Matrix-Valued Measures and Wishart Statistics for Target Tracking Applications

TL;DR

This paper tackles the challenge of validating multivariate target-tracking models, where scalar measures like NEES and NIS fail to capture cross-variable dependencies. It introduces matrix-valued statistics, notably the NEES matrix and the NIS matrix, and derives Wishart-based eigenvalue statistics to assess matrix credibility and conservativeness both offline and online. The approach enables robust offline evaluation of track fusion designs and online detection of filter process-model mismatch, demonstrated through two target-tracking scenarios and supported by exact and approximate eigenvalue distributions. The methods offer a principled, multivariate evaluation framework with practical implications for detector design, filter tuning, and broader estimation problems such as SLAM and localization.

Abstract

Ensuring sufficiently accurate models is crucial in target tracking systems. If the assumed models deviate too much from the truth, the tracking performance might be severely degraded. While the models are usually defined using multivariate conditions, the measures used to validate them are most often scalar-valued. In this paper, we propose matrix-valued measures for both offline and online assessment of target tracking systems. Recent results from Wishart statistics, and approximations thereof, are adapted and it is shown how these can be incorporated to infer statistical properties for the eigenvalues of the proposed measures. In addition, we relate these results to the statistics of the baseline measures. Finally, the applicability of the proposed measures are demonstrated using two important problems in target tracking: (i) distributed track fusion design; and (ii) filter model mismatch detection.

Figures (8)

• Figure 1: Motivating example. The estimation error is sampled from $\mathcal{N}_2(\mathbf{0},\bm{\Sigma})$ and $\mathbf{P}\neq\bm{\Sigma}$ is the covariance computed by the estimator. However, from the NEES it might be concluded that $\mathbf{P}=\bm{\Sigma}$ despite that $\lambda_{\min}(\bm{\Xi})=0.75$ and $\lambda_{\max}(\bm{\Xi})=1.25$.
• Figure 2: Expected values and inverse CDFs of $\lambda_{\min}$ and $\lambda_{\max}$ as functions of $n$. The curves are normalized with $n$.
• Figure 3: Switching dynamics example. At $k_{\text{switch}}$, the target dynamics switches from a CV model with isotropic random accelerations to a CV model with central random accelerations. The switch is captured by $\lambda_{\min}$ and $\lambda_{\max}$ for both $\hat{\bm{\Xi}}$ and $\hat{\bm{\Pi}}'$, but not clearly by $\bar{\lambda}$. The confidence intervals given by $F^{-1}$ have been normalized for comparison reason.
• Figure 4: NEES matrix results for the track fusion design. To evaluate if a track fusion method leads to conservative estimates, $\lambda_{\max}(\hat{\bm{\Xi}})$ is compared with $F_{\lambda_{\max}}^{-1}/M$. For convenience, also $\bar{\lambda}(\hat{\bm{\Xi}})=\text{NEES}/n_x$ and $\lambda_{\min}(\hat{\bm{\Xi}})$ are included.
• Figure 5: RMT results for the track fusion design. The RMT curves have been normalized by the CRLB.

Theorems & Definitions (6)

• Proposition 1
• proof
• Definition 1: The NEES Matrix
• Definition 2: The NIS Matrix---Single Run Statistics
• Definition 3: The NIS Matrix---MC Statistics
• Remark 1