SINGVA                 package:PTAk                 R Documentation

_O_p_t_i_m_i_s_a_t_i_o_n _a_l_g_o_r_i_t_h_m _R_P_V_S_C_C

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

     Computes the best rank-one approximation using the RPVSCC
     algorithm.

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

     SINGVA(X,test=1E-12,PTnam="vs111",Maxiter=1000,
                       verbose=getOption("verbose"),file=NULL,
                         smoothing=FALSE,smoo=list(NA),
                          modesnam=NULL,
                           Ini="Presvd",sym=NULL)

_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

    test: numerical value to stop optimisation

   PTnam: character giving the name of the k-modes Principal Tensor

 Maxiter: if `iter > Maxiter' prompts to carry on or not, then do it
          every other 200 iterations

 verbose: control printing

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

smoothing: logical to use smooth functiosns or not (see `SVDgen') 

    smoo: list of functions returning smoothed vectors (see `PTA3') 

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

     Ini: method used for initialisation of the algorithm (see
          `INITIA')

     sym: description of the symmetry of the tensor e.g. c(1,1,3,4,1)
          means the second mode and the fifth are identical to the
          first 

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

     The algorithm termed RPVSCC  in Leibovici(1993) is implemented to
     compute the first Principal Tensor (rank-one tensor with its
     singular value) of the given tensor `X'. According to the
     decomposition described in Leibovici(1993) and Leibovici and
     Sabatier(1998), the function gives a generalisation  to k modes of
     the best rank-one approximation issued from SVD whith 2 modes. It
     is identical to  the PCA-kmodes if only 1 dimension is asked in
     each space, and to PARAFAC/CANDECOMP if the rank of the
     approximation is fixed to 1. Then the methods differs, PTA-kmodes
     will look for best approximation according to the orthogonal rank
     (i.e. the rank-one tensors (of the decomposition) are orthogonal),
     PCA-kmodes will look for best approximation according to the space
     ranks (i.e.  ranks of every bilinear form deducted from the
     original tensor, 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).

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

     a `solutions.PTAk' object (without `datanam method')

_N_o_t_e:

     The algorithm was derived in generalising the transition formulae
     of SVD (Leibovici 1993), can also be understood as a
     generalisation of the power method (De Lathauwer et al. 2000). In
     this paper they also use a similar algorithm  to build bases in
     each space, reminiscent of three-modes and n-modes PCA
     (Kroonenberg(1980)), i.e. defining what they called a
     rank-(R1,R2,...,Rn) approximation (called here space ranks, see
     `PCAn'). RPVSCC stands for  Recherche de la Premire Valeur
     Singulire par Contraction Complte.

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

     Didier Leibovici didier@fmrib.ox.ac.uk

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

     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
     (1993) Facteurs  Mesures Rptes et Analyses Factorielles :
     applications  un suivi pidmiologique. Universit de Montpellier
     II. PhD Thesis in Mathmatiques et Applications (Biostatistiques).

     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.

     De Lathauwer L, De Moor B and Vandewalle J (2000) On the best
     rank-1 and rank-(R1,R2,...,Rn) approximation of higher-order
     tensors. SIAM J. Matrix Anal. Appl. 21,4:1324-1342.

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

     `INITIA', `PTAk', `PCAn', `CANDPARA'

