selectLambda {rospca}

Selection of sparsity parameter using IC

Description

Selection of the sparsity parameter for ROSPCA and SCoTLASS using BIC of Hubert et al. (2016), and for SRPCA using BIC of Croux et al. (2013).

Usage

selectLambda(X, k, kmax = 10, method = "ROSPCA", lmin = 0, lmax = 2, lstep = 0.02,
             alpha = 0.75, stand = TRUE, skew = FALSE, multicore = FALSE, 
             mc.cores = NULL, P = NULL, ndir = "all")

Arguments

X

An n by p matrix or data matrix with observations in the rows and variables in the columns.

k

Number of Principal Components (PCs).

kmax

Maximal number of PCs to be computed, only used when method = "ROSPCA" or method = "ROSPCAg". Default is 10.

method

PCA method to use: ROSPCA ("ROSPCA" or "ROSPCAg"), SCoTLASS ("SCoTLASS" or "SPCAg") or SRPCA ("SRPCA"). Default is "ROSPCA".

lmin

Minimal value of \lambda to look at, default is 0.

lmax

Maximal value of \lambda to look at, default is 2.

lstep

Difference between two consecutive values of \lambda, i.e. the step size, default is 0.02.

alpha

Robustness parameter for ROSPCA, default is 0.75.

stand

Logical indicating if the data should be standardised, default is TRUE.

skew

Logical indicating if the skewed version of ROSPCA should be applied, default is FALSE.

multicore

Logical indicating if multiple cores can be used, default is TRUE. Note that this is not possible for the Windows platform, so multicore is always FALSE there.

mc.cores

Number of cores to use if multicore=TRUE, default is NULL which corresponds to the number of cores minus 1.

P

True loadings matrix, a numeric matrix of size p by k. The default is NULL which means that no true loadings matrix is specified.

ndir

Number of directions used when computing the outlyingness (or the adjusted outlyingness when skew=TRUE) in rospca, see outlyingness and adjOutl for more details.

Details

We select an optimal value of \lambda for a certain method on a certain dataset by looking at an equidistant grid of \lambda values. For each value of \lambda, we apply the method on the dataset using this sparsity parameter, and compute an Information Criterion (IC). The optimal value of \lambda is then the one corresponding to the minimal IC. The ICs we consider are the BIC of for Hubert et al. (2016) for ROSPCA and SCoTLASS, and the BIC of Croux et al. (2013) for SRPCA. The BIC of Hubert et al. (2016) is defined as

BIC(\lambda)=\ln(1/(h_1p)\sum_{i=1}^{h_1} OD^2_{(i)}(\lambda))+df(\lambda)\ln(h_1p)/(h_1p),

where h_1 is the size of H_1 (the subset of observations that are kept in the non-sparse reweighting step) and OD_{(i)}(\lambda) is the ith smallest orthogonal distance for the model when using \lambda as the sparsity parameter. The degrees of freedom df(\lambda) are the number of non-zero loadings when \lambda is used as the sparsity parameter.

Value

A list with components:

opt.lambda

Value of \lambda corresponding to minimal IC.

min.IC

Minimal value of IC.

Lambda

Numeric vector containing the used values of \lambda.

IC

Numeric cector containing the IC values corresponding to all values of \lambda in Lambda.

loadings

Loadings obtained using method with sparsity parameter opt.lambda, a numeric matrix of size p by k.

fit

Fit obtained using method with sparsity parameter opt.lambda. This is a list containing the loadings (loadings), the eigenvalues (eigenvalues), the standardised data matrix used as input (Xst), the scores matrix (scores), the orthogonal distances (od) and the score distances (sd).

type

Type of IC used: BICod (BIC of Hubert et al. (2016)) or BIC (BIC of Croux et al. (2013)).

measure

A numeric vector containing the standardised angles between the true and the estimated loadings matrix for each value of \lambda if a loadings matrix is given. When no loadings matrix is given as input (P=NULL), measure is equal to NULL.

Author(s)

Tom Reynkens

References

Hubert, M., Reynkens, T., Schmitt, E. and Verdonck, T. (2016). “Sparse PCA for High-Dimensional Data with Outliers,” Technometrics, 58, 424–434.

Croux, C., Filzmoser, P., and Fritz, H. (2013), “Robust Sparse Principal Component Analysis,” Technometrics, 55, 202–214.

See Also

selectPlot, mclapply, angle

Examples

X <- dataGen(m=1, n=100, p=10, eps=0.2, bLength=4)$data[[1]]

sl <- selectLambda(X, k=2, method="ROSPCA", lstep=0.1)
selectPlot(sl)