pclust {relations}

Prototype-Based Partitions of Relations

Description

Compute prototype-based partitions of a relation ensemble by minimizing \sum w_b u_{bj}^m d(x_b, p_j)^e, the sum of the case-weighted and membership-weighted e-th powers of the dissimilarities between the elements x_b of the ensemble and the prototypes p_j, for suitable dissimilarities d and exponents e.

Usage

relation_pclust(x, k, method, m = 1, weights = 1,
                control = list())

Arguments

x

an ensemble of relations (see relation_ensemble()), or something which can be coerced to this.

k

an integer giving the number of classes to be used in the partition.

method

the consensus method to be employed, see relation_consensus().

m

a number not less than 1 controlling the softness of the partition (as the “fuzzification parameter” of the fuzzy c-means algorithm). The default value of 1 corresponds to hard partitions obtained from a generalized k-means problem; values greater than one give partitions of increasing softness obtained from a generalized fuzzy c-means problem.

weights

a numeric vector of non-negative case weights. Recycled to the number of elements in the ensemble given by x if necessary.

control

a list of control parameters. See Details.

Details

For m = 1, a generalization of the Lloyd-Forgy variant of the k-means algorithm is used, which iterates between reclassifying objects to their closest prototypes, and computing new prototypes as consensus relations (generalized “central relations”, Régnier (1965)) for the classes. This procedure was proposed in Gaul and Schader (1988) as the “Clusterwise Aggregation of Relations” (CAR).

For m > 1, a generalization of the fuzzy c-means recipe is used, which alternates between computing optimal memberships for fixed prototypes, and computing new prototypes as the consensus relations for the classes.

This procedure is repeated until convergence occurs, or the maximal number of iterations is reached.

Consensus relations are computed using relation_consensus().

Available control parameters are as follows.

maxiter

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

reltol

the relative convergence tolerance. Defaults to sqrt(.Machine$double.eps).

control

control parameters to be used in relation_consensus().

The dissimilarities d and exponent e are implied by the consensus method employed, and inferred via a registration mechanism currently only made available to built-in consensus methods. For the time being, all optimization-based consensus methods use the symmetric difference dissimilarity (see relation_dissimilarity()) for d and e = 1.

The fixed point approach employed is a heuristic which cannot 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.

Value

An object of class cl_partition().

References

S. Régnier (1965). Sur quelques aspects mathématiques des problèmes de classification automatique. ICC Bulletin, 4, 175–191.

W. Gaul and M. Schader (1988). Clusterwise aggregation of relations. Applied Stochastic Models and Data Analysis, 4, 273–282. doi:10.1002/asm.3150040406.