cl: Chen-Linkens Index

In zcebeci/fcvalid: Internal Validity Indexes for Fuzzy and Possibilistic Clustering

Chen-Linkens Index

Description

Computes the Chen-Linkens index (Chen & Linkens, 2004) in order to validate the result of a fuzzy and/or possibilistic clustering analysis.

Usage

cl(u, m, t=NULL, eta, tidx="f")

Arguments

u	an object of class ‘ppclust’ containing the clustering results from a fuzzy clustering algorithm in the package ppclust. Alternatively, a numeric data frame or matrix containing the data set.
t	a numeric data frame or matrix containing the cluster prototypes. It should be specified if u is not an object of ‘ppclust’ and the option e or g is assigned to tidx.
m	a number specifying the fuzzy exponent. It should be specified if x is not an object of ‘ppclust’.
eta	a number specifying the typicality exponent. It should be specified if x is not an object of ‘ppclust’ and tidx is either e or g.
tidx	a character specifying the type of index. The default is ‘f’ for fuzzy index. The other options are ‘e’ for extended and ‘g’ for generalized index.

Details

Chen-Linkens (CL) index consists of two terms. The first term reflects the compactness within a cluster. The second one indicates the separation between clusters (Chen & Linkens, 2004). The formula of CL index is:

I_{CL}=\frac{1}{n} ∑\limits_{i=1}^n \max\limits_j (u_{ij}) - \frac{1}{K} ∑\limits_{j=1}^{k-1} ∑\limits_{l=j+1}^k \Big[ \frac{1}{n} ∑\limits_{i=1}^n \min(u_{ij}, u_{il})\Big]

In the above equation K is a summation as follows:

K=∑\limits_{j=1}^{k-1} j

The optimal clustering is obtained at the maximum value of I_{CL}.

Value

cl	CL index value if tidx is ‘f’
cl.e	extended CL index value if tidx is ‘e’
cl.g	generalized CL index value if tidx is ‘g’

Author(s)

Zeynel Cebeci

References

Chen, M. Y. & Linkens, D. A. (2004). Rule-base self-generation and simplification for data-driven fuzzy models. Fuzzy Sets and Systems, 142(2):243-265. <doi:10.1016/S0165-0114(03)00160-X>

See Also

allindexes, apd, cs, cwb, fhv, fs, kpbm, kwon, mcd, mpc, pbm, pc, pe, sc, si, tss, ws, xb

Examples

# Load the dataset iris and use the first four feature columns 
data(iris)
x <- iris[,1:4]

# Run FCM algorithm in the package ppclust 
res.fcm <- ppclust::fcm(x, centers=3)

# Compute the CL index using res.fcm, which is a ppclust object
idx <- cl(res.fcm)
print(idx)
 
# Compute the XB index using X and U matrices
idx <- cl(u=res.fcm$u, m=2)
print(idx)

# Run UPFC algorithm in the package ppclust 
res.upfc <- ppclust::upfc(x, centers=3)
# Compute the generalized CL index using res.upfc, which is a ppclust object
idx <- cl(res.upfc, tidx="g")
print(idx)

zcebeci/fcvalid documentation built on Oct. 4, 2022, 9:01 p.m.