aplus              package:compositions              R Documentation

_A_m_o_u_n_t_s _a_n_a_l_y_s_e_d _i_n _l_o_g-_s_c_a_l_e

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

     A class to analyse positive amounts in a logistic framework.

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

               aplus(X,parts=1:NCOL(oneOrDataset(X)),total=NA)
               

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

       X: vector or dataset of positive numbers

   parts: vector containing the indices  xor names of the columns to be
          used

   total: a numeric vectors giving the total amounts of each dataset. 

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

     Many multivariate datasets essentially describe amounts of D
     different parts in a whole. When the whole is large in relation to
     the considered parts, such that they do not exclude each other, or
     when the total amount of each componenten is indeed determined by
     the phenomenon under investigation and not by sampling artifacts
     (such as dilution or sample preparation), then the parts can be
     treated as amounts rather than as a composition (cf. 'acomp',
     'rcomp'). 
      Like compositions, amounts have some important properties.
     Amounts are always positive. An amount of exactly zero essentially
     means that we have a substance of an other quality. Different
     amounts - spanning different orders of magnitude  -  are often
     given in different units (ppm, ppb, %) and conversion factors need
     not to be fixed (e.g. for ppm, g/l, vol.%, mass %, molar
     fraction). Often, these amounts are also taken as indicators of
     other non-measured components (e.g. K as indicator for potassium
     feldspar), which might be proportional to the measured amount. 
     However, in contrast to compositions, amounts themselves do
     matter. Amounts are typically heavily skewed and in many practical
     cases a log-transform makes their distribution roughly symmetric,
     even normal. 
      In full analogy to Aitchison's compositions, we introduce vector
     space operations for amounts: the perturbation 'perturbe.aplus' as
     a vector space addition (corresponding to change of units), the
     power transformation 'power.aplus' as scalar multiplication
     describing the law of mass action, and a distance 'dist' which is
     independent of the chosen units. The induced vector space is
     mapped isometrically to a classical R^D by a simple
     log-transformation called 'ilt', resembling classical log
     transform approaches.   
      The general approach in analysing aplus objects is thus to
     performe classical multivariate analysis on ilt-transformed
     coordinates and to backtransform or display the results in such a
     way that they can be interpreted in terms of the original amounts.         
      The class aplus is complemented by the 'rplus', allowing to
     analyse amounts directly as real numbers, and by the classes
     'acomp' and 'rcomp' to analyse the same data as compositions
     disregarding the total amounts, focusing on relative amounts only. 
      The classes rcomp, acomp, aplus, and rplus are designed as
     similar as possible in order to allow direct comparison between
     results achieved   by the different approaches. Especially the
     acomp simplex transforms 'clr', 'alr', 'ilr' are mirrored in the
     aplus class by the single bijective isometric transform 'ilt'

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

     a vector of class '"aplus"' representing a vector of amounts or a
     matrix of class '"aplus"' representing multiple vectors of
     amounts, each vector in one row.

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

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

     'ilt','acomp', 'rplus', 'princomp.aplus',  'plot.aplus',
     'boxplot.aplus', 'barplot.aplus', 'mean.aplus', 'var.aplus',
     'variation.aplus', 'cov.aplus', 'msd'

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

     data(SimulatedAmounts)
     plot(aplus(sa.lognormals))

