Performs a particular PTAk data as a ratio Observed/Expected
under complete independence with metrics as margins of the multiple
contingency table (in frequencies).
FCAk(X,nbPT=3,nbPT2=1,minpct=0.01,
smoothing=FALSE,smoo=rep(list(
function(u)ksmooth(1:length(u),u,kernel="normal",
bandwidth=3,x.points=(1:length(u)))$y),length(dim(X))),
verbose=getOption("verbose"),file=NULL,
modesnam=NULL,addedcomment="",chi2=TRUE,E=NULL, ...)a FCAk (inherits PTAk) object
a multiple contingency table (array) of order k
a number or a vector of dimension (k-2)
if 0 no 2-modes solutions will be computed, 1 =all, >1 otherwise
numerical 0-100 to control of computation of future solutions at this level and below
see SVDgen
see SVDgen
control printing
output printed at the prompt if NULL, or printed in the given file
character vector of the names of the modes, if NULL "mo 1"
..."mo k"
character string printed if printt after the title of the analysis
print the chi2 information when computing margins in FCAmet
if not NULL is an array with the same dimensions as X
any other arguments passed to SVDGen or other functions
Didier G. Leibovici
Gives the SVD-kmodes decomposition of the \(1+\chi^2/N\) of
the multiple contingency table of full count \(N=\sum X_{ijk...}\),
i.e. complete independence + lack of independence (including marginal
independences) as shown for example in Lancaster(1951)(see reference
in Leibovici(2000)). Noting \(P=X/N\), a PTAk of the
\((k+1)\)-uple is done, e.g. for a three way contingency table
\(k=3\) the 4-uple data and metrics is:
$$ ((D_I^{-1} \otimes D_J^{-1} \otimes D_K^{-1})P, \quad D_I, \quad D_J, \quad D_K)$$
where the metrics are diagonals of the corresponding margins. For
full description of arguments see PTAk. If E
is not NULL an FCAk-modes relatively to a model is
done (see Escoufier(1985) and therin reference
Escofier(1984) for a 2-way derivation), e.g. for a three way contingency table
\(k=3\) the 4-tuple data and metrics is:
$$ ((D_I^{-1} \otimes D_J^{-1} \otimes D_K^{-1})(P-E), \quad D_I, \quad D_J, \quad D_K)$$
If E was the complete independence (product of the margins)
then this would give an AFCk but without looking at the
marginal dependencies (i.e. for a three way table no two-ways lack of
independence are looked for).
Escoufier Y (1985) L'Analyse des correspondances : ses propri<e9>t<e9>s et ses extensions. ISI 45th session Amsterdam.
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
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")
Leibovici DG and Birkin MH (2013) Simple, multiple and multiway correspondence analysis applied to spatial census-based population microsimulation studies using R. NCRM Working Paper. NCRM-n^o 07/13, Id-3178 https://eprints.ncrm.ac.uk/id/eprint/3178
PTAk, FCAmet, summary.FCAk