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Selection-Induced Contraction of Innovation Statistics in Gated Kalman Filters

Barak Or

TL;DR

This paper analyzes how validation gating and nearest-neighbor data association alter innovation-based statistics in Kalman-tracking. It derives exact gate-conditioned first- and second-order moments, showing a dimension-dependent contraction of the innovation covariance via the factor $\gamma(\tau,m)$, and proves an unavoidable energy contraction from NN selection. In two dimensions, closed-form expressions quantify how gating and NN combine to bias NIS statistics away from their nominal chi-square references. The findings explain observed biases in post-gate diagnostics and provide gate-aware normalization and interpretation guidelines, without modifying the Kalman recursion or gating rules themselves.

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.

Selection-Induced Contraction of Innovation Statistics in Gated Kalman Filters

TL;DR

This paper analyzes how validation gating and nearest-neighbor data association alter innovation-based statistics in Kalman-tracking. It derives exact gate-conditioned first- and second-order moments, showing a dimension-dependent contraction of the innovation covariance via the factor , and proves an unavoidable energy contraction from NN selection. In two dimensions, closed-form expressions quantify how gating and NN combine to bias NIS statistics away from their nominal chi-square references. The findings explain observed biases in post-gate diagnostics and provide gate-aware normalization and interpretation guidelines, without modifying the Kalman recursion or gating rules themselves.

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.

Paper Structure

This paper contains 37 sections, 5 theorems, 58 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Kalman Filter Innovation Model
  5. Post-Gate Innovation Distribution
  6. Innovation Gating as a Statistical Selection Mechanism
  7. Distribution of the NIS and Ellipsoidal Gating
  8. Gate-Conditioned Innovation Moments
  9. Gate-Conditioned NIS Statistics
  10. Implications for Innovation-Based Diagnostics
  11. Gate-Aware Consistency and Interpretation of Innovation Statistics
  12. Classical use of NIS for consistency Validation
  13. Post-gate NIS is not chi-square
  14. Gate-aware interpretation of NIS statistics
  15. Implications for adaptive tuning and diagnostics
  16. ...and 22 more sections

Key Result

Proposition 1

If $\nu \sim \mathcal{N}(0,S)$ with $S \in \mathbb{S}_{++}^m$, then

Figures (2)

  • Figure 1: Gate-conditioned NIS distribution in the two-dimensional case. The nominal $\chi^2_2$ distribution is shown together with the truncated distribution induced by ellipsoidal validation gating. The gate-conditioned mean is systematically lower than the nominal reference value, illustrating the systematic contraction of innovation energy relative to the nominal reference induced by validation gating, even under ideal Kalman filter assumptions.
  • Figure 2: Effect of NN association on normalized innovation squared (NIS) statistics after gating. The distribution of $Z \mid \mathcal{A}$ is shown together with the distribution of the minimum NIS selected from $M=2,3,5$ independent in-gate measurements. As $M$ increases, the expected selected NIS decreases, illustrating the order-statistic energy contraction induced by nearest-neighbor association.

Theorems & Definitions (7)

  • Proposition 1: Distribution of the NIS
  • Proposition 2: Gate-Conditioned Innovation Moments
  • Corollary 1: Gate-Conditioned Mean NIS
  • Proposition 3: Post-gate NN energy contraction
  • Corollary 2: Impossibility of preserving nominal innovation energy under selection
  • proof
  • proof