http://www.mdpi.com/journal/entropy/special_issues/maximum-entropy/ http://www.mdpi.com/1099-4300/12/1/14/ This paper proposes a new method for estimating seismic wavelets. Suppose a seismic wavelet can be modeled by a formula with three free parameters (scale, frequency and phase). We can transform the estimation of the wavelet into determining these three parameters. The phase of the wavelet is estimated by constant-phase rotation to the seismic signal, while the other two parameters are obtained by the Higher-order Statistics (HOS) (fourth-order cumulant) matching method. In order to derive the estimator of the Higher-order Statistics (HOS), the multivariate scale mixture of Gaussians (MSMG) model is applied to formulating the multivariate joint probability density function (PDF) of the seismic signal. By this way, we can represent HOS as a polynomial function of second-order statistics to improve the anti-noise performance and accuracy. In addition, the proposed method can work well for short time series. http://www.mdpi.com/1099-4300/12/1/14/ Mon, 28 Dec 2009 00:00:00 CET Entropy 2009-12-28 12 1 Article 14 33 1099-4300 Estimation of Seismic Wavelets Based on the Multivariate Scale Mixture of Gaussians Model 2009-12-28 doi: 10.3390/e12010014 Jing-Huai Gao Bing Zhang http://www.mdpi.com/1099-4300/11/4/1123/ The existence of noise has great influence on the real features of observed time series, thus noise reduction in time series data is a necessary and significant task in many practical applications. When using traditional de-noising methods, the results often cannot meet the practical needs due to their inherent shortcomings. In the present paper, first a set of key but difficult wavelet de-noising problems are discussed, and then by applying information entropy theories to the wavelet de-noising process, i.e., using the principle of maximum entropy (POME) to describe the random character of the noise and using wavelet energy entropy to describe the degrees of complexity of the main series in original series data, a new entropy-based wavelet de-noising method is proposed. Analysis results of both several different synthetic series and typical observed time series data have verified the performance of the new method. A comprehensive discussion of the results indicates that compared with traditional wavelet de-noising methods, the new proposed method is more effective and universal. Furthermore, because it uses information entropy theories to describe the obviously different characteristics of noises and the main series in the series data is observed first and then de-noised, the analysis process has a more reliable physical basis, and the results of the new proposed method are more reasonable and are the global optimum. Besides, the analysis process of the new proposed method is simple and is easy to implement, so it would be more applicable and useful in applied sciences and practical engineering works. http://www.mdpi.com/1099-4300/11/4/1123/ Tue, 22 Dec 2009 00:00:00 CET Entropy 2009-12-22 11 4 Article 1123 1147 1099-4300 Entropy-Based Wavelet De-noising Method for Time Series Analysis 2009-12-22 doi: 10.3390/e11041123 Yan-Fang Sang Dong Wang Ji-Chun Wu Qing-Ping Zhu Ling Wang http://www.mdpi.com/1099-4300/11/4/1001/ The maximum entropy method is a theoretically sound approach to construct an analytical form for the probability density function (pdf) given a sample of random events. In practice, numerical methods employed to determine the appropriate Lagrange multipliers associated with a set of moments are generally unstable in the presence of noise due to limited sampling. A robust method is presented that always returns the best pdf, where tradeoff in smoothing a highly varying function due to noise can be controlled. An unconventional adaptive simulated annealing technique, called funnel diffusion, determines expansion coefficients for Chebyshev polynomials in the exponential function. http://www.mdpi.com/1099-4300/11/4/1001/ Fri, 04 Dec 2009 00:00:00 CET Entropy 2009-12-04 11 4 Article 1001 1024 1099-4300 Best Probability Density Function for Random Sampled Data 2009-12-04 doi: 10.3390/e11041001 Donald J. Jacobs http://www.mdpi.com/1099-4300/11/4/972/ We review here the difference between quantum statistical treatments and semiclassical ones, using as the main concomitant tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well. http://www.mdpi.com/1099-4300/11/4/972/ Thu, 03 Dec 2009 00:00:00 CET Entropy 2009-12-03 11 4 Review 972 992 1099-4300 Fisher Information and Semiclassical Treatments 2009-12-03 doi: 10.3390/e11040972 Flavia Pennini Gustavo Ferri Angelo Plastino http://www.mdpi.com/1099-4300/11/4/917/ The method of Generalized Maximum Entropy (GME), proposed in Golan, Judge and Miller (1996), is an information-theoretic approach that is robust to multicolinearity problem. It uses an objective function that is the sum of the entropies for coefficient distributions and disturbance distributions. This method can be generalized to the weighted GME (W-GME), where different weights are assigned to the two entropies in the