lav_matrix: Utility Functions: Matrices and Vectors

Description Usage Arguments Details References Examples

Description

Utility functions for Matrix and Vector operations.

Usage

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# matrix to vector
lav_matrix_vec(A)
lav_matrix_vecr(A)
lav_matrix_vech(S, diagonal = TRUE)
lav_matrix_vechr(S, diagonal = TRUE)

# matrix/vector indices
lav_matrix_vech_idx(n = 1L, diagonal = TRUE)
lav_matrix_vech_row_idx(n = 1L, diagonal = TRUE)
lav_matrix_vech_col_idx(n = 1L, diagonal = TRUE)
lav_matrix_vechr_idx(n = 1L, diagonal = TRUE)
lav_matrix_vechru_idx(n = 1L, diagonal = TRUE)
lav_matrix_diag_idx(n = 1L)
lav_matrix_diagh_idx(n = 1L)
lav_matrix_antidiag_idx(n = 1L)

# vector to matrix
lav_matrix_vech_reverse(x, diagonal = TRUE)
lav_matrix_vechru_reverse(x, diagonal = TRUE) 
lav_matrix_upper2full(x, diagonal = TRUE)
lav_matrix_vechr_reverse(x, diagonal = TRUE)
lav_matrix_vechu_reverse(x, diagonal = TRUE)
lav_matrix_lower2full(x, diagonal = TRUE)

# the duplication matrix
lav_matrix_duplication(n = 1L)
lav_matrix_duplication_pre(A = matrix(0,0,0))
lav_matrix_duplication_post(A = matrix(0,0,0))
lav_matrix_duplication_pre_post(A = matrix(0,0,0))
lav_matrix_duplication_ginv(n = 1L)
lav_matrix_duplication_ginv_pre(A = matrix(0,0,0))
lav_matrix_duplication_ginv_post(A = matrix(0,0,0))
lav_matrix_duplication_ginv_pre_post(A = matrix(0,0,0))

# the commutation matrix
lav_matrix_commutation(m = 1L, n = 1L)
lav_matrix_commutation_pre(A = matrix(0,0,0))
lav_matrix_commutation_mn_pre(A, m = 1L, n = 1L)

# sample statistics
lav_matrix_cov(Y, ybar = NULL)

# other matrix operations
lav_matrix_symmetric_sqrt(S = matrix(0,0,0))
lav_matrix_orthogonal_complement(A = matrix(0,0,0))
lav_matrix_bdiag(...)
lav_matrix_trace(..., check = TRUE)

Arguments

A

A general matrix.

S

A symmetric matrix.

Y

A matrix representing a (numeric) dataset.

diagonal

Logical. If TRUE, include the diagonal.

n

Integer. When it is the only argument, the dimension of a square matrix. If m is also provided, the number of column of the matrix.

m

Integer. The number of rows of a matrix.

x

Numeric. A vector.

ybar

Numeric. The mean vector of the matrix.

...

One or more matrices, or a list of matrices.

check

Logical. If check = TRUE, we check if the (final) matrix is square.

Details

These are a collection of lower-level matrix/vector related functions that are used throughout the lavaan code. They are made public per request of package developers. Below is a brief description of what they do:

The lav_matrix_vec function implements the vec operator (for 'vectorization') and transforms a matrix into a vector by stacking the columns of the matrix one underneath the other.

The lav_matrix_vecr function is similar to the lav_matrix_vec function but transforms a matrix into a vector by stacking the rows of the matrix one underneath the other.

The lav_matrix_vech function implements the vech operator (for 'half vectorization') and transforms a symmetric matrix into a vector by stacking the columns of the matrix one underneath the other, but eliminating all supradiagonal elements. If diagonal = FALSE, the diagonal elements are also eliminated.

The lav_matrix_vechr function is similar to the lav_matrix_vech function but transforms a matrix into a vector by stacking the rows of the matrix one underneath the other, eliminating all supradiagonal elements.

The lav_matrix_vech_idx function returns the vector indices of the lower triangular elements of a symmetric matrix of size n, column by column.

The lav_matrix_vech_row_idx function returns the row indices of the lower triangular elements of a symmetric matrix of size n.

The lav_matrix_vech_col_idx function returns the column indices of the lower triangular elements of a symmetric matrix of size n.

The lav_matrix_vechr_idx function returns the vector indices of the lower triangular elements of a symmetric matrix of size n, row by row.

The lav_matrix_vechu_idx function returns the vector indices of the upper triangular elements of a symmetric matrix of size n, column by column.

The lav_matrix_vechru_idx function returns the vector indices of the upper triangular elements of a symmetric matrix of size n, row by row.

