Outlier-robust Autocovariance Least Square Estimation via Iteratively Reweighted Least Square

Jiahong Li; Fang Deng

math.OCMar 9, 2026

Outlier-robust Autocovariance Least Square Estimation via Iteratively Reweighted Least Square

Authors: Jiahong Li, Fang Deng

Organizations: College of Robotics, Beijing Union University, Beijing 100101, China · Key Laboratory of Intelligent Control and Decision of Complex Systems, and also with the School of Automation, Beijing Institute of Technology, Beijing 100081, China

Abstract

The autocovariance least squares (ALS) method is a computationally efficient approach for estimating noise covariances in Kalman filters without requiring specific noise models. However, conventional ALS and its variants rely on the classic least mean squares (LMS) criterion, making them highly sensitive to measurement outliers and prone to severe performance degradation. To overcome this limitation, this paper proposes a novel outlier-robust ALS algorithm, termed ALS-IRLS, based on the iteratively reweighted least squares (IRLS) framework. Specifically, the proposed approach introduces a two-tier robustification strategy. First, an innovation-level adaptive thresholding mechanism is employed to filter out heavily contaminated data. Second, the outlier-contaminated autocovariance is formulated using an $ε$ -contamination model, where the standard LMS criterion is replaced by the Huber cost function. The IRLS method is then utilized to iteratively adjust data weights based on estimation deviations, effectively mitigating the influence of residual outliers. Comparative simulations demonstrate that ALS-IRLS reduces the root-mean-square error (RMSE) of noise covariance estimates by over two orders of magnitude compared to standard ALS. Furthermore, it significantly enhances downstream state estimation accuracy, outperforming existing outlier-robust Kalman filters and achieving performance nearly equivalent to the ideal Oracle lower bound in the presence of noisy and anomalous data.

Figures & tables

Parameter	Description	Value
$N_{\mathrm{mc}}$	Monte Carlo trials	100
$\bar{Q},\;\bar{R}$	True noise covariances	$5,\;3$
$\hat{Q}^{(0)},\;\hat{R}^{(0)}$	Initial estimates	$2,\;1$
$N$	Autocovariance lag window	15
$\tau$	Batch length	150
$T_{\mathrm{wu}}$	Warm-up length with outliers	1 500

TABLE I: Simulation Parameters

Fig. 1: Innovation-level outlier detection at Monte Carlo trial 6 ( $\tau=150$ , $\varepsilon=0.15$ , $\omega=8$ ). Left: KF innovations $e_{k}$ ; red triangles mark the 23 flagged time steps satisfying $|e_{k}|>3.5\,\hat{\sigma}_{e}$ , with the threshold shown as dashed lines. Right: Raw $\mathbf{b}$ versus cleaned $\mathbf{b}_{\mathrm{clean}}$ : the zero-lag entry decreases from $32.4$ to $4.4$ while higher-lag entries are unaffected, confirming that outlier contamination is confined to $\hat{C}_{e,0}$ .

Fig. 2: Partial-regression fitting comparison at Monte Carlo trial 6, batch 1 ( $N=15$ , $\tau=150$ , $\varepsilon=0.15$ , $\omega=8$ ). Horizontal axis: $\mathbf{A}\hat{\theta}$ ; vertical axis: $\mathbf{b}$ . Blue crosses: uncontaminated autocovariance entries; red circle: contaminated lag-0 entry ( $\hat{C}_{e,0}^{\mathrm{raw}}\approx 32.4$ , true $\approx 4.3$ ); red square: the same entry after removal by the innovation-cleaning step. Shaded band: $95\%$ prediction confidence interval. Standard ALS is displaced by the outlier; ALS-IRLS recovers an accurate fit with all retained entries inside the confidence band.

Fig. 3: IRLS Huber weights at Monte Carlo trial 6, batch 1 ( $N=15$ ). Blue bars: uncontaminated lags with $w_{j}\approx 1$ ; purple bar: down-weighted lag-0 entry ( $w_{0}\approx 0.03$ ). Dashed line at $w=1$ ; dotted line at $w=0.5$ . All lags beyond lag-0 retain $w_{j}\geq 0.95$ , demonstrating selective penalisation of the contaminated entry without distorting the remaining autocovariance structure.

Fig. 4: Monte Carlo scatter of noise covariance estimates $(\hat{Q},\hat{R})$ over $N_{\mathrm{mc}}=100$ trials ( $\varepsilon=0.15$ , $\omega=8$ , $N=15$ , $\tau=150$ ). Left: Joint scatter; true value marked by a black star. Right: Marginal histogram of $\hat{Q}$ ; dashed line at $\bar{Q}=5$ . ALS estimates are widely scattered at inflated values; ALS-IRLS estimates concentrate near $(\bar{Q},\bar{R})$ .

