xb: Xie-Beni Index
In zcebeci/fcvalid: Internal Validity Indexes for Fuzzy and Possibilistic Clustering

View source: R/xb.R

xb	R Documentation

Xie-Beni Index

Description

Computes the Xie-Beni (Xie & Beni, 1991) index to validate the result of a fuzzy and/or possibilistic clustering analysis.

Usage

xb(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

Xie-Beni index (XB) is a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c-partitions without assumptions as to the number of substructures inherent in the data (Xie & Beni, 1991). Therefore it is also called the compactness and separation validity function. The index is first representative of its category with its successors. The formula of XB index is:

I_{XB}=\frac{∑\limits_{i=1}^n ∑\limits_{j=1}^k u_{ij}^m \; d_{ij}^2}{n \; \Big(\min\limits_{j\neq l}(||\vec{v_j} - \vec{v}_l||^2)\Big)}

Since the smaller values of the index indicate the compact and well-separated clusters, the optimal clustering is obtained at the minimum value of I_{XB}.

Value

`xb`	XB index value if `tidx` is ‘f’
`xb.e`	extended XB index value if `tidx` is ‘e’
`xb.g`	generalized XB index value if `tidx` is ‘g’

Author(s)

Zeynel Cebeci

References

Xie, X.L. & Beni, G. (1991). A validity measure for fuzzy clustering, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(8):841-847. <doi:10.1109/34.85677>

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 XB index using res.fcm, which is a ppclust object
idx <- xb(res.fcm)
print(idx)
 
# Compute the XB index using X, U and V matrices
idx <- xb(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 <- xb(res.upfc, tidx="g")
print(idx)

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