fanny                package:cluster                R Documentation

_F_u_z_z_y _A_n_a_l_y_s_i_s _C_l_u_s_t_e_r_i_n_g

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

     fanny(x, k, diss = inherits(x, "dist"),
           memb.exp = 2, metric = "euclidean", stand = FALSE,
           maxit = 500, tol = 1e-15)

_A_r_g_u_m_e_n_t_s:

       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. 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.

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.

_D_e_t_a_i_l_s:

     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 to allow user specified
     'memb.exp'.

     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.

     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').

_V_a_l_u_e:

     an object of class '"fanny"' representing the clustering. See
     'fanny.object' for details.

_S_e_e _A_l_s_o:

     'agnes' for background and references; 'fanny.object',
     'partition.object', 'plot.partition', 'daisy', 'dist'.

_E_x_a_m_p_l_e_s:

     ## 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)
     ## Plot similar to Figure 6 in Stryuf et al (1996)
     plot(fanny(ruspini, 5))

