pcaL1 (version 1.5.7)

pcalp: 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)

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).

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.

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)