scholarly journals A Novel Perspective of the Kalman Filter from the Rényi Entropy

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 982
Author(s):  
Yarong Luo ◽  
Chi Guo ◽  
Shengyong You ◽  
Jingnan Liu
Keyword(s):  
Kalman Filter ◽  
Shannon Entropy ◽  
Unscented Kalman Filter ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Error Matrix ◽  
Gaussian Probability Density Function ◽  
Loosely Coupled ◽  
Temporal Derivative ◽  
New Perspective

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.

scholarly journals Shannon, Rényi, Tsallis Entropies and Onicescu Information Energy for Low-Lying Singly Excited States of Helium

Atoms ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 70 ◽  
Author(s):  
Jen-Hao Ou ◽  
Yew Kam Ho
Keyword(s):  
Excited States ◽  
Shannon Entropy ◽  
Electronic Structures ◽  
Tsallis Entropy ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Electron Delocalization ◽  
Information Theoretic ◽  
Molecular Systems ◽  
Information Energy

Knowledge of the electronic structures of atomic and molecular systems deepens our understanding of the desired system. In particular, several information-theoretic quantities, such as Shannon entropy, have been applied to quantify the extent of electron delocalization for the ground state of various systems. To explore excited states, we calculated Shannon entropy and two of its one-parameter generalizations, Rényi entropy of order α and Tsallis entropy of order α , and Onicescu Information Energy of order α for four low-lying singly excited states (1s2s 1 S e , 1s2s 3 S e , 1s3s 1 S e , and 1s3s 3 S e states) of helium. This paper compares the behavior of these three quantities of order 0.5 to 9 for the ground and four excited states. We found that, generally, a higher excited state had a larger Rényi entropy, larger Tsallis entropy, and smaller Onicescu information energy. However, this trend was not definite and the singlet–triplet reversal occurred for Rényi entropy, Tsallis entropy and Onicescu information energy at a certain range of order α .

Rényi Entropy and Rényi Divergence in Sequential Effect Algebra

2020 ◽  
Vol 27 (02) ◽  
pp. 2050008
Author(s):  
Zahra Eslami Giski
Keyword(s):  
Shannon Entropy ◽  
Effect Algebra ◽  
Sequential Effect ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Numerical Examples ◽  
Basic Properties ◽  
Rényi Divergence ◽  
Leibler Divergence ◽  
Entropy Measures

The aim of this study is to extend the results concerning the Shannon entropy and Kullback–Leibler divergence in sequential effect algebra to the case of Rényi entropy and Rényi divergence. For this purpose, the Rényi entropy of finite partitions in sequential effect algebra and its conditional version are proposed and the basic properties of these entropy measures are derived. In addition, the notion of Rényi divergence of a partition in sequential effect algebra is introduced and the basic properties of this quantity are studied. In particular, it is proved that the Kullback–Leibler divergence and Shannon’s entropy of partitions in a given sequential effect algebra can be obtained as limits of their Rényi divergence and Rényi entropy respectively. Finally, to illustrate the results, some numerical examples are presented.

scholarly journals Loosely Coupled INS/GPS Integration with Constant Lever Arm using Marginal Unscented Kalman Filter

2013 ◽  
Vol 67 (3) ◽  
pp. 419-436 ◽  
Author(s):  
Guobin Chang
Keyword(s):  
Kalman Filter ◽  
Unscented Kalman Filter ◽  
Performance Comparison ◽  
Observation Equation ◽  
Computationally Efficient ◽  
Loosely Coupled ◽  
Lever Arm ◽  
Computational Expense ◽  
Nonlinear Observation ◽  
Lever Arm Effect

