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NIR-EKF: Normalized Innovation Ratio-Based EKF for Robust State Estimation

Talha Nadeem, Khurram Ali, Muhammad Tahir

arXiv:2610.01817v1eess.SP

TL;DR

Outliers in sensor measurements can undermine state estimation, including when they occur across multiple dimensions. The paper develops a MAP-based EKF with an NIR-based outlier test. Its simulations report performance matching other robust techniques with lower computational demand.

  • Problem

    Outliers can make state estimates inaccurate, and measurements may contain outliers simultaneously across multiple dimensions.

    Source: §I · p. 1

  • Method

    The method uses MAP-based EKF inference for outlier scenarios and an NIR-based threshold to detect and discard measurements flagged as outliers.

    Source: §I · p. 1, A. MAP-EKF · p. 2, B. Exhaustive MAP-EKF · p. 2

  • Results

    The MAP-based methods match other robust techniques’ performance with less computational demand in the reported simulations.

    Source: §III · p. 4, §IV · p. 4

  • Takeaways & Limitations

    The simulations support MAP-based EKFs as efficient alternatives for robust state estimation with multidimensional outliers.

    Source: §III · p. 4, §IV · p. 4

Abstract

from arXiv · show

Sensors deployed in real-world conditions often produce measurements corrupted by outliers due to model uncertainties, changes in the surrounding environment, and/or data loss. As a result, managing these outliers becomes crucial for state estimation to avoid inaccurate estimations and a reduction in the reliability of results. To address this issue, we introduce a novel form of extended Kalman filter (EKF) based on the maximum a posteriori (MAP) principle for scenarios where outliers simultaneously occur in multiple dimensions. For detecting outliers during the filtering process, we introduce a novel variant of the normalized innovation ratio (NIR) test and embed it within the EKF framework. Our approach enhances the estimation accuracy and computational efficiency of state estimation process even when data from several sensors simultaneously contain outliers.

I. INTRODUCTION

Outliers can undermine Kalman-filter state estimates, while existing approaches face limits in single- or multiple-dimension settings. The paper proposes a MAP-based EKF with an NIR test for detecting multidimensional outliers.

I. INTRODUCTION

  • Outliers can make Kalman-filter predictions inaccurate, despite the filters’ widespread use under Gaussian-noise assumptions.The introduction identifies sensor issues, environmental disruptions, malicious activities, and data errors as sources of outliers.Source: §I · p. 1
  • Prior M-estimation, l1-norm, and H∞ methods mainly mitigate outliers in one measurement dimension.The cited discussion also notes that l1-norm performance can be limited in dynamic environments and H∞ filtering may introduce conservative bias.Source: §I · p. 1
  • Multidimensional outlier methods can require parameter tuning or iterative computation, and may trade computational efficiency against discarding valid data.The discussion names Monte Carlo, Student-t, and variational-Bayes approaches among the methods with these drawbacks.Source: §I · p. 1
  • The proposed MAP-based EKF models outliers across measurement dimensions and uses an NIR test to detect them during filtering.The authors describe the method as fast, robust, and straightforward to apply.Source: §I · p. 1

II. PROPOSED METHOD - NIR-EKF

The method represents the system with state and measurement vectors linked by process and observation functions. Its measurement model includes an added vector for outlier corruption.

II. PROPOSED METHOD - NIR-EKF

  • The system model defines a state vector x_k and a measurement vector y_k, with process dynamics f and observation function h.The functions map the state space to the state and measurement spaces, respectively.Source: §II · p. 2
  • The measurement model adds an outlier vector v_k to the observation and noise terms.The passage identifies v_k as the vector used to model measurements corrupted by outliers.Source: §II · p. 2
  • The state and process noise vectors are modeled by multivariate normal distributions with covariances Q_k and R_k.The passage assigns Q_k and R_k as the respective covariance matrices.Source: §II · p. 2

A. MAP-EKF

The MAP-EKF represents outlier occurrence through alternative measurement scenarios and forms a state posterior across those cases. The MAP estimate selects the candidate with the highest posterior probability.

A. MAP-EKF

  • The model includes a no-outlier event and events representing an outlier in an individual measurement dimension.For the no-outlier event, the outlier covariance matrix is zero, indicating no additional variability from outliers.Source: A. MAP-EKF · p. 2
  • The state posterior accounts for the possible events, each weighted by its prior probability.The passage denotes the prior for the no-outlier event by π_0 and for event I_i by π_i.Source: A. MAP-EKF · p. 2
  • At each time step, the MAP estimator selects the state estimate with the highest posterior probability.The method first produces candidate state estimates for the possible cases.Source: A. MAP-EKF · p. 2

B. Exhaustive MAP-EKF

The EMAP-EKF models outliers as subsets of sensors and evaluates every possible subset. It uses MAP estimation to select among the resulting candidate state estimates.

