pcalp: PCA-Lp

In pcaL1: L1-Norm PCA Methods

PCA-Lp

Description

Performs a principal component analysis using the greedy algorithms PCA-Lp(G) and PCA-Lp(L) given by Kwak (2014).

Usage

pcalp(X, projDim=1, p = 1.0, center=TRUE, projections="none", 
        initialize="l2pca",solution = "L", 
	epsilon = 0.0000000001, lratio = 0.02)

Arguments

X	data, must be in matrix or table form.
projDim	number of dimensions to project data into, must be an integer, default is 1.
p	p-norm use to measure the distance between points.
center	whether to center the data using the median, default is TRUE.
projections	whether to calculate reconstructions and scores using the L1 norm ("l1") the L2 norm ("l2") or not at all ("none", default).
initialize	method for initial guess for component. Options are: "l2pca" - use traditional PCA/SVD, "maxx" - use the point with the largest norm, "random" - use a random vector.
solution	method projection vector update. Options are: "G" - PCA-Lp(G) implementation: Gradient search, "L" - PCA-Lp(L) implementation: Lagrangian (default).
epsilon	for checking convergence.
lratio	learning ratio, default is 0.02. Suggested value 1/(nr. instances).

Details

The calculation is performed according to the algorithm described by Kwak (2014), an extension of the original Kwak(2008). The method is a greedy locally-convergent algorithm for finding successive directions of maximum Lp dispersion.

Value

'pcalp' returns a list with class "pcalp" containing the following components:

loadings	the matrix of variable loadings. The matrix has dimension ncol(X) x projDim. The columns define the projected subspace.
scores	the matrix of projected points. The matrix has dimension nrow(X) x projDim.
dispExp	the proportion of L1 dispersion explained by the loadings vectors. Calculated as the L1 dispersion of the score on each component divided by the L1 dispersion in the original data.
projPoints	the matrix of projected points in terms of the original coordinates. The matrix has dimension nrow(X) x ncol(X).

References

Kwak N. (2008) Principal component analysis based on L1-norm maximization, IEEE Transactions on Pattern Analysis and Machine Intelligence, 30: 1672-1680. DOI:10.1109/TPAMI.2008.114

Kwak N. (2014). Principal component analysis by Lp-norm maximization. IEEE transactions on cybernetics, 44(5), 594-609. DOI: 10.1109/TCYB.2013.2262936

Examples

  
##for 100x10 data matrix X, 
## lying (mostly) in the subspace defined by the first 2 unit vectors, 
## projects data into 1 dimension.
X <- matrix(c(runif(100*2, -10, 10), rep(0,100*8)),nrow=100) 
               + matrix(c(rep(0,100*2),rnorm(100*8,0,0.1)),ncol=10)
mypcalp <- pcalp(X, p = 1.5)

##projects data into 2 dimensions.
mypcalp <- pcalp(X, projDim=2, p = 1.5, center=FALSE, projections="l1")

## plot first two scores
plot(mypcalp$scores)

pcaL1 documentation built on Jan. 22, 2023, 1:55 a.m.