Model selection for Gaussian mixture model filtering and sensor scheduling

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2020-09-11

Authors

Tuggle, Kirsten Elizabeth

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Abstract

The use of Gaussian mixture model representations for nonlinear estimation is an attractive tool for object tracking and orbit determination. It is the potential for a reasonable balance between algorithm speed and estimator performance that lends these models to applications which necessitate consistent effectiveness in both. Performance of such filters relies on the ability to intelligently manage the number of mixture components, a notion equivalent to model selection among the class of approximating mixtures. The purpose of replacing a state density with a mixture approximation is clearly not better representation of this original density, which by definition is now only approximately represented. Rather, the goal is better representation of the mixture-generated density resulting from a nonlinear transformation of the state. Knowing that use of a compact, linearized estimator is typically desired, successful approximation of the transformed density is accomplished via anticipating and mitigating linearized solution weaknesses while still substantially capturing the original density. A popular method for re-approximation is the act of splitting individual Gaussian components into sub-mixtures. The current work first offers an automated splitting algorithm to address the less investigated measurement update step. A key aspect of the solution is quantification of not only prior uncertainty but also nonlinearity associated with particular regions of the state-space. Via these direct quantifications we pose and solve an optimization problem that yields a simple, optimal splitting direction, i.e., the dimension along which sub-mixture components are added. Automation of the scheme allows recursive splitting until a measure of information content loss signals acceptable error. We extend this new method from nonlinear inference to nonlinear prediction for coverage of both filtering phases. It is demonstrated via numerical simulations that this method automatically refines the number of components to better encompass the effects of the nonlinear transformations.

The second model selection domain in this dissertation is sensor scheduling. Many applications have emerged with a significant need for configurability of rapid, real-time, plug-and-play sensing systems. Even with modern computational power, the underlying guidance, navigation, and control tasks can quickly overwhelm processing ability as the problem complexity increases over time and/or degrees of freedom. This dissertation offers an efficient algorithm when selections must be made among multiple sensors at each time-step of a general dynamic linear system. To the author's knowledge, this algorithm is the first in the literature to address the case of time-correlated measurement or process noises and one of the first for spatially-correlated sensors. By deriving an expression of the true scheduling objective explicit in all schedules (which was previously considered infeasible), this method efficiently approximates the effects of accepting or rejecting each measurement on the basis of the underlying estimation and/or control problem.

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