objective function. We propose a data-driven method to select the weights in the entropy objective function. We use the least squares cross validation to derive the optimal weights. MonteCarlo simulations demonstrate that the proposedW-GME estimator is comparable to and often outperforms the conventional GME estimator, which places equal weights on the entropies of coefficient and disturbance distributions. http://www.mdpi.com/1099-4300/11/4/917/ Thu, 26 Nov 2009 00:00:00 CET Entropy 2009-11-26 11 4 Article 917 930 1099-4300 A Weighted Generalized Maximum Entropy Estimator with a Data-driven Weight 2009-11-26 doi: 10.3390/e11040917 Ximing Wu http://www.mdpi.com/1099-4300/11/4/867/ In this paper, we develop a general theory for the estimation of the transition probabilities of reversible Markov chains using the maximum entropy principle. A broad range of physical models can be studied within this approach. We use one-dimensional classical spin systems to illustrate the theoretical ideas. The examples studied in this paper are: the Ising model, the Potts model and the Blume-Emery-Griffiths model. http://www.mdpi.com/1099-4300/11/4/867/ Tue, 17 Nov 2009 00:00:00 CET Entropy 2009-11-17 11 4 Article 867 887 1099-4300 Maximum Entropy Estimation of Transition Probabilities of Reversible Markov Chains 2009-11-17 doi: 10.3390/e11040867 Erik Van der Straeten http://www.mdpi.com/1099-4300/11/4/854/ Maximum entropy (Maxent) modeling has great potential for identifying distributions and habitat selection of wildlife given its reliance on only presence locations. Recent studies indicate Maxent is relatively insensitive to spatial errors associated with location data, requires few locations to construct useful models, and performs better than other presence-only modeling approaches. Further advances are needed to better define model thresholds, to test model significance, and to address model selection. Additionally, development of modeling approaches is needed when using repeated sampling of known individuals to assess habitat selection. These advancements would strengthen the utility of Maxent for wildlife research and management. http://www.mdpi.com/1099-4300/11/4/854/ Mon, 16 Nov 2009 00:00:00 CET Entropy 2009-11-16 11 4 Review 854 866 1099-4300 Use of Maximum Entropy Modeling in Wildlife Research 2009-11-16 doi: 10.3390/e11040854 Roger A. Baldwin http://www.mdpi.com/1099-4300/11/4/766/ The high pay packages of U.S. CEOs have raised serious concerns about what would constitute a fair pay. Since the present economic models do not adequately address this fundamental question, we propose a new theory based on statistical mechanics and information theory. We use the principle of maximum entropy to show that the maximally fair pay distribution is lognormal under ideal conditions. This prediction is in agreement with observed data for the bottom 90%–95% of the working population. The theory estimates that the top 35 U.S. CEOs were overpaid by about 129 times their ideal salaries in 2008. We also provide an insight of entropy as a measure of fairness, which is maximized at equilibrium, in an economic system. http://www.mdpi.com/1099-4300/11/4/766/ Tue, 03 Nov 2009 00:00:00 CET Entropy 2009-11-03 11 4 Article 766 781 1099-4300 What is Fair Pay for Executives? An Information Theoretic Analysis of Wage Distributions 2009-11-03 doi: 10.3390/e11040766 Venkat Venkatasubramanian http://www.mdpi.com/1099-4300/11/4/713/ The efficiency of any application involving a liquid spray is known to be highly dependent on the spray characteristics, and mainly, on the drop-diameter distribution. There is therefore a crucial need of models allowing the prediction of this distribution. However, atomization processes are partially known and so far a universal model is not available. For almost thirty years, models based on the Maximum Entropy Formalism have been proposed to fulfill this task. This paper presents a review of these models emphasizing their similarities and differences, and discusses expectations of the use of this formalism to model spray drop-size distribution http://www.mdpi.com/1099-4300/11/4/713/ Mon, 02 Nov 2009 00:00:00 CET Entropy 2009-11-02 11 4 Review 713 747 1099-4300 The Maximum Entropy Formalism and the Prediction of Liquid Spray Drop-Size Distribution 2009-11-02 doi: 10.3390/e11040713 Christophe Dumouchel http://www.mdpi.com/1099-4300/11/4/675/ In this paper we present a simple model to describe a rather general system in a stationary non-equilibrium state, which is an open system traversed by a stationary flux. The probabilistic description is provided by a non-homogeneous Markov chain, which is not assumed on the basis of a model of the microscopic interactions but rather derived from the knowledge of the macroscopic fluxes traversing the system through a maximum entropy rate principle. http://www.mdpi.com/1099-4300/11/4/675/ Thu, 29 Oct 2009 00:00:00 CET Entropy 2009-10-29 11 4 Article 675 687 1099-4300 The Maximum Entropy Rate Description of a Thermodynamic System in a Stationary Non-Equilibrium State 2009-10-29 doi: 10.3390/e11040675 Marco Favretti