게이트링에 의한 혁신 통계 수축과 최근접 이웃 연관 효과

📝 Abstract

Validation gating is a fundamental component of classical Kalman-based tracking systems. Only measurements whose normalized innovation squared (NIS) falls below a prescribed threshold are considered for state update. While this procedure is statistically motivated by the chi-square distribution, it implicitly replaces the unconditional innovation process with a conditionally observed one, restricted to the validation event. This paper shows that innovation statistics computed after gating converge to gate-conditioned rather than unconditional nominal reference quantities. Under classical linear-Gaussian assumptions, we derive exact expressions for the first-and secondorder moments of the innovation conditioned on ellipsoidal gating, and show that gating induces a deterministic, dimensiondependent contraction of the innovation covariance relative to the nominal reference. The analysis is extended to nearest-neighbor (NN) association, which is shown to act as an additional statistical selection operator. We prove that selecting the minimum norm innovation among multiple in-gate measurements introduces an unavoidable energy contraction, implying that nominal innovation reference statistics cannot be preserved under nontrivial gating and association due to deterministic selection effects, even under perfectly matched linear-Gaussian Kalman filter assumptions. Closed-form results in the two-dimensional case quantify the combined effects and illustrate their practical significance.

💡 Analysis

Validation gating is a fundamental component of classical Kalman-based tracking systems. Only measurements whose normalized innovation squared (NIS) falls below a prescribed threshold are considered for state update. While this procedure is statistically motivated by the chi-square distribution, it implicitly replaces the unconditional innovation process with a conditionally observed one, restricted to the validation event. This paper shows that innovation statistics computed after gating converge to gate-conditioned rather than unconditional nominal reference quantities. Under classical linear-Gaussian assumptions, we derive exact expressions for the first-and secondorder moments of the innovation conditioned on ellipsoidal gating, and show that gating induces a deterministic, dimensiondependent contraction of the innovation covariance relative to the nominal reference. The analysis is extended to nearest-neighbor (NN) association, which is shown to act as an additional statistical selection operator. We prove that selecting the minimum norm innovation among multiple in-gate measurements introduces an unavoidable energy contraction, implying that nominal innovation reference statistics cannot be preserved under nontrivial gating and association due to deterministic selection effects, even under perfectly matched linear-Gaussian Kalman filter assumptions. Closed-form results in the two-dimensional case quantify the combined effects and illustrate their practical significance.

📄 Content

Kalman-based tracking systems are a cornerstone of modern aerospace, radar, and navigation applications, where reliable state estimation is required under uncertainty. A central role in such systems is played by innovation-based statistics, which are used for measurement validation, data association, consistency monitoring, and adaptive tuning. Among these, the normalized innovation squared (NIS) is particularly attractive due to its simple statistical characterization under nominal linear Gaussian assumptions: in the absence of additional selection mechanisms, the NIS follows a chi-square distribution whose moments provide natural reference values for diagnostic and tuning procedures.

In operational tracking systems, however, innovation statistics are rarely observed in their unconditional form. Prior to data association and state update, measurements are typically subjected to ellipsoidal validation gating, whereby only innovations whose Mahalanobis distance falls below a prescribed threshold are accepted. Validation gating is widely justified on statistical and practical grounds, as it limits computational B. Or is with metaor artificial intelligence, Haifa, Israel, and with the Google Reichman Tech School, Reichman University, Herzliya, Israel. E-mail: barakorr@gmail.com complexity and suppresses gross outliers. As a result, nearly all innovation-based diagnostics used in practice operate on a post-gate innovation stream rather than on the nominal innovation process assumed in classical Kalman filter theory.

Despite its ubiquity, the statistical consequences of validation gating are rarely treated explicitly. Once gating is applied, the innovations that enter association, filtering, and diagnostic logic are no longer samples from the nominal Gaussian distribution, but from a distribution truncated to the validation region. Consequently, innovation-based statistics computed after gating need not satisfy the classical reference properties associated with the chi-square law. In particular, empirical NIS statistics obtained from accepted measurements may systematically deviate from their nominal expectations even when the underlying system model and noise statistics are perfectly matched.

A closely related issue arises in data association. When multiple measurements fall inside the validation gate, NN association is commonly employed to select a single measurement for the state update. While computationally efficient and widely used, NN association further conditions the observed innovation through order-statistic selection, as it selects the innovation with minimum normalized distance among multiple candidates. The combined effect of validation gating and NN association therefore induces a multi-stage selection process whose impact on innovation statistics is not captured by nominal Kalman filter assumptions.

The objective of this paper is to isolate and characterize the statistical effects induced solely by validation gating and NN association, independent of clutter, false alarms, nonlinear dynamics, or modeling errors. Within the classical linear-Gaussian Kalman filtering framework, we provide an exact characterization of the innovation moments conditioned on ellipsoidal gating and show that validation gating induces a deterministic, dimension-dependent contraction of the innovation covariance relative to the nominal reference. We further show that NN association acts as an additional statistical selection operator and introduces an unavoidable, multiplicitydependent contraction of innovation energy through orderstatistic selection. Together, these results imply that nominal innovation statistics cannot be preserved under nontrivial gating and association, even when all Kalman filter assumptions are satisfied.

Innovation-based statistics are a fundamental component of Kalman filtering theory and practice, where quantities such as the NIS are routinely used for consistency monitoring, measurement validation, and adaptive tuning. Under nominal linear Gaussian assumptions, the statistical properties of the innovation process are well understood and documented in classical estimation references [1], [2], [3], [4], [5].

In practical tracking systems, innovation statistics are commonly employed in adaptive filtering and noise covariance estimation schemes, often relying on assumed chi-square properties of the NIS [6], [7], [8], [9]. Such approaches implicitly assume that the observed innovation sequence is representative of the nominal innovation distribution. However, in operational settings, innovation statistics are almost always evaluated after validation gating and data association, conditions under which these assumptions may no longer hold.

Validation gating and NN association are standard components of tracking systems in aerospace, radar, and navigation applications [4], [10], [11], [12]. Despite their widespread use, the statistical impact of these selection mechanisms on innovation-based dia

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