# Principal Tensor Analysis on 3 modes Performs a truncated SVD-**3**modes analysis with or without specific metrics, penalised or not. ```r PTA3(X,nbPT=2,nbPT2=1, smoothing=FALSE, smoo=list(function(u)ksmooth(1:length(u),u,kernel="normal", bandwidth=4,x.points=(1:length(u)))$y, function(u)smooth.spline(u,df=3)$y, NA), minpct=0.1,verbose=getOption("verbose"),file=NULL, modesnam=NULL,addedcomment="", ...) ``` ## Arguments - `X`: a tensor (as an array) of order **3**, if non-identity metrics are used `X` is a list with `data` as the array and `met` a list of metrics - `nbPT`: a number specifying the number of **3**modes Principal Tensors requested - `nbPT2`: if 0 no **2**-modes solutions will be computed, 1 =all, >1 otherwise - `smoothing`: logical to consider smoothing or not - `smoo`: a list of length 3 with lists of functions operating on vectors component for the appropriate dimension (see details) - `minpct`: numerical 0-100 to control of computation of future solutions at this level and below - `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 after the title of the analysis - `...`: any other arguments passed to SVDGen or 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 **3** modes. It is the same algorithm used for `PTAk` but simpler as the recursivity implied by the **k** modes analysis is reduced only to one level **i.e** for every **3**-modes Principal Tensors, 3 SVD are performed for every contracted product with one the three components of the **3**-modes Principal Tensors (see `APSOLU3`, `PTAk`). Recent work from Tamara G Kolda showed on an example that **orthogonal rank** decompositions are not necesseraly nested. This makes PTA-**3**modes a model with nested decompositions not giving the exact **orthogonal rank**. So PTA-**3**modes will look for best approximation according to orthogonal tensors in a nested approximmation process. PTA3 decompositions is "a" generalisation of SVD but not the ... With the `smoothing` option `smoo` contain a list of (lists) of functions to apply on vectors of component (within the algorithm, see `SVDgen`). For a given dimension (1,2,or 3) a list of functions is given. If this list consists only of one function (no list needed) this function will be used at any level all the time : if one want to smooth only for the first Principal Tensor, put `list(function,NA)`. Now you start to understand this list will have a maximum length of `nbPT` and the corresponding function will be used for the corresponding **3**mode Principal Tensor. To smooth differently the associated solutions one have to put another level of nested lists otherwise the function given at the **3**mode level will be used for all. These rules are te same for `PTAk`. ## Returns a `PTAk` object ## 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 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 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. tools:::Rd_expr_doi("10.18637/jss.v034.i10") ## 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 extends the analysis towards functional data analysis, and or outliers "protection" is theoretically valid for tensors belonging to a tensor product of separable Hilbert spaces (**e.g.** Sobolev spaces) (see references in `PTAk`, `SVDgen`). ## Author(s) Didier G. Leibovici ## See Also `SVDgen`, `FCAk`, `PTAk`, `summary.PTAk` ## Examples ```r # example using Zone_climTUN dataset # # library(maptools) # library(RColorBrewer) # Yl=brewer.pal(11,"PuOr") # data(Zone_climTUN) ## in fact a modified version of plot.Map was used # plot(Zone_climTUN,ol=NA,auxvar=Zone_climTUN$att.data$PREC_OCTO) ##indicators 84 +3 to repeat # Zone_clim<-Zone_climTUN$att.data[,c(2:13,15:26,28:39,42:53,57:80,83:95,55:56)] # Zot <-Zone_clim[,85:87] ;temp <-colnames(Zot) # Zot <- as.matrix(Zot)%x%t(as.matrix(rep(1,12))) # colnames(Zot) <-c(paste(rep(temp [1],12),1:12),paste(rep(temp [2],12),1:12), # paste(rep(temp [3],12),1:12)) # Zone_clim <-cbind(Zone_clim[,1:84],Zot) # Zone3w <- array(as.vector(as.matrix(Zone_clim)),c(2599,12,10)) ## preprocessing #Zone3w<-Multcent(dat=Zone3w,bi=NULL,by=3,centre=mean, # centrebyBA=c(TRUE,FALSE),scalebyBA=c(TRUE,FALSE)) # Zone3w.PTA3<-PTA3(Zone3w,nbPT=3,nbPT2=3) ## summary and plot # summary(Zone3w.PTA3) #plot(Zone3w.PTA3,mod=c(2,3),nb1=1,nb2=11,lengthlabels=5,coefi=list(c(1,1,1),c(1,-1,-1))) #plot(Zone_climTUN,ol=NA,auxvar=Zone3w.PTA3[[1]]$v[1,],nclass=30) #plot(Zone_climTUN,ol=NA,auxvar=Zone3w.PTA3[[1]]$v[11,],nclass=30) ############## cat(" A little fun using iris3 and matching randomly 15 for each iris sample!","\n") cat(" then performing a PTA-3modes. If many draws are done, plots") cat(" show the stability of the first and third Principal Tensors.","\n") cat("iris3 is centered and reduced beforehand for each original variables.","\n") # demo function # source(paste(R.home(),"/library/PTAk/demo/PTA3.R",sep="")) # demo.PTA3(bootn=10,show=5,openX11s=FALSE) ```

PTA3 function