cluster (version 1.10.5)

fanny: Fuzzy Analysis Clustering

Description

Computes a fuzzy clustering of the data into k clusters.

Usage

fanny(x, k, diss = inherits(x, "dist"),
      memb.exp = 2, metric = "euclidean", stand = FALSE,
      maxit = 500, tol = 1e-15)

Arguments

x

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

In case of a matrix or data frame, each row corresponds to an observation, and each column corresponds to a variable. All variables

k

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

diss

logical flag: if TRUE (default for dist or dissimilarity objects), then x is assumed to be a dissimilarity matrix. If FALSE, then x is treated as a matrix of observations by variables.

memb.exp

number $r$ strictly larger than 1 specifying the membership exponent used in the fit criterion; see the Details below. Default: 2 which used to be hardwired inside FANNY.

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 manhat

stand

logical; if true, 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 me

maxit, tol

maximal number of iterations and default tolerance for convergence (relative convergence of the fit criterion) for the FANNY algorithm. The defaults maxit = 500 and tol = 1e-15 used to be hardwired inside the algor

Value

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

Details

In a fuzzy clustering, each observation is spread out over the various clusters. Denote by $u_{iv}$ 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) (see the references in daisy) and has been extended to allow user specified memb.exp.

Fanny aims to minimize the objective function $$\sum _{v=1}^k\frac{\sum _{i=1}^n\sum _{j=1}^n{u}_{iv}^r{u}_{jv}^{r}d(i,j)}{2\sum _{j=1}^n{u}_{jv}^r}$$ where $n$ is the number of observations, $k$ is the number of clusters, $r$ is the membership exponent memb.exp and $d(i,j)$ is the dissimilarity between observations $i$ and $j$. Note that $r \to 1$ gives increasingly crisper clusterings whereas $r \to \infty$ leads to complete fuzzyness. K&R(1990), p.191 note that values too close to 1 can lead to slow convergence.

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).

See Also

agnes for background and references; fanny.object, partition.object, plot.partition, daisy, dist.

Examples

## generate 10+15 objects in two clusters, plus 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( 3,3.2,0.5), rnorm( 3,3.2,0.5)))
fannyx <- fanny(x, 2)
## Note that observations 26:28 are "fuzzy" (closer to # 2):
fannyx
summary(fannyx)
plot(fannyx)

(fan.x.15 <- fanny(x, 2, memb.exp = 1.5)) # 'crispier' for obs. 26:28
(fanny(x, 2, memb.exp = 3))               # more fuzzy in general

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