Selection-Induced Contraction of Innovation Statistics in Gated Kalman Filters
For practitioners using Kalman filters with gating, this work reveals a fundamental bias in innovation statistics that must be accounted for to avoid incorrect inference.
This paper shows that validation gating in Kalman filters causes innovation statistics to converge to gate-conditioned quantities, not nominal ones, and proves that nominal statistics cannot be preserved under nontrivial gating and association. Exact expressions for first- and second-order moments under ellipsoidal gating are derived, and the combined effects with NN association are quantified in the 2D case.
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.