fanny                package:cluster                R Documentation

Fuzzy Analysis Clustering

Description:

     Computes a fuzzy clustering of the data into 'k' clusters.

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 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 of length
          n*(n-1)/2 is allowed (where n is the number of observations),
          and will be interpreted in the same way as the output of the
          above-mentioned functions.  Missing values (NAs) are not
          allowed. 

       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 manhattan distances are the sum of absolute differences.
          If 'x' is already a dissimilarity matrix, then this argument
          will be ignored. 

   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 mean
          absolute deviation.  If 'x' is already a dissimilarity
          matrix, then this argument will be ignored.

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] (SUM_(i,j) u(i,v)^2 u(j,v)^2 d(i,j)) / (2 SUM_j u(j,v)^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))

