Entropy

244 KiB

Open AccessArticle

Some Further Results on the Minimum Error Entropy Estimation

by Badong Chen and Jose C. Principe

Entropy 2012, 14(5), 966-977; https://doi.org/10.3390/e14050966 - 21 May 2012

Cited by 20 | Viewed by 5610

The minimum error entropy (MEE) criterion has been receiving increasing attention due to its promising perspectives for applications in signal processing and machine learning. In the context of Bayesian estimation, the MEE criterion is concerned with the estimation of a certain random variable [...] Read more.

The minimum error entropy (MEE) criterion has been receiving increasing attention due to its promising perspectives for applications in signal processing and machine learning. In the context of Bayesian estimation, the MEE criterion is concerned with the estimation of a certain random variable based on another random variable, so that the error’s entropy is minimized. Several theoretical results on this topic have been reported. In this work, we present some further results on the MEE estimation. The contributions are twofold: (1) we extend a recent result on the minimum entropy of a mixture of unimodal and symmetric distributions to a more general case, and prove that if the conditional distributions are generalized uniformly dominated (GUD), the dominant alignment will be the MEE estimator; (2) we show by examples that the MEE estimator (not limited to singular cases) may be non-unique even if the error distribution is restricted to zero-mean (unbiased). Full article

► Show Figures

Figure 1

214 KiB

Open AccessArticle

On Chirality of the Vorticity of the Universe

by Davor Palle

Entropy 2012, 14(5), 958-965; https://doi.org/10.3390/e14050958 - 16 May 2012

Cited by 14 | Viewed by 5655

Abstract

The presence of dark energy in the Universe challenges the Einstein’s theory of gravity at cosmic scales. It motivates the inclusion of rotational degrees of freedom in the Einstein–Cartan gravity, representing the minimal and the most natural extension of the General Relativity. One [...] Read more.

The presence of dark energy in the Universe challenges the Einstein’s theory of gravity at cosmic scales. It motivates the inclusion of rotational degrees of freedom in the Einstein–Cartan gravity, representing the minimal and the most natural extension of the General Relativity. One can, consequently, expect the violation of the cosmic isotropy by the rotating Universe. We study chirality of the vorticity of the Universe within the Einstein–Cartan cosmology. The role of the spin of fermion species during the evolution of the Universe is studied by averaged spin densities and Einstein–Cartan equations. It is shown that spin density of the light Majorana neutrinos acts as a seed for vorticity at early stages of the evolution of the Universe. Its chirality can be evaluated in the vicinity of the spacelike infinity. It turns out that vorticity of the Universe has right-handed chirality. Full article

(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)

283 KiB

Open AccessArticle

Applying Multiscale Entropy to the Complexity Analysis of Rainfall-Runoff Relationships

by Chien-Ming Chou

Entropy 2012, 14(5), 945-957; https://doi.org/10.3390/e14050945 - 16 May 2012

Cited by 35 | Viewed by 6427

Abstract

This paper presents a novel framework for the complexity analysis of rainfall, runoff, and runoff coefficient (RC) time series using multiscale entropy (MSE). The MSE analysis of RC time series was used to investigate changes in the complexity of rainfall-runoff processes due to [...] Read more.

This paper presents a novel framework for the complexity analysis of rainfall, runoff, and runoff coefficient (RC) time series using multiscale entropy (MSE). The MSE analysis of RC time series was used to investigate changes in the complexity of rainfall-runoff processes due to human activities. Firstly, a coarse graining process was applied to a time series. The sample entropy was then computed for each coarse-grained time series, and plotted as a function of the scale factor. The proposed method was tested in a case study of daily rainfall and runoff data for the upstream Wu–Tu watershed. Results show that the entropy measures of rainfall time series are higher than those of runoff time series at all scale factors. The entropy measures of the RC time series are between the entropy measures of the rainfall and runoff time series at various scale factors. Results also show that the entropy values of rainfall, runoff, and RC time series increase as scale factors increase. The changes in the complexity of RC time series indicate the changes of rainfall-runoff relations due to human activities and provide a reference for the selection of rainfall-runoff models that are capable of dealing with great complexity and take into account of obvious self-similarity can be suggested to the modeling of rainfall-runoff processes. Moreover, the robustness of the MSE results were tested to confirm that MSE analysis is consistent and the same results when removing 25% data, making this approach suitable for the complexity analysis of rainfall, runoff, and RC time series. Full article

