PTAk {PTAk}

Principal Tensor Analysis on k modes

Description

Performs a truncated SVD-kmodes analysis with or without specific metrics, penalised or not.

Usage

 PTAk(X,nbPT=2,nbPT2=1,minpct=0.1,
                smoothing=FALSE,
                   smoo=list(NA),
                    verbose=getOption("verbose"),file=NULL,
                     modesnam=NULL,addedcomment="", ...)

Arguments

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

nbPT

integer vector of length (k-2) specifying the maximum number of Principal Tensors requested for the (3,...,k-1, k) modes levels (see details), if it is not a vector every levels would have the same given nbPT value

nbPT2

if 0 no 2-modes solutions will be computed, 1 =all, >1 otherwise

minpct

numerical 0-100 to control of computation of future solutions at this level and below

smoothing

see PTA3, 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 if printt after the title of the analysis

...

any other arguments passed to other functions

Details

According to the decomposition described in Leibovici(1993) and Leibovici and Sabatier(1998) the function gives a generalisation of the SVD (2 modes) to k modes. The algorithm is recursive, calling APSOLUk which calls PTAk for (k-1). nbPT, nbPT2 and minpct control the number of Principal Tensors desired. For example nbPT=c(2,4,3) means a tensor of order 5 is analysed, the maximum number of 5-modes PT is set to 3, for each of them one sets a maximum of 4 associated 4-modes (for each of the five components), for each of these later a maximum of 2 associated 3-modes PT is asked (for each of the four components). Then nbPT2 complete for 2-modes associated or not. Overall minpct controls to carry on the algorithm at any level and lower, i.e. stops if 100(vs^2/ssx)<minpct (where vs is the singular value, and ssx is the total sum of squares of the tensor X or the "metric transformed" X). Putting a 0 at a given level in nbPT obviously automatically puts 0 in nbPT at lower levels. Putting high values in nbPT allows control only on minpct helping to reach the full decomposition. All these controls allow to truncate the full decomposition in a level-controlled fashion. Notice the full decomposition always contains any possible choice of truncation, i.e. the solutions are not dependant on the truncation scheme (Generalised Eckart-Young Theorem).
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.

Value

a PTAk object which consist of a list of lists. Each mode has a list in which is listed:

$v

matrix of components for the given mode

$iter

vector of iterations numbers where maximum was reach

$test

vector of test values at maximum

$modesnam

name of the mode

$v

matrix of components for the given mode

The last mode list has also some additional information on the analysis done:

$d

vector of singular values

$pct

percentage of sum of squares for each quared singular value

$ssX

vector of local sum of squares i.e. of the current tensor with the rescursive algorithm

$vsnam

vector of names given to the singular value according to a recursive data dependent scheme

$datanam

data reference

$method

call applied: could be PTAk or CANDPARA or PCAn or even SVDgen, with parameters choices

$addedcomment

the addedcomment (repeated) given in the call

You will notice that methods other than PTAk may not have all list elements but the essential ones such as: $v, $d, $ssX, and may also have additional ones like $coremat for PCAn (the core array).

Note

The use of metrics (diagonal or not) allows flexibility of analysis like in 2 modes e.g. correspondence analysis, discriminant analysis, robust analysis. Smoothing option extending the analysis towards functional data analysis is theoretically valid for Principal Tensors belonging to a tensor product of separable Hilbert spaces (e.g. Sobolev spaces) see Leibovici and El Maach (1997).

Author(s)

Didier G. Leibovici

References

Leibovici D(1993) Facteurs <e0> Mesures R<e9>p<e9>t<e9>es et Analyses Factorielles : applications <e0> un suivi <e9>pid<e9>miologique. Universit<e9> de Montpellier II. PhD Thesis in Math<e9>matiques et Applications (Biostatistiques).

Leibovici D and El Maache H (1997) Une d<e9>composition en Valeurs Singuli<e8>res d'un <e9>l<e9>ment d'un produit Tensoriel de k espaces de Hilbert S<e9>parables. Compte Rendus de l'Acad<e9>mie des Sciences tome 325, s<e9>rie I, Statistiques (Statistics) & Probabilit<e9>s (Probability Theory): 779-782.

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.

Leibovici D (2008) A Simple Penalised algorithm for SVD and Multiway functional methods. (to be submitted in the futur)

Leibovici DG (2010) Spatio-temporal Multiway Decomposition using Principal Tensor Analysis on k-modes:the R package PTAk. Journal of Statistical Software, 34(10), 1-34. doi:10.18637/jss.v034.i10

See Also

REBUILD, FCAk, PTA3 summary.PTAk

Examples

  # don <- array((1:3)%x%rnorm(6*4)%x%(1:10),c(10,4,6,3))

don <- array(1:360,c(5,4,6,3))
 don <- don + rnorm(360,1,2)

 dimnames(don) <- list(paste("s",1:5,sep=""),paste("T",1:4,sep=""),
          paste("t",1:6,sep=""),c("young","normal","old"))
   # hypothetic data on learning curve at different age and period of year

 ones <-list(list(v=rep(1,5)),list(v=rep(1,4)),list(v=rep(1,6)),list(v=rep(1,3)))

 don <- PROJOT(don,ones)
 don.sol <- PTAk(don,nbPT=1,nbPT2=2,minpct=0.01,
               verbose=TRUE,
                modesnam=c("Subjects","Trimester","Time","Age"),
                 addedcomment="centered on each mode")

don.sol[[1]] # mode Subjects results and components
don.sol[[2]] # mode Trimester results and components
don.sol[[3]] # mode Time results and components
don.sol[[4]] # mode Age results and  components with additional information on the call

 summary(don.sol,testvar=2)
  plot(don.sol,mod=c(1,2,3,4),nb1=1,nb2=NULL,
     xlab="Subjects/Trimester/Time/Age",main="Best rank-one approx" )
  plot(don.sol,mod=c(1,2,3,4),nb1=4,nb2=NULL,
      xlab="Subjects/Trimester/Time/Age",main="Associated to Subject vs1111")

#  demo function
 # demo.PTAk()