Optimal Solution of the Nearest Correlation Matrix Problem by Minimization of the Maximum Norm

16 Pages Posted: 9 Aug 2004 Last revised: 14 Oct 2010

Date Written: August 6, 2004

Abstract

The nearest correlation matrix problem is to find a valid (positive semidefinite) correlation matrix, R(m,m), that is nearest to a given invalid (non-positive semidefinite) or pseudo-correlation matrix, Q(m,m); m larger than 2. In the literature on this problem, 'nearest' is invariably defined in the sense of the least Frobenius norm. Research works of Rebonato and Jaeckel (1999), Higham (2002), Anjos et al. (2003), Grubisic and Pietersz (2004), Pietersz, and Groenen (2004), etc. use Frobenius norm explicitly or implicitly.

However, it is not necessary to define 'nearest' in this conventional sense. The thrust of this paper is to define 'nearest' in the sense of the least maximum norm (LMN) of the deviation matrix (R-Q), and to obtain R nearest to Q. The LMN provides the overall minimum range of deviation of the elements of R from those of Q.

We also append a computer program (source codes in FORTRAN) to find the LMN R from a given Q. Presently we use the random walk search method for optimization. However, we suggest that more efficient methods based on the Genetic algorithms may replace the random walk algorithm of optimization.

Keywords: Nearest correlation matrix problem, Frobenius norm, maximum norm, LMN correlation matrix, positive semidefinite, negative semidefinite, positive definite, random walk algorithm, Genetic algorithm, computer program, source codes, FORTRAN, simulation

JEL Classification: C15, C63, C87, C88

Suggested Citation

Mishra, Sudhanshu K., Optimal Solution of the Nearest Correlation Matrix Problem by Minimization of the Maximum Norm (August 6, 2004). Available at SSRN: https://ssrn.com/abstract=573241 or http://dx.doi.org/10.2139/ssrn.573241

Sudhanshu K. Mishra (Contact Author)

North-Eastern Hill University (NEHU) ( email )

NEHU Campus
Shillong, 793022
India
03642550102 (Phone)

HOME PAGE: http://www.nehu-economics.info

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