# Orthogonal Tensor projection Orthogonal-tensoriel projection of a tensor according to a rank-1 tensor, or a to bigger structure defined by kronecker product of matrices. ```r 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. $$ (P_{S1} \otimes P_{S2} \otimes \ldots \otimes P_{Sk})(X) $$ . ## Returns A tensor with dimensions as `X` ## Author(s) Didier G. Leibovici GeotRycs@gmail.com ## References 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 (2000) **Multiway Multidimensional Analysis for Pharmaco-EEG Studies**. [http://www.fmrib.ox.ac.uk/analysis/techrep/tr00dl2/tr00dl2.pdf](http://www.fmrib.ox.ac.uk/analysis/techrep/tr00dl2/tr00dl2.pdf) ## Note For PTA-**k**modes 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-**k**modes (PTAk on Instrumental Variables Leibovici(1993). (see other references for tensorial product of linear operators in Leibovici(2000) **e.g.** Dauxois et al.(1994)) ## See Also `PTAk` ## Examples ```r 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,...) } ```

PROJOT function