cmat$C The I\times I C matrix of scalar products (or R_V between distance matrices).

• res.Cmat$vectors The eigenvectors of the C matrix

• res.Cmat$alpha * The \alpha weights

• res.Cmat$value The eigenvalues of the C matrix

• res.Cmat$G The factor scores for the C matrix

• res.Cmat$ctr The contributions for res.Cmat$G,

• res.Cmat$cos2 The squared cosines for res.Cmat$G

• res.Cmat$d2 The squared Euclidean distance for res.Cmat$G.

res.Splus

Results for the between observation analysis.

• res.Splus$SCP an I\times I\times K array. Contains the (normalized if needed) cross product matrices corresponding to the distance matrices.

• res.Splus$Splus * The compromise (optimal linear combination of the SCP's').

• res.Splus$eigValues * The eigenvalues of the compromise).

• res.Splus$eigVectors * The eigenvectors of the compromise).

• res.Splus$tau * The percentage of explained inertia of the eigenValues).

• res.Splus$ProjectionMatrix The projection matrix used to compute factor scores and partial factor scores.

• res.Splus$F The factor scores for the observations.

• res.Splus$ctr The contributions for res.Cmat$F.

• res.Splus$cos2 The squared cosines for res.Cmat$F.

• res.Splust$d2 The squared Euclidean distance for res.Cmat$F.

• res.Splus$PartialF an I \times \code{nf2keep} \times K array. Contains the partial factors for the distance matrices.

Author(s)

Hervé Abdi #@seealso GraphDistatisAll GraphDistatisBoot #GraphDistatisCompromise # GraphDistatisPartial #GraphDistatisRv DistanceFromSort #BootFactorScores BootFromCompromise #as help,

References

Abdi, H., Valentin, D., O'Toole, A.J., & Edelman, B. (2005). DISTATIS: The analysis of multiple distance matrices. Proceedings of the IEEE Computer Society: International Conference on Computer Vision and Pattern Recognition. (San Diego, CA, USA). pp. 42–47.

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.

Abdi, H., Dunlop, J.P., & Williams, L.J. (2009). How to compute reliability estimates and display confidence and tolerance intervals for pattern classifiers using the Bootstrap and 3-way multidimensional scaling (DISTATIS). NeuroImage, 45, 89–95.

Abdi, H., Williams, L.J., Valentin, D., & Bennani-Dosse, M. (2012). STATIS and DISTATIS: Optimum multi-table principal component analysis and three way metric multidimensional scaling. Wiley Interdisciplinary Reviews: Computational Statistics, 4, 124–167.

The R_V coefficient is described in

Abdi, H. (2007). RV coefficient and congruence coefficient. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 849–853.

Abdi, H. (2010). Congruence: Congruence coefficient, RV coefficient, and Mantel Coefficient. In N.J. Salkind, D.M., Dougherty, & B. Frey (Eds.): Encyclopedia of Research Design. Thousand Oaks (CA): Sage. pp. 222–229.

(These papers are available from https://personal.utdallas.edu/~herve/)

Examples

# 1. Load the DistAlgo data set 
# (available from the DistatisR package).
data(DistAlgo)
# DistAlgo is a 6*6*4 Array (face*face*Algorithm)
#------------------------------------------------------------------
# 2. Call the DISTATIS routine with the array 
#  of distance (DistAlgo) as parameter
DistatisAlgo <- distatis(DistAlgo)