# Convex Clustering

### Description

The data given by `x`

is clustered by an algorithm.

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, if no cluster center has changed during the last
iteration or the maximum number of iterations (given by
`iter.max`

) is reached.

If `verbose`

is `TRUE`

, only for `"kmeans"`

method,
displays for each iteration the number of the iteration and the
numbers of cluster indices which have changed since the last iteration
is given.

If `dist`

is `"euclidean"`

, the distance between the cluster
center and the data points is the Euclidian distance (ordinary kmeans
algorithm). If `"manhattan"`

, the distance between the cluster
center and the data points is the sum of the absolute values of the
distances of the coordinates.

If `method`

is `"kmeans"`

, then we have the kmeans
clustering method, which works by repeatedly moving all cluster
centers to the mean of their Voronoi sets. If `"hardcl"`

we have
the On-line Update (Hard Competitive learning) method, which works by
performing an update directly after each input signal, and if
`"neuralgas"`

we have the Neural Gas (Soft Competitive learning)
method, that sorts for each input signal the units of the network
according to the distance of their reference vectors to input signal.

If `rate.method`

is `"polynomial"`

, the polynomial learning
rate is used, that means *1/t*, where *t* stands for the
number of input data for which a particular cluster has been the
winner so far. If `"exponentially decaying"`

, the exponential
decaying learning rate is used according to
*par1*{(par2/par1)}^{(iter/itermax)}*
where *par1* and *par2* are the initial and final values of
the learning rate.

The parameters `rate.par`

of the learning rate, where
if `rate.method`

is `"polynomial"`

then by default
`rate.par=1.0`

, otherwise `rate.par=(0.5,1e-5)`

.

### Usage

1 2 |

### Arguments

`x` |
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 |

`dist` |
If |

`method` |
If |

`rate.method` |
If |

`rate.par` |
The parameters of the learning rate. |

### Value

`cclust`

returns an object of class `"cclust"`

.

`centers` |
The final cluster centers. |

`initcenters` |
The initial cluster centers. |

`ncenters` |
The number of the centers. |

`cluster` |
Vector containing the indices of the clusters where the data points are assigned to. |

`size` |
The number of data points in each cluster. |

`iter` |
The number of iterations performed. |

`changes` |
The number of changes performed in each iteration step with the Kmeans algorithm. |

`dist` |
The distance measure used. |

`method` |
The algorithm method being used. |

`rate.method` |
The learning rate being used by the Hardcl clustering method. |

`rate.par` |
The parameters of the learning rate. |

`call` |
Returns a call in which all of the arguments are specified by their names. |

`withinss` |
Returns the sum of square distances within the clusters. |

### Author(s)

Evgenia Dimitriadou

### See Also

`predict.cclust`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
## 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<-cclust(x,2,20,verbose=TRUE,method="kmeans")
plot(x, col=cl$cluster)
## a 3-dimensional example
x<-rbind(matrix(rnorm(150,sd=0.3),ncol=3),
matrix(rnorm(150,mean=1,sd=0.3),ncol=3),
matrix(rnorm(150,mean=2,sd=0.3),ncol=3))
cl<-cclust(x,6,20,verbose=TRUE,method="kmeans")
plot(x, col=cl$cluster)
## assign classes to some new data
y<-rbind(matrix(rnorm(33,sd=0.3),ncol=3),
matrix(rnorm(33,mean=1,sd=0.3),ncol=3),
matrix(rnorm(3,mean=2,sd=0.3),ncol=3))
ycl<-predict(cl, y)
plot(y, col=ycl$cluster)
``` |