PPbigdata (version 1.0.0)

wsph: Spherize/Whiten data with observational weights

Description

This function performs PCA sphering/whitening transformation on data with observational weights.

Usage

wsph(data,weight)

Value

A list containing the following components:

data_wsph

The spherized data matrix that has a zero weighted mean for each column, and a weighted covariance that equals identity matrix.

data_wcen

The centered data matrix that has a zero weighted mean for each column. It's obtained by extracting the weighted sample mean from the original data matrix.

wmean

Vector of length ncol(data). It's the weighted sample mean of the original data matrix.

wcov

Matrix of size ncol(data) by ncol(data). It's the weighted sample covariance matrix of the original data matrix.

wsph_proj

Matrix of size ncol(data) by ncol(data). It's the sphering/whitening matrix for the transformation. The spherized data matrix data_wsph is obtained by multiplying the centered data matrix with weights data_wcen with this sphering/whitening matrix.

Arguments

data

A data matrix (of class matrix, or data.frame) containing only entries of class numeric.

weight

Vector of length nrow(data) of weights for each observation in the dataset. Must be of class numeric or integer or table. If NULL, the default value is a vector of 1 with length nrow(data), i.e., weights equal 1 for all observations.

Author

Yajie Duan, Javier Cabrera

Details

This function performs PCA sphering/whitening transformation on data with observational weights. Specifically, weighted sample mean and weighted sample covariance matrix are firstly calculated. Next, data are centered with weights, and spectral decomposition of the weighted covariance matrix is conducted to obtain its eigenvalues and eigenvectors. Based on them, the PCA sphering/whitening transformation is performed to obtain a spherized data matrix considering the observational weights. The spherized data matrix has a zero weighted mean for each column, and a weighted covariance that equals identity matrix.

References

Cabrera, J., & McDougall, A. (2002). Statistical consulting. Springer Science & Business Media.

Examples

 dataset = matrix(rnorm(300),100,3)

 # assign random weights to observations
 weight = sample(1:20,100,replace = TRUE)

 # spherize the dataset with observational weights
 res = wsph(dataset,weight)

 # spherized data matrix considering the observation weights
 res$data_wsph

 # The spherzied data matrix has a zero weighted sample mean for each column
 1/sum(weight)*t(as.matrix(weight))%*%as.matrix(res$data_wsph)

 # The spherzied data matrix has a weighted covariance that equals identity matrix
 1/sum(weight)*t(as.matrix(res$data_wsph))%*%diag(weight)%*%as.matrix(res$data_wsph)