l1pca {pcaL1}

L1-PCA

Description

Performs a principal component analysis using the algorithm L1-PCA given by Ke and Kanade (2005).

Usage

   l1pca(X, projDim=1, center=TRUE, projections="l1", 
	 initialize="l2pca", tolerance=0.0001, iterations=10)

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.

center

whether to center the data using the median, default is TRUE.

projections

Whether to calculate reconstructions and scores using the L1 ("l1", default) or L2 ("l2") norm.

initialize

initial guess for loadings matrix. Options are: "l2pca" - use traditional PCA/SVD, "random" - use a randomly-generated matrix. The user can also provide a matrix as an initial guess.

tolerance

sets the convergence tolerance for the algorithm, default is 0.0001.

iterations

sets the number of iterations to run before returning the result, default is 10.

Details

The calculation is performed according to the linear programming-based algorithm described by Ke and Kanade (2005). The method is a locally-convergent algorithm for finding the L1-norm best-fit subspace by alternatively optimizing the scores and the loadings matrix at each iteration. Linear programming instances are solved using Clp (http://www.coin-or.org)

Value

'l1pca' returns a list with class "l1pca" containing the following components:

loadings

the matrix of variable loadings. The matrix has dimension ncol(X) x projDim. The columns defined 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 (reconstructions). The matrix has dimension nrow(X) x ncol(X).

References

Ke Q. and Kanade T. (2005) Robust L1 norm factorization in the presence of outliers and missing data by alternative convex programming, IEEE Conference on Computer Vision and Pattern Recognition. DOI:10.1109/CVPR.2005.309

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)
myl1pca <- l1pca(X)

##projects data into 2 dimensions.
myl1pca <- l1pca(X, projDim=2, center=FALSE, 
                 tolerance=0.00001, iterations=20)

## plot first two scores
plot(myl1pca$scores)