A loosely coupled Inertial Navigation System (INS) and Global Positioning System (GPS) are studied, particularly considering the constant lever arm effect. A five-element vector, comprising a craft's horizontal velocities in the navigation frame and its position in the earth-centred and earth-fixed frame, is observed by GPS, and in the presence of lever arm effect, the nonlinear observation equation from the state vector to the observation vector is established and addressed by the correction stage of an unscented Kalman filter (UKF). The conditionally linear substructure in the nonlinear observation equation is exploited, and a computationally efficient refinement of the UKF called marginalized UKF (MUKF) is investigated to incorporate this substructure where fewer sigma points are needed, and the computational expense is cut down while the high accuracy and good applicability of the UKF are retained. A performance comparison between UKF and MUKF demonstrates that the MUKF can achieve, if not better, at least a comparable performance to the UKF, but at a lower computational expense.

Hybrid extended-unscented Kalman filters for continuous-time nonlinear fractional-order systems involving process and measurement noises

2020 ◽  
Vol 42 (9) ◽  
pp. 1618-1631
Author(s):  
Xiaojiao Chen ◽  
Zhe Gao ◽  
Ruicheng Ma ◽  
Xiaomin Huang
Keyword(s):  
Kalman Filter ◽  
Fractional Order ◽  
Continuous Time ◽  
Kalman Filters ◽  
Unscented Kalman Filter ◽  
Nonlinear Function ◽  
Fractional Order Systems ◽  
Error Matrix ◽  
Current Time ◽  
Measurement Noises

Hybrid extended-unscented Kalman filters (HEUKFs) for continuous-time nonlinear fractional-order systems with process and measurement noises are investigated in this paper. The Grünwald-Letnikov difference and the fractional-order average derivative (FOAD) method are adopted to discretize the investigated nonlinear fractional-order system, and the nonlinear functions in the system description are coped with the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). The first-order Taylor expansion used in the EKF method is performed for the nonlinear function at the current time. Meanwhile, the unscented transformation used in the UKF is also concerned for the nonlinear function at the previous time. By using the HEUKF designed in this paper, the third-order approximations for the nonlinear function can be achieved to enhance the accuracy of state estimation and estimation error matrix. Finally, numerical examples are provided to illustrate the effectiveness of the proposed HEUKF for nonlinear fractional-order systems.

scholarly journals Rényi Entropy and Rényi Divergence in the Intuitionistic Fuzzy Case

2018 ◽  
Vol 72 (1) ◽  
pp. 77-105
Author(s):  
Beloslav Riečan ◽  
Dagmar Markechová
Keyword(s):  
Shannon Entropy ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Numerical Examples ◽  
Rényi Divergence ◽  
Intuitionistic Fuzzy ◽  
Leibler Divergence

Abstract Our objective in this paper is to define and study the Rényi entropy and the Rényi divergence in the intuitionistic fuzzy case. We define the Rényi entropy of order of intuitionistic fuzzy experiments (which are modeled by IF-partitions) and its conditional version and we examine their properties. It is shown that the suggested concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of IF-partitions. In addition, we introduce and study the concept of Rényi divergence in the intuitionistic fuzzy case. Specifically, relationships between the Rényi divergence and Kullback-Leibler divergence and between the Rényi divergence and the Rényi entropy in the intuitionistic fuzzy case are studied. The results are illustrated with several numerical examples.

scholarly journals A Novel Unscented Kalman Filter Strategy to Enhance Navigation System Performance

2020 ◽  
Vol 8 (5) ◽  
pp. 2446-2453
Keyword(s):  
Kalman Filter ◽  
Navigation System ◽  
Unscented Kalman Filter ◽  
Ground Truth ◽  
Estimation Algorithm ◽  
Large Error ◽  
Loosely Coupled ◽  
Grade Span ◽  
Complementary Nature ◽  
Navigation Information

