A Non-Linear Filtering Algorithm Based on Alpha-Divergence Minimization

Abstract

A non-linear filtering algorithm based on the alpha-divergence is proposed, which uses the exponential family distribution to approximate the actual state distribution and the alpha-divergence to measure the approximation degree between the two distributions; thus, it provides more choices for similarity measurement by adjusting the value of α during the updating process of the equation of state and the measurement equation in the non-linear dynamic systems. Firstly, an α -mixed probability density function that satisfies the normalization condition is defined, and the properties of the mean and variance are analyzed when the probability density functions p ( x ) and q ( x ) are one-dimensional normal distributions. Secondly, the sufficient condition of the alpha-divergence taking the minimum value is proven, that is when α ≥ 1 , the natural statistical vector’s expectations of the exponential family distribution are equal to the natural statistical vector’s expectations of the α -mixed probability state density function. Finally, the conclusion is applied to non-linear filtering, and the non-linear filtering algorithm based on alpha-divergence minimization is proposed, providing more non-linear processing strategies for non-linear filtering. Furthermore, the algorithm’s validity is verified by the experimental results, and a better filtering effect is achieved for non-linear filtering by adjusting the value of α .

Keywords

• alpha-divergence
• Kullback–Leibler divergence
• non-linear filtering
• exponential family distribution