B. Exhaustive MAP-EKF

  • The set Γ contains all 2^m distinct sensor subsets that may have outliers, with the subsets mutually exclusive and exhaustive.Source: B. Exhaustive MAP-EKF · p. 2
  • The EMAP-EKF checks all 2^m possible combinations of measurement outliers and uses MAP estimation to select the most appropriate state estimate.Here, m is the number of measurements.Source: B. Exhaustive MAP-EKF · p. 2
  • For each outlier subset γ, I_γ denotes an outlier in that subset, with nonzero entries of 0_γ and the diagonal of Σ_γ restricted to its sensors.Source: B. Exhaustive MAP-EKF · p. 2

C. Sequential MAP-EKF

The SMAP-EKF processes measurements sequentially, using two filters per dimension to form candidate estimates and MAP selection to choose between them. This sequential approach has O(m) time complexity rather than the EMAP-EKF’s O(2^m).

C. Sequential MAP-EKF

  • The sequential estimator requires O(m) time, compared with the EMAP-EKF’s O(2^m) computational cost.It uses two assessments per measurement.Source: C. Sequential MAP-EKF · p. 2
  • For each measurement, two filters accommodate possible outliers and produce a pair of candidate estimates, from which MAP selects the most likely.Source: C. Sequential MAP-EKF · p. 2
  • The measurements are processed sequentially, with each element informing the prediction for the next; the resulting method is called SMAP-EKF.Source: C. Sequential MAP-EKF · p. 2
  • The selection criterion uses the ith component of the observation function evaluated at the preceding sequential estimate.Source: C. Sequential MAP-EKF · p. 2

D. Reduced SMAP-EKF

The RSMAP-EKF simplifies the sequential MAP probability comparison by assuming equal prior probabilities for the two alternatives. It performs two evaluations per measurement and has O(m) computational cost.

D. Reduced SMAP-EKF

  • With both π₀ and π_C in (12) and (13) assumed to equal 0.5, the key determinant between the two probabilities lies in comparing the Gaussian terms.Source: D. Reduced SMAP-EKF · p. 3
  • When a measurement closely matches nominal characteristics, the MAP estimator selects the nominal estimate; otherwise it selects the alternative estimate.Source: D. Reduced SMAP-EKF · p. 3
  • The RSMAP-EKF performs two evaluations per measurement and has computational cost O(m).It is reported as faster than the SMAP-EKF because of its simplified probability function and assumptions.Source: D. Reduced SMAP-EKF · p. 3

E. NIR-EKF

The NIR-EKF uses a normalized innovation ratio test to identify outliers while processing measurements. The test provides a threshold-based detection rule, and the estimator has O(m) computational cost.

E. NIR-EKF

  • The NIR test normalizes the difference between actual and predicted measurements, and measurements exceeding a set threshold are flagged and discarded.Source: E. NIR-EKF · p. 3
  • The NIR-EKF has O(m) computational cost and is reported as more efficient than estimators using multiple hypotheses per measurement.The authors attribute the speedup to using a simple outlier detection test.Source: E. NIR-EKF · p. 3
  • The NIR-EKF applies the test while processing measurements individually to identify outliers in the measurement vector.Source: E. NIR-EKF · p. 3

III. SIMULATION RESULTS

The simulations compare MAP-based robust estimators with established robust EKFs in target tracking. At larger measurement counts, the MAP-based methods match other robust techniques’ performance with lower computational demand.

III. SIMULATION RESULTS

  • The target-tracking comparison uses EKF as a common inference tool and includes R-EKF, SOR-EKF, and ROR-EKF.Source: §III · p. 3
  • The detection threshold φ_k=10^6 balances a higher threshold's risk of missing subtle outliers against a lower threshold's risk of flagging normal variations.Source: §III · p. 3
  • Outliers are modeled with Bernoulli probability λ and strength α=800; evaluation averages 100 Monte Carlo runs over 1,000 steps using TRMSE and elapsed time.Source: §III · p. 3
  • For m∈{2, 4, 6, 8, 10}, EMAP-EKF has the lowest TRMSE, while SMAP-EKF, RSMAP-EKF, and NIR-EKF perform similarly.Source: §III · p. 3, Figure 1 · p. 4
  • Fig. 2's per-run timing comparison identifies EMAP-EKF as unsuitable for computation-limited applications.Source: Figure 2 · p. 4, §III · p. 4
  • At m∈{200, 400, 600, 800, 1000}, the MAP-based methods match other robust techniques’ performance with lower computational demand.Figs. 3 and 4 report TRMSE and elapsed time, respectively.Source: Figure 3 · p. 4, §III · p. 4

IV. CONCLUSION

The conclusion presents MAP-based EKFs, including NIR-EKF, for handling simultaneous outliers across multiple measurement dimensions. The authors report improved estimation quality and greater computational efficiency than other estimators and robust filters.

IV. CONCLUSION

  • The proposed MAP-based EKFs handle outliers occurring simultaneously in multiple measurement dimensions, leading to NIR-EKF.Source: §IV · p. 4
  • The authors report greater estimation accuracy and computational efficiency than other estimators and state-of-the-art robust filters.Source: §IV · p. 4
  • Fig. 4 plots elapsed time against measurement count for λ values from 0 to 0.8.Source: Figure 4 · p. 4
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