cluster (version 1.3-2)

fanny: Fuzzy Analysis

Description

Returns a list representing a fuzzy clustering of the data into k clusters.

Usage

fanny(x, k, diss = FALSE, metric = "euclidean", stand = FALSE)

Arguments

x

data matrix or dataframe, or dissimilarity matrix, depending on the value of the diss argument.

In case of a matrix or dataframe, each row corresponds to an observation, and each column corresponds to a variable. All variables must be numer

k

integer, the number of clusters. It is required that 0 < k < n/2 where n is the number of observations.

diss

logical flag: if TRUE, then x will be considered as a dissimilarity matrix. If FALSE, then x will be considered as a matrix of observations by variables.

metric

character string specifying the metric to be used for calculating dissimilarities between observations. The currently available options are "euclidean" and "manhattan". Euclidean distances are root sum-of-squares of differences, and manhattan distances ar

stand

logical flag: if TRUE, then the measurements in x are standardized before calculating the dissimilarities. Measurements are standardized for each variable (column), by subtracting the variable's mean value and dividing by the variable's mean

Value

• an object of class "fanny" representing the clustering. See fanny.object for details.

BACKGROUND

Cluster analysis divides a dataset into groups (clusters) of observations that are similar to each other. Partitioning methods like pam, clara, and fanny require that the number of clusters be given by the user. Hierarchical methods like agnes, diana, and mona construct a hierarchy of clusterings, with the number of clusters ranging from one to the number of observations.

Details

In a fuzzy clustering, each observation is "spread out" over the various clusters. Denote by u(i,v) the membership of observation i to cluster v. The memberships are nonnegative, and for a fixed observation i they sum to 1. The particular method fanny stems from chapter 4 of Kaufman and Rousseeuw (1990). Compared to other fuzzy clustering methods, fanny has the following features: (a) it also accepts a dissimilarity matrix; (b) it is more robust to the spherical cluster assumption; (c) it provides a novel graphical display, the silhouette plot (see plot.partition).

Fanny aims to minimize the objective function $$\sum_{v=1}^k \frac{\sum_{i=1}^n\sum_{j=1}^n u_{iv}^2 u_{jv}^2 d(i,j)}{ 2 \sum_{j=1}^n u_{jv}^2}$$ where n is the number of observations, k is the number of clusters and d(i,j) is the dissimilarity between observations i and j.

References

Kaufman, L. and Rousseeuw, P.J. (1990). Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.

Anja Struyf, Mia Hubert & Peter J. Rousseeuw (1996): Clustering in an Object-Oriented Environment. Journal of Statistical Software, 1. http://www.stat.ucla.edu/journals/jss/ Struyf, A., Hubert, M. and Rousseeuw, P.J. (1997). Integrating Robust Clustering Techniques in S-PLUS, Computational Statistics and Data Analysis, 26, 17-37.

See Also

fanny.object, daisy, partition.object, plot.partition, dist.

Examples

## generate 25 objects, divided into two clusters, and 3 objects lying
## between those clusters. 
x <- rbind(cbind(rnorm(10,0,0.5), rnorm(10,0,0.5)),
           cbind(rnorm(15,5,0.5), rnorm(15,5,0.5)),
           cbind(rnorm(3,3.5,0.5), rnorm(3,3.5,0.5)))
fannyx <- fanny(x, 2)
fannyx
summary(fannyx)
plot(fannyx)

data(ruspini)
## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))