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Prepare orthogonal matrix

Source: R/prepare_orthogonal_matrix.R
prepare_orthogonal_matrix.Rd

Calculate the orthogonal matrix U_Gamma for decomposition in Theorem 1 from references.

Usage

prepare_orthogonal_matrix(perm, perm_size = NULL, basis = NULL)

Arguments

perm

An object of a gips_perm or anything a gips_perm() can handle. It can also be of a gips class, but it will be interpreted as the underlying gips_perm.

perm_size

Size of a permutation. Required if perm is neither gips_perm nor gips.

basis

A matrix with basis vectors in COLUMNS. Identity by default.

Value

A square matrix of size perm_size by perm_size with columns from vector elements \(v_k^{(c)}\) according to Theorem 6 from references.

Details

Given X - a matrix invariant under the permutation perm. Call Gamma the permutations cyclic group: \(\Gamma = <perm> = \{perm, perm^2, ...\}\).

Then, \(U_\Gamma\) is such an orthogonal matrix, which block-diagonalizes X.

To be more precise, the matrix t(U_Gamma) %*% X %*% U_Gamma has a block-diagonal structure, which is ensured by Theorem 1 from references.

The formula for U_Gamma can be found in Theorem 6 from references.

A nice example is demonstrated in the Block Decomposition - [1], Theorem 1 section of vignette("Theory", package="gips") or its pkgdown page.

References

Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek, Hélène Massam. "Model selection in the space of Gaussian models invariant by symmetry." The Annals of Statistics, 50(3) 1747-1774 June 2022. arXiv link; doi:10.1214/22-AOS2174

See also

  • project_matrix() - A function used in examples to show the properties of prepare_orthogonal_matrix().

  • Block Decomposition - [1], Theorem 1 section of vignette("Theory", package = "gips") or its pkgdown page - A place to learn more about the math behind the gips package and see more examples of prepare_orthogonal_matrix().

Examples

gperm <- gips_perm("(1,2,3)(4,5)", 5)
U_Gamma <- prepare_orthogonal_matrix(gperm)

number_of_observations <- 10
X <- matrix(rnorm(5 * number_of_observations), number_of_observations, 5)
S <- cov(X)
X <- project_matrix(S, perm = gperm) # this matrix in invariant under gperm

block_decomposition <- t(U_Gamma) %*% X %*% U_Gamma
round(block_decomposition, 5) # the non-zeros only on diagonal and [1,2] and [2,1]
#>         [,1]    [,2]    [,3]    [,4]   [,5]
#> [1,] 0.41684 0.11768 0.00000 0.00000 0.0000
#> [2,] 0.11768 1.09027 0.00000 0.00000 0.0000
#> [3,] 0.00000 0.00000 0.92989 0.00000 0.0000
#> [4,] 0.00000 0.00000 0.00000 0.92989 0.0000
#> [5,] 0.00000 0.00000 0.00000 0.00000 0.9588