colless             package:apTreeshape             R Documentation

_C_o_m_p_u_t_e _t_h_e _C_o_l_l_e_s_s' _s_h_a_p_e _s_t_a_t_i_s_t_i_c _o_n _t_r_e_e _d_a_t_a

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

     'colless' computes the Colless' index of a tree and provides
     standardized values according to the Yule and PDA models.

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

     colless(tree, norm = NULL)

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

    tree: An object of class '"treeshape"' on which the Colless' index
          is computed.

    norm: A character string equals to 'NULL' (default) for no
          normalization or one of '"pda"' or '"yule"'.

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

     The Colless' index Ic computes the sum of absolute values |L-R| at
     each node of the tree where L (resp. R) is the size of the left
     (resp. right) daughter clade at the node.

     The mean and standard deviation of the Colless's statistic Ic have
     been computed by Blum et al (2005). Under the Yule model the
     standardized index 

              Iyule = (Ic-n*log(n)-n(gamma-1-log(2)))/n

     converges in distribution (gamma is the Euler constant). The
     limiting distribution is non Gaussian and is characterized as a
     functional fixed-point equation solution. Under the PDA model, the
     standardization is different 

                         Ipda = Ic / n^(3/2)

     and converges in distribution to the Airy distribution (See
     Flajolet and Louchard (2001)). Standardized indices are useful
     when one wishes to compare trees with different sizes. The
     'colless' function returns the value of the unnormalized index
     (default) or one of the standardized statistics (Yule or PDA).

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

     An object of class 'numeric' which is the Colless' index of the
     tree.

_A_u_t_h_o_r(_s):

     Michael Blum <michael.blum@imag.fr>
      Nicolas Bortolussi <nicolas.bortolussi@imag.fr> 
      Eric Durand <eric.durand@imag.fr>
      Olivier Francois <olivier.francois@imag>

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

     Mooers, A. O. and Heard, S. B. (1997) Inferring Evolutionnary
     Process from Phylogenetic Tree Shape. _The Quarterly Review of
     Biology_, *72*, 31 - 54. 

     Blum, M., Francois, O. and Janson, S. The mean, variance and
     limiting distribution of two statistics sensitive to phylogenetic
     tree balance; manuscript available from 
      <URL: http://www-timc.imag.fr/Olivier.Francois/bfj.pdf>.

     Flajolet, P. and Louchard, O. (2001) Analytic Variations on the
     Airy Distribution. _Algorithmica_, *31*, 361 - 377.

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

     'sackin'

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

     ## Colless' index for a randomly generated PDA tree (unnormalized value)
     tpda<-rtreeshape(1,tip.number=70,model="pda")
     colless(tpda[[1]],norm="pda")
       
     ## Histogram of Colless' indices for randomly generated Yule trees
     main="Colless' indices for randomly generated Yule trees"
     xlab="Colless' indices"
     hist(sapply(rtreeshape(300,tip.number=50,model="yule"),FUN=colless,norm="yule"),
           freq=FALSE,main=main,xlab=xlab)
       
     ## Change the number of tips
     hist(sapply(rtreeshape(300,tip.number=100,model="yule"),FUN=colless,norm="yule"),
           freq=FALSE,main=main,xlab=xlab)

