Exact Consistency Tests for Gaussian Mixture Filters using Normalized Deviation Squared Statistics

TL;DR

The paper addresses dynamic consistency testing for discrete-time Gaussian-mixture (GM) state estimators under non-Gaussian uncertainties. It derives that the normalized deviation squared statistic for a GM pdf, $q(x)=(x-\bar{\mu})^T\bar{\Sigma}^{-1}(x-\bar{\mu})$, has an exact distribution as a mixture of generalized chi-square distributions, enabling precise threshold computation with existing tools. The authors validate the theory through static GM tests and dynamic GM filter experiments, and discuss practical considerations such as mixture-size growth and condensation. The work provides a rigorous, exact alternative to sampling-based tests for GM posterior validation with broad relevance to robotics, navigation, and related dynamic estimation applications.

Abstract

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.

Figures (4)

• Figure 1: (a) multi-modal GM pdf $p(x)$ with moment-matched Gaussian; (b) $q(x)$ histogram for 500 samples; (c) empirical vs. theoretical $q(x)$ CDFs, showing strong agreement between empirical and generalized chi-squared mixture, and strong disagreement with 1 DOF simple chi-square ($q(x)$ pdf for moment-matched Gaussian).
• Figure 2: Empirical CDF from 1000 samples of $Q(x)$ for $M=3$ multivariate GMs in $n=2$ dimensions.
• Figure 3: GM priors and typical data for 1D localization example: (a) prior $p(x_0)$; (b) process noise pdf $p(w_k)$; (c) measurement noise pdf $p(v_k)$; (d) ground truth sample trajectory and measurements. (e)-(f) GM filter-estimated posterior pdf (magenta lines) at selected times ($k=25,30,40,100$), showing true platform state (black cross) and $y_k$ data (red x).
• Figure 4: GM filter MMSE estimation error (blue) and $2\sigma$ (black dash) for (a) nominal and (b) mismatched cases.