Find the additive tree distance or centroid distance minimizing least squares distance (Euclidean dissimilarity) to a given dissimilarity object.
ls_fit_addtree(x, method = c("SUMT", "IP", "IR"), weights = 1,
control = list())
ls_fit_centroid(x)An object of class "cl_addtree" containing the optimal additive
tree distances.
a dissimilarity object inheriting from class
"dist".
a character string indicating the fitting method to be
employed. Must be one of "SUMT" (default), "IP", or
"IR", or a unique abbreviation thereof.
a numeric vector or matrix with non-negative weights
for obtaining a weighted least squares fit. If a matrix, its
numbers of rows and columns must be the same as the number of
objects in x, and the lower diagonal part is used.
Otherwise, it is recycled to the number of elements in x.
a list of control parameters. See Details.
See as.cl_addtree for details on additive tree distances
and centroid distances.
With \(L(d) = \sum w_{ij} (x_{ij} - d_{ij})^2\), the problem to be
solved by ls_fit_addtree is minimizing \(L\) over all
additive tree distances \(d\). This problem is known to be NP
hard.
We provide three heuristics for solving this problem.
Method "SUMT" implements the SUMT (Sequential
Unconstrained Minimization Technique, Fiacco and McCormick, 1968)
approach of de Soete (1983). Incomplete dissimilarities are currently
not supported.
Methods "IP" and "IR" implement the Iterative
Projection and Iterative Reduction approaches of Hubert and Arabie
(1995) and Roux (1988), respectively. Non-identical weights and
incomplete dissimilarities are currently not supported.
See ls_fit_ultrametric for details on these methods and
available control parameters.
It should be noted that all methods are heuristics which can not be guaranteed to find the global minimum. Standard practice would recommend to use the best solution found in “sufficiently many” replications of the base algorithm.
ls_fit_centroid finds the centroid distance \(d\) minimizing
\(L(d)\) (currently, only for the case of identical weights). This
optimization problem has a closed-form solution.
A. V. Fiacco and G. P. McCormick (1968). Nonlinear programming: Sequential unconstrained minimization techniques. New York: John Wiley & Sons.
L. Hubert and P. Arabie (1995). Iterative projection strategies for the least squares fitting of tree structures to proximity data. British Journal of Mathematical and Statistical Psychology, 48, 281--317. tools:::Rd_expr_doi("10.1111/j.2044-8317.1995.tb01065.x").
M. Roux (1988). Techniques of approximation for building two tree structures. In C. Hayashi and E. Diday and M. Jambu and N. Ohsumi (Eds.), Recent Developments in Clustering and Data Analysis, pages 151--170. New York: Academic Press.
G. de Soete (1983). A least squares algorithm for fitting additive trees to proximity data. Psychometrika, 48, 621--626. tools:::Rd_expr_doi("10.1007/BF02293884").