The Nearest Correlation Matrix Problem: Solution by Differential Evolution Method of Global Optimization

Abstract

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 authors to solve the nearest correlation matrix problem by majorization, hypersphere decomposition, semi-definite programming, or geometric programming, etc. In this paper we propose to obtain the nearest valid correlation matrix by the differential evaluation method of global optimization. We may draw some conclusions from the exercise in this paper. First, the ‘nearest correlation matrix problem may be solved satisfactorily by the evolutionary algorithm like the differential evolution method/Particle Swarm Optimizer. Other methods such as the Particle Swarm method also may be used. Secondly, these methods are easily amenable to choice of the norm to minimize. Absolute, Frobenius or Chebyshev norm may easily be used. Thirdly, the ‘complete the correlation matrix problem’ can be solved (in a limited sense) by these methods. Fourthly, one may easily opt for weighted norm or un-weighted norm minimization. Fifthly, 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 and thus they are on the borderline of non-semi-positive-definiteness. One finds difficulty in rounding off their elements even at 6th or 7th places after decimal, without running the risk of making the rounded off matrix non-positive definite. 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 method proposed by us does it very well.