Selection-Induced Contraction of Innovation Statistics in Gated Kalman Filters
A statistical analysis of how validation gating and nearest-neighbor association reshape innovation statistics in Kalman-based tracking systems.
Abstract
Validation gating is central to Kalman-based tracking, but it changes the statistical population that downstream diagnostics observe. This paper shows that normalized innovation squared (NIS) statistics computed after ellipsoidal gating converge to gate-conditioned, not nominal unconditional, reference values. Under classical linear-Gaussian assumptions, the work derives exact first- and second-order moments for gate-conditioned innovations and proves that validation gating induces deterministic, dimension-dependent contraction of innovation covariance. The analysis is extended to nearest-neighbor association, showing that selecting the minimum-norm in-gate innovation introduces additional order-statistic contraction even when the filter model is perfectly matched. Monte Carlo experiments and an end-to-end tracking study demonstrate that nominal NIS-based tuning can learn biased measurement-noise scales, while gate-aware correction remains close to the true scale.
Key Innovations
Gate-conditioned innovation statistics
Derives exact moments for innovations observed after ellipsoidal validation gating, replacing nominal unconditional references with gate-aware references.
Order-statistic view of nearest-neighbor association
Shows that nearest-neighbor association creates unavoidable innovation-energy contraction by selecting the minimum-norm accepted candidate.
Gate-aware diagnostics and tuning
Explains why post-gate NIS should use corrected references, preventing biased measurement-noise adaptation in tracking systems.