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 during the updating process of the equation of state and the measurement equation in the non-linear dynamic systems. Firstly, an -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 and are one-dimensional normal distributions. Secondly, the sufficient condition of the alpha-divergence taking the minimum value is proven, that is when , the natural statistical vector’s expectations of the exponential family distribution are equal to the natural statistical vector’s expectations of the -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 .
Highlights
The analysis and design of non-linear filtering algorithms are of enormous significance because non-linear dynamic stochastic systems have been widely used in practical systems, such as navigation system [1], simultaneous localization and mapping [2], and so on
After the α-mixed probability density function is defined by q( x ) and p( x ), we prove that the sufficient condition for alpha-divergence minimization is when α ≥ 1 and the expected value of the natural statistical vector of q( x ) is equivalent to the expected value of the natural statistical vector of the α-mixed probability density function
A non-linear filtering algorithm based on the alpha-divergence minimization has been proposed by applying the above two points to the non-linear filtering
Summary
The analysis and design of non-linear filtering algorithms are of enormous significance because non-linear dynamic stochastic systems have been widely used in practical systems, such as navigation system [1], simultaneous localization and mapping [2], and so on. In order to measure the similarity between the hypothetical state distribution density function and the actual one, we need to select a measurement method to ensure the effectiveness of the above methods. It can be used to measure the similarity between the hypothetical state distribution density function and the actual one for the non-linear filtering. Compared with the Kullback–Leibler divergence (the KL divergence), the alpha-divergence provides more choices for measuring the similarity between the hypothetical state distribution density function and the Sensors 2018, 18, 3217; doi:10.3390/s18103217 www.mdpi.com/journal/sensors. Adjusting the value of parameter α in the function can ensure the interesting properties of similarity measurement Another choice of α characterizes different learning principles, in the sense that the model distribution is more inclusive (α → ∞) or more exclusive (α → −∞) [5]. The experiments show that the proposed method can achieve better performance by using a proper α value
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