get_nearest_psd_matrix: Approximates a symmetric, real matrix by the nearest positive...

In bschneidr/svrep: Tools for Creating, Updating, and Analyzing Survey Replicate Weights

Approximates a symmetric, real matrix by the nearest positive semidefinite matrix.

Description

Approximates a symmetric, real matrix by the nearest positive semidefinite matrix in the Frobenius norm, using the method of Higham (1988). For a real, symmetric matrix, this is equivalent to "zeroing out" negative eigenvalues. See the "Details" section for more information.

Usage

get_nearest_psd_matrix(X)

Arguments

X	A symmetric, real matrix with no missing values.

Details

Let A denote a symmetric, real matrix which is not positive semidefinite. Then we can form the spectral decomposition A=\Gamma \Lambda \Gamma^{\prime}, where \Lambda is the diagonal matrix whose entries are eigenvalues of A. The method of Higham (1988) is to approximate A with \tilde{A} = \Gamma \Lambda_{+} \Gamma^{\prime}, where the ii-th entry of \Lambda_{+} is \max(\Lambda_{ii}, 0).

Value

The nearest positive semidefinite matrix of the same dimension as X.

References

- Higham, N. J. (1988). "Computing a nearest symmetric positive semidefinite matrix." Linear Algebra and Its Applications, 103, 103–118.

Examples

X <- matrix(
  c(2, 5, 5,
    5, 2, 5,
    5, 5, 2),
  nrow = 3, byrow = TRUE
)
get_nearest_psd_matrix(X)

bschneidr/svrep documentation built on Feb. 11, 2025, 4:24 a.m.