clue (version 0.3-27)

pclust: Prototype-Based Partitions of Clusterings

Description

Compute prototype-based partitions of a cluster 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

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

Arguments

x

an ensemble of partitions or hierarchies, or something coercible to that (see cl_ensemble).

k

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

method

the consensus method to be employed, see cl_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

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.

Value

• An object of class "cl_partition" representing the obtained secondary partition by an object of class "cl_pclust", which is a list containing at least the following components.
• prototypes
  a cluster ensemble with the $k$ prototypes.
• membership
  an object of class "cl_membership" with the membership values $u_{bj}$.
• cluster
  the class ids of the nearest hard partition.
• silhouette
  Silhouette information for the partition, see silhouette.
• validity
  precomputed validity measures for the partition.
• m
  the softness control argument.
• call
  the matched call.
• d
  the dissimilarity function $d = d(x, p)$ employed.
• e
  the exponent $e$ employed.

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 clusterings for the classes. Gaul und Schader (1988) introduced this procedure for Clusterwise Aggregation of Relations (with the same domains), containing equivalence relations, i.e., hard partitions, as a special case.

For $m > 1$, a generalization of the fuzzy $c$-means recipe (e.g., Bezdek (1981)) is used, which alternates between computing optimal memberships for fixed prototypes, and computing new prototypes as the suitably weighted consensus clusterings for the classes.

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

Consensus clusterings are computed using cl_consensus.

Available control parameters are as follows. [object Object],[object Object],[object Object]

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. The default methods compute Least Squares Euclidean consensus clusterings, i.e., use Euclidean dissimilarity $d$ and $e = 2$.

The fixed point approach employed is a heuristic which cannot be guaranteed to find the global minimum (as this is already true for the computation of consensus clusterings). Standard practice would recommend to use the best solution found in sufficiently many replications of the base algorithm.

References

J. C. Bezdek (1981). Pattern recognition with fuzzy objective function algorithms. New York: Plenum.

W. Gaul and M. Schader (1988). Clusterwise aggregation of relations. Applied Stochastic Models and Data Analysis, 4:273--282.

See Also

kmeans, cmeans.

Examples

## Use a precomputed ensemble of 50 k-means partitions of the
## Cassini data.
data("CKME")
CKME <- CKME[1 : 30]		# for saving precious time ...
diss <- cl_dissimilarity(CKME)
hc <- hclust(diss)
plot(hc)
## This suggests using a partition with three classes, which can be
## obtained using cutree(hc, 3).  Could use cl_consensus() to compute
## prototypes as the least squares consensus clusterings of the classes,
## or alternatively:
set.seed(123)
x1 <- cl_pclust(CKME, 3, m = 1)
x2 <- cl_pclust(CKME, 3, m = 2)
## Agreement of solutions.
cl_dissimilarity(x1, x2)
table(cl_class_ids(x1), cl_class_ids(x2))