The fuzzy version of the known *k*means clustering algorithm as
well as an on-line variant (Unsupervised Fuzzy Competitive learning).

```
cmeans(x, centers, iter.max = 100, verbose = FALSE,
dist = "euclidean", method = "cmeans", m = 2,
rate.par = NULL, weights = 1, control = list())
```

An object of class `"fclust"`

which is a list with components:

- centers
the final cluster centers.

- size
the number of data points in each cluster of the closest hard clustering.

- cluster
a vector of integers containing the indices of the clusters where the data points are assigned to for the closest hard clustering, as obtained by assigning points to the (first) class with maximal membership.

- iter
the number of iterations performed.

- membership
a matrix with the membership values of the data points to the clusters.

- withinerror
the value of the objective function.

- call
the call used to create the object.

- x
The data matrix where columns correspond to variables and rows to observations.

- centers
Number of clusters or initial values for cluster centers.

- iter.max
Maximum number of iterations.

- verbose
If

`TRUE`

, make some output during learning.- dist
Must be one of the following: If

`"euclidean"`

, the mean square error, if`"manhattan"`

, the mean absolute error is computed. Abbreviations are also accepted.- method
If

`"cmeans"`

, then we have the \(c\)-means fuzzy clustering method, if`"ufcl"`

we have the on-line update. Abbreviations are also accepted.- m
A number greater than 1 giving the degree of fuzzification.

- rate.par
A number between 0 and 1 giving the parameter of the learning rate for the on-line variant. The default corresponds to \(0.3\).

- weights
a numeric vector with non-negative case weights. Recycled to the number of observations in

`x`

if necessary.- control
a list of control parameters. See

**Details**.

Evgenia Dimitriadou and Kurt Hornik

The data given by `x`

is clustered by generalized versions of the
fuzzy *c*-means algorithm, which use either a fixed-point or an
on-line heuristic for minimizing the objective function
$$\sum_i \sum_j w_i u_{ij}^m d_{ij},$$
where \(w_i\) is the weight of observation \(i\), \(u_{ij}\) is
the membership of observation \(i\) in cluster \(j\), and
\(d_{ij}\) is the distance (dissimilarity) between observation
\(i\) and center \(j\). The dissimilarities used are the sums of
squares (`"euclidean"`

) or absolute values (`"manhattan"`

)
of the element-wise differences.

If `centers`

is a matrix, its rows are taken as the initial cluster
centers. If `centers`

is an integer, `centers`

rows of
`x`

are randomly chosen as initial values.

The algorithm stops when the maximum number of iterations (given by
`iter.max`

) is reached, or when the algorithm is unable to reduce
the current value `val`

of the objective function by
`reltol * (abs(val) * reltol)`

at a step. The relative
convergence tolerance `reltol`

can be specified as the
`reltol`

component of the list of control parameters, and
defaults to `sqrt(.Machine$double.eps)`

.

If `verbose`

is `TRUE`

, each iteration displays its number
and the value of the objective function.

If `method`

is `"cmeans"`

, then we have the \(c\)-means
fuzzy clustering method, see for example Bezdek (1981). If
`"ufcl"`

, we have the On-line Update (Unsupervised Fuzzy
Competitive Learning) method due to Chung and Lee (1992), see also Pal
et al (1996). This method works by performing an update directly
after each input signal (i.e., for each single observation).

The parameters `m`

defines the degree of fuzzification. It is
defined for real values greater than 1 and the bigger it is the more
fuzzy the membership values of the clustered data points are.

J. C. Bezdek (1981).
*Pattern recognition with fuzzy objective function algorithms*.
New York: Plenum.

Fu Lai Chung and Tong Lee (1992).
Fuzzy competitive learning.
*Neural Networks*, **7**(3), 539--551.

Nikhil R. Pal, James C. Bezdek, and Richard J. Hathaway (1996).
Sequential competitive learning and the fuzzy c-means clustering
algorithms.
*Neural Networks*, **9**(5), 787--796.

```
# a 2-dimensional example
x<-rbind(matrix(rnorm(100,sd=0.3),ncol=2),
matrix(rnorm(100,mean=1,sd=0.3),ncol=2))
cl<-cmeans(x,2,20,verbose=TRUE,method="cmeans",m=2)
print(cl)
# a 3-dimensional example
x<-rbind(matrix(rnorm(150,sd=0.3),ncol=3),
matrix(rnorm(150,mean=1,sd=0.3),ncol=3),
matrix(rnorm(150,mean=2,sd=0.3),ncol=3))
cl<-cmeans(x,6,20,verbose=TRUE,method="cmeans")
print(cl)
```

Run the code above in your browser using DataCamp Workspace