sfsmisc (version 1.0-0)

posdefify: Find a Close Positive Definite Matrix

Description

From a matrix m, construct a "close" positive definite one.

Usage

posdefify(m, method = c("someEVadd", "allEVadd"), symmetric, eps.ev = 1e-07)

Arguments

m

a numeric (square) matrix.

method

a string specifying the method to apply; can be abbreviated.

symmetric

logical, simply passed to eigen.

eps.ev

number specifying the tolerance to use, see Details below.

Value

• a matrix of the same dimensions and the same diagonal (i.e. diag) as m but with the property to be positive definite.

Details

We form the eigen decomposition $$m = V \Lambda V'$$ where $\Lambda$ is the diagonal matrix of eigenvalues, $\Lambda_{j,j} = \lambda_j$, with decreasing eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$.

When the smallest eigenvalue $\lambda_n$ are less than Eps <- eps.ev * abs(lambda[1]), i.e., negative or almost zero, some or all eigenvalues are replaced by positive (>= Eps) values, $\tilde\Lambda_{j,j} = \tilde\lambda_j$. Then, $\tilde m = V \tilde\Lambda V'$ is computed and rescaled in order to keep the original diagonal (where that is >= Eps).

References

Section 4.4.2 of Gill, P.~E., Murray, W. and Wright, M.~H. (1981) Practical Optimization, Academic Press.

Cheng, Sheung Hun and Higham, Nick (1998) A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization; SIAM J. Matrix Anal. Appl., 19, 1097--1110.

Knol DL, ten Berge JMF (1989) Least-squares approximation of an improper correlation matrix by a proper one. Psychometrika 54, 53--61.

Highham (2002) Computing the nearest correlation matrix - a problem from finance; IMA Journal of Numerical Analysis 22, 329--343.

Lucas (2001) Computing nearest covariance and correlation matrices. A thesis submitted to the University of Manchester for the degree of Master of Science in the Faculty of Science and Engeneering.

See Also

eigen on which the current methods rely.

Examples

set.seed(12)
 m <- matrix(round(rnorm(25),2), 5, 5); m <- 1+ m + t(m); diag(m) <- diag(m) + 4
 m
 posdefify(m)
 1000 * zapsmall(m - posdefify(m))