dual-methods            package:kappalab            R Documentation

_T_h_e _d_u_a_l (_o_r _c_o_n_j_u_g_a_t_e) _t_r_a_n_s_f_o_r_m

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

     Computes the dual (also called the _conjugate_) of a set function.
     The dual of the dual of a game gives the original game back.

_M_e_t_h_o_d_s:


     _o_b_j_e_c_t = "_c_a_p_a_c_i_t_y"  Returns an object of class 'capacity'.

     _o_b_j_e_c_t = "_c_a_r_d._c_a_p_a_c_i_t_y"  Returns an object of class
          'card.capacity'.

     _o_b_j_e_c_t = "_c_a_r_d._s_e_t._f_u_n_c"  Returns an object of class 'card.game'.

     _o_b_j_e_c_t = "_s_e_t._f_u_n_c"  Returns an object of class 'game'. 

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

     T. Murofushi and M. Sugeno (2000), _Fuzzy measures and fuzzy
     integrals_, in: M. Grabisch, T. Murofushi, and M. Sugeno Eds,
     Fuzzy Measures and Integrals: Theory and Applications,
     Physica-Verlag, pages 3-41.

_S_e_e _A_l_s_o:

     'capacity-class',  
      'card.capacity-class',  
      'set.func-class', 
      'card.set.func-class'.

_E_x_a_m_p_l_e_s:

     ## a game
     mu <- game(c(0,-7:7))
     mu

     ## its dual 
     dual(mu)
     ## and mu again 
     dual(dual(mu))

     ## a similar example with the upper capacity
     mu <- capacity(c(0,rep(1,15)))
     mu
     dual(mu)
     dual(dual(mu))

     ## a similar example with an object of class card.capacity
     mu <- upper.capacity(6)
     mu
     dual(mu)
     dual(dual(mu))

     ## the dual of a set function is a game
     mu <- set.func(-7:8)
     mu
     dual(mu)
     mu <- card.set.func(-2:5)
     dual(mu)

