Exact Consistency Tests for Gaussian Mixture Filters using Normalized Deviation Squared Statistics
This provides a method for consistency testing in mixture-based estimators, which is incremental but useful for applications like estimator tuning and validation.
The paper tackles the problem of evaluating dynamic consistency in Gaussian mixture filters, deriving an exact result that shows normalized deviation squared statistics follow mixtures of generalized chi-square distributions, with numerical demonstrations on static and dynamic examples.
We consider the problem of evaluating dynamic consistency in discrete time probabilistic filters that approximate stochastic system state densities with Gaussian mixtures. Dynamic consistency means that the estimated probability distributions correctly describe the actual uncertainties. As such, the problem of consistency testing naturally arises in applications with regards to estimator tuning and validation. However, due to the general complexity of the density functions involved, straightforward approaches for consistency testing of mixture-based estimators have remained challenging to define and implement. This paper derives a new exact result for Gaussian mixture consistency testing within the framework of normalized deviation squared (NDS) statistics. It is shown that NDS test statistics for generic multivariate Gaussian mixture models exactly follow mixtures of generalized chi-square distributions, for which efficient computational tools are available. The accuracy and utility of the resulting consistency tests are numerically demonstrated on static and dynamic mixture estimation examples.