FKM.gk: Gustafson and Kessel - like fuzzy k-means

In fclust: Fuzzy Clustering

Gustafson and Kessel - like fuzzy k-means

Description

Performs the Gustafson and Kessel - like fuzzy k-means clustering algorithm.
Differently from fuzzy k-means, it is able to discover non-spherical clusters.

Usage

 FKM.gk (X, k, m, vp, RS, stand, startU, index, alpha, conv, maxit, seed)

Arguments

X	Matrix or data.frame
k	An integer value or vector specifying the number of clusters for which the index is to be calculated (default: 2:6)
m	Parameter of fuzziness (default: 2)
vp	Volume parameter (default: rep(1,k))
RS	Number of (random) starts (default: 1)
stand	Standardization: if stand=1, the clustering algorithm is run using standardized data (default: no standardization)
startU	Rational start for the membership degree matrix U (default: no rational start)
index	Cluster validity index to select the number of clusters: "PC" (partition coefficient), "PE" (partition entropy), "MPC" (modified partition coefficient), "SIL" (silhouette), "SIL.F" (fuzzy silhouette), "XB" (Xie and Beni) (default: "SIL.F")
alpha	Weighting coefficient for the fuzzy silhouette index SIL.F (default: 1)
conv	Convergence criterion (default: 1e-9)
maxit	Maximum number of iterations (default: 1e+6)
seed	Seed value for random number generation (default: NULL)

Details

If startU is given, the argument k is ignored (the number of clusters is ncol(startU)).
If startU is given, the argument RS is ignored (the algorithm is run using the rational start) and therefore value, cput and iter refer to such a rational start.
If a cluster covariance matrix becomes singular, then the algorithm stops and the element of value is NaN.
The Babuska et al. variant in FKM.gkb is recommended.

Value

Object of class fclust, which is a list with the following components:

U	Membership degree matrix
H	Prototype matrix
F	Array containing the covariance matrices of all the clusters
clus	Matrix containing the indexes of the clusters where the objects are assigned (column 1) and the associated membership degrees (column 2)
medoid	Vector containing the indexes of the medoid objects (NULL for FKM.gk)
value	Vector containing the loss function values for the RS starts
criterion	Vector containing the values of the cluster validity index
iter	Vector containing the numbers of iterations for the RS starts
k	Number of clusters
m	Parameter of fuzziness
ent	Degree of fuzzy entropy (NULL for FKM.gk)
b	Parameter of the polynomial fuzzifier (NULL for FKM.gk)
vp	Volume parameter (default: rep(1,max(k)). If k is a vector, for each group the first k element of vpare considered.
delta	Noise distance (NULL for FKM.gk)
gam	Weighting parameter for the fuzzy covariance matrices (NULL for FKM.gk)
mcn	Maximum condition number for the fuzzy covariance matrices (NULL for FKM.gk)
stand	Standardization (Yes if stand=1, No if stand=0)
Xca	Data used in the clustering algorithm (standardized data if stand=1)
X	Raw data
D	Dissimilarity matrix (NULL for FKM.gk)
call	Matched call

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Gustafson E.E., Kessel W.C., 1978. Fuzzy clustering with a fuzzy covariance matrix. Proceedings of the IEEE Conference on Decision and Control, pp. 761-766.

See Also

FKM.gkb, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, unemployment

Examples

## Not run: 
## unemployment data
data(unemployment)
## Gustafson and Kessel-like fuzzy k-means, fixing the number of clusters
clust=FKM.gk(unemployment,k=3,RS=10)
## Gustafson and Kessel-like fuzzy k-means, selecting the number of clusters
clust=FKM.gk(unemployment,k=2:6,RS=10)

## End(Not run)

fclust documentation built on Nov. 5, 2025, 7:10 p.m.