cluster (version 2.0.7)

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 = c("euclidean", "manhattan", "SqEuclidean"),
      stand = FALSE, iniMem.p = NULL, cluster.only = FALSE,
      keep.diss = !diss && !cluster.only && n < 100,
      keep.data = !diss && !cluster.only,
      maxit = 500, tol = 1e-15, trace.lev = 0)

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 must be numeric. Missing values (NAs) are allowed.

In case of a dissimilarity matrix, x is typically the output of daisy or dist. Also a vector of length n*(n-1)/2 is allowed (where n is the number of observations), and will be interpreted in the same way as the output of the above-mentioned functions. Missing values (NAs) are not allowed.

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. Options are "euclidean" (default), "manhattan", and "SqEuclidean". Euclidean distances are root sum-of-squares of differences, and manhattan distances are the sum of absolute differences, and "SqEuclidean", the squared euclidean distances are sum-of-squares of differences. Using this last option is equivalent (but somewhat slower) to computing so called “fuzzy C-means”. If x is already a dissimilarity matrix, then this argument will be ignored.

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 mean absolute deviation. If x is already a dissimilarity matrix, then this argument will be ignored.

iniMem.p

numeric \(n \times k\) matrix or NULL (by default); can be used to specify a starting membership matrix, i.e., a matrix of non-negative numbers, each row summing to one.

cluster.only

logical; if true, no silhouette information will be computed and returned, see details.

keep.diss, keep.data

logicals indicating if the dissimilarities and/or input data x should be kept in the result. Setting these to FALSE can give smaller results and hence also save memory allocation time.

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 algorithm.

trace.lev

integer specifying a trace level for printing diagnostics during the C-internal algorithm. Default 0 does not print anything; higher values print increasingly more.

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 by Martin Maechler to allow user specified memb.exp, iniMem.p, maxit, tol, etc.

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. Further note that even the default, \(r = 2\) can lead to complete fuzzyness, i.e., memberships \(u_{iv} \equiv 1/k\). In that case a warning is signalled and the user is advised to chose a smaller memb.exp (\(=r\)).

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

# NOT RUN {
## 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)
f4 <- fanny(ruspini, 4)
stopifnot(rle(f4$clustering)$lengths == c(20,23,17,15))
plot(f4, which = 1)
## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))
# }