Find a sequence of ultrametrics with sum minimizing square distance (Euclidean dissimilarity) to a given dissimilarity object.
ls_fit_sum_of_ultrametrics(x, nterms = 1, weights = 1,
control = list())a dissimilarity object inheriting from or coercible to class
"dist".
an integer giving the number of ultrametrics to be fitted.
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.
A list of objects of class "cl_ultrametric" containing
the fitted ultrametric distances.
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
maxiteran integer giving the maximal number of iteration steps to be performed. Defaults to 100.
epsa nonnegative number controlling the iteration,
which stops when the maximal change in all \(u(k)\) is less than
eps.
Defaults to \(10^{-6}\).
reltolthe relative convergence tolerance. Iteration
stops when the relative change in the criterion function is less
than reltol.
Defaults to \(10^{-6}\).
methoda character string indicating the fitting method to be employed by the individual least squares fits.
controla list of control parameters to be used by the
method of ls_fit_ultrametric employed. By default,
if the SUMT method method is used, 10 inner
SUMT runs are performed for each refitting.
It should be noted that the method used is a heuristic which can not be guaranteed to find the global minimum.
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.