e1071 (version 1.5-20)

cmeans: Fuzzy C-Means Clustering

Description

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

Usage

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

Arguments

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.

Value

• 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.

Details

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.

References

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.

Examples

# 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)