2000 character limit reached
Exact Consistency Tests for Gaussian Mixture Filters using Normalized Deviation Squared Statistics (2312.17420v2)
Published 29 Dec 2023 in stat.ME, cs.CV, cs.RO, cs.SY, eess.SY, and stat.AP
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.
- Yaakov Bar-Shalom, X Rong Li and Thiagalingam Kirubarajan “Estimation with applications to tracking and navigation: theory algorithms and software” John Wiley & Sons, 2001
- Ferdinand Heijden “Consistency checks for particle filters” In IEEE Transactions on Pattern Analysis and Machine Intelligence 28.1 IEEE, 2005, pp. 140–145
- Petar M Djuric and Joaquín Míguez “Assessment of nonlinear dynamic models by Kolmogorov-Smirnov statistics” In IEEE Transactions on Signal Processing 58.10 IEEE, 2010, pp. 5069–5079
- Pavel Ivanov, Simo Ali-Löytty and Robert Piché “Evaluating the consistency of estimation” In International Conference on Localization and GNSS 2014 (ICL-GNSS 2014), 2014, pp. 1–5 IEEE
- “Nonlinear Bayesian estimation using Gaussian sum approximations” In IEEE Transactions on Automatic Control 17.4 IEEE, 1972, pp. 439–448
- Jonathan R Schoenberg, Mark Campbell and Isaac Miller “Posterior representation with a multi-modal likelihood using the gaussian sum filter for localization in a known map” In Journal of Field Robotics 29.2 Wiley Online Library, 2012, pp. 240–257
- Friedrich Faubel, John McDonough and Dietrich Klakow “The split and merge unscented Gaussian mixture filter” In IEEE Signal Processing Letters 16.9 IEEE, 2009, pp. 786–789
- Jayesh H Kotecha and Petar M Djuric “Gaussian sum particle filtering” In IEEE Transactions on Signal Processing 51.10 IEEE, 2003, pp. 2602–2612
- “Adaptive Gaussian sum filter for nonlinear Bayesian estimation” In IEEE Transactions on Automatic Control 56.9 IEEE, 2011, pp. 2151–2156
- Andrew R Runnalls “Kullback-Leibler approach to Gaussian mixture reduction” In IEEE Transactions on Aerospace and Electronic Systems 43.3 IEEE, 2007, pp. 989–999
- “Bridging the ensemble Kalman filter and particle filters: the adaptive Gaussian mixture filter” In Computational Geosciences 15 Springer, 2011, pp. 293–305
- “Discrete and continuous, probabilistic anticipation for autonomous robots in urban environments” In IEEE Transactions on Robotics 30.2 IEEE, 2013, pp. 461–474
- “Particle Gaussian mixture filters-I” In Automatica 98 Elsevier, 2018, pp. 331–340
- Mark L Psiaki “Gaussian mixture nonlinear filtering with resampling for mixand narrowing” In IEEE Transactions on Signal Processing 64.21 IEEE, 2016, pp. 5499–5512
- “Negative information for occlusion reasoning in dynamic extended multiobject tracking” In IEEE Transactions on Robotics 31.2 IEEE, 2015, pp. 425–442
- Mark L Psiaki “Gaussian-mixture Kalman filter for orbit determination using angles-only data” In Journal of Guidance, Control, and Dynamics 40.9 American Institute of AeronauticsAstronautics, 2017, pp. 2341–2347
- Kyle J DeMars, Yang Cheng and Moriba K Jah “Collision probability with Gaussian mixture orbit uncertainty” In Journal of Guidance, Control, and Dynamics 37.3 American Institute of AeronauticsAstronautics, 2014, pp. 979–985
- “Gaussian mixture filter for multipath assisted positioning” In Proceedings of the 29th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2016), 2016, pp. 1263–1269
- “Robust mobile location estimation in NLOS environment using GMM, IMM, and EKF” In IEEE Systems Journal 13.3 IEEE, 2018, pp. 3490–3500
- “Collaborative human-autonomy semantic sensing through structured POMDP planning” In Robotics and Autonomous Systems 140 Elsevier, 2021, pp. 103753
- “Weak in the NEES?: Auto-tuning Kalman filters with Bayesian optimization” In 2018 21st International Conference on Information Fusion (FUSION), 2018, pp. 1072–1079 IEEE
- “A linear-time kernel goodness-of-fit test” In Advances in Neural Information Processing Systems 30, 2017
- Qiang Liu, Jason Lee and Michael Jordan “A kernelized Stein discrepancy for goodness-of-fit tests” In International Conference on Machine Learning, 2016, pp. 276–284 PMLR
- Supawadee Wichitchan, Weixin Yao and Chun Yu “A new class of multivariate goodness of fit tests for multivariate normal mixtures” In Communications in Statistics-Simulation and Computation 51.11 Taylor & Francis, 2022, pp. 6635–6648
- “Distribution of quadratic form in Gaussian mixture variables and an application in relay networks” In IEEE 6th Workshop on Signal Processing Advances in Wireless Communications, 2005., 2005, pp. 490–494 IEEE
- P. Ivanov “Consistency of Estimation”, 2014 URL: http://URN.fi/URN:NBN:fi:tty-201405131158
- Arakaparampil M Mathai and Serge B Provost “Quadratic forms in random variables: theory and applications” New York: Marcel Dekker, 1992
- Abhranil Das and Wilson S Geisler “A method to integrate and classify normal distributions” In Journal of Vision 21.10 The Association for Research in VisionOphthalmology, 2021, pp. 1–1