R/data-block-autocorrelation.r
cov_block_autocorrelation.RdThis 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.
cov_block_autocorrelation(num_blocks, block_size, rho, sigma2 = 1)
| 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 |
autocorrelated covariance matrix
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.