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Optimal solution of the nearest correlation matrix problem by minimization of the maximum norm

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  • Mishra, SK

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 (negative 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.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 1783.

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Date of creation: 06 Aug 2004
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Handle: RePEc:pra:mprapa:1783

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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;

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References

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  1. Raoul Pietersz & Patrick J. F. Groenen, 2005. "Rank Reduction of Correlation Matrices by Majorization," Finance 0502006, EconWPA.
  2. Mishra, SK, 2004. "Optimal solution of the nearest correlation matrix problem by minimization of the maximum norm," MPRA Paper 1783, University Library of Munich, Germany.
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Cited by:
  1. Sudhanshu K Mishra, 2013. "Global Optimization of Some Difficult Benchmark Functions by Host-Parasite Coevolutionary Algorithm," Economics Bulletin, AccessEcon, vol. 33(1), pages 1-18.
  2. Mishra, SK, 2004. "Optimal solution of the nearest correlation matrix problem by minimization of the maximum norm," MPRA Paper 1783, University Library of Munich, Germany.
  3. Mishra, SK, 2007. "Completing correlation matrices of arbitrary order by differential evolution method of global optimization: A Fortran program," MPRA Paper 2000, University Library of Munich, Germany.

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