Fuzzy Cluster Indexes (Validity/Performance Measures)

Description

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.

Usage

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

Arguments

Details

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.

gath.geva: Gath and Geva introduced 2 main criteria for comparing and finding optimal partitions based on the heuristics that a better clustering assumes clear separation between the clusters, minimal volume of the clusters and maximal number of data points concentrated in the vicinity of the cluster centroids. These indexes are only for the cmeans clustering algorithm valid. For the first, the ``fuzzy hypervolume'' we have: F_{HV}=sum_{j=1}^{c}{[det(F_j)]}^{1/2}, where F_j=frac{sum_{i=1}^N u_{ij}(x_i-v_j)(x_i-v_j)^T}{sum_{i=1}^{N}u_{ij}}, for the case when the defuzzification parameter is 2. For the second, the ``average partition density'': D_{PA}=frac{1}{k}sum_{j=1}^kfrac{S_j}{{[det(F_j)]}^{1/2}}, where S_j=sum_{i=1}^N u_{ij}. Moreover, the ``partition density'' which expresses the general partition density according to the physical definition of density is calculated by: P_D=frac{S}{F_{HV}}, where S=sum_{j=1}^ksum_{i=1}^N u_{ij}.

xie.beni: This index is a function of the data set and the centroids of the clusters. Xie and Beni explained this index by writing it as a ratio of the total variation of the partition and the centroids $(U,V)$ and the separation of the centroids vectors. The minimum values of this index under comparison support the best partitions. u_{XB}(U,V;X)=frac{sum_{j=1}^ksum_{i=1}^Nu_{ij}^2{||x_i-v_j||}^2}{N(min_{jneq l}{{||v_j-v_l||}^2})}

fukuyama.sugeno: This index consists of the difference of two terms, the first combining the fuzziness in the membership matrix with the geometrical compactness of the representation of the data set via the prototypes, and the second the fuzziness in its row of the partition matrix with the distance from the $i$th prototype to the grand mean of the data. The minimum values of this index also propose a good partition. u_{FS}(U,V;X)=sum_{i=1}^{N}sum_{j=1}^k (u_{ij}^2)^q(||x_i-v_j||^2-||v_j-bar v||^2)

partition.coefficient: An index which measures the fuzziness of the partition but without considering the data set itself. It is a heuristic measure since it has no connection to any property of the data. The maximum values of it imply a good partition in the meaning of a least fuzzy clustering. F(U;k)=frac{tr (UU^T)}{N}=frac{<U,U>}{N}=frac{||U||^2}{N}

• 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.

partition.entropy: It is a measure that provides information about the membership matrix without also considering the data itself. The minimum values imply a good partition in the meaning of a more crisp partition. H(U;k)=sum_{i=1}^{N} h(u_i)/N, where h(u)=-sum_{j=1}^{k} u_j,log _a (u_j) the Shannon's entropy.

• 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].

proportion.exponent: It is a measure P(U;k) of fuzziness adept to detect structural variations in the partition matrix as it becomes more fuzzier. A crisp cluster in the partition matrix can drive it to infinity when the partition coefficient and the partition entropy are more sensitive to small changes when approaching a hard partition. Its evaluation does not also involve the data or the algorithm used to partition them and its maximum implies the optimal partition but without knowing what maximum is a statistically significant maximum.

• 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.

separation.index (known as CS Index): This index identifies unique cluster structure with well-defined properties that depend on the data and a measure of distance. It answers the question if the clusters are compact and separated, but it rather seems computationally infeasible for big data sets since a distance matrix between all the data membership values has to be calculated. It also presupposes that a hard partition is derived from the fuzzy one.
D_1(U;k;X,d)=min_{i+1,<=q,l,<=q,k-1}<=ft{min_{1,<=q,j,<=q,k}<=ft{frac{dis(u_j,u_l)}{max_{1<=q m<=q k}{dia(u_m)}}right}right}, where dia is the diameter of the subset, dis the distance of two subsets, and d a metric. U is a CS partition of X Leftrightarrow D_1>1. When this holds then U is unique.

Value

Returns a vector with the validity measures values.

Author(s)

Evgenia Dimitriadou

References

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).

See Also

cmeans

Examples

# 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