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Lambert function and a new non-extensive form of entropy

IMA JOURNAL OF APPLIED MATHEMATICS 2007年第6期72卷 785-800页

作者： Shafee, Fariel Princeton Univ Dept Phys Princeton NJ USA

We propose a new way of defining entropy of a system, which gives a general form that is non-extensive like Tsallis entropy, but is linearly dependent on component entropies, like Renyi entropy, which is extensive. This entropy has a conceptually novel but simple origin and is mathematically easy to define by a very simple expression involving a derivative. It leads to a probability distribution function involving the Lambert function resulting from optimizing the entropy, which has hitherto never appeared in this context, and is somewhat more complex than the Shannon or Boltzmann form, but is nevertheless mathematically quite tractable. We have compared it numerically with the Tsallis and Shannon entropies. We have also considered constraints on the energy spectra imposed by the properties of the Lambert function, which are absent in the Shannon form. It may turn out to be a more appropriate candidate in a physical situation where the probability distribution does not suit any of the previously defined forms, especially when the probability density function sought is expected to be stiffer than that resulting from maximizing the other entropies. We consider the problem of defining free energy and other thermodynamic functions when the entropy is given as a general function of the probability distribution, including that for non-extensive forms. We then find that the free energy, which is central to the determination of all other quantities of interest in a thermodynamic context, can be obtained uniquely, at least numerically, even when it is the root of a transcendental equation. In particular, we examine the cases of the Tsallis form and the new form proposed by us. We compare the free energy, the internal energy and the specific heat of a simple system of two energy states for each of these forms and find significant departures for some quantities, while some others are less sensitive to the parametrization.

关键词： Lambert function non-extensive entropy probability distribution function free energy

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Breakpoint detection in non-stationary runoff time series under uncertainty

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JOURNAL OF HYDROLOGY 2020年 590卷

作者： Farsi, Niloofar Mahjouri, Najmeh Ghasemi, Hamid KN Toosi Univ Technol Fac Civil Engn Tehran Iran

Most conventional hydrological time series models, used for forecasting or synthetic data generation, are based on the stationary hypothesis. Hydrologic variables such as surface runoff discharge exhibit non-stationary behavior due to the effects of climate change or human activities. In this paper, a new methodology is proposed to analyze the uncertainty of the number and location of breakpoints in runoff time series. The non-stationary runoff time series is simulated by segmenting the series into stationary pieces, which follow an Auto-Regressive Moving Average (ARMA) process, based on the minimum description length (MDL) principle. The length of each piece and the order of the piecewise ARMA model are optimized using the Genetic Algorithm (GA). Several runoff time series with variable lengths are generated based on the observed data and the probability distribution function(s) (PDFs) of breakpoint location(s) are constructed. An optimal set of breakpoint locations are found for each generated runoff time series by minimizing the objective function based on the MDL. Finally, the PDF(s) of breakpoint location(s) is (are) constructed using the results of the mentioned optimization model. To evaluate the proposed methodology, it is applied to the case study of Zayandehrud River in Iran. According to the obtained breakpoint PDF for the runoff time series, three breakpoints around years of 1987, 1996 and 2006 are identified. The results are analyzed by investigating the time series of agricultural lands derived using remote sensing data and studying the impacts of interbasin water transfer projects implemented in the study area. Overall, investigating human-induced changes in the study area confirms the obtained shape of the PDF of runoff breakpoint and shows the good performance of the developed method for finding the number and location of breakpoints in the runoff time series.

关键词： Breakpoint Uncertainty Non-stationarity Runoff time series probability distribution function

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STOCHASTIC MODELING AND ANALYSIS OF PROPAGATION DELAYS IN GAAS ADDERS

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IEEE TRANSACTIONS ON COMPUTERS 1991年第1期40卷 31-45页

作者： PERUNICIC, B LAKHANI, S MILUTINOVIC, V PURDUE UNIV SCH ELECT ENGN W LAFAYETTE IN 47907 USA UNIV BELGRADE SCH ELECT ENGN YU-11000 BELGRADE YUGOSLAVIA

There are a number of important differences between GaAs and Silicon which are relevant for computer architecture/design. One difference, which may have an important impact on computer architecture design, is that GaAs gates, when exposed to high radiation levels, may change their propagation delay. In other words, gates continue to function, but with different gate delays. Typically, at the design time, certain clock or other margins are included, and the problem is to determine the probability with which a given margin will work at the post-radiation time. In this paper, we introduce an original methodology which solves the problem for combinational circuits of arbitrary topology (without loops). We demonstrate its capabilities through the modeling and analysis of propagation delays in GaAs adders. This methodology is directly applicable to a number of analogous problems in other fields.

