scholarly journals Rényi Entropy and Rényi Divergence in Product MV-Algebras

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 587 ◽  
Author(s):  
Dagmar Markechová ◽  
Beloslav Riečan
Keyword(s):  
Shannon Entropy ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Rényi Divergence ◽  
New Concepts ◽  
Leibler Divergence ◽  
Mv Algebra ◽  
The Relationship ◽  
Mv Algebras

This article deals with new concepts in a product MV-algebra, namely, with the concepts of Rényi entropy and Rényi divergence. We define the Rényi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovičová (Soft Comput.2000, 4, 41–44). Moreover, we introduce and study the notion of Rényi divergence in a product MV-algebra. It is proven that the Kullback–Leibler divergence of states on a given product MV-algebra introduced by Markechová and Riečan in (Entropy2017, 19, 267) can be obtained as the limit of their Rényi divergence. In addition, the relationship between the Rényi entropy and the Rényi divergence as well as the relationship between the Rényi divergence and Kullback–Leibler divergence in a product MV-algebra are examined.

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 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 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 .

scholarly journals Study of MV-algebras via derivations

2019 ◽  
Vol 27 (3) ◽  
pp. 259-278
Author(s):  
Jun Tao Wang ◽  
Yan Hong She ◽  
Ting Qian
Keyword(s):  
Boolean Algebra ◽  
Direct Product ◽  
Galois Connection ◽  
Product Representation ◽  
Mv Algebra ◽  
One To One ◽  
The Relationship ◽  
Mv Algebras ◽  
Derivation Pair

AbstractThe main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

scholarly journals Fast Tuning of Topic Models: An Application of Rényi Entropy and Renormalization Theory

Proceedings ◽  
2019 ◽  
Vol 46 (1) ◽  
pp. 5 ◽  
Author(s):  
Sergei Koltcov ◽  
Vera Ignatenko ◽  
Sergei Pashakhin
Keyword(s):  
Latent Dirichlet Allocation ◽  
Parameter Tuning ◽  
Significant Loss ◽  
Optimal Number ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Grid Search ◽  
Renormalization Procedure ◽  
Number Of Clusters ◽  
Leibler Divergence

In practice, the critical step in building machine learning models of big data (BD) is costly in terms of time and the computing resources procedure of parameter tuning with a grid search. Due to the size, BD are comparable to mesoscopic physical systems. Hence, methods of statistical physics could be applied to BD. The paper shows that topic modeling demonstrates self-similar behavior under the condition of a varying number of clusters. Such behavior allows using a renormalization technique. The combination of a renormalization procedure with the Rényi entropy approach allows for fast searching of the optimal number of clusters. In this paper, the renormalization procedure is developed for the Latent Dirichlet Allocation (LDA) model with a variational Expectation-Maximization algorithm. The experiments were conducted on two document collections with a known number of clusters in two languages. The paper presents results for three versions of the renormalization procedure: (1) a renormalization with the random merging of clusters, (2) a renormalization based on minimal values of Kullback–Leibler divergence and (3) a renormalization with merging clusters with minimal values of Rényi entropy. The paper shows that the renormalization procedure allows finding the optimal number of topics 26 times faster than grid search without significant loss of quality.

scholarly journals Indicators of wavefunction (de)localisation for avoided crossing in a quadrupole quantum billiard

2021 ◽  
Author(s):  
Kyu-Won Park ◽  
Juman Kim ◽  
Jisung Seo ◽  
Songky Moon ◽  
Kabgyun Jeong
Keyword(s):  
Root Mean Square ◽  
Image Contrast ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Mean Square ◽  
Inverse Participation Ratio ◽  
Participation Ratio ◽  
Avoided Crossing ◽  
Quantum Billiard ◽  
The Relationship

Abstract The relationship between wavefunction (de)localisation and avoided crossing in a quadrupole billiard is analysed. The following three-types of measures are employed for wavefunction (de)localisation: inverse participation ratio, inverse of Rényi entropy, and root-mean-square (RMS) image contrast. All these measures exhibit minimal values at the centre of the avoided crossing, where the wavefunction is maximally delocalised. Our results indicate that these quantities can be sufficient for the indication of wavefunction (de)localisation.

scholarly journals Study of Quantile-Based Cumulative Renyi Information Measure

2020 ◽  
Vol 9 (4) ◽  
pp. 886-909
Author(s):  
Rekha ◽  
Vikas Kumar
Keyword(s):  
Order Statistic ◽  
Random Variable ◽  
Information Measure ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Reliability Measure ◽  
Extreme Order Statistic ◽  
The Relationship

In this paper, we proposed a quantile version of cumulative Renyi entropy for residual and past lifetimes and study their properties. We also study quantile-based cumulative Renyi entropy for extreme order statistic when random variable untruncated or truncated in nature. Some characterization results are studied using the relationship between proposed information measure and reliability measure. We also examine it in relation to some applied problems such as weighted and equillibrium models.

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 Correlation functions and quantum measures of descendant states

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Enrico M. Brehm ◽  
Matteo Broccoli
Keyword(s):  
Differential Operator ◽  
Correlation Functions ◽  
Recursive Formula ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Two Dimensional ◽  
Computer Implementation ◽  
Point Function ◽  
Rényi Divergence ◽  
Holographic Description

Abstract We discuss a computer implementation of a recursive formula to calculate correlation functions of descendant states in two-dimensional CFT. This allows us to obtain any N-point function of vacuum descendants, or to express the correlator as a differential operator acting on the respective primary correlator in case of non-vacuum descendants. With this tool at hand, we then study some entanglement and distinguishability measures between descendant states, namely the Rényi entropy, trace square distance and sandwiched Rényi divergence. Our results provide a test of the conjectured Rényi QNEC and new tools to analyse the holographic description of descendant states at large c.

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.

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