Strictly positive definite correlation functions
John T. Dolloff, Brian Lofy, Alan Sussman, Charles Taylor · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2006
Sufficient conditions for strictly positive definite correlation functions are developed. These functions are associated with wide-sense stationary stochastic processes and provide practical models for various errors affecting tracking, fusion, and general estimation problems. In particular, the expected magnitude and temporal correlation of a stochastic error process are modeled such that the covariance matrix corresponding to a set of errors sampled (measured) at different times is positive definite (invertible) - a necessary condition for many applications. The covariance matrix is generated using the strictly positive definite correlation function and the sample times. As a related benefit, a large covariance matrix can be naturally compressed for storage and dissemination by a few parameters that define the specific correlation function and the sample times. Results are extended to wide-sense homogeneous multi-variate (vector-valued) random fields. Corresponding strictly positive definite correlation functions can statistically model fiducial (control point) errors including their inter-fiducial spatial correlations. If an estimator does not model correlations, its estimates are not optimal, its corresponding accuracy estimates (a posteriori error covariance) are unreliable, and it may diverge. Finally, results are extended to approximate error covariance matrices corresponding to non-homogeneous, multi-variate random fields (a generalization of non-stationary stochastic processes). Examples of strictly positive definite correlation functions and corresponding error covariance matrices are provided throughout the paper.