| fanny {cluster} | R Documentation |
Computes a fuzzy clustering of the data into k clusters.
fanny(x, k, diss = inherits(x, "dist"),
memb.exp = 2, metric = "euclidean", stand = FALSE,
maxit = 500, tol = 1e-15)
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.
The currently available options are "euclidean" and "manhattan".
Euclidean distances are root sum-of-squares of differences, and
manhattan distances are the sum of absolute differences.
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. |
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. |
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) (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] (SUM_(i,j) u(i,v)^r u(j,v)^r d(i,j)) / (2 SUM_j u(j,v)^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 -> 1 gives increasingly crisper
clusterings whereas r -> Inf 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).
an object of class "fanny" representing the clustering.
See fanny.object for details.
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)
## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))