# fanny

##### Fuzzy Analysis Clustering

Computes a fuzzy clustering of the data into `k`

clusters.

- Keywords
- cluster

##### Usage

`fanny(x, k, diss = inherits(x, "dist"), metric = "euclidean", stand = FALSE)`

##### 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

- 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. - 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 manhat
- 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 me

##### Details

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).
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 $$\sum_{v=1}^k \frac{\sum_{i=1}^n\sum_{j=1}^n u_{iv}^2 u_{jv}^2 d(i,j)}{ 2 \sum_{j=1}^n u_{jv}^2}$$ where $n$ is the number of observations, $k$ is the number of clusters and $d(i,j)$ is the dissimilarity between observations $i$ and $j$.

##### 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

```
## 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)
data(ruspini)
## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))
```

*Documentation reproduced from package cluster, version 1.4-1, License: GPL version 2 or later*