Fuzzy Analysis
DESCRIPTION:
Returns a list representing a fuzzy clustering of the data into k clusters.
USAGE: 
fanny(x, k, diss = F, metric = "euclidean", stand = F)
REQUIRED ARGUMENTS:
x:
data matrix or dataframe, or dissimilarity matrix, depending on the value of the diss argument.

In case of a matrix or dataframe, 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 with length n*(n-1)/2 is allowed (where n is the number of objects), and will be interpreted in the same way as the output of the above-mentioned functions. Missing values (NAs) are not allowed.

k:
integer, the number of clusters.

OPTIONAL ARGUMENTS:
diss:
logical flag: if TRUE, then x will be considered as a dissimilarity matrix. If FALSE, then x will be considered as a matrix of observations by variables.

metric:
character string specifying the metric to be used for calculating dissimilarities between objects. 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 flag: if TRUE, then 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.

VALUE:
an object of class "fanny" representing the clustering. See fanny.object for details.
DETAILS:
In a fuzzy clustering, each object is "spread out" over the various clusters. Denote by u(i,v) the membership of object i to cluster v. The memberships are nonnegative, and for a fixed object i they sum to 1. The particular method fanny stems from chapter 4 of Kaufman and Rousseeuw (1990). 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).

Fanny aims to minimize the objective function n n 2 2 k sum sum u (i,v) u (j,v) d(i,j) i=1 j=1 sum ------------------------------------ n 2 v=1 2 sum u (j,v) j=1 where n is the number of objects, k is the number of clusters and d(i,j) is the dissimilarity between objects i and j.

BACKGROUND:
Cluster analysis divides a dataset into groups (clusters) of objects that are similar to each other. Partitioning methods like pam, clara, and fanny require that the number of clusters be given by the user. Hierarchical methods like agnes, diana, and mona construct a hierarchy of clusterings, with the number of clusters ranging from one to the number of objects.

REFERENCES:
Kaufman, L. and Rousseeuw, P.J. (1990). Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.
SEE ALSO:
daisy , dist , fanny.object , partition.object , plot.partition .
EXAMPLES: 
# generate 25 objects, divided into two clusters,
# and 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(3,3.5,0.5), rnorm(3,3.5,0.5)))


fannyx <- fanny(x, 2)
fannyx
summary(fannyx)
plot(fannyx)


fanny(daisy(x, metric = "manhattan"), 2, diss = T)