► Show Figures

Graphical abstract

1638 KiB

Open AccessArticle

Entropy Concept for Paramacrosystems with Complex States

by Yuri S. Popkov

Entropy 2012, 14(5), 924-944; https://doi.org/10.3390/e14050924 - 10 May 2012

Cited by 1 | Viewed by 5103

Abstract

Consideration is given to macrosystems called paramacrosystems with states of finite capacity and distinguishable and undistinguishable elements with stochastic behavior. The paramacrosystems fill a gap between Fermi and Einstein macrosystems. Using the method of the generating functions, we have obtained expressions for probabilistic [...] Read more.

Consideration is given to macrosystems called paramacrosystems with states of finite capacity and distinguishable and undistinguishable elements with stochastic behavior. The paramacrosystems fill a gap between Fermi and Einstein macrosystems. Using the method of the generating functions, we have obtained expressions for probabilistic characteristics (distribution of the macrostate probabilities, physical and information entropies) of the paramacrosystems. The cases with equal and unequal prior probabilities for elements to occupy the states with finite capacities are considered. The unequal prior probabilities influence the morphological properties of the entropy functions and the functions of the macrostate probabilities, transforming them in the multimodal functions. The examples of the paramacrosystems with two-modal functions of the entropy and distribution of the macrostate probabilities are presented. The variation principle does not work for such cases. Full article

(This article belongs to the Special Issue Concepts of Entropy and Their Applications)

► Show Figures

Figure 1

260 KiB

Open AccessArticle

An Entropic Estimator for Linear Inverse Problems

by Amos Golan and Henryk Gzyl

Entropy 2012, 14(5), 892-923; https://doi.org/10.3390/e14050892 - 10 May 2012

Cited by 5 | Viewed by 7214

Abstract

In this paper we examine an Information-Theoretic method for solving noisy linear inverse estimation problems which encompasses under a single framework a whole class of estimation methods. Under this framework, the prior information about the unknown parameters (when such information exists), and constraints [...] Read more.

In this paper we examine an Information-Theoretic method for solving noisy linear inverse estimation problems which encompasses under a single framework a whole class of estimation methods. Under this framework, the prior information about the unknown parameters (when such information exists), and constraints on the parameters can be incorporated in the statement of the problem. The method builds on the basics of the maximum entropy principle and consists of transforming the original problem into an estimation of a probability density on an appropriate space naturally associated with the statement of the problem. This estimation method is generic in the sense that it provides a framework for analyzing non-normal models, it is easy to implement and is suitable for all types of inverse problems such as small and or ill-conditioned, noisy data. First order approximation, large sample properties and convergence in distribution are developed as well. Analytical examples, statistics for model comparisons and evaluations, that are inherent to this method, are discussed and complemented with explicit examples. Full article

(This article belongs to the Special Issue Concepts of Entropy and Their Applications)

207 KiB

Open AccessArticle

Second Law Constraints on the Dynamics of a Mixture of Two Fluids at Different Temperatures

by A. D. Kirwan, Jr.

Entropy 2012, 14(5), 880-891; https://doi.org/10.3390/e14050880 - 9 May 2012

Cited by 5 | Viewed by 4891

Abstract

Constitutive laws for multi-component fluids (MCF) is one of the thorniest problems in science. Two questions explored here are: how to ensure that these relations reduce to accepted forms when all but one of the constituents vanishes; and what constraints does the Second [...] Read more.

