From Physics to Economics: An Econometric Example Using Maximum Relative Entropy

Econophysics, is based on the premise that some ideas and methods from physics can be applied to economic situations. We intend to show in this paper how a physics concept such as entropy can be applied to an economic problem. In so doing, we demonstrate how information in the form of observable data and moment constraints are introduced into the method of Maximum relative Entropy (MrE). A general example of updating with data and moments is shown. Two specific econometric examples are solved in detail which can then be used as templates for real world problems. A numerical example is compared to a large deviation solution which illustrates some of the advantages of the MrE method.Comment: This paper has been accepted in Physica A. 19 Pages, 3 Figure

Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis

Background: For the kinetic models used in contrast-based medical imaging, the assignment of the arterial input function named AIF is essential for the estimation of the physiological parameters of the tissue via solving an optimization problem. Objective: In the current study, we estimate the AIF relayed on the modified maximum entropy method. The effectiveness of several numerical methods to determine kinetic parameters and the AIF is evaluated-in situations where enough information about the AIF is not available. The purpose of this study is to identify an appropriate method for estimating this function. Materials and Methods: The modified algorithm is a mixture of the maximum entropy approach with an optimization method, named the teaching-learning method. In here, we applied this algorithm in a Bayesian framework to estimate the kinetic parameters when specifying the unique form of the AIF by the maximum entropy method. We assessed the proficiency of the proposed method for assigning the kinetic parameters in the dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI), when determining AIF with some other parameter-estimation methods and a standard fixed AIF method. A previously analyzed dataset consisting of contrast agent concentrations in tissue and plasma was used. Results and Conclusions: We compared the accuracy of the results for the estimated parameters obtained from the MMEM with those of the empirical method, maximum likelihood method, moment matching ("method of moments"), the least-square method, the modified maximum likelihood approach, and our previous work. Since the current algorithm does not have the problem of starting point in the parameter estimation phase, it could find the best and nearest model to the empirical model of data, and therefore, the results indicated the Weibull distribution as an appropriate and robust AIF and also illustrated the power and effectiveness of the proposed method to estimate the kinetic parameters

Maximum-entropy moment-closure for stochastic systems on networks

Moment-closure methods are popular tools to simplify the mathematical analysis of stochastic models defined on networks, in which high dimensional joint distributions are approximated (often by some heuristic argument) as functions of lower dimensional distributions. Whilst undoubtedly useful, several such methods suffer from issues of non-uniqueness and inconsistency. These problems are solved by an approach based on the maximisation of entropy, which is motivated, derived and implemented in this article. A series of numerical experiments are also presented, detailing the application of the method to the Susceptible-Infective-Recovered model of epidemics, as well as cautionary examples showing the sensitivity of moment-closure techniques in general.Comment: 20 pages, 7 figure

Adaptive meshless point collocation methods: investigation and application to geometrically non-linear solid mechanics

Conventional mesh-based methods for solid mechanics problems suffer from issues resulting from the use of a mesh, therefore, various meshless methods that can be grouped into those based on weak or strong forms of the underlying problem have been proposed to address these problems by using only points for discretisation. Compared to weak form meshless methods, strong form meshless methods have some attractive features because of the absence of any background mesh and avoidance of the need for numerical integration, making the implementation straightforward. The objective of this thesis is to develop a novel numerical method based on strong form point collocation methods for solving problems with geometric non-linearity including membrane problems. To address some issues in existing strong form meshless methods, the local maximum entropy point collocation method is developed, where the basis functions possess some advantages such as the weak Kronecker-Delta property on boundaries. r- and h-adaptive strategies are investigated in the proposed method and are further combined into a novel rh-adaptive approach, achieving the prescribed accuracy with the optimised locations and limited number of points. The proposed meshless method with h-adaptivity is then extended to solve geometrically non-linear problems described in a Total Lagrangian formulation, where h-adaptivity is again employed after the initial calculation to improve the accuracy of the solution effciently. This geometrically non-linear method is finally developed to analyse membrane problems, in which the out-of-plane deformation for membranes complicates the governing PDEs and the use of hyperelastic materials makes the computational modelling of membrane problems challenging. The Newton-Raphson arc-length method is adopted here to capture the snap-through behaviour in hyperelastic membrane problems. Several numerical examples are presented for each proposed algorithm to validate the proposed methodology and suggestions are made for future work leading on from the findings of this thesis

Calculation of Densities of States and Spectral Functions by Chebyshev Recursion and Maximum Entropy

We present an efficient algorithm for calculating spectral properties of large sparse Hamiltonian matrices such as densities of states and spectral functions. The combination of Chebyshev recursion and maximum entropy achieves high energy resolution without significant roundoff error, machine precision or numerical instability limitations. If controlled statistical or systematic errors are acceptable, cpu and memory requirements scale linearly in the number of states. The inference of spectral properties from moments is much better conditioned for Chebyshev moments than for power moments. We adapt concepts from the kernel polynomial approximation, a linear Chebyshev approximation with optimized Gibbs damping, to control the accuracy of Fourier integrals of positive non-analytic functions. We compare the performance of kernel polynomial and maximum entropy algorithms for an electronic structure example.Comment: 8 pages RevTex, 3 postscript figure

A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry

We develop a high-order kinetic scheme for entropy-based moment models of a one-dimensional linear kinetic equation in slab geometry. High-order spatial reconstructions are achieved using the weighted essentially non-oscillatory (WENO) method, and for time integration we use multi-step Runge-Kutta methods which are strong stability preserving and whose stages and steps can be written as convex combinations of forward Euler steps. We show that the moment vectors stay in the realizable set using these time integrators along with a maximum principle-based kinetic-level limiter, which simultaneously dampens spurious oscillations in the numerical solutions. We present numerical results both on a manufactured solution, where we perform convergence tests showing our scheme converges of the expected order up to the numerical noise from the numerical optimization, as well as on two standard benchmark problems, where we show some of the advantages of high-order solutions and the role of the key parameter in the limiter

Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis

The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating distribution by minimizing the Kullback-Leibler information (relative entropy) of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints. We study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases. We prove that (i) the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with, and (ii) the approximate discrete distribution weakly converges to the given continuous distribution. Moreover, we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem.Comment: 20 pages, 14 figure