ls_fit_sum_of_ultrametrics: Least Squares Fit of Sums of Ultrametrics to Dissimilarities

Description

Find a sequence of ultrametrics with sum minimizing square distance (Euclidean dissimilarity) to a given dissimilarity object.

Usage

ls_fit_sum_of_ultrametrics(x, nterms = 1, weights = 1, control = list())

Arguments

a dissimilarity object inheriting from or coercible to class "dist".

nterms

an integer giving the number of ultrametrics to be fitted.

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

A list of objects of class "cl_ultrametric" containing the fitted ultrametric distances.

Details

The problem to be solved is minimizing the criterion function $$L(u(1), \dots, u(n)) = \sum_{i,j} w_{ij} \left(x_{ij} - \sum_{k=1}^n u_{ij}(k)\right)^2$$ over all $u(1), \ldots, u(n)$ satisfying the ultrametric constraints.

We provide an implementation of the iterative heuristic suggested in Carroll & Pruzansky (1980) which in each step $t$ sequentially refits the $u(k)$ as the least squares ultrametric fit to the residuals $x - \sum_{l \ne k} u(l)$ using ls_fit_ultrametric.

Available control parameters include [object Object],[object Object],[object Object],[object Object],[object Object]

It should be noted that the method used is a heuristic which can not be guaranteed to find the global minimum.

References

J. D. Carroll and S. Pruzansky (1980). Discrete and hybrid scaling models. In E. D. Lantermann and H. Feger (eds.), Similarity and Choice. Bern (Switzerland): Huber.