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"), memb.exp = 2,
           metric = c("euclidean", "manhattan", "SqEuclidean"),
           stand = FALSE, iniMem.p = NULL, cluster.only = FALSE,
           keep.diss = !diss && !cluster.only && n < 100,
           keep.data = !diss && !cluster.only,
           maxit = 500, tol = 1e-15, trace.lev = 0)

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. 

memb.exp: number r strictly larger than 1 specifying the _membership
          exponent_ used in the fit criterion; see the 'Details' below.
          Default: '2' which used to be hardwired inside FANNY.

  metric: character string specifying the metric to be used for
          calculating dissimilarities between observations.  Options
          are '"euclidean"' (default), '"manhattan"', and
          '"SqEuclidean"'.  Euclidean distances are root sum-of-squares
          of differences, and manhattan distances are the sum of
          absolute differences, and '"SqEuclidean"', the _squared_
          euclidean distances are sum-of-squares of differences.  Using
          this last option is equivalent (but somewhat slower) to
          computing so called "fuzzy C-means". 
           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.

iniMem.p: numeric n * k matrix or 'NULL' (by default); can be used to
          specify a starting 'membership' matrix, i.e., a matrix of
          non-negative numbers, each row summing to one. 

cluster.only: logical; if true, no silhouette information will be
          computed and returned, see details.

keep.diss, keep.data: logicals indicating if the dissimilarities and/or
          input data 'x' should be kept in the result.  Setting these
          to 'FALSE' can give smaller results and hence also save
          memory allocation _time_.

maxit, tol: maximal number of iterations and default tolerance for
          convergence (relative convergence of the fit criterion) for
          the FANNY algorithm.  The defaults 'maxit = 500' and 'tol =
          1e-15' used to be hardwired inside the algorithm.

trace.lev: integer specifying a trace level for printing diagnostics
          during the C-internal algorithm. Default '0' does not print
          anything; higher values print increasingly more.

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) (see the references in 'daisy')
     and has been extended by Martin Maechler to allow user specified
     'memb.exp', 'iniMem.p', 'maxit', 'tol', etc.

     Fanny aims to minimize the objective function

 SUM_[v=1..k] (SUM_(i,j) u(i,v)^r u(j,v)^r d(i,j)) / (2 SUM_j u(j,v)^r)

     where n is the number of observations, k is the number of
     clusters, r is the membership exponent 'memb.exp' and d(i,j) is
     the dissimilarity between observations i and j. 
      Note that r -> 1 gives increasingly crisper clusterings whereas r
     -> Inf leads to complete fuzzyness.  K&R(1990), p.191 note that
     values too close to 1 can lead to slow convergence.  Further note
     that even the default, r = 2 can lead to complete fuzzyness, i.e.,
     memberships u(i,v) == 1/k.  In that case a warning is signalled
     and the user is advised to chose a smaller 'memb.exp' (=r).

     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').

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 10+15 objects in two clusters, plus 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( 3,3.2,0.5), rnorm( 3,3.2,0.5)))
     fannyx <- fanny(x, 2)
     ## Note that observations 26:28 are "fuzzy" (closer to # 2):
     fannyx
     summary(fannyx)
     plot(fannyx)

     (fan.x.15 <- fanny(x, 2, memb.exp = 1.5)) # 'crispier' for obs. 26:28
     (fanny(x, 2, memb.exp = 3))               # more fuzzy in general

     data(ruspini)
     f4 <- fanny(ruspini, 4)
     stopifnot(rle(f4$clustering)$lengths == c(20,23,17,15))
     plot(f4, which = 1)
     ## Plot similar to Figure 6 in Stryuf et al (1996)
     plot(fanny(ruspini, 5))

