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17 May 2006Strictly positive definite correlation functions
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.
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John Dolloff, Brian Lofy, Alan Sussman, Charles Taylor, "Strictly positive definite correlation functions," Proc. SPIE 6235, Signal Processing, Sensor Fusion, and Target Recognition XV, 62351A (17 May 2006); https://doi.org/10.1117/12.663967