scholarly journals Entropy Based Genetic Association Tests and Gene-Gene Interaction Tests

2011 ◽  
Vol 10 (1) ◽  
Author(s):  
Mariza de Andrade ◽  
Xin Wang
Keyword(s):  
Genetic Association ◽  
Shannon Entropy ◽  
Statistical Power ◽  
Genetic Model ◽  
Gene Interaction ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Entropy Measure ◽  
P Value ◽  
The Impact

In the past few years, several entropy-based tests have been proposed for testing either single SNP association or gene-gene interaction. These tests are mainly based on Shannon entropy and have higher statistical power when compared to standard χ2 tests. In this paper, we extend some of these tests using a more generalized entropy definition, Rényi entropy, where Shannon entropy is a special case of order 1. The order λ (>0) of Rényi entropy weights the events (genotype/haplotype) according to their probabilities (frequencies). Higher λ places more emphasis on higher probability events while smaller λ (close to 0) tends to assign weights more equally. Thus, by properly choosing the λ, one can potentially increase the power of the tests or the p-value level of significance. We conducted simulation as well as real data analyses to assess the impact of the order λ and the performance of these generalized tests. The results showed that for dominant model the order 2 test was more powerful and for multiplicative model the order 1 or 2 had similar power. The analyses indicate that the choice of λ depends on the underlying genetic model and Shannon entropy is not necessarily the most powerful entropy measure for constructing genetic association or interaction tests.

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 Gene-Based Nonparametric Testing of Interactions Using Distance Correlation Coefficient in Case-Control Association Studies

Genes ◽  
2018 ◽  
Vol 9 (12) ◽  
pp. 608
Author(s):  
Yingjie Guo ◽  
Chenxi Wu ◽  
Maozu Guo ◽  
Xiaoyan Liu ◽  
Alon Keinan
Keyword(s):  
Correlation Coefficient ◽  
Statistical Power ◽  
Association Studies ◽  
Gene Interaction ◽  
P Value ◽  
Genome Wide Association Studies ◽  
Nucleotide Polymorphisms ◽  
Real World Data ◽  
Distance Correlation ◽  
The Difference

Among the various statistical methods for identifying gene–gene interactions in qualitative genome-wide association studies (GWAS), gene-based methods have recently grown in popularity because they confer advantages in both statistical power and biological interpretability. However, most of these methods make strong assumptions about the form of the relationship between traits and single-nucleotide polymorphisms, which result in limited statistical power. In this paper, we propose a gene-based method based on the distance correlation coefficient called gene-based gene-gene interaction via distance correlation coefficient (GBDcor). The distance correlation (dCor) is a measurement of the dependency between two random vectors with arbitrary, and not necessarily equal, dimensions. We used the difference in dCor in case and control datasets as an indicator of gene–gene interaction, which was based on the assumption that the joint distribution of two genes in case subjects and in control subjects should not be significantly different if the two genes do not interact. We designed a permutation-based statistical test to evaluate the difference between dCor in cases and controls for a pair of genes, and we provided the p-value for the statistic to represent the significance of the interaction between the two genes. In experiments with both simulated and real-world data, our method outperformed previous approaches in detecting interactions accurately.

scholarly journals P427 A hybrid approach of handling missing data in inflammatory bowel disease (IBD) trials: results from VISIBLE 1 and VARSITY

2020 ◽  
Vol 14 (Supplement_1) ◽  
pp. S388-S389
Author(s):  
J Chen ◽  
S Hunter ◽  
K Kisfalvi ◽  
R A Lirio
Keyword(s):  
Sensitivity Analysis ◽  
Missing Data ◽  
Statistical Power ◽  
Hybrid Approach ◽  
Missing At Random ◽  
P Value ◽  
Two Phase ◽  
Treatment Difference ◽  
Mayo Score ◽  
The Impact

Abstract Background Missing data is common in IBD trials. Depending on the volume and nature of missing data, it can reduce statistical power for detecting treatment difference, introduce potential bias and invalidate conclusions. Non-responder imputation (NRI), where patients (patients) with missing data are considered treatment failures, is widely used to handle missing data for dichotomous efficacy endpoints in IBD trials. However, it does not consider the mechanisms leading to missing data and can potentially underestimate the treatment effect. We proposed a hybrid (HI) approach combining NRI and multiple imputation (MI) as an alternative to NRI in the analyses of two phase 3 trials of vedolizumab (VDZ) in patients with moderate-to-severe UC – VISIBLE 11 and VARSITY2. Methods VISIBLE 1 and VARSITY assessed efficacy using dichotomous endpoints based on complete Mayo score. Full methodologies reported previously.1,2 Our proposed HI approach is aimed at imputing missing Mayo scores, instead of imputing the missing dichotomous efficacy endpoint. To assess the impact of dropouts for different missing data mechanisms (categorised as ‘missing not at random [MNAR]’ and ‘missing at random [MAR]’, HI was implemented as a potential sensitivity analysis, where dropouts owing to safety or lack of efficacy were imputed using NRI (assuming MNAR) and other missing data were imputed using MI (assuming MAR). For MI, each component of the Mayo score was imputed via a multivariate stepwise approach using a fully conditional specification ordinal logistic method. Missing baseline scores were imputed using baseline characteristics data. Missing scores from each subsequent visit were imputed using all previous visits in a stepwise fashion. Fifty imputation datasets were computed for each component of Mayo score. The complete Mayo score and relevant efficacy endpoints were derived subsequently. The analysis was performed within each imputed dataset to determine treatment difference, 95% CI and p-value, which were then combined via Rubin’s rules3. Results Tables 1 and 2 show a comparison of efficacy in the two studies using the primary NRI analysis vs. the alternative HI approach for handling missing data. Conclusion HI and NRI approaches can provide consistent efficacy analyses in IBD trials. The HI approach can serve as a useful sensitivity analysis to assess the impact of dropouts under different missing data mechanisms and evaluate the robustness of efficacy conclusions. Reference

Rényi entropy measure of noise-aided information transmission in a binary channel

2010 ◽  
Vol 81 (5) ◽  
Author(s):  
François Chapeau-Blondeau ◽  
David Rousseau ◽  
Agnès Delahaies
Keyword(s):  
Information Transmission ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Entropy Measure ◽  
Binary Channel

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 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 Low Complexity Estimation Method of Rényi Entropy for Ergodic Sources

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 657 ◽  
Author(s):  
Young-Sik Kim
Keyword(s):  
Estimation Method ◽  
Low Complexity ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Entropy Measure ◽  
General Representation ◽  
Iterative Estimation ◽  
Random Samples ◽  
Complexity Estimation ◽  
Special Case

Since the entropy is a popular randomness measure, there are many studies for the estimation of entropies for given random samples. In this paper, we propose an estimation method of the Rényi entropy of order α . Since the Rényi entropy of order α is a generalized entropy measure including the Shannon entropy as a special case, the proposed estimation method for Rényi entropy can detect any significant deviation of an ergodic stationary random source’s output. It is shown that the expected test value of the proposed scheme is equivalent to the Rényi entropy of order α . After deriving a general representation of parameters of the proposed estimator, we discuss on the particular orders of Rényi entropy such as α → 1 , α = 1 / 2 , and α = 2 . Because the Rényi entropy of order 2 is the most popular one, we present an iterative estimation method for the application with stringent resource restrictions.

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.

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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