Description Usage Arguments Details Value References

View source: R/covariance-matrices.r

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.

1 2 | ```
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 |

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`

.

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

autocorrelated covariance matrix

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

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