The lav_matrix_diag_idx function returns the vector indices of the diagonal elements of a symmetric matrix of size n.

The lav_matrix_diagh_idx function returns the vector indices of the lower part of a symmetric matrix of size n.

The lav_matrix_antidiag_idx function returns the vector indices of the anti diagonal elements a symmetric matrix of size n.

The lav_matrix_vech_reverse function (alias: lav_matrix_vechru_reverse and lav_matrix_upper2full) creates a symmetric matrix, given only upper triangular elements, row by row. If diagonal = FALSE, an diagonal with zero elements is added.

The lav_matrix_vechr_reverse (alias: lav_matrix_vechu_reverse and lav_matrix_lower2full) creates a symmetric matrix, given only the lower triangular elements, row by row. If diagonal = FALSE, an diagonal with zero elements is added.

The lav_matrix_duplication function generates the duplication matrix for a symmetric matrix of size n. This matrix duplicates the elements in vech(S) to create vec(S) (where S is symmetric). This matrix is very sparse, and should probably never be explicitly created. Use one of the functions below.

The lav_matrix_duplication_pre function computes the product of the transpose of the duplication matrix and a matrix A. The A matrix should have n*n rows, where n is an integer. The duplication matrix is not explicitly created.

The lav_matrix_duplication_post function computes the product of a matrix A with the duplication matrix. The A matrix should have n*n columns, where n is an integer. The duplication matrix is not explicitly created.

The lav_matrix_duplication_pre_post function first pre-multiplies a matrix A with the transpose of the duplication matrix, and then post multiplies the result again with the duplication matrix. A must be square matrix with n*n rows and columns, where n is an integer. The duplication matrix is not explicitly created. multiplies a matrix A with the

The lav_matrix_duplication_ginv function computes the generalized inverse of the duplication matrix. The matrix removes the duplicated elements in vec(S) to create vech(S). This matrix is very sparse, and should probably never be explicitly created. Use one of the functions below.

The lav_matrix_duplication_ginv_pre function computes the product of the generalized inverse of the duplication matrix and a matrix A with n*n rows, where n is an integer. The generalized inverse of the duplication matrix is not explicitly created.

The lav_matrix_duplication_ginv_post function computes the product of a matrix A (with n*n columns, where n is an integer) and the transpose of the generalized inverse of the duplication matrix. The generalized inverse of the duplication matrix is not explicitly created.

The lav_matrix_duplication_ginv_pre_post function first pre-multiplies a matrix A with the transpose of the generalized inverse of the duplication matrix, and the post multiplies the result again with the transpose of the generalized inverse matrix. The matrix A must be square with n*n rows and columns, where n is an integer. The generalized inverse of the duplication matrix is not explicitly created.

The lav_matrix_commutation function computes the commutation matrix which is a permutation matrix which transforms vec(A) (with m rows and n columns) into vec(t(A)).

The lav_matrix_commutation_pre function computes the product of the commutation matrix with a matrix A, without explicitly creating the commutation matrix. The matrix A must have n*n rows, where n is an integer.

The lav_matrix_commutation_mn_pre function computes the product of the commutation matrix with a matrix A, without explicitly creating the commutation matrix. The matrix A must have m*n rows, where m and n are integers.

The lav_matrix_cov function computes the sample covariance matrix of its input matrix, where the elements are divided by N (the number of rows).

The lav_matrix_symmetric_sqrt function computes the square root of a positive definite symmetric matrix (using an eigen decomposition). If some of the eigenvalues are negative, they are silently fixed to zero.

The lav_matrix_orthogonal_complement function computes an orthogonal complement of the matrix A, using a qr decomposition.

The lav_matrix_bdiag function constructs a block diagonal matrix from its arguments.

The lav_matrix_trace function computes the trace (the sum of the diagonal elements) of a single (square) matrix, or if multiple matrices are provided (either as a list, or as multiple arguments), we first compute their product (which must result in a square matrix), and then we compute the trace; if check = TRUE, we check if the (final) matrix is square.

References

Magnus, J. R. and H. Neudecker (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

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# upper elements of a 3 by 3 symmetric matrix (row by row)
x <- c(30, 16, 5, 10, 3, 1)
# construct full symmetric matrix
S <- lav_matrix_upper2full(x)

# compute the normal theory `Gamma' matrix given a covariance
# matrix (S), using the formula: Gamma = 2 * D^{+} (S %x% S) t(D^{+})
Gamma.NT <- 2 * lav_matrix_duplication_ginv_pre_post(S %x% S)
Gamma.NT

yrosseel/lavaan documentation built on Dec. 14, 2018, 6:24 p.m.