OUTLIER-INSENSITIVE KALMAN FILTERING USING NUV PRIORS Shunit Truzman Guy Revach Nir Shlezinger and Itzik Klein ABSTRACT

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OUTLIER-INSENSITIVE KALMAN FILTERING USING NUV PRIORS

Shunit Truzman, Guy Revach, Nir Shlezinger, and Itzik Klein

ABSTRACT

The Kalman ﬁlter (KF) is a widely-used algorithm for track-

ing the latent state of a dynamical system from noisy ob-

servations. For systems that are well-described by linear

Gaussian state space models, the KF minimizes the mean-

squared error (MSE). However, in practice, observations are

corrupted by outliers, severely impairing the KF’s perfor-

mance. In this work, an outlier-insensitive KF is proposed,

where robustness is achieved by modeling each potential out-

lier as a normally distributed random variable with unknown

variance (NUV). The NUVs variances are estimated online,

using both expectation-maximization (EM) and alternating

maximization (AM). The former was previously proposed

for the task of smoothing with outliers and was adapted here

to ﬁltering, while both EM and AM obtained the same per-

formance and outperformed the other algorithms, the AM

approach is less complex and thus requires 40% less run-

time. Our empirical study demonstrates that the MSE of our

proposed outlier-insensitive KF outperforms previously pro-

posed algorithms, and that for data clean of outliers, it reverts

to the classic KF, i.e., MSE optimality is preserved.

Index Terms—Kalman ﬁlter, outliers, AM

1. INTRODUCTION

State estimation of dynamical systems from noisy observa-

tions plays a key role in various scientiﬁc and technological

ﬁelds such as radar target tracking, complex image process-

ing, navigation, and positioning [1]. The celebrated Kalman

ﬁlter (KF) [2] is an efﬁcient recursive state estimation algo-

rithm that is mean-squared error (MSE) optimal for dynami-

cal systems obeying a linear Gaussian state space (SS) model.

However, the quadratic form of its objective, i.e., MSE, makes

it sensitive to deviations from nominal noise. Thus, the KF is

severely impaired when outliers are present in the measure-

ments [3,4]. As sensory data is often populated with outliers,

robustness to outliers is essential [5–7]. A common approach

for dealing with outliers is to detect and then disregard inﬂu-

ential observations. Such detection can be achieved using ap-

propriate statistical diagnostics [8] on the posterior distribu-

S. Truzman and I. Klein are with the Hatter Dept. of Marine Technolo-

gies, University of Haifa, Haifa, Israel, (e-mail: shunitruzman@gmail.com,

kitzik@univ.haifa.ac.il ). G. Revach is with the Institute for Signal and In-

formation Processing (ISI), D-ITET, ETH Z¨

urich, Switzerland (e-mail: gre-

vach@ethz.ch). N. Shlezinger is with the School of ECE, Ben-Gurion Uni-

versity of the Negev, Beer Sheva, Israel (e-mail: nirshl@bgu.ac.il). S. Truz-

man. is supported by the Maurice Hatter Foundation. We thank Hans-Andrea

Loeliger for helpful discussions.

tion, e.g., Z-test [9] and χ2-test [10,11]. The main drawbacks

of these approaches are that they need to be carefully tuned

for a required false alarm, and that potentially useful outlier

information is not accounted for in the estimation process. Al-

ternatively, one can limit the effect of outliers by reweighting

the covariance of the observation noise at each data sample

when estimating the current state [5,6]. These techniques re-

quire careful tuning of multiple hyperparameters to operate

reliably as well. A different approach formulates the KF as a

linear regression problem, detecting outliers via a sparsifying

`1-penalty [4,12], tackled via optimization techniques, which

may be computationally complex.

In this work, a new approach for outlier insensitive

Kalman-ﬁltering (OIKF) is proposed that leverages ideas

from sparse Bayesian learning [13]. Here, each potential

outlier is modeled as a normally distributed random variable

with unknown variance (NUV) [7,14,15], i.e., an additive

component on top of the observation noise. NUV incorpo-

ration effectively yields a modiﬁed overall sparsity-aware

objective [14], which is shown to yield a robust outlier detec-

tion statistical test with a relatively low false alarm. When

an outlier is reliably detected, we incorporate its variance

into the overall covariance matrix of the observation noise,

thereby balancing its contribution to the information fusion

(i.e., update) step in the KF, and the outlier information is

used in the state estimation process.

To estimate the NUV online, we ﬁrst adapt the expectation-

maximization (EM) algorithm [16–18], which was previously

proposed for ofﬂine smoothing [7], to the online ﬁltering task.

EM is based on computing second-order moments, i.e., the

full state observation posterior covariance. We then present an

implementation, which has not been considered to date, using

asimpler alternating maximization (AM) algorithm [19–21].

Unlike EM, AM uses only ﬁrst-order moments as an empiri-

cal surrogate for outlier detection, without sacriﬁcing perfor-

mance and with improved robustness. We evaluate the OIKF

for tracking based on the white noise acceleration (WNA) mo-

tion model with outliers [22]. We empirically demonstrate the

superiority of our proposed algorithm compared to previous

robust variants of the KF, achieving improved performances

with low complexity operations.

The rest of this paper is organized as follows: Section 2

formulates the system and the KF with outliers. Section 3

presents the OIKF with its estimation for the outlier’s vari-

ance, and Section 4empirically evaluates our methods.

arXiv:2210.06083v1 [eess.SP] 12 Oct 2022

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OUTLIER-INSENSITIVEKALMANFILTERINGUSINGNUVPRIORSShunitTruzman,GuyRevach,NirShlezinger,andItzikKleinABSTRACTTheKalmanlter(KF)isawidely-usedalgorithmfortrack-ingthelatentstateofadynamicalsystemfromnoisyob-servations.Forsystemsthatarewell-describedbylinearGaussianstatespacemodels,theKFminimizesthemean...

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OUTLIER-INSENSITIVE KALMAN FILTERING USING NUV PRIORS Shunit Truzman Guy Revach Nir Shlezinger and Itzik Klein ABSTRACT

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