Towards robust inference for Bayesian filtering of linear Gaussian dynamical systems subject to additive change

State space models enjoy wide popularity in mathematical and statistical modelling across disciplines and research fields. Frequent solutions to problems of estimation and forecasting of a latent signal such as the celebrated Kalman filter hereby rely on a set of strong assumptions such as linearity of system dynamics and Gaussianity of noise terms. We investigate fallacy in mis-specification of the noise terms, that is signal noise and observation noise, regarding heavy tailedness in that the true dynamic frequently produces observation outliers or abrupt jumps of the signal state due to realizations of these heavy tails not considered by the model. We propose a formalisation of observation noise mis-specification in terms of Huber’s ε-contamination as well as a computationally cheap solution via generalised Bayesian posteriors with a diffusion Stein divergence loss resulting in the diffusion score matching Kalman filter - a modified algorithm akin in complexity to the regular Kalman filter. For this new filter interpretations of novel terms, stability and an ensemble variant are discussed. Regarding signal noise mis-specification, we propose a formalisation in the frame work of change point detection and join ideas from the popular CUSUM algo- rithm with ideas from Bayesian online change point detection to combine frequent reliability constraints and online inference resulting in a Gaussian mixture model variant of multiple Kalman filters. We hereby exploit open-end sequential probability ratio tests on the evidence of Kalman filters on observation sub-sequences for aggregated inference under notions of plausibility. Both proposed methods are combined to investigate the double mis-specification problem and discussed regarding their capabilities in reliable and well-tuned uncertainty quantification. Each section provides an introduction to required terminology and tools as well as simulation experiments on the popular target tracking task and the non-linear, chaotic Lorenz-63 system to showcase practical performance of theoretical considerations.

Level thesis:degree_level

master

Grantor dc:publisher

Universität Potsdam

Year

2024

Author dc:creator

• Reimann, Hans

Contributors dc:contributor

• Reich, Sebastian
• Carpentier, Alexandra

• statistical model Mis-specification
• robust filtering
• state space change point detection
• Bayesian filtering
• Bayesian modelling
• Bayes'sche Modellierung
• robustes Filtern
• Zustandsraum Wechselpunktdetektierung
• Mis-Spezifizierung statistischer Modelle

Statement dc:rights

• Keine öffentliche Lizenz: Unter Urheberrechtsschutz

Repository record source_url

https://publishup.uni-potsdam.de/frontdoor/index/index/docId/64946

OAI identifier oai:identifier

oai:kobv.de-opus4-uni-potsdam:64946