R: Least Squares Fit of Sums of Ultrametrics to Dissimilarities

ls_fit_sum_of_ultrametrics {clue} | R Documentation

## 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

`x` | 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. Otherwise, it is recycled to the number of elements in `x`.
`control` | a list of control parameters. See Details.

### Details

The problem to be solved is minimizing the criterion function

L(u(1), …, u(n)) = ∑_{i,j} w_{ij} ≤ft(x_{ij} - ∑_{k=1}^n u_{ij}(k)\right)^2

over all u(1), …, 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 - ∑_{l \ne k} u(l) using `ls_fit_ultrametric`.

Available control parameters include

`maxiter`

  an integer giving the maximal number of iteration steps to be performed. Defaults to 100.

`eps`

  a nonnegative number controlling the iteration, which stops when the maximal change in all u(k) is less than `eps`. Defaults to 10^{-6}.

`reltol`

  the relative convergence tolerance. Iteration stops when the relative change in the criterion function is less than `reltol`. Defaults to 10^{-6}.

`method`

  a character string indicating the fitting method to be employed by the individual least squares fits.

`control`

  a 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.

### Value

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

### 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.

[Package clue version 0.3-54 Index]