Mishra, SK (2004): Optimal solution of the nearest correlation matrix problem by minimization of the maximum norm.

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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 pseudocorrelation 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 (RQ), 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.
Item Type:  MPRA Paper 

Institution:  NorthEastern Hill University, Shillong (India) 
Original Title:  Optimal solution of the nearest correlation matrix problem by minimization of the maximum norm 
Language:  English 
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 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C61  Optimization Techniques; Programming Models; Dynamic Analysis C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology; Computer Programs > C87  Econometric Software C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology; Computer Programs > C88  Other Computer Software C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology; Computer Programs > C82  Methodology for Collecting, Estimating, and Organizing Macroeconomic Data 
Item ID:  1783 
Depositing User:  Sudhanshu Kumar Mishra 
Date Deposited:  13. Feb 2007 
Last Modified:  12. Feb 2013 12:22 
References:  · AlSubaihi, AA (2004). “Simulating Correlated Multivariate Pseudorandom Numbers”, At www.jstatsoft.org/counter.php?id=85& url=v09/i04/paper.pdf&ct=1 · Anjos, MF, NJ Higham, PL Takouda and H Wolkowicz (2003) “A Semidefinite Programming Approach for the Nearest Correlation Matrix Problem”, Preliminary Research Report, Dept. of Combanitorics & Optimization, Waterloo, Ontario. · Goldberg, DE (1989). Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley, Reading, Mass. · Grubisic, I and R Pietersz (2004) “Efficient Rank Reduction of Correlation Matrices”, Working Paper Series, SSRN, http://ssrn.com/abstract=518563 · Higham, NJ (2002). “Computing the Nearest Correlation Matrix – A Problem from Finance”, IMA Journal of Numerical Analysis, 22, pp. 329343. · Holland, J (1975). Adaptation in Natural and Artificial Systems, Univ. of Michigan Press, Ann Arbor. · Kaiser, HF and K Dichman (1962). “Sample and Population Score Matrices and Sample Correlation Matrices from an Arbitrary Population Correlation Matrix”, Psychometrica, 27(2), pp. 179182. · Krishnamurthy, EV and SK Sen (1976). ComputerBased Numerical Algorithms, Affiliated EastWest Press, New Delhi. · Mishra, SK (2004) “Optimal Solution of the Nearest Correlation Matrix Problem by Minimization of the Maximum Norm”, Social Science Research Network (SSRN) at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=573241 August 9, 2004. · Pietersz, R (2004) Personal communication with the author, dated August 26 & 27, 2004. · Pietersz, R and PJF Groenen (2004) “Rank Reduction of Correlation Matrices by Majorization”, Econometric Institute Report EI 200411, Erasmus Univ., Rotterdam. · Rebonato, R and P Jäckel (1999) “ The Most General Methodology to Create a Valid Correlation Matrix for Risk Management and Option Pricing Purposes”, Quantitative Research Centre, NatWest Group, www.rebonato.com/CorrelationMatrix.pdf · Wright, AH (1991). “Genetic Algorithms for Real Parameter Optimization”, in GJE Rawlings (ed) Foundations of Genetic Algorithms, Morgan Kauffmann Publishers, San Mateo, CA, pp. 205218. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/1783 