The principle of the maximum entropy method

Abstract

The theoretical background of the maximum entropy method (MEM) when it is applied to restore the electron or nuclear densities from diffraction data is described. In MEM, the concept of “entropy” is introduced to deal with any incompleteness in an observation in a proper way. An incompleteness causes some ambiguities in the results to some extent. The essence of the method is to find a solution which necessarily agrees with the observation, leaving the measure of ambiguities (entropy) maximum. A few results for simple structures with typical types of chemical bonding are also presented.     

Time-dependent radiative transfer using Chapman-Enskog maximum entropy method

The time-dependent problems of radiative transfer involve a coupling between radiation and material energy fields and are nonlinear because of proposed temperature dependence of the medium characteristics in semi-infinite medium with Rayleigh anisotropic scattering. By means of the limited flux, Chapman-Enskog and maximum entropy technique the time-dependent radiative transfer equation has been solved explicitly. The maximum entropy method is used to solve the resulting differential equation for radiative energy density. The calculations are carried out for temperature (normalized dimensionless) Θ(x,τ), radiative energy density and net flux with Rayleigh and anisotropic scattering for different space at different times.     

Recovery of lifetime distributions from frequency-domain fluorometry data by means of the quantified maximum entropy method

The new quantified version of the maximum entropy method allows one to recover lifetime distributions with a precise statement of the accuracy of position, surface, and broadness of peaks in the distribution. Applications to real data (2,6-ANS in aqueous solutions of sodium dodecyl sulfate micelles of -cyclodextrin) are presented.Died on September 1, 1993.     

Susceptibilities of QCD Vacuum from Renormalized Dyson-Schwinger Equations

The pion and tensor vacuum susceptibilities are calculated in the framework of the renormalizable Dyson-Schwinger equations. A comparison with the results of other nonperturbative QCD approaches is given.     

最大熵法在风廓线雷达谱分析中的应用研究

为了解现代谱分析技术在风廓线雷达中应用的可行性,通过采用实测的和模拟的风廓线雷达回波信号,对比研究了FFT法与最大熵法的谱分析效果。结果表明:(1)当回波信号比较强时,两者都可以得到较好的谱分析效果;但是当回波信号较弱时,最大熵法分析效果优于FFT法,最大熵法对地杂波具有较好的抑制能力。(2)最大熵谱比较光滑,表明最大熵法对随机白噪声也有一定的抑制作用。(3)最大熵法的递推阶数对谱分析结果有一定影响,最终预测误差准则确定的递推阶数一般偏小,采用15阶的递推阶数进行最大熵法分析取得了较好结果。由于风廓线雷达回波通常都较弱,因此研究结果可望用于改善信号处理效果。     

Intermittency from maximum entropy distribution

A maximum entropy distribution has been formulated in which the imposed constraint contains a stochastic (rather than a deterministic) variable. The distribution depends on the observational bin size through the smoothing of population by intrabin averaging. Moments of fluctuations calculated with this distribution give bin-size dependences (intermittency exponents) that agree reasonably with those obtained from the size dependence in nuclear multifragmentation. The exponents depend on the spread of the stochastic mechanism (supposed to be a cascading, multiplicative process) and on the magnitude of the constraint imposed. An information-theoretic interpretation is provided for the relation between statistical and mechanism-induced (dynamic) fluctuations.     

Scalar Susceptibility of QCD from Dyson-Schwinger Approach

In quantum chromodynamics （QCD）, the scalar susceptibility represents the modification of the quark condensate, to a small perturbation of the parameter responsible for the explicit breaking of the symmetry, i.e., the current quark mass. By studying the linear response of the dressed quark propagator to the presence of a nonzero quark mass, we derive a model-independent formula for the scalar susceptibility, which contains the dressed quark propagator G（p） and the dressed scalar vertex F（p, 0）. The numerical values of the scalar susceptibility Xs are calculated within the framework of the rainbow-ladder approximation of the Dyson-Schwinger approach by employing two typical forms of model gluon propagator.     

