Post by Sapphire Capital on Jul 11, 2008 22:08:36 GMT 4
A Note on Solution of the Nearest Correlation Matrix Problem by von Neumann Matrix Divergence
S. K. MISHRA
North-Eastern Hill University (NEHU)
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Abstract:
In the extant literature a suggestion has been made to solve the nearest correlation matrix problem by a modified von Neumann approximation. In this paper it has been shown that obtaining the nearest positive semi-definite matrix from a given negative definite correlation matrix by such method is either infeasible or suboptimal. First, if a given matrix is already positive semi-definite, there is no need to obtain any other positive semi-definite matrix closest to it. When the given matrix is negative definite (Q), then only we seek a positive semi-definite matrix closest to it. Then the proposed procedure fails as we cannot find log(Q). But, if we replace negative eigenvalues of Q by a zero/near-zero values, we obtain a positive semi-definite matrix, but it is not nearest to the Q matrix; there are indeed other procedures to obtain better approximation. However, the modified von Neumann approximation method yields results (although sub-optimal) and is, perhaps, one of the fastest method most suitable to dealing with larger matrices. Yet, we provide an alternative algorithm (and a Fortran program)to obtain a positive (semi-)definite matrix that performs (speed as well as accuracy-wise) much better.
papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1117383_code353253.pdf?abstractid=1106882&mirid=4
S. K. MISHRA
North-Eastern Hill University (NEHU)
--------------------------------------------------------------------------------
Abstract:
In the extant literature a suggestion has been made to solve the nearest correlation matrix problem by a modified von Neumann approximation. In this paper it has been shown that obtaining the nearest positive semi-definite matrix from a given negative definite correlation matrix by such method is either infeasible or suboptimal. First, if a given matrix is already positive semi-definite, there is no need to obtain any other positive semi-definite matrix closest to it. When the given matrix is negative definite (Q), then only we seek a positive semi-definite matrix closest to it. Then the proposed procedure fails as we cannot find log(Q). But, if we replace negative eigenvalues of Q by a zero/near-zero values, we obtain a positive semi-definite matrix, but it is not nearest to the Q matrix; there are indeed other procedures to obtain better approximation. However, the modified von Neumann approximation method yields results (although sub-optimal) and is, perhaps, one of the fastest method most suitable to dealing with larger matrices. Yet, we provide an alternative algorithm (and a Fortran program)to obtain a positive (semi-)definite matrix that performs (speed as well as accuracy-wise) much better.
papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1117383_code353253.pdf?abstractid=1106882&mirid=4