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

TL;DR

A novel outlier-robust ALS algorithm, termed ALS-IRLS, based on the iteratively reweighted least squares (IRLS) framework is proposed, which 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.

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 (8)

• Figure 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 $\hat{C}_{e,0}$.
• Figure 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 ($\hat{C}_{e,0}^{\mathrm{raw}}\approx32.4$, true $\approx4.3$); 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.
• Figure 3: IRLS Huber weights at Monte Carlo trial 6, batch 1 ($N=15$). Blue bars: uncontaminated lags with $w_j\approx1$; purple bar: down-weighted lag-0 entry ($w_0\approx0.03$). Dashed line at $w=1$; dotted line at $w=0.5$. All lags beyond lag-0 retain $w_j\geq0.95$, demonstrating selective penalisation of the contaminated entry without distorting the remaining autocovariance structure.
• Figure 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 $(\bar{Q},\bar{R})$.
• Figure 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.

Theorems & Definitions (2)

• proof
• proof