dissimilarity           package:relations           R Documentation

_D_i_s_s_i_m_i_l_a_r_i_t_y _B_e_t_w_e_e_n _R_e_l_a_t_i_o_n_s

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

     Compute the dissimilarity between (ensembles of) relations.

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

     relation_dissimilarity(x, y = NULL, method = "symdiff", ...)

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

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

       y: 'NULL' (default), or as for 'x'.

  method: a character string specifying one of the built-in methods for
          computing dissimilarity, or a function to be taken as a
          user-defined method.  If a character string, its lower-cased
          version is matched against the lower-cased names of the
          available built-in methods using 'pmatch'.  See *Details* for
          available built-in methods.

     ...: further arguments to be passed to methods.

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

     Available built-in methods are as follows.  Apart from the
     symmetric difference distance, these are applicable to
     endorelations only.


     '"_s_y_m_d_i_f_f"' symmetric difference distance. This computes the
          cardinality of the symmetric difference of two relations,
          i.e., the number of tuples contained in exactly one of two
          relations.  For preference relations, this coincides with the
          _Kemeny-Snell_ metric (Kemeny and Snell, 1962).  For linear
          orders, it gives Kendall's tau metric (Diaconis, 1988).

          Can also be referred to as '"SD"' or '"manhattan"'.


     '"_e_u_c_l_i_d_e_a_n"' the Euclidean distance between the incidences.


     '"_C_S"' Cook-Seiford distance, a generalization of the distance
          function of Cook and Seiford (1978).  Let the generalized
          ranks of an object a in the (first) domain of an endorelation
          R be defined as the number of objects b dominating a (i.e.,
          for which a R b and not b R a), plus half the number of
          objects b equivalent to a (i.e., for which a R b and b R a). 
          For preference relations, this gives the usual Kendall ranks
          arranged according to decreasing preference (and averaged for
          ties).  Then the generalized Cook-Seiford distance is defined
          as the l_1 distance between the generalized ranks.  For
          linear orders, this gives Spearman's footrule metric
          (Diaconis, 1988).


     '"_C_K_S"' Cook-Kress-Seiford distance, a generalization of the
          distance function of Cook, Kress and Seiford (1986).  For
          each pair of objects a and b in an endorelation R, we can
          have a R b and not b R a or vice versa (cases of strict
          preference, a R b and b R a (the case of indifference), or
          neither a R b nor b R a (the case of incomparability). 
          (Only the last two are possible if a = b.)  The distance by
          Cook, Kress and Seiford puts indifference as the metric
          centroid between both preference cases and incomparability
          (i.e., indifference is at distance one from the other three,
          and each of the other three is at distance two from the
          others).  The generalized Cook-Kress-Seiford distance is the
          paired comparison distance (i.e., a metric) based on these
          distances between the four paired comparison cases.  (Formula
          3 in the reference must be slightly modified for the
          generalization from partial rankings to arbitrary
          endorelations.)


     '"_s_c_o_r_e"' score-based distance.  This computes Delta(s(x), s(y))
          for suitable score and distance functions s and Delta,
          respectively.  These can be specified by additional arguments
          'score' and 'Delta'.  If 'score' is a character string, it is
          taken as the method for 'relation_scores'.  Otherwise, if
          given it must be a function giving the score function itself.
           If 'Delta' is a number p >= 1, the usual l_p distance is
          used. Otherwise, it must be a function given the distance
          function. The defaults correspond to using the default
          relation scores and p = 1, which for linear orders gives
          Spearman's footrule distance.


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

     If 'y' is 'NULL', an object of class 'dist' containing the
     dissimilarities between all pairs of elements of 'x'.  Otherwise,
     a matrix with the dissimilarities between the elements of 'x' and
     the elements of 'y'.

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

     W. D. Cook, M. Kress and L. M. Seiford (1986), Information and
     preference in partial orders: a bimatrix representation.
     _Psychometrika_ *51*/2, 197-207.

     W. D. Cook and L. M. Seiford (1978), Priority ranking and
     consensus formation. _Management Science_, *24*/16, 1721-1732.

     P. Diaconis (1988), _Group Representations in Probability and
     Statistics_. Institute of Mathematical Statistics: Hayward, CA.

     J. G. Kemeny and J. L. Snell (1962), _Mathematical Models in the
     Social Sciences_, chapter Preference Rankings: An Axiomatic
     Approach. MIT Press: Cambridge.

