homals {homals}

Homogeneity Analysis

Description

This function performs a homogeneity analysis, aka a multiple correspondence analysis, but with many additional options. Variables can be grouped into sets, in order to emulate regression analysis and canonical analysis. For each variable there are, in addition, rank constraints on the category quantifications (or transformations) and level constraints (which allows one to treat a variable as nominal, ordinal, or numerical).

Usage

homals(data, ndim = 2, rank = ndim, level = "nominal", sets = 0, active = TRUE, 
       eps = 1e-06, itermax = 1000, verbose = 0)

Arguments

data

Data in data frame.

ndim

Number of dimensions to be extracted.

rank

Which quantification ranks. Default is number of dimensions ndim

level

Which quantification levels. Possible values are "nominal", "ordinal", "numerical", and "polynomial" which can be defined as single character (if all variable are of the same level) or as vector which length corresponds to the number of variables.

sets

List of vectors of indices.

active

Which variables are active for computation (TRUE means all)

eps

Iteration precision eigenvalues.

itermax

Maximum number of iterations.

verbose

0 for no iteration output, 1 prints outer iterations, 2 prints outer and inner iterations, 3 prints outer and inner and innermost iterations.

Details

The general idea of homogeneity analysis is to make a joint plot in p-space of all objects (or individuals) and the categories of all variables. The technique, in its most simple form, makes the graph plot in such a way that the sum of squares of the length of the nm lines is a small as possible, subject to a normalization of the object scores (their n x p coordinate matrix must be orthonormal).

Rank constraints require the category quantifications of the categories of a variable to lie in a subspace of p-space. Requiring rank equal to one for all variables reduces homogeneity analysis to principal component analysis (with optimal scaling of the variables).

Sets of variables are incorporated by using additivity restrictions on the category quantifications (i.e. we code the variables within a set interactively, but then use quantifications based on main effects only).

By combining the various types of restrictions we obtain far-reaching generalizations of principal component analysis, canonical analysis, discriminant analysis and regression analysis.

Value

Returns an object of class "homals".

datname

Name of the data file.

catscores

List of category scores for each variable.

scoremat

Array containing the reproduced data matrix based on category scores.

objscores

Matrix with object scores on each dimension.

cat.centroids

List of category centroids.

ind.mat

Dummy coded indicator matrix.

loadings

List of loadings of the variables on each dimension.

low.rank

List of lower rank quantifications.

discrim

Matrix with discrimination measures for each variable on each dimension.

niter

Number of iterations.

eigenvalues

Final eigenvalues.

loss

Value of the loss function.

rank.vec

Vector with ranks for each variable.

active

Vector with active/inactive variables.

Author(s)

Jan de Leeuw, Patrick Mair

References

de Leeuw, J., & Mair, P. (2009). Gifi methods for optimal scaling in R: The package homals. Journal of Statistical Software, 31(4), 1-20, https://www.jstatsoft.org/v31/i04/.

Gifi, A. (1990). Nonlinear Multivariate Analysis. New York: Wiley.

See Also

plot.homals

Examples

  data(galo)
  
  ##simple homals 2D-homals solution (School not active)
  res <- homals(galo, active = c(rep(TRUE, 4), FALSE))
  
  ##predict IQ 
  res <- homals(galo, active = c(rep(TRUE, 4), FALSE), sets = list(c(1,3,4),2,5))
  
  ##Single ranks for each variable (non-linear PCA)
  res <- homals(galo, active = c(rep(TRUE, 4), FALSE), sets = list(c(1,3,4),2,5))
  
  ##Nonlinear canonical correlation (2 sets of variables)
  res <- homals(galo, active = c(rep(TRUE, 4), FALSE), sets = list(c(1,2),c(3,4),5))
  
  ##regression (linear)
  data(neumann)
  res <- homals(neumann, sets = list(3, 1:2), level = "numerical", rank = 1)
  
  ##regression (monotone)
  res <- homals(neumann, sets = list(3, 1:2), level = "ordinal", rank = 1)
  
  ##3-dimensional senate solution
  data(senate)
  res <- homals(senate, active = c(FALSE, rep(TRUE, 20)), ndim = 3)
  
  ##Predicting Species in iris dataset (polynomial level constraints for predictors)
  data(iris)
  res <- homals(iris, sets = list(1:4,5), level = c(rep("polynomial",4),"nominal"), 
  rank = 2, itermax = 1000)