Computes a fuzzy clustering of the data into k clusters.
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)an object of class "fanny" representing the clustering.
See fanny.object for details.
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.
integer giving the desired number of clusters. It is required that \(0 < k < n/2\) where \(n\) is the number of observations.
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.
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.
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.
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.
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.
logical; if true, no silhouette information will be computed and returned, see details.
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.
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.
integer specifying a trace level for printing
diagnostics during the C-internal algorithm.
Default 0 does not print anything; higher values print
increasingly more.
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).
agnes for background and references;
fanny.object, partition.object,
plot.partition, daisy, dist.
## 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))
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