Single-particle spectral density of the unitary Fermi gas: Novel approach based on the operator product expansion,sum rules and the maximum entropy method

Making use of the operator product expansion, we derive a general class of sum rules for the imaginary part of the single-particle self-energy of the unitary Fermi gas. The sum rules are analyzed numerically with the help of the maximum entropy method, which allows us to extract the single-particle spectral density as a function of both energy and momentum. These spectral densities contain basic information on the properties of the unitary Fermi gas, such as the dispersion relation and the superfluid pairing gap, for which we obtain reasonable agreement with the available results based on quantum Monte-Carlo simulations.     

Dynamical Running Mass of Quark in the Dyson-Schwinger Equation Approach

Based on the Dyson-Schwinger equations of QCD in the "rainbow" approximation, the fully dressed quarkpropagator Sf(p) is investigated, and then an algebraic parametrization form of the propagator is obtained as a solutionof the equations. The dressed quark amplitudes Af and Bf built up the fully dressed quark propagator and the dynamicalrunning masses Mf defined by Af and Bf for light quarks u, d and s are calculated, respectively. Using the predictedrunning masses Mf, quark condensates <0|q(0)q(0)|0> = -(0.255 GeV)a for u, d quarks, and <0|s s|0> = 0.8<0|q(0)q(0)]0)for s quark, and experimental pion decay constant fπ = 0.093 GeV, the masses of Goldstone bosons K, π, and η are alsoevaluated. The numerical results show that the masses of quarks are dependent on their momentum p2. The fully dressedquark amplitudes Af and Bf have correct behaviors which can be used for many purposes in our future researches onnonperturbative QCD.     

The Dyson-Schwinger Method of the NJL Model Beyond the Mean Field Approximation

By making use of the generating functional method we derive the Dyson-Schwinger equations in the NJL model and obtain the revised quark gap equation due to the meson dressing effects (usually called beyond the mean-field correction), and finally analyze its influence on the meson properties as well as the nucleon sigma term.     

Chapman-Enskog-maximum entropy method on time-dependent neutron transport equation

The time-dependent neutron transport equation in semi and infinite medium with linear anisotropic and Rayleigh scattering is proposed. The problem is solved by means of the flux-limited, Chapman-Enskog-maximum entropy for obtaining the solution of the time-dependent neutron transport. The solution gives the neutron distribution density function which is used to compute numerically the radiant energy density E(x,t), net flux F(x,t) and reflectivity R_f. The behaviour of the approximate flux-limited maximum entropy neutron density function are compared with those found by other theories. Numerical calculations for the radiant energy, net flux and reflectivity of the proposed medium are calculated at different time and space.     

Nonuniform sampling and maximum entropy reconstruction applied to the accurate measurement of residual dipolar couplings

Residual dipolar couplings (RDC) provide important global restraints for accurate structure determination by NMR. We show that nonuniform sampling in combination with maximum entropy reconstruction (MaxEnt) is a promising strategy for accelerating and potentially enhancing the acquisition of RDC spectra. Using MaxEnt-processed spectra of nonuniformly sampled data sets that are reduced up to one fifth relative to uniform sampling, accurate 13C'-13Calpha RDCs can be obtained that agree with an RMS of 0.67 Hz with those derived from uniformly sampled, Fourier transformed spectra. While confirming that frequency errors in MaxEnt spectra are very slight, an unexpected class of systematic errors was found to occur in the 6th significant figure of 13C' chemical shifts of doublets obtained by MaxEnt reconstruction. We show that this error stems from slight line shape perturbations and predict it should be encountered in other nonlinear spectral estimation algorithms. In the case of MaxEnt reconstruction, the error can easily be rendered systematic by straightforward optimization of MaxEnt reconstruction parameters and self-cancels in obtaining RDCs from nonuniformly sampled, MaxEnt reconstructed spectra.     

