fclustIndex {e1071} | R Documentation |

Calculates the values of several fuzzy validity measures. The values of the indexes can be independently used in order to evaluate and compare clustering partitions or even to determine the number of clusters existing in a data set.

fclustIndex(y, x, index = "all")

`y` |
An object of a fuzzy clustering result of class `"fclust"` |

`x` |
Data matrix |

`index` |
The validity measures used: `"gath.geva"` , `"xie.beni"` ,
`"fukuyama.sugeno"` , `"partition.coefficient"` ,
`"partition.entropy"` , `"proportion.exponent"` ,
`"separation.index"` and `"all"` for all the indexes. |

The validity measures and a short description of them follows, where
*N* is the number of data points, *u_{ij}* the values of the
membership matrix, *v_j* the centers of the clusters and *k*
te number of clusters.

*F(U;k)*shows the fuzziness or the overlap of the partition and depends on*kN*elements.*1/k<=q F(U;k)<=q 1*, where if*F(U;k)=1*then*U*is a hard partition and if*F(U;k)=1/k*then*U=[1/k]*is the centroid of the fuzzy partion space*P_{fk}*. The converse is also valid.

*H(U;k)*shows the uncertainty of a fuzzy partition and depends also on*kN*elements. Specifically,*h(u_i)*is interpreted as the amount of fuzzy information about the membership of*x_i*in*k*classes that is retained by column*u_j*. Thus, at*U=[1/k]*the most information is withheld since the membership is the fuzziest possible.*0<=q H(U;k)<=q log_a(k)*, where for*H(U;k)=0**U*is a hard partition and for*H(U;k)=log_a(k)**U=[1/k]*.

*0<=q P(U;k)<infty*, since the*[0,1]*values explode to*[0,infty)*due to the natural logarithm. Specifically,*P=0*when and only when*U=[1/k]*, while*Prightarrowinfty*when any column of*U*is crisp.*P(U;k)*can easily explode and it is good for partitions with large column maximums and at detecting structural variations.

Returns a vector with the validity measures values.

Evgenia Dimitriadou

James C. Bezdek, *Pattern Recognition with Fuzzy Objective
Function Algorithms*, Plenum Press, 1981, NY.

L. X. Xie and G. Beni, *Validity measure for fuzzy
clustering*, IEEE Transactions on Pattern Analysis and Machine
Intelligence, vol. **3**, n. 8, p. 841-847, 1991.

I. Gath and A. B. Geva, *Unsupervised Optimal Fuzzy
Clustering*, IEEE Transactions on Pattern Analysis and Machine
Intelligence, vol. **11**, n. 7, p. 773-781, 1989.

Y. Fukuyama and M. Sugeno, *A new method of choosing the
number of clusters for the fuzzy $c$-means method*, Proc. 5th Fuzzy
Syst. Symp., p. 247-250, 1989 (in japanese).

# a 2-dimensional example x<-rbind(matrix(rnorm(100,sd=0.3),ncol=2), matrix(rnorm(100,mean=1,sd=0.3),ncol=2)) cl<-cmeans(x,2,20,verbose=TRUE,method="cmeans") resultindexes <- fclustIndex(cl,x, index="all") resultindexes

[Package *e1071* version 1.5-2 Index]