The Extended Kalman Filter (EKF) is the most widely estimation algorithm used for nonlinear system such as a navigation system to fuse an inertial navigation system (INS) with Global Positioning System (GPS) which its information has complementary nature to get more accurate navigation information. Unfortunately, the performance of INS/GPS fusion using EKF is degraded due to the linearization error and GPS error. Therefore, a new algorithm is developed to overcome these issues. This algorithm uses the sampling-based Unscented Kalman Filter (UKF) to solve the linearization problem, and ignore the GPS reading when there is a large error in its measurements. The new algorithm is named Adaptive Loosely Coupled Unscented Kalman Filter (ALCUKF). The ALCUKFbased INS/GPS systems are presented for two different datasets. The first dataset is acquired using a high-end tactical-grade SPAN unit featuring Novatel HG1700 IMU module. The second dataset is acquired from a MEMS-based SCC1300-D04 IMU unit from VTI. The results of the new method are compared against reference ground truth trajectories and measured quantitatively using the Root Mean Square Error (RMSE). The ALCUKF increasedthe navigation system performancesignificantly when compared with EKF for both datasets as shown in the paper.

Combining Renyi Entropy and EWMA to Detect Common Attacks in Network

2016 ◽  
Vol 30 (10) ◽  
pp. 1650021 ◽  
Author(s):  
Ruoyu Yan
Keyword(s):  
Shannon Entropy ◽  
Research Work ◽  
True Positive Rate ◽  
Training Data ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Network Attacks ◽  
Ewma Control Chart ◽  
Positive Rate ◽  
Real Traffic

How to timely and precisely identify attack behaviors in network without dealing with a large number of traffic features and historical data, such as training data, is an important research work in the field of network security. In this paper, firstly, the differences between Renyi entropy and Shannon entropy are analyzed and compared. In order to capture network traffic changes exactly, Renyi entropy instead of Shannon entropy is proposed to measure selected traffic features. Then EWMA control chart theory is used to check Renyi entropy time series for detecting and screening anomalies. And three kinds of network attacks are also analyzed and characterized by behavior feature vector for attack identification. Finally a feature similarity-based method is used to identify attacks. The experimental results of real traffic traces show that the proposed method has good capability to detect and identify these attacks with less computation cost. To evaluate attack identification method conveniently, an approach is proposed to generate simulated attack traffics. Compared with Shannon entropy-based method, the experiments on simulation traffics show that Renyi entropy-based method has much higher overall accuracy, average precision and average true positive rate. Further comparison indicates the proposed method has more powerful performance to detect attacks than PCA-based method.

scholarly journals A new perspective and extension of the Gaussian Filter

2016 ◽  
Vol 35 (14) ◽  
pp. 1731-1749 ◽  
Author(s):  
Manuel Wüthrich ◽  
Sebastian Trimpe ◽  
Cristina Garcia Cifuentes ◽  
Daniel Kappler ◽  
Stefan Schaal
Keyword(s):  
Kalman Filter ◽  
Unscented Kalman Filter ◽  
Estimation Accuracy ◽  
Affine Function ◽  
Gaussian Filter ◽  
Actual Measurement ◽  
Difference Filter ◽  
Small Change ◽  
New Perspective ◽  
Nonlinear Observation

The Gaussian Filter (GF) is one of the most widely used filtering algorithms; instances are the Extended Kalman Filter, the Unscented Kalman Filter and the Divided Difference Filter. The GF represents the belief of the current state by a Gaussian distribution, whose mean is an affine function of the measurement. We show that this representation can be too restrictive to accurately capture the dependences in systems with nonlinear observation models, and we investigate how the GF can be generalized to alleviate this problem. To this end, we view the GF as the solution to a constrained optimization problem. From this new perspective, the GF is seen as a special case of a much broader class of filters, obtained by relaxing the constraint on the form of the approximate posterior. On this basis, we outline some conditions which potential generalizations have to satisfy in order to maintain the computational efficiency of the GF. We propose one concrete generalization which corresponds to the standard GF using a pseudo measurement instead of the actual measurement. Extending an existing GF implementation in this manner is trivial. Nevertheless, we show that this small change can have a major impact on the estimation accuracy.

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