cpt               package:compositions               R Documentation

_C_e_n_t_e_r_e_d _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 centered planar  transform of a (dataset of)
     compositions and its inverse.

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

               cpt( x )
               cpt.inv( z )
               

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

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

       z: the cpt-transform of a composition or a data matrix of
          cpt-transforms of compositions. It is checked that the z sum
          up to 0.

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

     The cpt-transform maps a composition in the D-part real-simplex
     isometrically to a D-1 dimensional euclidian vector space,
     identified with a plane parallel  to the simplex but passing
     through the origin. However the transformation is not injective
     and does not even reach the whole plane. Thus resulting covariance
     matrices are always singular. 


     The data can then be analysed in this transformed space by all
     classical multivariate analysis tools not relying on a full rank
     of the covariance matrix. See 'ipt' and 'apt' for alternatives.
     The interpretation of the results is relitevly easy since the
     relation of each  transformed component to the original parts is
     preserved.

     The centered planar transform is given by

                      cpt(x)_i := clo(x)_i - 1/D

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

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

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

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

     'clr','apt','ipt'

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

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

