Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inverse Cubature and Quadrature Kalman filters

Himali Singh, Kumar Vijay Mishra, Arpan Chattopadhyay

TL;DR

This work tackles inverse cognition in highly nonlinear settings by developing inverse cubature, quadrature, and cubature-quadrature Kalman filters (I-CKF, I-QKF, I-CQKF) and an RKHS-based CKF for unknown dynamics. The proposed methods leverage efficient numerical integration to approximate Bayesian recursions, with rigorous stability and consistency analyses ensuring reliable inverse state estimation. Empirical results across target tracking, Lorenz dynamics, and FM demodulation demonstrate that I-CKF and I-QKF often surpass mismatched inverse filters and approach or beat the recursive Cramér-Rao lower bound, albeit with higher computation for QKF-based variants. RKHS-CKF is highlighted as a powerful tool when forward models are unknown, enabling joint state and parameter learning in both forward and inverse roles. The framework broadens the applicability of inverse filtering to nonlinear, non-Gaussian, and unknown-model scenarios with practical implications for cognitive radar and adversarial settings.

Abstract

Recent research in inverse cognition with cognitive radar has led to the development of inverse stochastic filters that are employed by the target to infer the information the cognitive radar may have learned. Prior works addressed this inverse cognition problem by proposing inverse Kalman filter (I-KF) and inverse extended KF (I-EKF), respectively, for linear and non-linear Gaussian state-space models. However, in practice, many counter-adversarial settings involve highly non-linear system models, wherein EKF's linearization often fails. In this paper, we consider the efficient numerical integration techniques to address such non-linearities and, to this end, develop inverse cubature KF (I-CKF), inverse quadrature KF (I-QKF), and inverse cubature-quadrature KF (I-CQKF). For the unknown system model case, we develop reproducing kernel Hilbert space (RKHS)-based CKF. We derive the stochastic stability conditions for the proposed filters in the exponential-mean-squared-boundedness sense and prove the filters' consistency. Numerical experiments demonstrate the estimation accuracy of our I-CKF, I-QKF, and I-CQKF with the recursive Cramér-Rao lower bound as a benchmark.

Inverse Cubature and Quadrature Kalman filters

TL;DR

This work tackles inverse cognition in highly nonlinear settings by developing inverse cubature, quadrature, and cubature-quadrature Kalman filters (I-CKF, I-QKF, I-CQKF) and an RKHS-based CKF for unknown dynamics. The proposed methods leverage efficient numerical integration to approximate Bayesian recursions, with rigorous stability and consistency analyses ensuring reliable inverse state estimation. Empirical results across target tracking, Lorenz dynamics, and FM demodulation demonstrate that I-CKF and I-QKF often surpass mismatched inverse filters and approach or beat the recursive Cramér-Rao lower bound, albeit with higher computation for QKF-based variants. RKHS-CKF is highlighted as a powerful tool when forward models are unknown, enabling joint state and parameter learning in both forward and inverse roles. The framework broadens the applicability of inverse filtering to nonlinear, non-Gaussian, and unknown-model scenarios with practical implications for cognitive radar and adversarial settings.

Abstract

Recent research in inverse cognition with cognitive radar has led to the development of inverse stochastic filters that are employed by the target to infer the information the cognitive radar may have learned. Prior works addressed this inverse cognition problem by proposing inverse Kalman filter (I-KF) and inverse extended KF (I-EKF), respectively, for linear and non-linear Gaussian state-space models. However, in practice, many counter-adversarial settings involve highly non-linear system models, wherein EKF's linearization often fails. In this paper, we consider the efficient numerical integration techniques to address such non-linearities and, to this end, develop inverse cubature KF (I-CKF), inverse quadrature KF (I-QKF), and inverse cubature-quadrature KF (I-CQKF). For the unknown system model case, we develop reproducing kernel Hilbert space (RKHS)-based CKF. We derive the stochastic stability conditions for the proposed filters in the exponential-mean-squared-boundedness sense and prove the filters' consistency. Numerical experiments demonstrate the estimation accuracy of our I-CKF, I-QKF, and I-CQKF with the recursive Cramér-Rao lower bound as a benchmark.
Paper Structure (18 sections, 3 theorems, 41 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 18 sections, 3 theorems, 41 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. System model
  3. Inverse Filter Models
  4. Inverse CKF
  5. Non-Gaussian systems
  6. Inverse QKF
  7. Inverse CQKF
  8. Continuous-discrete framework
  9. Complex-valued systems
  10. RKHS-CKF
  11. Performance Analyses
  12. Stability
  13. Consistency
  14. Numerical experiments
  15. I-CKF for target tracking
  16. ...and 3 more sections

Key Result

Proposition 1

Consider the system eqn:state transition x and eqn:observation y with forward CKF. Forward CKF's estimation error $\widetilde{\mathbf{x}}_{k}$ is exponentially bounded in the mean-squared sense and bounded with probability one if the following holds true. C1. There exist positive real numbers $\bar{ C2.$\mathbf{U}^{x}_{k}$ and $\mathbf{F}_{k}$ are non-singular for every $k\geq 0$. C3. The constant

Figures (3)

  • Figure 1: An illustration of defender-attacker interaction. The defender's true state at $k$-th time step is $\mathbf{x}_{k}$. The attacker observes $\mathbf{x}_{k}$ as $\mathbf{y}_{k}$ through observation function $h(\cdot)$ with measurement noise $\mathbf{v}_{k}$. With $\mathbf{y}_{k}$ as input, the attacker's forward filter computes state estimate $\hat{\mathbf{x}}_{k}$, which the defender observes as $\mathbf{a}_{k}$ through observation function $g(\cdot)$ with measurement noise $\bm{\epsilon}_{k}$. Finally, the defender's inverse filter provides estimate $\hat{\newline {\hat{\mathbf{x}}}}_{k}$ of $\hat{\mathbf{x}}_{k}$ with $\mathbf{x}_{k}$ and $\mathbf{a}_{k}$ as inputs.
  • Figure 2: Time-averaged estimation error of forward and inverse (a) CKFs for target tracking system (averaged over $250$ runs); (b) UKFs, (c) QKFs and (d) CQKFs for Lorenz system (averaged over $50$ runs).
  • Figure 3: Time-averaged RMSE for (a) forward filters, (b) I-UKFs, (c) I-CKFs, and (d) I-QKFs for FM demodulator (averaged over $500$ runs).

Theorems & Definitions (13)

  • Remark 1: Differences with I-KF and I-EKF
  • Remark 2: Differences with I-UKF
  • Remark 3: Computational complexity
  • Remark 4: Simplification to I-UKF
  • Definition 1: Exponential mean-squared boundedness reif1999stochastic
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark 5: Practical bounds on system functions
  • ...and 3 more