FKM.ent.noise: Fuzzy k-means with entropy regularization and noise cluster

In fclust: Fuzzy Clustering

Fuzzy k-means with entropy regularization and noise cluster

Description

Performs the fuzzy k-means clustering algorithm with entropy regularization and noise cluster.
The entropy regularization allows us to avoid using the artificial fuzziness parameter m. This is replaced by the degree of fuzzy entropy ent, related to the concept of temperature in statistical physics. An interesting property of the fuzzy k-means with entropy regularization is that the prototypes are obtained as weighted means with weights equal to the membership degrees (rather than to the membership degrees at the power of m as is for the fuzzy k-means).
The noise cluster is an additional cluster (with respect to the k standard clusters) such that objects recognized to be outliers are assigned to it with high membership degrees.

Usage

 
 FKM.ent.noise (X, k, ent, delta, 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)
ent	Degree of fuzzy entropy (default: 1)
delta	Noise distance (default: average Euclidean distance between objects and prototypes from FKM.ent using the same values of k and m)
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 first element of value, cput and iter refer to the rational start.
The default value for ent is in general not reasonable if FKM.ent is run using raw data.
The update of the membership degrees requires the computation of exponential functions. In some cases, this may produce NaN values and the algorithm stops. Such a problem is usually solved by running FKM.ent using standardized data (stand=1).

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 (NULL for FKM.ent.noise)
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.ent.noise)
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 (NULL for FKM.ent.noise)
ent	Degree of fuzzy entropy
b	Parameter of the polynomial fuzzifier (NULL for FKM.ent.noise)
vp	Volume parameter (NULL for FKM.ent.noise)
delta	Noise distance
gam	Weighting parameter for the fuzzy covariance matrices (NULL for FKM.ent.noise)
mcn	Maximum condition number for the fuzzy covariance matrices (NULL for FKM.ent.noise)
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.ent.noise)
call	Matched call

Author(s)

Paolo Giordani, Maria Brigida Ferraro, Alessio Serafini

References

Dave' R.N., 1991. Characterization and detection of noise in clustering. Pattern Recognition Letters, 12, 657-664.
Li R., Mukaidono M., 1995. A maximum entropy approach to fuzzy clustering. Proceedings of the Fourth IEEE Conference on Fuzzy Systems (FUZZ-IEEE/IFES '95), pp. 2227-2232.
Li R., Mukaidono M., 1999. Gaussian clustering method based on maximum-fuzzy-entropy interpretation. Fuzzy Sets and Systems, 102, 253-258.

See Also

FKM.ent, Fclust, Fclust.index, print.fclust, summary.fclust, plot.fclust, butterfly

Examples

## butterfly data
data(butterfly)
## fuzzy k-means with entropy regularization and noise cluster, fixing the number of clusters
clust=FKM.ent.noise(butterfly,k = 2, RS=5,delta=3)
## fuzzy k-means with entropy regularization and noise cluster, selecting the number of clusters
clust=FKM.ent.noise(butterfly,RS=5,delta=3)

fclust documentation built on Nov. 16, 2022, 5:10 p.m.