bootSVD (version 1.1)

bootSVD_LD: Calculate bootstrap distribution of \(n\)-dimensional PCs

Description

bootSVD_LD Calculates the bootstrap distribution of the principal components (PCs) of a low dimensional matrix. If the score matrix is inputted, the output of bootSVD_LD can be used to to calculate bootstrap standard errors, confidence regions, or the full bootstrap distribution of the high dimensional components. Most users may want to instead consider using bootSVD, which also calculates descriptions of the high dimensional components. Note that bootSVD calls bootSVD_LD.

Usage

bootSVD_LD(UD, DUt = t(UD), bInds = genBootIndeces(B = 1000, n =
  dim(DUt)[2]), K, warning_type = "silent", verbose = getOption("verbose"),
  centerSamples = TRUE)

Arguments

UD

(optional) a (\(n\) by \(n\)) matrix of scores, were rows denote individuals, and columns denote measurements in the PC space.

DUt

the transpose of UD. If both UD and UDt are entered and t(UD)!=DUt, the DUt argument will override the UD argument.

bInds

a (\(B\) by \(n\)) matrix of bootstrap indeces, where B is the number of bootstrap samples, and n is the sample size. Each row should be an indexing vector that can be used to generate a new bootstrap sample (i.e. sample(n,replace=TRUE)). The matrix of bootstrap indeces is taken as input, rather than being calculated within bootSVD_LD, so that this method can be more easily compared against traditional bootstrap SVD methods on the exact same bootstrap samples. The bInds matrix can be calculated using the helper function genBootIndeces).

K

the number of PCs to be estimated.

warning_type

passed to qrSVD, when taking the SVD of the low dimensional bootstrap score matrices.

verbose

if TRUE, a progress bar will appear.

centerSamples

whether each bootstrap sample should be centered before calculating the SVD.

Value

For each bootstrap matrix \((DU')^b\), let \(svd(DU')=:A^b D^b U^b\), where \(A^b\) and \(U^b\) are (\(n\) by \(n\)) orthonormal matrices, and \(D^b\) is a (\(n\) by \(n\)) diagonal matrix \(K\). Here we calculate only the first K columns of \(A^b\), but all n columns of \(U^b\). The results are stored as a list containing

As

a B-length list of the (n by K) matrices containing the first K PCs from each bootstrap sample. This list is indexed by b, with the \(b^{th}\) element containing the results from the \(b^{th}\) bootstrap sample.

ds

a B-length list of vectors, indexed by the bootstrap index b, with each vector containing the singular values of the corresponding bootstrap sample.

Us

a B-length list, indexed by the bootstrap index b, of the (\(n\) by \(n\)) matrices \(U^b\).

time

The computation time required for the procedure, taken using system.time.

If the score matrix is inputted to bootSVD_LD, the results can be transformed to get the PCs on the original space by multiplying each matrix A^b by the PCs of the original sample, V (see As2Vs). The bootstrap scores of the original sample are equal to U^b D^b.

Examples

# NOT RUN {
#use small n, small B, for a quick illustration
set.seed(0)
Y<-simEEG(n=100, centered=TRUE, wide=TRUE) 
svdY<-fastSVD(Y)
DUt<- tcrossprod(diag(svdY$d),svdY$u)
bInds<-genBootIndeces(B=50,n=dim(DUt)[2])
bootSVD_LD_output<-bootSVD_LD(DUt=DUt,bInds=bInds,K=3,verbose=interactive())
# }