zeta-methods        package:kappalab        R Documentation(latin1)

_T_h_e _z_e_t_a _t_r_a_n_s_f_o_r_m

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

     Computes the zeta transform of a set function given under the form
     of its Mobius transform. The zeta transform is the inverse of the
     Mobius transform.

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



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

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

     _o_b_j_e_c_t = "_M_o_b_i_u_s._g_a_m_e"  Returns an object of class 'game'.

     _o_b_j_e_c_t = "_M_o_b_i_u_s._s_e_t._f_u_n_c"  Returns an object of class 'set.func'.

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

     G-C. Rota (1964), _On the foundations of combinatorial theory. I.
     Theory of Mobius functions_, Z. Wahrscheinlichkeitstheorie und
     Verw. Gebiete 2, pages 340-368.

     A. Chateauneuf and J-Y. Jaffray (1989), _Some characterizations of
     lower  probabilities and other monotone capacities through the use
     of Mobius inversion_, Mathematical Social Sciences 17, pages
     263-283.

     M. Grabisch, J-L. Marichal and M. Roubens (2000), _Equivalent
     representations of set functions_, Mathematics of Operations
     Research 25:2, pages 157-178.

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

     'capacity-class',  
      'card.set.func-class',  
      'game-class', 
      'set.func-class', 
      'Mobius.capacity-class',  
      'Mobius.card.set.func-class',  
      'Mobius.game-class', 
      'Mobius.set.func-class',  
      'Mobius-methods'.

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

     ## the Mobius transform of a capacity
     a <- Mobius.capacity(c(rep(0,15),1),4,4)
     a

     ## its zeta transform
     zeta(a)

     ## let us check
     Mobius(zeta(a))

     ## a similar example with a Mobius.card.set.func object
     mu <- card.set.func(-3:4)
     a <- Mobius(mu)
     zeta(a)

