Inverse Cubature and Quadrature Kalman filters

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 Cram\'{e}r-Rao lower bound as a benchmark.