A Novel Perspective of the Kalman Filter from the Rényi Entropy

Abstract

Rényi entropy as a generalization of the Shannon entropy allows for different averaging of probabilities of a control parameter α. This paper gives a new perspective of the Kalman filter from the Rényi entropy. Firstly, the Rényi entropy is employed to measure the uncertainty of the multivariate Gaussian probability density function. Then, we calculate the temporal derivative of the Rényi entropy of the Kalman filter’s mean square error matrix, which will be minimized to obtain the Kalman filter’s gain. Moreover, the continuous Kalman filter approaches a steady state when the temporal derivative of the Rényi entropy is equal to zero, which means that the Rényi entropy will keep stable. As the temporal derivative of the Rényi entropy is independent of parameter α and is the same as the temporal derivative of the Shannon entropy, the result is the same as for Shannon entropy. Finally, an example of an experiment of falling body tracking by radar using an unscented Kalman filter (UKF) in noisy conditions and a loosely coupled navigation experiment are performed to demonstrate the effectiveness of the conclusion.

Kalman Filter from the Mutual Information Perspective

“…The classical Kalman filter can be derived as a best linear unbiased estimate [1] and it is easy understand it from the probabilistic perspective [2]. Recently, Kalman filter has also been presented using the methods of maximum relative entropy [3] and the temporal derivative of the Rényi entropy [4], which go beyond the general Bayesian filter. More and more evidences show that Kalman filter can be regarded as a direct extension of information theory.…”

Data Science: Measuring Uncertainties

“…Time series is one of the most prominent areas in data science, and some of the articles published here propose solutions with practical motivations in this area [ 5 , 6 , 7 , 8 ]. As mentioned before, this Special Issue encouraged articles on the foundations of measuring uncertainty [ 9 , 10 , 11 , 12 ].…”

“…Ref. [ 11 ] considered a Kalman filter and a Rényi entropy. The Rényi entropy was employed to measure the uncertainty of the multivariate Gaussian probability density function.…”