Fluorescence lifetime characterization of bacteria using total lifetime distribution analysis with the maximum entropy method

Total lifetime distribution analysis was employed to obtain fluorescence lifetime profiles of the intrinsic fluorescence ofPseudomonas fluorescens, Escherichia coli, Bacillus subtilis, andStaphylococcus epidermidis. The lifetimes were measured using a multiharmonic Fourier transform phase-modulation fluorometer which can simultaneously measure the phase shift and demodulation at many modulation frequencies. The 364-nm line from an argon-ion laser and the 325- and 442-nm lines from a helium-cadmium laser were used for sample excitation. Broad emission windows were used to capture as much of the bacterial emission as possible for the lifetime measurements. The maximum entropy method was used to recover lifetime profiles from the multifrequency phasemodulation data. At all three excitation wavelengths, the bacteria exhibited three lifetime components, in the ranges of 0.5-1, 2–3, and 4–8 ns. Using 325-nm excitation, a fourth component, in the range of 9–14 ns, was recovered in all of the bacteria; using 364-nm excitation, the fourth component was resolved only in the two Gram-negative bacteria (P. fluorescens andE. coli). Excitation at 364 nm provided the most reproducible lifetime profiles and showed some differences among the four bacteria.     

The criteria for selecting a method for unfolding neutron spectra based on the information entropy theory

To further expand the application of an artificial neural network in the field of neutron spectrometry, the criteria for choosing between an artificial neural network and the maximum entropy method for the purpose of unfolding neutron spectra was presented. The counts of the Bonner spheres for IAEA neutron spectra were used as a database, and the artificial neural network and the maximum entropy method were used to unfold neutron spectra; the mean squares of the spectra were defined as the differences between the desired and unfolded spectra. After the information entropy of each spectrum was calculated using information entropy theory, the relationship between the mean squares of the spectra and the information entropy was acquired. Useful information from the information entropy guided the selection of unfolding methods. Due to the importance of the information entropy, the method for predicting the information entropy using the Bonner spheres' counts was established. The criteria based on the information entropy theory can be used to choose between the artificial neural network and the maximum entropy method unfolding methods. The application of an artificial neural network to unfold neutron spectra was expanded.     

Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support

The maximum entropy (MaxEnt) principle is a versatile tool for statistical inference of the probability density function (pdf) from its moments as a least-biased estimation among all other possible pdf’s. It maximizes Shannon entropy, satisfying the moment constraints. Thus, the MaxEnt algorithm transforms the original constrained optimization problem to the unconstrained dual optimization problem using Lagrangian multipliers. The Classic Moment Problem (CMP) uses algebraic power moments, causing typical conventional numerical methods to fail for higher-order moments (m>5–10)$(m>5''–''10)$ due to different sensitivities of Lagrangian multipliers and unbalanced nonlinearities. Classic MaxEnt algorithms overcome these difficulties by using orthogonal polynomials, which enable roughly the same sensitivity for all Lagrangian multipliers. In this paper, we employ an idea based on different principles, using Fup_n${\mathit{Fup}}_n$ basis functions with compact support, which can exactly describe algebraic polynomials, but only if the Fup order-n   is greater than or equal to the polynomial’s order. Our algorithm solves the CMP with respect to the moments of only low order Fup₂${\mathit{Fup}}_2$ basis functions, finding a Fup₂${\mathit{Fup}}_2$ optimal pdf with better balanced Lagrangian multipliers. The algorithm is numerically very efficient due to localized properties of Fup₂${\mathit{Fup}}_2$ basis functions implying a weaker dependence between Lagrangian multipliers and faster convergence. Only consequences are an iterative scheme of the algorithm where power moments are a sum of Fup₂${\mathit{Fup}}_2$ and residual moments and an inexact entropy upper bound. However, due to small residual moments, the algorithm converges very quickly as demonstrated on two continuous pdf examples – the beta distribution and a bi-modal pdf, and two discontinuous pdf examples – the step and double Dirac pdf. Finally, these pdf examples present that Fup MaxEnt algorithm yields smaller entropy value than classic MaxEnt algorithm, but differences are very small for all practical engineering purposes.     

