Computing a Near Positive Definite Covariance Matrix from a Nonpositive Definite Matrix

• Activity Number: 136
• Type: Contributed
• Date/Time: Monday, August 2, 2010 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Jose H. Guardiola*+ and Rachel Larreta and Pablo Tarazaga and Hassan Elsalloukh
• Companies: Texas A&M University and Texas A&M University and Texas A&M University and University of Arkansas at Little Rock
• Keywords: Positive Definite ; Covariance Matrix ; Cholesky Decomposition ; Spatial Bootstrap ; Spatial Statistics ; Gershgorin bounds

Abstract

Covariance matrices in spatial statistics are estimated using procedures that may result in a non positive definite matrix. The estimated covariance matrix is often not suitable for further computation even that still may contain useful information. This paper proposes some methods that approximate a non-positive definite matrix to a near positive definite covariance matrix by five different methods, these methods are: the Cheng and Higham algorithm, adding a multiple of the identity, adjusting the lower Gershgorin bounds, replacing non positive eigenvalues, and substituting small diagonal entries with a tolerance. These methods produce a suitable positive definite matrix that can be used for further computations. The proposed methods are then applied to a spatial bootstrap procedure that requires a positive definite matrix and these procedures are timed and compared.