Find a Close Positive Definite Matrix
From a matrix m, construct a "close" positive definite
one.
posdefify(m, method = c("someEVadd", "allEVadd"),
symmetric = TRUE, eigen.m = eigen(m, symmetric= symmetric),
eps.ev = 1e-07)m |
a numeric (square) matrix. |
method |
a string specifying the method to apply; can be abbreviated. |
symmetric |
logical, simply passed to |
eigen.m |
the |
eps.ev |
number specifying the tolerance to use, see Details below. |
We form the eigen decomposition
m = V L V'
where L is the diagonal matrix of eigenvalues, L[j,j] = l[j], with decreasing eigenvalues l[1] >= l[2] >= ... >= l[n].
When the smallest eigenvalue l[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,
L~[j,j] = l~[j].
Then, m~ = V L~ V' is computed
and rescaled in order to keep the original diagonal (where that is
>= Eps).
a matrix of the same dimensions and the “same” diagonal
(i.e. diag) as m but with the property to
be positive definite.
Martin Maechler, July 2004
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.
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))
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