Particle Filter and UKF Filter: One-Dimensional State Estimation Algorithms

In the field of state estimation, Particle Filter (PF) and Unscented Kalman Filter (UKF) are two widely used algorithms. Each has its advantages and limitations, but combining their strengths can enhance estimation accuracy for one-dimensional state variables in certain scenarios.

Characteristics of Particle Filter Particle Filter approximates the probability distribution of state variables using Monte Carlo methods. It doesn't rely on linearity assumptions, making it suitable for non-Gaussian noise and nonlinear systems. However, with insufficient particles, it may suffer from sample degeneracy - where a few particles dominate the weight distribution, reducing estimation accuracy. In implementation, the PF algorithm typically involves three main steps: prediction update (propagating particles through system dynamics), weight calculation (based on measurement likelihood), and resampling (to address weight degeneracy).

Characteristics of UKF Filter UKF captures the mean and covariance of nonlinear systems through Unscented Transform, avoiding the linearization errors of Extended Kalman Filter (EKF). UKF offers higher computational efficiency but may underperform in strongly nonlinear or high-dimensional systems. The algorithm implementation involves selecting sigma points around the current estimate, propagating these points through the nonlinear system, and then reconstructing the Gaussian approximation from the transformed points.

Hybrid Algorithm Approach The hybrid algorithm combining PF and UKF typically uses UKF to generate proposal distributions that guide PF's sampling process. In practical implementation, UKF first predicts the state mean and covariance, which then serve as the importance density function for particle generation. This strategy reduces particle degeneracy while improving sampling efficiency. Code implementation would involve initializing particles, running UKF prediction, and using the UKF output to inform the particle resampling process.

Application Scenarios This hybrid algorithm is particularly suitable for one-dimensional state estimation problems, such as single-variable position prediction in target tracking or financial time series analysis. In one-dimensional cases, computational complexity is lower, UKF's Unscented Transform can more accurately approximate nonlinear characteristics, while PF enhances robustness against non-Gaussian noise. The algorithm can be implemented using sequential Monte Carlo methods with UKF-based proposal distributions.

By integrating the advantages of both filters, the hybrid algorithm improves estimation accuracy and stability while maintaining real-time performance capabilities.