fcvalid: Internal Validity Indexes for Fuzzy and Possibilistic Clustering

View source: R/sc.R

sc	R Documentation

Separation/Compactness Ratio

Description

Computes the Separation-Compactness (Zahid et al, 1999) ratio index in order to validate the result of a fuzzy and/or possibilistic clustering analysis.

Usage

sc(x, u, v, m, t=NULL, eta, tidx="f")

Arguments

`x`	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.
`u`	a numeric data frame or matrix containing the fuzzy membership values. It should be specified if `x` is not an object of ‘ppclust’.
`v`	a numeric data frame or matrix containing the cluster prototypes. It should be specified if `x` is not an object of ‘ppclust’.
`t`	a numeric data frame or matrix containing the cluster prototypes. It should be specified if `x` 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

The formula of Separation-Compactness (SC) index is:

I_{SC}= SC_1 - SC_2

In the above equation:

SC_1 is a measure of fuzzy separation and is calculated as follows:

SC_1 = \frac{∑\limits_{j=1}^k ||\vec{v}_j - \bar{v}||^2/k}{∑\limits_{j=1}^k \Big(∑\limits_{i=1}^n (u_{ij})^m ||\vec{x}_i - \vec{v}_j||^2 / ∑\limits_{i=1}^n u_{ij} \Big)}

SC_2 is a measure of fuzzy compactness and is calculated as follows:

SC_2 = \frac{∑\limits_{j=1}^{k-1} ∑\limits_{l=1}^{k-j} \Big( ∑\limits_{i=1}^n (min(u_{ij}, u_{il})^2 / n_{jz} \Big)}{∑\limits_{i=1}^n \max\limits_{1 ≤q j ≤q k} (u_{ij})^2 / ∑\limits_{i=1}^n \max\limits_{1 ≤q j ≤q k} u_{ij}}

Where: z=j+l.

Larger values of SC_1 indicate that the fuzzy partition is considered well-separated and compact while smaller values of SC_2 indicate compact and well-separated fuzzy clusters correspond to the partioning. The maximum of I_{SC} is expected to detect well-defined fuzzy partition (Zahid et al, 1999).

Value

`sc`	SC index value if `tidx` is ‘f’
`sc.e`	extended SC index value if `tidx` is ‘e’
`sc.g`	generalized SC index value if `tidx` is ‘g’

Author(s)

Zeynel Cebeci

References

Zahid, N., Limouri, M. & Essaid, A. (1999). A new cluster-validity for fuzzy clustering. Pattern Recognition, 32(7): 1089-1097. <doi:10.1016/S0031-3203(98)00157-5>

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 SC index using res.fcm, which is a ppclust object
idx <- sc(res.fcm)
print(idx)
 
# Compute the SC index using X, U and V matrices
idx <- sc(res.fcm$x, res.fcm$u, res.fcm$v)
print(idx)

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

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