alr               package:compositions               R Documentation

_A_d_d_i_t_i_v_e _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 additive log ratio transform of a (dataset of)
     composition(s), and its inverse.

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

               alr( x  )
               alr.inv( z )
               

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

       x: a composition, not necessarily closed

       z: the alr-transform of a composition, thus a (D-1)-dimensional
          real vector

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

     The alr-transform maps a composition in the D-part
     Aitchison-simplex non-isometrically to a D-1 dimensonal euclidian
     vector, treating the last part as common denominator of the
     others. The data can then be analysed in this transformation by
     all classical multivariate analysis tools not relying on a
     distance. The interpretation of the results is relatively simple,
     since the relation to the original D-1 first parts is preserved.
     However distance is an extremely relevant concept in most types of
     analysis, where a 'clr' or 'ilr' transformation should be
     preferred.

     The additive logratio transform is given by

                       alr(x)_i := ln(x_i/x_D)

     .

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

     'alr' gives the additive log ratio transform; accepts a
     compositional dataset 'alr.inv' gives a closed composition with
     the given alr-transform; accepts a dataset

_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:

     'clr','alr','apt', <URL: http://ima.udg.es/Activitats/CoDaWork03>

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

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

