PCAmixdata (version 2.1)

PCArot: Varimax rotation in PCAmix

Description

Orthogonal rotation in PCAmix by maximization of the varimax function expressed in terms of PCAmix squared loadings (correlation ratios for qualitative variables and squared correlations for quantitative variables). PCArot includes the ordinary varimax rotation in Principal Component Analysis (PCA) and rotation in Multiple Correspondence Analysis (MCA) as special cases.

Usage

PCArot(obj, dim, itermax = 100, graph = TRUE)

Arguments

obj

an object of class PCAmix.

dim

number of rotated Principal Components.

itermax

maximum number of iterations in the Kaiser's practical optimization algorithm based on successive pairwise planar rotations.

graph

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

Value

• theta
  angle of rotation if dim is equal to 2.
• T
  matrix of rotation.
• eig
  variances of the ndim dimensions after rotation.
• scores
  a n by ndim numerical matrix which contains the rotated scores of the n observations.
• scores.stand
  a n by dim numerical matrix which contains the rotated standardized scores of the n observations.
• sload
  a p by ndim numerical matrix which contains the rotated squared loadings of the p variables. For quantitative variables (resp. qualitative), the rotated squared loadings are the squared correlations (resp. the correlation ratios) with the rotated standardized scores.
• categ.coord
  'NULL' if X.quali is 'NULL'. Otherwise a m by dim numerical matrix which contains the coordinates of the m categories after rotation. The coordinates of the categories after rotation are the averages of the rotated standardized scores of the observations in those categories.
• quanti.cor
  'NULL' if X.quanti is 'NULL'. Otherwise a p1 by dim numerical matrix which contains the coordinates (the loadings) of the p1 quantitative variables after rotation. The coordinates of the quantitative variables after rotation are correlations with the rotated standardized scores.
• quali.eta2
  'NULL' if X.quali is 'NULL'. Otherwise a p2 by dim numerical matrix which contains the squared loadings of the p2 quantitative variables after rotation. The squared loadings of the qualitative variables after rotation are correlation ratio with the rotated standardized scores.
• coef
  Coefficients of the linear combinations of the quantitative variables and the categories for constructing the rotated principal components of PCAmix.
• V
  The rotated standardized loadings.

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.

See Also

plot.PCAmix, summary.PCAmix,PCAmix,predict.PCAmix

Examples

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

rot<-PCArot(pca,3)
rot
rot$eig #percentages of variances after rotation

plot(rot,choice="ind",quali=wine[,1],
	posleg="bottomleft", main="Rotated scores")
plot(rot,choice="var",main="Squared loadings after rotation")
plot(rot,choice="categ",main="Categories after rotation")
plot(rot,choice="cor",main="Correlation circle after rotation")



#PCA:
data(decathlon)
quali<-decathlon[,13]
pca<-PCAmix(decathlon[,1:10], graph=FALSE)

rot<-PCArot(pca,3)
plot(rot,choice="ind",quali=quali,cex=0.8,
	posleg="topright",main="Scores after rotation")
plot(rot, choice="var", main="Squared correlations after rotation")
plot(rot, choice="cor", main="Correlation circle after rotation")

#MCA
data(flower)
mca <- PCAmix(X.quali=flower[,1:4],graph=FALSE)

rot<-PCArot(mca,2)
plot(rot,choice="ind",main="Scores after rotation")
plot(rot, choice="var", main="Correlation ratios after rotation")
plot(rot, choice="categ", main="Categories after rotation")