Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique

Abstract

For <i>n</i>-dimensional real-valued matrix <i>A</i>, the computation of nearest correlation matrix; that is, a symmetric, positive semi-definite, unit diagonal and off-diagonal entries between <inline-formula><math display="inline"><semantics><mrow><mo>&minus;</mo><mn>1</mn></mrow></semantics></math></inline-formula> and 1 is a problem that arises in the finance industry where the correlations exist between the stocks. The proposed methodology presented in this article computes the admissible perturbation matrix and a perturbation level to shift the negative spectrum of perturbed matrix to become non-negative or strictly positive. The solution to optimization problems constructs a gradient system of ordinary differential equations that turn over the desired perturbation matrix. Numerical testing provides enough evidence for the shifting of the negative spectrum and the computation of nearest correlation matrix.