# Generalisation of Correspondence Analysis for k-way tables Performs a particular `PTAk` data as a ratio Observed/Expected under complete independence with metrics as margins of the multiple contingency table (in frequencies). ```r 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, ...) ``` ## Arguments - `X`: a multiple contingency table (array) of order **k** - `nbPT`: a number or a vector of dimension **(k-2)** - `nbPT2`: if 0 no 2-modes solutions will be computed, 1 =all, >1 otherwise - `minpct`: numerical 0-100 to control of computation of future solutions at this level and below - `smoothing`: see `SVDgen` - `smoo`: see `SVDgen` - `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 if `printt` after the title of the analysis - `chi2`: print the chi2 information when computing margins in `FCAmet` - `E`: if not `NULL` is an array with the same dimensions as X - `...`: any other arguments passed to SVDGen or other functions ## Details Gives the SVD-**k**modes 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). ## Returns a `FCAk` (inherits `PTAk`) object ## Author(s) Didier G. Leibovici ## References Escoufier Y (1985) **L'Analyse des correspondances : ses proprits et ses extensions**. ISI 45th session Amsterdam. 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) 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](https://eprints.ncrm.ac.uk/id/eprint/3178) ## See Also `PTAk`, `FCAmet`, `summary.FCAk` ## Examples ```r # try the demo # demo.FCAk() ```

FCAk function