Limits of Kalman Filter application in heavy tailed problems

TL;DR

This paper investigates how Kalman Filter estimates perform with heavy-tailed distributions, revealing significant errors when deviations from Gaussian assumptions occur, and compares MLE and EM methods for parameter estimation.

Contribution

It analyzes the impact of heavy-tailed noise on Kalman Filter accuracy and compares MLE and EM estimation methods under these conditions.

Findings

Large deviations from Gaussian noise increase estimation errors

EM estimation accounts for noise covariance better than MLE in heavy-tailed cases

Kalman Filter performance degrades with heavy-tailed noise distributions

Abstract

In this paper we consider the behavior of Kalman Filter state estimates in the case of distribution with heavy tails .The simulated linear state space models with Gaussian measurement noises were used. Gaussian noises in state equation are replaced by components with alpha-stable distribution with di erent parameters alpha and beta. We consider the case when "all parameters are known" and two methods of parameters estimation are compared: the maximum likelihood estimator (MLE) and the expectation- maximization algorithm (EM). It was shown that in cases of large deviation from Gaussian distribution the total error of states estimation rises dramatically. We conjecture that it can be explained by underestimation of the state equation noises covariance matrix that can be taken into account through the EM parameters estimation and ignored in the case of ML estimation.