PCAn                  package:PTAk                  R Documentation

_P_r_i_n_c_i_p_a_l _C_o_m_p_o_n_e_n_t _A_n_a_l_y_s_i_s _o_n _n _m_o_d_e_s

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

     Performs the Tucker_n_ model using a space version of RPVSCC
     ('SINGVA').

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

      PCAn(X,dim=c(2,2,2,3),test=1E-12,Maxiter=400,
                   smoothing=FALSE,smoo=list(NA),
                     verbose=getOption("verbose"),file=NULL,
                       modesnam=NULL,addedcomment="")

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

       X: a tensor (as an array) of order _k_, if non-identity metrics
          are used 'X' is a list with 'data'  as the array and 'met' a
          list of metrics

     dim: a vector of  numbers specifying the dimensions in each space 

    test: control of convergence

 Maxiter: maximum number of iterations allowed for convergence

smoothing: see 'SVDgen'

    smoo: see 'PTA3'

 verbose: control printing

    file: output printed at the prompt if 'NULL', or printed in the
          given  'file'

modesnam: character vector of the names of the modes, if 'NULL' "'mo
          1'" ..."'mo k'"

addedcomment: character string printed after the title of the analysis

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

     Looking for the best rank-one tensor approximation (LS) the three
     methods described in the package are equivalent. If the number of
     tensors looked for is greater then one the  methods differs:
     PTA-_k_modes will "look" for "best" approximation according to the
     _orthogonal rank_ (_i.e._ the rank-one tensors are orthogonal),
     PCA-_k_modes will look for best approximation according to the
     _space ranks_ (_i.e._ the rank of every bilinear form, that is the
     number of components in each space), PARAFAC/CANDECOMP will look
     for best approximation according to the _rank_ (_i.e._ the
     rank-one tensors are not necessarily orthogonal). For the sake of
     comparisons the PARAFAC/CANDECOMP method and the PCA-_n_modes are
     also in the  package but complete functionnality  of the use these
     methods  and more complete packages may be fetched at the www site
     quoted below. 
      Recent work from Tamara G Kolda showed on an example that
     _orthogonal rank_ decompositions are not necesseraly nested. This
     makes PTA-_k_modes a model with nested decompositions not giving
     the exact _orthogonal rank_. So PTA-_k_modes will look for best
     approximation according to orthogonal tensors in a nested
     approximmation process.

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

     a 'PCAn' (inherits 'PTAk') object

_N_o_t_e:

     The use of metrics (diagonal or not) and smoothing extend
     flexibility of analysis.

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

     Didier Leibovici c3s2i@free.fr

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

     Caroll J.D  and Chang J.J (1970) _Analysis of individual
     differences in multidimensional   scaling via n-way generalization
     of "Eckart-Young" decomposition_. Psychometrika 35,283-319.

     Harshman R.A (1970) _Foundations of the PARAFAC procedure: models
     and conditions for "an explanatory" multi-mode factor analysis_.
     UCLA Working Papers in Phonetics, 16,1-84.

     Kroonenberg P (1983) _Three-mode Principal Component Analysis:
     Theory and Applications_. DSWO press. Leiden.(related references
     in <URL: http://www.fsw.leidenuniv.nl/~kroonenb/>)

     Leibovici D and Sabatier R (1998) _A Singular Value Decomposition
     of a k-ways array for a Principal Component Analysis of multi-way
     data, the PTA-k_. Linear Algebra and its Applications,
     269:307-329.

     Kolda T.G (2003)_ A Counterexample to the Possibility of an
     Extension of the Eckart-Young Low-Rank Approximation Theorem for
     the Orthogonal Rank Tensor Decomposition_. SIAM J. Matrix
     Analysis, 24(2):763-767, Jan. 2003.