关键词： ADDERS CARRY-LOOK-AHEAD ADDER CARRY-SELECT ADDER FULL ADDER probability distribution function PROPAGATION DELAY RAYLEIGH RIPPLE-CARRY ADDER STOCHASTIC MODELING

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Effect of percent non-detects on estimation bias in censored distributions

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JOURNAL OF HYDROLOGY 2004年第1-4期297卷 74-94页

作者： Zhang, Z Lennox, WC Panu, US Lakehead Univ Dept Civil Engn Thunder Bay ON P7B 5E1 Canada Univ Waterloo Dept Civil Engn Waterloo ON N2L 3G1 Canada Stantec Consulting Ltd Kitchener ON N2H 6M7 Canada

Uniqueness of the problem surrounding non-detects has been a concern alike to researchers and statisticians dealing with summary statistics while analyzing censored data. To incorporate non-detects in the estimation process, a simple substitution by the MDL (method detection limit) and the maximum likelihood estimation method are routinely implemented as standard methods by US-EPA laboratories. In situations where numerical standards are set at or near the MDL by regulatory agencies, it is prudent and important to closely investigate both the variability in test measurements and the estimation bias, because an inference based on biased estimates could entail significant liabilities. Variability is understood to be not only inevitable but also an inherent and integral part of any chemical analysis or test. In situations where regulatory agencies fail to account for the inherently present variability of test measurements, there is a need for regulated facilities to seek remedial action merely as a consequence of inadequate statistical procedure. This paper utilizes a mathematical approach to derive the bias functions and resulting bias curves are developed to investigate the censored samples from a variety of probability distributions such as normal, log-normal, gamma, and Gumbel distributions. Finally, the bias functions and bias curves are also compared to the results obtained by using Monte Carlo simulations. (C) 2004 Elsevier B.V. All rights reserved.

关键词： bias non-detects censoring dirac function heaviside function probability distribution function

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Fragmented edge structure coding for Chinese writer identification

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NEUROCOMPUTING 2012年 86卷 45-51页

作者： Wen, Jing Fang, Bin Chen, JunLin Tang, YuanYan Chen, HengXin Chongqing Univ Coll Comp Chongqing 400044 Peoples R China

As a branch of biometrics, handwriting-based writer identification is an active research topic in pattern recognition. This paper presents a theoretically very simple, yet efficient, multiresolution approach to off-line, text-independent Chinese writer identification based on edge structure code (ESC) distribution feature and nonparametric discrimination of sample. ESC distribution feature is based on probability distribution function, which characterizes the frequent structures distribution of edge fragments on multiple scales. Experiments were conducted on an open HIT-MW database which is widely used for performance evaluation. Experimental results show that the proposed method was able to improve the identification accuracy. Crown Copyright (C) 2012 Published by Elsevier B.V. All rights reserved.

关键词： Text-independent writer identification Edge fragments Encoding feature probability distribution function Chi-square distance Pattern recognition

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Statistics of phase turbulence II

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PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 2003年第3-4期317卷 391-400页

作者： Shibata, H Sojo Univ Fac Engn Dept Gen Educ Kumamoto 8600082 Japan

The system described by the Kuramoto-Sivashinsky equation (SKSE) is shown to have a short- and a long-time regimes. The probability distribution function (PDF) for the absolute value of the solution is subjected to different scaling laws in the two time regimes. The local expansion rate (LER) and its two-time correlation function are calculated. The time regime where the correlation for the LER remains is shorter than the one for the absolute value of the solution for the SKSE. The fluctuation in the LER becomes small as the damping constant is increased. The PDF for the LER is also given in this paper. (C) 2003 Elsevier Science B.V. All rights reserved.

关键词： Kuramoto-Sivashinsky equation local expansion rate phase turbulence two-time correlation function probability distribution function

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Time delay systems with distribution dependent dynamics

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ANNUAL REVIEWS IN CONTROL 2007年第1期31卷 17-26页

作者： Banks, H. T. Dediu, Sava Nguyen, Hoan K. N Carolina State Univ Ctr Res Sci Compuatat Raleigh NC 27695 USA State & Appl Math Sci Inst Res Triangle Pk NC 27709 USA Ctr Naval Anal Alexandria VA 22311 USA

General delay dynamical systems in which uncertainty is present in the form of probability measure dependent dynamics are considered. Several motivating examples arising in biology are discussed. A functional analytic framework for investigating well-posedness (existence, uniqueness and continuous dependence of solutions), inverse problems, sensitivity analysis and approximations of the measures for computational purposes is surveyed. (c) 2007 Elsevier Ltd. All rights reserved.

