clr               package:compositions               R Documentation

_C_e_n_t_e_r_e_d _l_o_g _r_a_t_i_o _t_r_a_n_s_f_o_r_m

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

     Compute the centered log ratio transform of a (dataset of)
     composition(s) and its inverse.

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

               clr( x )
               clr.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 clr-transform of a composition or a data matrix of
          clr-transforms of compositions, not necessarily centered
          (i.e. summing up to zero)

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

     The clr-transform maps a composition in the D-part
     Aitchison-simplex isometrically to a D-1 dimensonal euclidian
     vector subspace: consequently, the transformation is not injective
     and only yields vectors which elements sum up to 0. Thus resulting
     covariance matrices are always singular. 


     The data can then be analysed in this transformation by all
     classical multivariate analysis tools not relying on a full rank
     of the covariance. See 'ilr' and 'alr' for alternatives. The
     interpretation of the results is relatively easy since the
     relation between each original part and a transformed variable is
     preserved.

     The centered logratio transform is given by

                clr(x) := (*ln* _xi_ - mean(*ln* x_j)

     The image of the 'clr' is given by the vectors with entries
     summing to 0. This hyperplane is also called the clr-plane.

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

     'clr' gives the centered log ratio transform, 'clr.inv' gives
     closed compositions with the given clr-transforms

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

     Aitchison, J. (1986) _The Statistical Analysis of Compositional
     Data_ Monographs on Statistics and Applied Probability. Chapman &
     Hall Ltd., London (UK). 416p.

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

     'ilr','alr','apt'

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

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

