A Robust Kalman Filter for Heavy-Tailed Measurement Noise Based on Gamma Pearson VII Mixture Distribution

This study develops an enhanced state estimation framework by integrating the Kalman filtering mechanism with a Gamma Pearson VII (GaPV) hybrid probability model to address non-stationary heavy-tailed noise characteristics in measurement systems. This innovative methodology synthesizes Gaussian morphological properties with Gamma’s supplementary parameters within the Pearson VII structural framework, establishing a unified probabilistic modeling paradigm. The proposed filter adaptively optimizes the Pearson VII formulation’s mean vector and covariance matrix, enabling precise characterization of data features in challenging environments exhibiting non-stationary behavior and heavy-tailed noise distributions. An Inverse Wishart distribution is introduced as the prior distribution of the measurement noise covariance to estimate it adaptively in response to unknown time-varying measurement noise. The variational Bayesian method is used to calculate the state and parameters jointly. Simulation results demonstrate that, compared to traditional Kalman filters based on the Gaussian assumption, the proposed filter exhibits higher precision and stronger robustness when dealing with non-stationary heavy-tailed measurement noise (HTMN). In summary, the Kalman Filter based on the Gamma Pearson VII distribution provides a new and effective tool for state estimation in complex noise environments, mainly showing significant advantages in applications requiring high precision and robustness.