pclust            package:relations            R Documentation(utf8)

_P_r_o_t_o_t_y_p_e-_B_a_s_e_d _P_a_r_t_i_t_i_o_n_s _o_f _R_e_l_a_t_i_o_n_s

_D_e_s_c_r_i_p_t_i_o_n:

     Compute prototype-based partitions of a relation ensemble by
     minimizing sum w_b u_{bj}^m d(x_b, p_j)^e, the sum of the
     case-weighted and membership-weighted e-th powers of the
     dissimilarities between the elements x_b of the ensemble and the
     prototypes p_j, for suitable dissimilarities d and exponents e.

_U_s_a_g_e:

     relation_pclust(x, k, method, m = 1, weights = 1, control = list())

_A_r_g_u_m_e_n_t_s:

       x: an ensemble of relations, or something which can be coerced
          to this (see 'relation_ensemble').

       k: an integer giving the number of classes to be used in the
          partition.

  method: the consensus method to be employed, see
          'relation_consensus'.

       m: a number not less than 1 controlling the softness of the
          partition (as the "fuzzification parameter" of the fuzzy
          c-means algorithm).  The default value of 1 corresponds to
          hard partitions obtained from a generalized k-means problem;
          values greater than one give partitions of increasing
          softness obtained from a generalized fuzzy c-means problem.

 weights: a numeric vector of non-negative case weights. Recycled to
          the number of elements in the ensemble given by 'x' if
          necessary.

 control: a list of control parameters.  See *Details*.

_D_e_t_a_i_l_s:

     For m = 1, a generalization of the Lloyd-Forgy variant of the
     k-means algorithm is used, which iterates between reclassifying
     objects to their closest prototypes, and computing new prototypes
     as consensus relations (generalized "central relations", Régnier
     (1965)) for the classes.  This procedure was proposed in Gaul and
     Schader (1988) as the "Clusterwise Aggregation of Relations"
     (CAR).

     For m > 1, a generalization of the fuzzy c-means recipe is used,
     which alternates between computing optimal memberships for fixed
     prototypes, and computing new prototypes as the consensus
     relations for the classes.

     This procedure is repeated until convergence occurs, or the
     maximal number of iterations is reached.

     Consensus relations are computed using 'relation_consensus'.

     Available control parameters are as follows.

     '_m_a_x_i_t_e_r' an integer giving the maximal number of iterations to be
          performed.  Defaults to 100.

     '_r_e_l_t_o_l' the relative convergence tolerance.  Defaults to
          'sqrt(.Machine$double.eps)'.

     '_c_o_n_t_r_o_l' control parameters to be used in 'relation_consensus'.

     The dissimilarities d and exponent e are implied by the consensus
     method employed, and inferred via a registration mechanism
     currently only made available to built-in consensus methods.  For
     the time being, all optimization-based consensus methods use the
     symmetric difference dissimilarity (see 'relation_dissimilarity')
     for d and e = 1.

     The fixed point approach employed is a heuristic which cannot be
     guaranteed to find the global minimum.  Standard practice would
     recommend to use the best solution found in "sufficiently many"
     replications of the base algorithm.

_V_a_l_u_e:

     An object of class 'cl_partition'.

_R_e_f_e_r_e_n_c_e_s:

     S. Régnier (1965). Sur quelques aspects mathématiques des
     problèmes de classification automatique. _ICC Bulletin_, *4*,
     175-191.

     W. Gaul and M. Schader (1988). Clusterwise aggregation of
     relations. _Applied Stochastic Models and Data Analysis_, *4*,
     273-282.

