cov_block_autocorrelation: Generates a p-dimensional block-diagonal covariance matrix with autocorrelated blocks.

Description

This function generates a $p \times p$ covariance matrix with autocorrelated blocks. The autocorrelation parameter is rho. There are num_blocks blocks each with size, block_size. The variance, sigma2, is constant for each feature and defaulted to 1.

Usage

cov_block_autocorrelation(num_blocks, block_size, rho,
    sigma2 = 1)

Arguments

num_blocks

the number of blocks in the covariance matrix

block_size

the size of each square block within the covariance matrix

rho

the autocorrelation parameter. Must be less than 1 in absolute value.

sigma2

the variance of each feature

Value

autocorrelated covariance matrix

Details

The autocorrelated covariance matrix is defined as: $$\Sigma = \Sigma^{(\rho)} \oplus \Sigma^{(-\rho)} \oplus \ldots \oplus \Sigma^{(\rho)},$$ where $\oplus$ denotes the direct sum and the $(i,j)$th entry of $\Sigma^{(\rho)}$ is $$\Sigma_{ij}^{(\rho)} = { \rho^{|i - j|} }.$$

The matrix $\Sigma^{(\rho)}$ is the autocorrelated block discussed above.

The value of rho must be such that $|\rho| < 1$ to ensure that the covariance matrix is positive definite.

The size of the resulting matrix is $p \times p$, where p = num_blocks * block_size.

The block-diagonal covariance matrix with autocorrelated blocks was popularized by Guo et al. (2007) for studying classification of high-dimensional data.

References

Guo, Y., Hastie, T., & Tibshirani, R. (2007). "Regularized linear discriminant analysis and its application in microarrays," Biostatistics, 8, 1, 86-100.