关键词： inverse dynamics problem probability distribution function sensitivity analysis biomedical systems

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Hydrologic controls on basin- scale distribution of benthic invertebrates

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WATER RESOURCES RESEARCH 2014年第4期50卷 2903-2920页

作者： Ceola, Serena Bertuzzo, Enrico Singer, Gabriel Battin, Tom J. Montanari, Alberto Rinaldo, Andrea Univ Bologna Dept DICAM Bologna Italy Ecole Polytech Fed Lausanne Sch Architecture Civil & Environm Engn Lab Ecohydrol Lausanne Switzerland Univ Vienna Dept Limnol & Oceanog Vienna Austria Leibniz Inst Freshwater Ecol & Inland Fisheries Dept Ecohydrol Berlin Germany WasserCluster Lunz Interuniv Ctr Aquat Ecosyst Res Lunz am See Austria Univ Padua Dipartimento Ingn Civile Edile & Ambientale Padua Italy

Streamflow variability is a major determinant of basin-scale distributions of benthic invertebrates. Here we present a novel procedure based on a probabilistic approach aiming at a spatially explicit quantitative assessment of benthic invertebrate abundance as derived from near-bed flow variability. Although the proposed approach neglects ecological determinants other than hydraulic ones, it is nevertheless relevant in view of its implications on the predictability of basin-scale patterns of organisms. In the present context, aquatic invertebrates are considered, given that they are widely employed as sensitive indicators of fluvial ecosystem health and human-induced perturbations. Moving from the analytical characterization of site-specific probability distribution functions of streamflow and bottom shear stress, we achieve a spatial extension to an entire stream network. Bottom shear stress distributions, coupled with habitat suitability curves derived from field studies, are used to produce maps of invertebrate suitability to shear stress conditions. Therefore, the proposed framework allows one to inspect the possible impacts on river ecology of human-induced perturbations of streamflow variability. We apply this framework to an Austrian river network for which rainfall and streamflow time series, river network hydraulic properties, and local information on invertebrate abundance for a limited number of sites are available. A comparison between observed species density versus modeled suitability to shear stress is also presented. Although the proposed strategy focuses on a single controlling factor and thus represents an ecological minimal model, it allows derivation of important implications for water resource management and fluvial ecosystem protection.

关键词： hydrology bottom shear stress streamflow benthic invertebrate probability distribution function

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Novel statistical-thermodynamic methods to predict protein-ligand binding positions using probability distribution functions

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PROTEINS-STRUCTURE function AND BIOINFORMATICS 2006年第1期62卷 202-208页

作者： Ruvinsky, AM Kozintsev, AV Algodign LLC Force Field Lab Moscow 103379 Russia

We present two novel methods to predict native protein-ligand binding positions. Both methods identify the native binding position as the most probable position corresponding to a maximum of a probability distribution function (PDF) of possible binding positions in a protein active site. Possible binding positions are the origins of clusters composed, on the basis of root-mean square deviations (RMSD), from the multiple ligand positions determined by a docking algorithm. The difference between the methods lies in the ways the PDF is derived. To validate the suggested methods, we compare the averaged RMSD of the predicted ligand docked positions relative to the experimentally determined positions for a set of 135 PDB protein-ligand complexes. We demonstrate that the suggested methods improve docking accuracy by as much as 21-24% in comparison with a method that simply identifies the binding position as the energy top-scored ligand position.

关键词： protein-ligand binding docking probability distribution function cluster

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Combining Mexican hat wavelet and spread spectrum for adaptive watermarking and its statistical detection using medical images

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MULTIMEDIA TOOLS AND APPLICATIONS 2019年第10期78卷 12647-12661页

作者： Chauhan, D. S. Singh, A. K. Adarsh, A. Kumar, B. Saini, J. P. Feroze Gandhi Inst Engn & Technol Dept Elect & Commun Engn Raebareli UP India Jaypee Univ Informat Technol Dept Comp Sci & Engn Solan HP India Motilal Nehru Natl Inst Technol Dept Elect & Commun Engn Allahabad UP India Bundelkhand Inst Engn & Technol Dept Elect Engn Jhansi UP India

This paper present a secure medical image watermarking technique applying spread-spectrum concept in wavelet transform domain is proposed. In the first step, discrete wavelet transform(DWT) decomposes the cover medical image into four frequency sub-bands using Mexican hat as mother wavelet and then corresponding to each pixel of the binary watermark a pair of Pseudo-Noise (PN) is embedded into a horizontal (HL) and a vertical (LH) sub-band. In order to maintain the imperceptibility of the watermarked image, strength of the generated PN sequence pair is adjusted according to specified document to watermark ratio (DWR). For the extraction the watermark, statistical profile of DWT coefficients of watermarked image is determined and the obtained probability distribution function (pdf) is utilized for designing the watermark detection procedure. Proposed detector considers the best fitted Cauchy statistical model of heavy-tailed family, which accurately models the non-Gaussian DWT coefficients of an image. The robustness of the method is examined for various kinds of attacks with varying watermark to document ratio. Further, experimental results show that the proposed technique offer more robustness than other state-of-the-art method.

关键词： DWT Spread-spectrum probability distribution function Cauchy statistical model Medical image LRT WDR PSNR NC

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