PCAmixdata (version 2.1)

PCAmix: Principal Component Analysis for a mixture of qualitative and quantitative variables

Description

PCAmix is a principal component method for a mixture of qualitative and quantitative variables. PCAmix includes the ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. Squared loadings are correlation ratios for qualitative variables and squared correlation for quantitative variables. Missing values are replaced by means for quantitative variables and by zeros in the indicator matrix for qualitative variables. Note that when all the p variables are qualitative, the scores of the n observations are equal to the usual scores of MCA times square root of p and the eigenvalues are then equal to the usual eigenvalues of MCA times p.

Usage

PCAmix(X.quanti = NULL, X.quali = NULL, ndim = 5,
    weight.col = NULL, weight.row = NULL, graph = TRUE)

Arguments

X.quanti

a numeric matrix of data, or an object that can be coerced to such a matrix (such as a numeric vector or a data frame with all numeric columns).

X.quali

a categorical matrix of data, or an object that can be coerced to such a matrix (such as a character vector, a factor or a data frame with all factor columns).

ndim

number of dimensions kept in the results (by default 5).

graph

boolean, if TRUE the following graphs are displayed for the first two dimensions of PCAmix: plot of the observations (the scores), plot of the variables (squared loadings) plot of the correlation circle (if quantitative variables are available

weight.col

a vector of weights for the quantitatives variables and for the indicator of qualitatives variables

weight.row

a vector of weights for the individuals

Value

• eig
  eigenvalues (i.e. variances) of the Principal Components (PC).
• scores
  scores a n by ndim numerical matrix which contains the scores of the n observations on the ndim first Principal Components (PC).
• scores.stand
  a n by ndim numerical matrix which contains the standardized scores of the n observations on the ndim first Principal Components (PC).
• sload
  a p by ndim numerical matrix which contains the squared loadings of the p variables on the ndim first PC. For quantitative variables (resp. qualitative), squared loadings are the squared correlations (resp. the correlation ratios) with the PC scores.
• categ.coord
  'NULL' if X.quali is 'NULL' . Otherwise a m by ndim numerical matrix which contains the coordinates of the m categories of the qualitative variables on the ndim first PC. The coordinates of the categories are the averages of the standardized PC scores of the observations in those categories.
• quanti.cor
  'NULL' if X.quanti is 'NULL'. Otherwise a p1 by ndim numerical matrix which contains the coordinates (the loadings) of the p1 quantitative variables on the ndim first PC. The coordinates of the quantitative variables are correlations with the PC scores.
• quali.eta2
  'NULL' if X.quali is 'NULL' . Otherwise a p2 by ndim numerical matrix which contains the squared loadings of the p2 qualitative variables on the ndim first PC. The squared loadings of the qualitative variables are correlation ratios with the PC scores.
• res.ind
  Results for the individuals (coord,contrib in percentage,cos2)
• res.quanti
  Results for the quantitatives variables (coord,contrib in percentage,cos2)
• res.categ
  Results for the categories of the categorials variables (coord,contrib in percentage,cos2)
• coef
  Coefficients of the linear combinations of the quantitative variables and the categories for constructing the principal components of PCAmix.
• V
  The standardized loadings.
• rec
  Results of the fonction recod(X.quanti,X.quali).
• M
  Metric used in the svd for the weights of the variables.

References

{Chavent, M., Kuentz, V., Saracco, J. (2011), Orthogonal Rotation in PCAMIX. Advances in Classification and Data Analysis, Vol. 6, pp. 131-146. Kiers, H.A.L., (1991), Simple structure in Component Analysis Techniques for mixtures of qualitative and quantitative variables, Psychometrika, 56, 197-212.}

Examples

#PCAMIX:
data(wine)
X.quanti <- wine[,c(3:29)]
X.quali <- wine[,c(1,2)]
pca<-PCAmix(X.quanti,X.quali,ndim=4)
pca<-PCAmix(X.quanti,X.quali,ndim=4,graph=FALSE)
pca$eig

#Scores on dim 1-2
plot(pca,choice="ind",quali=wine[,1],
    posleg="bottomleft",main="Scores")
#Scores on dim 2-3
plot(pca,choice="ind",axes=c(2,3),quali=wine[,1],
    posleg="bottomleft",main="Scores")
#Other graphics
plot(pca,choice="var",main="Squared loadings")
plot(pca,choice="categ",main="Categories")
plot(pca,choice="cor",xlim=c(-1.5,2.5),
    main="Correlation circle")
#plot with standardized scores:
plot(pca,choice="ind",quali=wine[,1],stand=TRUE,
    posleg="bottomleft",main="Standardized Scores")
plot(pca,choice="var",stand=TRUE,main="Squared loadings")
plot(pca,choice="categ",stand=TRUE,main="Categories")
plot(pca,choice="cor",stand=TRUE,main="Correlation circle")


#PCA:
data(decathlon)
quali<-decathlon[,13]
pca<-PCAmix(decathlon[,1:10])
pca<-PCAmix(decathlon[,1:10], graph=FALSE)
plot(pca,choice="ind",quali=quali,cex=0.8,
    posleg="topright",main="Scores")
plot(pca, choice="var",main="Squared correlations")
plot(pca, choice="cor",main="Correlation circle")


#MCA
data(flower)
mca <- PCAmix(X.quali=flower[,1:4])
mca <- PCAmix(X.quali=flower[,1:4],graph=FALSE)
plot(mca,choice="ind",main="Scores")
plot(mca,choice="var",main="Correlation ratios")
plot(mca,choice="categ",main="Categories")

#Missing values
data(vnf)
PCAmix(X.quali=vnf)
vnf2<-na.omit(vnf)
PCAmix(X.quali=vnf2)