Constitutive laws for multi-component fluids (MCF) is one of the thorniest problems in science. Two questions explored here are: how to ensure that these relations reduce to accepted forms when all but one of the constituents vanishes; and what constraints does the Second Law impose on the dynamics of viscous fluids at different temperatures? The analysis suggests an alternative to the metaphysical principles for MCF proposed by Truesdell [1]. Full article

(This article belongs to the Special Issue Concepts of Entropy and Their Applications)

482 KiB

Open AccessArticle

Entropic Approach to Multiscale Clustering Analysis

by Manlio De Domenico and Antonio Insolia

Entropy 2012, 14(5), 865-879; https://doi.org/10.3390/e14050865 - 9 May 2012

Cited by 4 | Viewed by 6170

Abstract

Recently, a novel method has been introduced to estimate the statistical significance of clustering in the direction distribution of objects. The method involves a multiscale procedure, based on the Kullback–Leibler divergence and the Gumbel statistics of extreme values, providing high discrimination power, even [...] Read more.

Recently, a novel method has been introduced to estimate the statistical significance of clustering in the direction distribution of objects. The method involves a multiscale procedure, based on the Kullback–Leibler divergence and the Gumbel statistics of extreme values, providing high discrimination power, even in presence of strong background isotropic contamination. It is shown that the method is: (i) semi-analytical, drastically reducing computation time; (ii) very sensitive to small, medium and large scale clustering; (iii) not biased against the null hypothesis. Applications to the physics of ultra-high energy cosmic rays, as a cosmological probe, are presented and discussed. Full article

(This article belongs to the Special Issue Concepts of Entropy and Their Applications)

► Show Figures

Figure 1

111 KiB

Open AccessArticle

A Comment on the Relation between Diffraction and Entropy

by Michael Baake and Uwe Grimm

Entropy 2012, 14(5), 856-864; https://doi.org/10.3390/e14050856 - 7 May 2012

Cited by 5 | Viewed by 5887

Abstract

Diffraction methods are used to detect atomic order in solids. While uniquely ergodic systems with pure point diffraction have zero entropy, the relation between diffraction and entropy is not as straightforward in general. In particular, there exist families of homometric systems, which are [...] Read more.

Diffraction methods are used to detect atomic order in solids. While uniquely ergodic systems with pure point diffraction have zero entropy, the relation between diffraction and entropy is not as straightforward in general. In particular, there exist families of homometric systems, which are systems sharing the same diffraction, with varying entropy. We summarise the present state of understanding by several characteristic examples. Full article

(This article belongs to the Special Issue Concepts of Entropy and Their Applications)

108 KiB

Open AccessArticle

A Phase Space Diagram for Gravity

by Xavier Hernandez

Entropy 2012, 14(5), 848-855; https://doi.org/10.3390/e14050848 - 4 May 2012

Cited by 7 | Viewed by 7826

Abstract

In modified theories of gravity including a critical acceleration scale a₀, a critical length scale r_M = (GM/a₀)^1/2 will naturally arise with the transition from the Newtonian to the dark matter mimicking regime occurring for systems larger [...] Read more.

In modified theories of gravity including a critical acceleration scale a₀, a critical length scale r_M = (GM/a₀)^1/2 will naturally arise with the transition from the Newtonian to the dark matter mimicking regime occurring for systems larger than r_M. This adds a second critical scale to gravity, in addition to the one introduced by the criterion v < c of the Schwarzschild radius, r_S= 2GM/c². The distinct dependencies of the two above length scales give rise to non-trivial phenomenology in the (mass, length) plane for astrophysical structures, which we explore here. Surprisingly, extrapolation to atomic scales suggests gravity should be at the dark matter mimicking regime there. Full article

(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Entropy, Volume 14, Issue 5 (May 2012) – 9 articles , Pages 848-977

Further Information

Guidelines

MDPI Initiatives

Follow MDPI