oeli (version 0.5.2)

diff_cov: Difference and un-difference covariance matrix

Description

These functions difference and un-difference a covariance matrix with respect to row ref.

Usage

diff_cov(cov, ref = 1)
undiff_cov(cov_diff, ref = 1)
delta(ref = 1, dim)

Value

A (differenced or un-differenced) covariance matrix.

Arguments

cov, cov_diff

A (differenced) covariance matrix of dimension dim (or dim - 1, respectively).

ref

An integer between 1 and dim, the reference row for differencing that maps cov to cov_diff, see details. By default, ref = 1.

dim

An integer, the dimension.

Details

For differencing: Let \(\Sigma\) be a covariance matrix of dimension \(n\). Then $$\tilde{\Sigma} = \Delta_k \Sigma \Delta_k'$$ is the differenced covariance matrix with respect to row \(k = 1,\dots,n\), where \(\Delta_k\) is a difference operator that depends on the reference row \(k\). More precise, \(\Delta_k\) the identity matrix of dimension \(n\) without row \(k\) and with \(-1\)s in column \(k\). It can be computed with delta(ref = k, dim = n).

For un-differencing: The "un-differenced" covariance matrix \(\Sigma\) cannot be uniquely computed from \(\tilde{\Sigma}\). For a non-unique solution, we add a column and a row of zeros at column and row number \(k\) to \(\tilde{\Sigma}\), respectively, and add \(1\) to each matrix entry to make the result a proper covariance matrix.

Examples

n <- 3
Sigma <- sample_covariance_matrix(dim = n)
k <- 2

# build difference operator
delta(ref = k, dim = n)

# difference Sigma
(Sigma_diff <- diff_cov(Sigma, ref = k))

# un-difference Sigma
undiff_cov(Sigma_diff, ref = k)