Difference and un-difference covariance matrix

These functions difference and un-difference random vectors and covariance matrices.

Usage

diff_cov(cov, ref = 1)

undiff_cov(cov_diff, ref = 1)

delta(ref = 1, dim)

M(ranking = seq_len(dim), dim)

Arguments

cov, cov_diff

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

ref

[integer(1)]
The reference row between 1 and dim for differencing that maps cov to cov_diff, see details.

dim

[integer(1)]
The matrix dimension.

ranking

[integer()]
The integers 1 to dim in arbitrary order.

Value

A (differenced or un-differenced) covariance matrix.

Details

Assume \(x \sim N(0, \Sigma)\) is a multivariate normally distributed random vector of dimension \(n\). We may want to consider the differenced vector $$\tilde x = (x_1 - x_k, x_2 - x_k, \dots, x_n - x_k)',$$ excluding the \(k\)th element (hence, \(\tilde x\) is of dimension \((n - 1) \times 1\)). Formally, \(\tilde x = \Delta_k x\), where \(\Delta_k\) is a difference operator that depends on the reference row \(k\). More precise, \(\Delta_k\) is the identity matrix of dimension \(n\) without row \(k\) and with \(-1\)s in column \(k\). The difference operator \(\Delta_k\) can be computed via delta(ref = k, dim = n).

Then, \(\tilde x \sim N(0, \tilde \Sigma)\), where $$\tilde{\Sigma} = \Delta_k \Sigma \Delta_k'$$ is the differenced covariance matrix with respect to row \(k = 1,\dots,n\). The differenced covariance matrix \(\tilde \Sigma\) can be computed via diff_delta(Sigma, ref = k).

Since \(\Delta_k\) is a non-bijective mapping, \(\Sigma\) cannot be uniquely restored from \(\tilde \Sigma\). However, it is possible to compute a non-unique solution \(\Sigma_0\), such that \(\Delta_k \Sigma_0 \Delta_k = \tilde \Sigma\). For such a non-unique solution, we add a column and a row of zeros at column and row number \(k\) to \(\tilde{\Sigma}\), respectively. An "un-differenced" covariance matrix \(\Sigma_0\) can be computed via undiff_delta(Sigma_diff, ref = k).

As a alternative to \(\Delta_k\), the function M() returns a matrix for taking differences such that the resulting vector is negative.

See also

Other matrix helpers: check_correlation_matrix(), check_covariance_matrix(), check_transition_probability_matrix(), cov_to_chol(), insert_matrix_column(), matrix_diagonal_indices(), matrix_indices(), sample_correlation_matrix(), sample_covariance_matrix(), sample_transition_probability_matrix(), stationary_distribution()

Examples

n <- 4
Sigma <- sample_covariance_matrix(dim = n)
k <- 2
x <- c(1, 3, 2, 4)

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

# difference vector
delta_k %*% x
#>      [,1]
#> [1,]   -2
#> [2,]   -1
#> [3,]    1

# difference Sigma
(Sigma_diff <- diff_cov(Sigma, ref = k))
#>          [,1]     [,2]      [,3]
#> [1,] 7.598265 1.545601  5.552754
#> [2,] 1.545601 3.668382  5.568672
#> [3,] 5.552754 5.568672 10.721597

# un-difference Sigma
(Sigma_0 <- undiff_cov(Sigma_diff, ref = k))
#>          [,1] [,2]     [,3]      [,4]
#> [1,] 7.598265    0 1.545601  5.552754
#> [2,] 0.000000    0 0.000000  0.000000
#> [3,] 1.545601    0 3.668382  5.568672
#> [4,] 5.552754    0 5.568672 10.721597

# difference again
Sigma_diff_2 <- diff_cov(Sigma_0, ref = k)
all.equal(Sigma_diff, Sigma_diff_2)
#> [1] TRUE

# difference such that the resulting vector is negative
M(ranking = order(x, decreasing = TRUE), dim = n) %*% x
#>      [,1]
#> [1,]   -1
#> [2,]   -1
#> [3,]   -1