addtree: Least Squares Fit of Additive Tree Distances to Dissimilarities

Description

Find the additive tree distance or centroid distance minimizing least squares distance (Euclidean dissimilarity) to a given dissimilarity object.

Usage

ls_fit_addtree(x, method = c("SUMT", "IP", "IR"), weights = 1,
               control = list())
ls_fit_centroid(x)

Arguments

a dissimilarity object inheriting from class "dist".

method

a character string indicating the fitting method to be employed. Must be one of "SUMT" (default), "IP", or "IR", or a unique abbreviation thereof.

weights

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.

control

a list of control parameters. See Details.

Value

An object of class "cl_addtree" containing the optimal additive tree distances.

encoding

UTF-8

Details

Additive tree distances are object dissimilarities $d$ satisfying the so-called additive tree conditions, also known as four-point conditions $d_{ij} + d_{kl} \le \max(d_{ik} + d_{jl}, d_{il} + d_{jk})$ for all quadruples $i, j, k, l$. Equivalently, for each such quadruple, the largest two values of the sums $d_{ij} + d_{kl}$, $d_{ik} + d_{jl}$, and $d_{il} + d_{jk}$ must be equal. Centroid distances are additive tree distances where the inequalities in the four-point conditions are strengthened to equalities (such that all three sums are equal), and can be represented as $d_{ij} = g_i + g_j$, i.e., as sums of distances from a centroid. See, e.g., Barthélémy and Guénoche (1991) for more details on additive tree 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.

There is a plot method for the fitted additive tree distance.

References

J.-P. Barthélémy and A. Guénoche (1991). Trees and proximity representations. Chichester: John Wiley & Sons. ISBN 0-471-92263-3.

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.

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.