apt               package:compositions               R Documentation

_A_d_d_i_t_i_v_e _p_l_a_n_a_r _t_r_a_n_s_f_o_r_m

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

     Compute the additive planar  transform of a (dataset of)
     compositions and its inverse.

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

               apt( x )
               apt.inv( z )
               

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

       x: a composition or a matrix of compositions, not necessarily
          closed

       z: the apt-transform of a composition or  a matrix of
          alr-transforms of compositions

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

     The apt-transform maps a composition in the D-part real-simplex
     linearly to a D-1 dimensional euclidian vector. Although the
     transformation does not reach the whole R^{D-1}, resulting
     covariance matrices are typically of full rank. 


     The data can then be analysed in this transformation by all
     classical multivariate analysis tools not relying on distances.
     See 'cpt' and 'ipt' for alternatives. The interpretation of the
     results is easy since the relation to the first D-1 original
     variables is preserved.

     The additive planar transform is given by

                  apt(x)_i := clo(x)_i, i=1,...,D-1

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

     'apt' gives the centered planar transform, 'apt.inv' gives closed
     compositions with the given apt-transforms

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

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

     'alr','cpt','ipt'

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

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

