• Genomics & Informatics
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• v.20 no.2
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• pp.17.1-17.11
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• 2022

Genetic associations have been quantified using a number of statistical measures. Entropy-based mutual information may be one of the more direct ways of estimating the association, in the sense that it does not depend on the parametrization. For this purpose, both the entropy and conditional entropy of the phenotype distribution should be obtained. Quantitative traits, however, do not usually allow an exact evaluation of entropy. The estimation of entropy needs a probability density function, which can be approximated by kernel density estimation. We have investigated the proper sequence of procedures for combining the kernel density estimation and entropy estimation with a probability density function in order to calculate mutual information. Genotypes and their interactions were constructed to set the conditions for conditional entropy. Extensive simulation data created using three types of generating functions were analyzed using two different kernels as well as two types of multifactor dimensionality reduction and another probability density approximation method called m-spacing. The statistical power in terms of correct detection rates was compared. Using kernels was found to be most useful when the trait distributions were more complex than simple normal or gamma distributions. A full-scale genomic dataset was explored to identify associations using the 2-h oral glucose tolerance test results and γ-glutamyl transpeptidase levels as phenotypes. Clearly distinguishable single-nucleotide polymorphisms (SNPs) and interacting SNP pairs associated with these phenotypes were found and listed with empirical p-values.

• Journal of the Korean Data and Information Science Society
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• v.16 no.3
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• pp.705-712
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• 2005

Entropy is the basic concept of information theory. It is well defined for random varibles with known probability density function(pdf). For given data with unknown pdf, entropy should be estimated. Usually, estimation of entropy is based on the approximations. In this paper, we consider a kernel based approximation and compare it to the cumulant approximation method for several distributions. Monte carlo simulation for various sample size is conducted.

• Communications for Statistical Applications and Methods
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• v.18 no.4
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• pp.457-465
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• 2011

Sample entropy (Vasicek, 1976) has poor performance, and several nonparametric entropy estimators have been proposed as alternatives. In this paper, we consider a piecewise uniform density function based on quantiles, which enables us to evaluate entropy in each interval, and study the poor performance of the sample entropy in terms of the poor estimation of lower and upper quantiles. Then we propose some improved entropy estimators by simply modifying the quantile estimators, and compare their performances with some existing estimators.

• The Pure and Applied Mathematics
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• v.23 no.3
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• pp.205-221
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• 2016

We formulate the additive entropy of a natural number in terms of the additive partition function, and show that its multiplicative entropy is directly related to the multiplicative partition function. We give a practical formula for the multiplicative entropy of natural numbers with two prime factors. We use this formula to analyze the comparative density of additive and multiplicative entropy, prove that this density converges to zero as the number tends to infinity, and empirically observe this asymptotic behavior.

• Journal of applied mathematics & informatics
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• v.41 no.1
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• pp.123-132
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• 2023

We define an operator version of the relative Rényi entropy as the generalization of relative von Neumann entropy, and provide its fundamental properties and the bounds for its trace value. Moreover, we see an effect of the relative Rényi entropy under tensor product, and show the sub-additivity for density matrices.

• Journal of the Korean Statistical Society
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• v.13 no.2
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• pp.87-106
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• 1984

It ${X_t}$ is a sequence of independent identically distributed normal random variables, then the conditional probability density of $X_1, X_2, \cdots, X_n$ given the first p+1 sample autocovariances converges to the maximum entropy probability density satisfying the corresponding covariance constraints as the length of the sample sequence tends to infinity. This establishes that the maximum entropy probability density and the associated Gaussian autoregressive process arise naturally as the answers of conditional limit problems.

• Journal of Mechanical Science and Technology
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• v.19 no.12
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• pp.2253-2262
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• 2005

The focus of this work is placed on the analysis of the mixture formation mechanism under the evaporative diesel spray of impinging and free conditions. As an experimental parameter, ambient gas density was selected. Effects of density variation of ambient gas on liquid and vapor-phase inside structure of evaporation diesel spray were investigated. Ambient gas density was changed between ${\rho}a=5.0\;kg/m^3$ and $12.3\;kg/m^3$. In the case of impinging spray, the spray spreading to the radial direction is larger due to the decrease of drag force of ambient gas in the case of the low density than that of the high density. On the other hand, in the case of free spray, in accordance with the increase in the ambient gas density, the liquid-phase length is getting short due to the increase in drag force of ambient gas. In order to examine the homogeneity of mixture consisted of vapor-phase fuel and ambient gas in the spray, image analysis was conducted with statistical thermodynamics based on the non-dimensional entropy (S) method. In the case of application of entropy analysis to diesel spray, the entropy value always increases. The entropy of higher ambient density is higher than that of lower ambient gas density during initial injection period.

• International Journal of Contents
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• v.11 no.2
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• pp.57-62
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• 2015

Relative entropy is a divergence measure between two probability density functions of a random variable. Assuming that the random variable has only two alphabets, the relative entropy becomes a cross-entropy error function that can accelerate training convergence of multi-layer perceptron neural networks. Also, the n-th order extension of cross-entropy (nCE) error function exhibits an improved performance in viewpoints of learning convergence and generalization capability. In this paper, we derive a new divergence measure between two probability density functions from the nCE error function. And the new divergence measure is compared with the relative entropy through the use of three-dimensional plots.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.33 no.3
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• pp.201-206
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• 2009

The focus of this work is placed on the analysis of the mixture formation process under the evaporative diesel-free spray conditions. In order to examine homogeneity of mixture within the vapor phase region of the injected spray, image analysis was carried out based on the entropy of statistical thermodynamics. As an experimental parameter, the injection pressure and ambient gas density were selected, and effects of the injection pressure and density variation of ambient gas on the mixture formation process in the evaporative diesel spray were investigated. In the case of application of the thermodynamic entropy analysis to evaporative diesel spray, the value of the dimensionless entropy always increases with increase in time from injection start. Consequently, the dimensionless entropy in the case of the higher injection pressure is higher than that of lower injection pressure during initial injection period.

• Journal of the Korean Institute of Electrical and Electronic Material Engineers
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• v.14 no.1
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• pp.48-53
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• 2001

The crystal structure of ZnS fabricated by gas-liquid phase reaction was refined by the Rietveld program using X-ray diffraction data. The R-weighted pattern (R$\sub$wp/) of ZnS powder was 10.85%. The fraction of HCP phase was closely related with extra amount of H$_2$S gas. The lattice parameters and crystalline size were changed by the relative ratio of multi-phase. The luminescence property of ZnS:Cu, Al green phosphors prepared by conventional methods was good in the range of 91∼94% and 150∼190${\AA}$, respectively. According to the maximum entropy electron density(MEED) methods, any defects in (001) plane of cubic phase were not found. We suggest that both the Rietveld and maximum entropy density methods may be useful tools for studying luminescence mechanism of other phosphors materials.