Maximum approximate likelihood estimation of general continuous-time state-space models

Continuous-time state-space models (SSMs) are flexible tools for analysing irregularly sampled sequential observations that are driven by an underlying state process. Corresponding applications typically involve restrictive assumptions concerning linearity and Gaussianity to facilitate inference on the model parameters via the Kalman filter. In this contribution, we provide a general continuous-time SSM framework, allowing both the observation and the state process to be non-linear and non-Gaussian. Statistical inference is carried out by maximum approximate likelihood estimation, where multiple numerical integration within the likelihood evaluation is performed via a fine discretization of the state process. The corresponding reframing of the SSM as a continuous-time hidden Markov model, with structured state transitions, enables us to apply the associated efficient algorithms for parameter estimation and state decoding. We illustrate the modelling approach in a case study using data from a longitudinal study on delinquent behaviour of adolescents in Germany, revealing temporal persistence in the deviation of an individual's delinquency level from the population mean.

Variational Bayesian-Based Improved Maximum Mixture Correntropy Kalman Filter for Non-Gaussian Noise

The maximum correntropy Kalman filter (MCKF) is an effective algorithm that was proposed to solve the non-Gaussian filtering problem for linear systems. Compared with the original Kalman filter (KF), the MCKF is a sub-optimal filter with Gaussian correntropy objective function, which has been demonstrated to have excellent robustness to non-Gaussian noise. However, the performance of MCKF is affected by its kernel bandwidth parameter, and a constant kernel bandwidth may lead to severe accuracy degradation in non-stationary noises. In order to solve this problem, the mixture correntropy method is further explored in this work, and an improved maximum mixture correntropy KF (IMMCKF) is proposed. By derivation, the random variables that obey Beta-Bernoulli distribution are taken as intermediate parameters, and a new hierarchical Gaussian state-space model was established. Finally, the unknown mixing probability and state estimation vector at each moment are inferred via a variational Bayesian approach, which provides an effective solution to improve the applicability of MCKFs in non-stationary noises. Performance evaluations demonstrate that the proposed filter significantly improves the existing MCKFs in non-stationary noises.

Radar Imaging Statistics of Non-Gaussian Rough Surface: A Physics-Based Simulation Study

This paper investigates the radar image statistics of rough surfaces by simulating the scattered signal’s dependence on the surface roughness. Statistically, the roughness characteristics include the height probability density (HPD) and, to the second-order, the power spectral density (PSD). We simulated the radar backscattered signal by computing the far-field scattered field from the rough surface within the antenna beam volume in the context of synthetic aperture radar (SAR) imaging. To account for the non-Gaussian height distribution, we consider microscopic details of the roughness on comparable radar wavelength scales to include specularly, singly, and multiply scatterers. We introduce surface roughness index (RSI) to distinguish the statistical characteristics of rough surfaces with different height distributions. Results suggest that increasing the RMS height does not impact the Gaussian HPD surface but significantly affects the Weibull surface. The results confirm that as the radar frequency increases, or reaches a relatively larger roughness, the surface’s HPD causes significant changes in incoherent scattering due to more frequent multiple scattering contributions. As a result, the speckle move further away from the Rayleigh model. By examining individual RSI, we see that the Gaussian HPD surface is much less sensitive to RMS height than the Weibull HPD surface. We demonstrate that to retrieve the surface parameters (both dielectric and roughness) from the estimated RCS, less accuracy is expected for the non-Gaussian surface than the Gaussian surface under the same conditions. Therefore, results drawn from this study are helpful for system performance evaluations, parameters estimation, and target detection for SAR imaging of a rough surface.