Statistical approach of the irreversible entropy variation

The constructal theory is based on the thought that flow architecture is a consequence of a principle of maximization of flow access, in time, and in flow configuration that is free to be realised. The principle of maxima for the variation of the entropy due to irreversibility represents a general principle of investigation for the stability of open systems, by which it is possible to predict some macroscopic shapes, originated by the spatial organisation, in Nature, both in living and in non-living objects. Its statistical meaning has recently been introduced. Here a statistical and dynamical interpretation for the entropy due to irreversibility is proposed as a foundation of the constructal theory.     

Gottfried sum rule from maximum entropy method quark distributions with DGLAP evolution and with DGLAP evolution with GLR-MQ-ZRS corrections

A new method to test the valence quark distribution of nucleons obtained from the maximum entropy method using the Gottfried sum rule by performing the DGLAP equations with GLR-MQ-ZRS corrections and the original leading-order/next-to-leading-order(LO/NLO) DGLAP equations is outlined. The test relies on knowledge of the unpolarized electron–proton structure function F_2~(ep) and the electron–neutron structure function F_2~(en) and the assumption that Bjorken scaling is satisfied. In this work, the original Gottfried summation value obtained by the integrals of the structure function at different Q~2 is in accordance with the theoretical value of 1/3 under the premise of light-quark flavor symmetry of the nucleon sea, whether it results from dynamical evolution equations or from global quantum chromodynamics fits of PDFs. Finally, we present the summation value of the LO/NLO DGLAP global fits of PDFs under the premise of light-quark flavor asymmetry of the nucleon sea. According to analysis of the original Gottfried summation value with two evolution equations at different Q~2, we find that the valence quark distributions of nucleons obtained by using the maximum entropy method are effective and reliable.     

A maximum entropy mean field method for driven diffusive systems

We introduce a method of generating systematic mean field (MF) approximations for the nonequilibrium steady state of ferromagnetic Ising driven diffusive systems (DDS), based on the maximum entropy principle due to Jaynes. In the phase coexistence region, MF approximations to the master equation do not provide a closed system of equations in the MF variables. This can be traced to the conservation of the order parameter by the stochastic dynamics. Our maximum entropy mean field (MEMF) approximation method is applicable to high temperatures as well to the low-temperature phase coexistence region. It is based on a derivation of a generalized variational free energy from the maximum entropy principle, with the MF evolution equations playing the role of constraints. In the phase coexistence region this free energy is nonconvex and is interpreted by means of a Maxwell construction. We use a pair-level variant of the MEMF approximation to calculate quantities of interest for the ferro-magnetic Ising DDS on a square lattice. Results of calculations with several different choices of transition rates satisfying local detailed balance are discussed and compared with those obtained by other methods.     

Application of maximum entropy method for the study of electron density distribution in SrS, BaS and PuS using powder X-ray data

A study of the electronic structure of the three sulphides, SrS, BaS and PuS has been carried out in this work, using the powder X-ray intensity data from JCPDS powder diffraction data base. The statistical approach, MEM (maximum entropy method) is used for the analysis of the data for the electron density distribution in these materials and an attempt has been made to understand the bonding between the metal atom and the sulphur atom. The mid-bond electron density is found to be maximum for PuS among these three sulphides, being 0.584 e/Å3 at 2.397 Å. SrS is found to have the lowest electron density at the mid-bond (0.003 e/ų) at 2.118 Å from the origin leaving it more ionic than the other two sulphides studied in this work. The two-dimensional electron density maps on (100) and (110) planes and the one-dimensional profiles along the bonding direction [111] are used for these analyses. The overall and individual Debye-Waller factors of atoms in these systems have also been studied and analyzed. The refinements of the observed X-ray data were carried out using standard softwares and also a routine written by the author