fanny
Fuzzy Analysis Clustering
Computes a fuzzy clustering of the data into k
clusters.
 Keywords
 cluster
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 = 1e15, 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 ofdaisy
ordist
. Also a vector of length n*(n1)/2 is allowed (where n is the number of observations), and will be interpreted in the same way as the output of the abovementioned 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
ordissimilarity
objects), thenx
is assumed to be a dissimilarity matrix. If FALSE, thenx
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 sumofsquares of differences, and manhattan distances are the sum of absolute differences, and"SqEuclidean"
, the squared euclidean distances are sumofsquares of differences. Using this last option is equivalent (but somewhat slower) to computing so called “fuzzy Cmeans”. Ifx
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. Ifx
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 startingmembership
matrix, i.e., a matrix of nonnegative 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 toFALSE
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
andtol = 1e15
used to be hardwired inside the algorithm.  trace.lev
 integer specifying a trace level for printing
diagnostics during the Cinternal algorithm.
Default
0
does not print anything; higher values print increasingly more.
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
).
Value
an object of class "fanny"
representing the clustering.
See fanny.object
for details.
See Also
agnes
for background and references;
fanny.object
, partition.object
,
plot.partition
, daisy
, dist
.
Examples
library(cluster)
## 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))