PTA3 {PTAk}

Principal Tensor Analysis on 3 modes

Description

Performs a truncated SVD-3modes analysis with or without specific metrics, penalised or not.

Usage

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 3modes 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-3modes a model with nested decompositions not giving the exact orthogonal rank. So PTA-3modes 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 3mode Principal Tensor. To smooth differently the associated solutions one have to put another level of nested lists otherwise the function given at the 3mode level will be used for all. These rules are te same for PTAk.

Value

a PTAk object

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

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 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. doi:10.18637/jss.v034.i10

See Also

SVDgen, FCAk, PTAk, summary.PTAk

Examples

 # 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)