Student-t process quadratures for filtering of non-linear systems with heavy-tailed noise

Abstract

Abstract-The aim of this article is to design a moment transformation for Student-t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable whose variance provides information about the incurred integration error. Advantage of the Student-t process quadrature over the traditional Gaussian process quadrature, is that the integral variance depends also on the function values, allowing for a more robust modelling of the integration error. The moment transform is applied in nonlinear sigma-point filtering and evaluated on two numerical examples, where it is shown to outperform the state-of-the-art moment transforms.

Classical quadrature rules via Gaussian processes

“…Deisenroth et al (2009Deisenroth et al ( , 2012 were first to propose the use of Gaussian process based numerical integration in this context. Since then, different versions and extensions of the idea have been studied in Särkkä et al (2014); Prüher and Šimandl (2016); Särkkä et al (2016); Prüher et al (2017); . An interesting connection is that a part of this approach, perhaps most clearly outlined by Prüher and Straka (2018, Sec. IV) (who use the term Gaussian process moment transform), closely resembles WSABI in that Gaussian integrals of f 2 are essentially approximated by placing a GP prior on f .…”

Laplace ℓ1 robust Student’s T-filter for attitude estimation of satellites

“…The maximum posterior estimation of x k is then obtained by solving the following cost function, which is derived and transformed from (27).…”

“…Many investigations focusing on this method have been presented in recent years. [27][28][29] To deal with the unknown modeling error, the maximum correntropy criterion is considered here to update the STF's measurement covariance. This can compensate for the covariance of the unknown modeling error to suppress it for the robust estimation design.…”

Double-Layer Cubature Kalman Filter for Nonlinear Estimation

“…We first consider the following univariate nonstationary growth model, which has been also used in [36,37] for its high nonlinearity. Its state space model can be written as:xk+1=0.5xk+prefixsin(0.04πk)+1+ωk zk+1=0.2xk+12+vk+1 where process noise ωk obeys gamma distribution Ga()3,2, and measurement noise vk+1 obeys Gaussian distribution with mean of 0 with variance of R=10−5.…”