Selection-Induced Contraction of Innovation Statistics in Gated Kalman Filters

TL;DR

This paper analyzes how gating in Kalman filters affects innovation statistics, showing it causes contraction of covariance and impacts the statistical properties of measurements, with implications for tracking accuracy.

Contribution

It provides exact analytical expressions for innovation moments under gating and reveals how gating and association induce covariance contraction and statistical bias.

Findings

Gating causes a deterministic contraction of innovation covariance.

Association further influences the statistical properties of innovation.

Closed-form results quantify the impact in two-dimensional scenarios.

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 nominal quantities. Under classical linear--Gaussian assumptions, we derive exact expressions for the first- and second-order moments of the innovation conditioned on ellipsoidal gating, and show that gating induces a deterministic, dimension-dependent contraction of the innovation covariance. The analysis is extended to 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 statistics cannot be preserved under nontrivial gating and association. Closed-form results in the two-dimensional case quantify the combined effects and illustrate their practical significance.