iit               package:compositions               R Documentation

_I_s_o_m_e_t_r_i_c _i_d_e_n_t_i_t_y _t_r_a_n_s_f_o_r_m

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

     Compute the isometric identity  transform of a vector (dataset) of
     amounts and its inverse.

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

               iit( x )
               iit.inv( z  )
               

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

       x: a vector or data matrix of amounts

       z: the iit-transform of a vector or  data.matrix of
          iit-transforms of amounts

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

     The iit-transform maps D amounts (considered in a real geometry)
     isometrically to a D dimensonal euclidian vector. The 'iit' is
     part of the 'rplus' framework. Despite its trivial operation, it
     is present to achieve maximal analogy between the 'aplus' and the
     'rplus' framework. 
       The data can then be analysed in this transformated space by all
     classical multivariate analysis tools. The interpretation of the
     results is easy since the relation to the original variables is
     preserved. However results may be inconsistent, since the
     multivariate analysis tools disregard the positivity condition and
     the inner laws of amounts.

     The isometric identity transform is a simple identity given by

                           iit(x)_i :=  x_i

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

     'ilt' gives the isometric identity transform, i.e. simply the
     input striped of the "rplus" class attribute, 'ipt.inv' gives
     amounts with class "rplus" with the given iit, i.e. simply the
     argument checked to be a valid "rplus" object, and with this class
     attribute.

_N_o_t_e:

     'iit' can be used to unclass amounts.

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

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

     'ilt', 'ilr,' 'rplus'

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

     (tmp <- iit(c(1,2,3)))
     iit.inv(tmp)
     iit.inv(tmp) - c(1,2,3) # 0
     data(Hydrochem)
     cdata <- Hydrochem[,6:19]
     pairs(iit(cdata)) 