True values: $\bar{Q}=5$ , $\bar{R}=3$ .
Method	$\mathrm{RMSE}(Q)$	$\mathrm{RMSE}(R)$	$\mathbb{E}[\hat{Q}]$	$\mathbb{E}[\hat{R}]$
ALS	48.5	107.3	53.5	110.2
ALS-IRLS	$\mathbf{0.38}$	$\mathbf{0.17}$	$\mathbf{5.02}$	$\mathbf{3.01}$

TABLE II: Noise Covariance Estimation Performance (

N_{\mathrm{mc}}=100

\varepsilon=0.15

\omega=8

)

Fig. 5: Noise covariance estimation RMSE ( $\varepsilon=0.15$ , $\omega=8$ ). ALS-IRLS reduces $\mathrm{RMSE}(Q)$ and $\mathrm{RMSE}(R)$ by more than two orders of magnitude relative to ALS, confirming that innovation-level outlier removal is essential for reliable autocovariance-based identification under $\varepsilon$ -contamination.

Fig. 6: Noise covariance estimation RMSE versus outlier rate $\varepsilon$ ( $\omega=8$ , $N=15$ , $\tau=150$ ). Left: $\mathrm{RMSE}(Q)$ . Right: $\mathrm{RMSE}(R)$ . ALS-IRLS maintains near-constant low RMSE; standard ALS degrades sharply as contaminated $\hat{C}_{e,0}$ increasingly dominates the least-squares solution.

Fig. 7: Noise covariance estimation RMSE versus lag window $N$ ( $\varepsilon=0.15$ , $\omega=8$ , $\tau_{\mathrm{use}}=\max(150,3N)$ ). Left: $\mathrm{RMSE}(Q)$ . Right: $\mathrm{RMSE}(R)$ . Dashed vertical line: default $N=15$ . ALS-IRLS achieves near-minimum RMSE at the default setting; ALS degrades uniformly due to the persistent bias in $\hat{C}_{e,0}$ .

Method	$\mathrm{RMSE}_{\mathrm{state}}$	Covariance source
Oracle KF	$1.80$	True $(\bar{Q},\bar{R})$
KF + ALS-IRLS (proposed)	$\mathbf{1.97}$	Online estimate
Student- $t$ KF [ 54 ]	$4.12$	Fixed $(0.3,\;0.1)$
MCKF [ 55 ]	$6.38$	Fixed $(0.3,\;0.1)$
KF + ALS	$14.25$	Contaminated estimate

TABLE III: State Estimation RMSE on the Outlier-Free Evaluation Phase (

T_{\mathrm{ev}}=500

N_{\mathrm{mc}}=100

)

Fig. 8: Mean state estimation RMSE on the $500$ -step outlier-free evaluation phase ( $N_{\mathrm{mc}}=100$ , $\varepsilon=0.15$ , $\omega=8$ ). Oracle KF (black) defines the lower bound. KF + ALS-IRLS (red) achieves RMSE within $9\%$ of the Oracle, outperforming all compared methods. Student- $t$ KF [ 54 ] and MCKF [ 55 ] incur $2.3\times$ and $3.5\times$ higher RMSE owing to misspecified $Q_{\mathrm{st}}=0.3$ , confirming that accurate covariance estimation outweighs filter-level robustness when the noise model is unknown. KF + ALS yields the largest RMSE due to the severely inflated $\hat{Q}_{\mathrm{ALS}}\approx 53$ .

Covariance Estimation

Covariance estimation develops methods for learning dependence and uncertainty structure from noisy, incomplete, or high-dimensional data.

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Sensor measurements are frequently corrupted by outliers and non-Gaussian noise. These imperfections in the sensor data can cause classical state estimators to generate biased and unreliable state and uncertainty estimates. Robust estimators reject or downweight outliers but do not perform measurement covariance estimation, whereas joint state and covariance estimators assume Gaussian residuals and fixed loss shape parameters. Integrating these two capabilities into a single framework is an opportunity to simultaneously estimate both state and covariance in the presence of outliers. This paper proposes a unified Block-Coordinate Descent framework that combines a norm-aware adaptive robust loss, an Iteratively Reweighted Least-Squares state update, and a Minimum Weighted Covariance Determinant covariance estimator, yielding a self-tuning joint state and covariance estimator. The framework is evaluated in a Monte-Carlo simulation and on real-world ultra-wideband localization experiments in cluttered non-line-of-sight environments. Results show that the proposed estimator consistently recovers the true inlier measurement covariance and matches or exceeds the state estimation accuracy of all baselines, without requiring any manual parameter tuning.

Abstract

Covariance Estimation

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