Power transform         package:compositions         R Documentation

_P_o_w_e_r _t_r_a_n_s_f_o_r_m _i_n _A_i_t_c_h_i_s_o_n_s _s_i_m_p_l_e_x

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

     The Aitchison Simplex with its two operations perturbation as + 
     and power transform as * is a vector space. This vector space is
     represented by these operations.

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

     power.acomp(x,s)
     x\method{acomp}{*}y
     x\method{acomp}{/}y
               

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

       x: an acomp composition or dataset of compositions (or a number
          or a numeric vector)

       y: a numeric vector of size 1 or nrow(x)

       s: a numeric vector of size 1 or nrow(x)

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

     The power transform is the basic multiplication operation of the
     Aitchison simplex seen as a vector space. It is defined as:

                  (x*y)_i:=  clo( (x_i^{y_i})_i )_i

     The division operation is just the multiplication with 1/y.

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

     An '"acomp"' vector or matrix.

_N_o_t_e:

     For '*' the arguments x and y can be exchanged. Note that  this
     definition generalizes the power by a scalar, since 'y' or  's'
     may be given as a scalar, or as a vector with as many components
     as the composition in 'acomp' 'x'. The result is then a matrix 
     where each row corresponds to the composition powered by one of
     the scalars  in the vector.

_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.

     Aitchison, J, C. Barcel'o-Vidal, J.J. Egozcue, V. Pawlowsky-Glahn
     (2002) A consise guide to the algebraic geometric structure of the
     simplex, the sample space for compositional data analysis, _Terra
     Nostra_, Schriften der Alfred Wegener-Stiftung, 03/2003

     Pawlowsky-Glahn, V. and J.J. Egozcue (2001) Geometric approach to
     statistical analysis on the simplex. _SERRA_ *15*(5), 384-398

     <URL: http://ima.udg.es/Activitats/CoDaWork03>

     <URL: http://ima.udg.es/Activitats/CoDaWork05>

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

     'ilr','clr', 'alr',

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

     acomp(1:5)* -1 + acomp(1:5)
     data(SimulatedAmounts)
     cdata <- acomp(sa.lognormals)
     plot( tmp <- (cdata-mean(cdata))/msd(cdata) )
     class(tmp)
     mean(tmp)
     msd(tmp)
     var(tmp)

