Resolving the Nearest Correlation Matrix Problem

Correlation matrices have many applications, particularly in marketing and financial economics - such as in risk management, option pricing and to forecast demand for a group of products in order to realize savings by properly managing inventories, etc. Various methods have been proposed by different researchers to solve the nearest correlation matrix problem by majorization, hypersphere decomposition, semi-definite programming, geometric programming or random walk methods etc (Mishra, 2004). However, this problem can be solved by Global Optimization methods such as Differential Evaluation (DE) or Repulsive Particle Swarm (RPS) method. Alternatively, the problem may be solved by the Neumann-Dykstra (ND) Method also.

The methods exploiting the global optimization procedures are easily amenable to choice of the norm to minimize. Absolute, Frobenius or Chebyshev norm may easily be used. Additionally, the "complete the correlation matrix problem" can be solved (in a limited sense) by these methods (Mishra, 2007). Further, one may easily opt for weighted norm or un-weighted norm minimization. We also note that minimization of absolute norm to obtain nearest correlation matrices appears to give better results.

In solving the nearest correlation matrix problem the resulting valid correlation matrices are often near-singular. One finds difficulty in rounding off their elements even at 6th or 7th places after decimal; the rounded off matrix may become non-positive-semidefinite. Such matrices are, therefore, difficult to handle. It is possible to obtain more robust positive definite valid correlation matrices by constraining the determinant (the product of eigenvalues) of the resulting correlation matrix to take on a value significantly larger than zero. But this can be done only at the cost of a compromise on the criterion of "nearness." The methods proposed here do the job very well. The Fortran programs (source codes) for all the three methods (using DE, RPS and ND) are provided. Any one of the programs may be compiled by a suitable FORTRAN compiler (e.g. Microsoft Fortran Compiler or or FORCE Fortran 77 compiler; note that FORCE Compilers are free downloadable).

One has also to make a data file, such as mycor.txt, which should be stored in the same directory (folder) where the nearest correlation matrix program is compiled and its executable version lies stored. The data file will contain only the invalid correlation matrix in the following format (for example).

1.0 0.9 0.7
0.9 1.0 0.3
0.7 0.3 1.0

We may also note that the method using DE as an optimizer gives more accurate results.

References :
  • Mishra, S.K. (2004) "Optimal Solution of the Nearest Correlation Matrix Problem by Minimization of the Maximum Norm". Available at SSRN: Download.
  • Mishra, S.K. (2007) "Completing Correlation Matrices of Arbitrary Order by Differential Evolution Method of Global Optimization: A Fortran Program". Available at SSRN: Download.
  • Mishra, S.K. (2008) "The nearest correlation matrix problem: A Solution by differential evolution method of global optimization", Journal of Quanatitative Economics, 6(1 & 2), 2008, pp. 240-262. Available at SSRN: Download


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