PROJOT {PTAk}

Orthogonal Tensor projection

Description

Orthogonal-tensoriel projection of a tensor according to a rank-1 tensor, or a to bigger structure defined by kronecker product of matrices.

Usage

 PROJOT(X,solu,numo=1,bortho=TRUE,Ortho=TRUE,metrics=NULL)

Arguments

X

a tensor(as an array) of any order

solu

an object like a solutions.PTAk object with at least v

numo

a vector of numbers or a list of vectors (length the order of the tensor) identifying for each space the structure to project onto, if NULL for a specific space then no projection is done for this space

bortho

list of logicals saying if the structures are othogonal or not.

Ortho

list of logicals telling the projectors on each space to be on the structure or on its orthogonal.

metrics

NULL or list of metrics (either diagonal or not) for each entry of X

Details

This function computes the tensorial orthogonal projection of X onto the tensorial structure defined by solu and numo. For each space (involved in the tensorial product where from X belongs), one defined the projector onto solu[[i]]$v[numo,] (or on its orthogonal if Ortho[[i]]==TRUE), then the result is the image of X by the tensorial product of the projectors, i.e.

(P_{S1} \otimes P_{S2} \otimes \ldots \otimes P_{Sk})(X)

.

Value

A tensor with dimensions as X

Note

For PTA-kmodes the projection used is only on rank-one tensors (Principal Tensors), i.e. numo is a number. The code here can be used for any structure (on each spaces) and constitutes the projector onto a tensorial structure, and can define the PTAIV-kmodes (PTAk on Instrumental Variables Leibovici(1993). (see other references for tensorial product of linear operators in Leibovici(2000) e.g. Dauxois et al.(1994))

Author(s)

Didier G. Leibovici [e-mail]

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 (2000) Multiway Multidimensional Analysis for Pharmaco-EEG Studies. http://www.fmrib.ox.ac.uk/analysis/techrep/tr00dl2/tr00dl2.pdf

See Also

PTAk

Examples

don <- array(1:360,c(5,4,6,3))
 don <- don + rnorm(360,10,2)

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

 apply(donfc,c(2,3,4),mean)
 apply(donfc,c(1),mean)

  # implementation de PTAIVk with obvious settings
 PTAIVk <- function(X,STruct,...)
         {X <- PROJOT(X$data,STruct,numo=Struct[[1]]$numo,Ortho=Struct[[1]]$Ortho,metrics=X$met)
         PTAk(X,...)
         }