Performs the Tuckern model using a space version of RPVSCC (SINGVA).
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 PCAn (inherits PTAk) object
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
a vector of numbers specifying the dimensions in each space
control of convergence
maximum number of iterations allowed for convergence
see SVDgen
see PTA3
control printing
output printed at the prompt if NULL, or printed in the given file
character vector of the names of the modes, if NULL
"mo 1" ..."mo k"
character string printed after the title of the analysis
Didier G. Leibovici
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-kmodes will "look" for "best" approximation according to the
orthogonal rank (i.e. the rank-one tensors are
orthogonal), PCA-kmodes 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-nmodes 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-kmodes a model with
nested decompositions not giving the exact orthogonal rank.
So PTA-kmodes will look for best approximation according to orthogonal tensors in a nested approximmation process.
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. (There was a maintained (by Pieter) library of contributions to multiway analysis ...))
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.