Generates a p \times p block-diagonal covariance matrix with autocorrelated blocks.

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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

1
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

Details

The autocorrelated covariance matrix is defined as:

Σ = Σ^{(ρ)} \oplus Σ^{(-ρ)} \oplus … \oplus Σ^{(ρ)},

where \oplus denotes the direct sum and the (i,j)th entry of Σ^{(ρ)} is

Σ_{ij}^{(ρ)} = \{ ρ^{|i - j|} \}.

The matrix Σ^{(ρ)} is the autocorrelated block discussed above.

The value of rho must be such that |ρ| < 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.

Value

autocorrelated covariance matrix

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