Use Of Fractional Moments Research Articles

Probability Distributions Approximation via Fractional Moments and Maximum Entropy: Theoretical and Computational Aspects

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance of the analogous procedure based on integer moments. No attention has been paid to two formal aspects concerning fractional moments, such as conditions for the existence of the maximum entropy approximation based on them or convergence in entropy of this approximation to F. This paper aims to fill this gap by providing proofs of these two fundamental results. In fact, convergence in entropy can be involved in the optimal selection of the order of fractional moments for accelerating the convergence of the MaxEnt approximation to F, to clarify the entailment relationships of this type of convergence with other types of convergence useful in statistical applications, and to preserve some important prior features of the underlying F distribution.

High Order Moment Model for Polydisperse Evaporating Sprays towards Interfacial Geometry Description

In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluationof the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization.

An effective approach for probabilistic lifetime modelling based on the principle of maximum entropy with fractional moments

The paper presents a fractional moment method for probabilistic lifetime modelling of uncertain engineering systems. A novel feature of the method is the use of fractional moments, as opposed to integer moments commonly used so far in the structural reliability literature. The fractional moments are calculated from a small, simulated sample of remaining useful life of the system. And the fractional exponents that are used to model the system lifetime distribution are determined through the entropy maximization process, rather than assigned by an analyst in prior. Together with the theory of copula, the efficiency and accuracy of the proposed method are illustrated by the probabilistic lifetime modelling of several dynamical and discontinuous stochastic systems.

System reliability analysis of the robotic manipulator with random joint clearances

The paper presents an innovative method for computing reliability of a robotic manipulator such that the positional error in the entire trajectory remains within acceptable limits. This problem is equivalent to a series system reliability analysis that can be solved using the extreme value distribution of the positional error. The principle of maximum entropy is applied to derive this distribution. A novel feature of the analysis is the use of fractional moments, instead of integer moments commonly used in the entropy literature. The fractional moments are obtained from a small, simulated sample of positional error. Examples are presented to illustrate accuracy and efficiency of the proposed method.

Hausdorff moment problem and fractional moments

Hausdorff moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, in this work a stable algorithm to obtain centered moments from integer moments is found. The algorithm transforms a direct method into an iterative Jacobi method which converges in a finite number of steps, as the iteration Jacobi matrix has null spectral radius. The centered moments are needed to calculate fractional moments from integer moments. As an application few fractional moments are used to solve finite Hausdorff moment problem via maximum entropy technique. Fractional moments represent a remedy to ill-conditioning coming from an high number of integer moments involved in recovering procedure.

Stieltjes moment problem and fractional moments

Stieltjes moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, a determinate Stieltjes moment problem, whose corresponding Hamburger moment problem is determinate too, is investigated in the setup of Maximum Entropy. Condition number in entropy calculation is provided endowing both Stieltjes moment problem existence conditions and Hamburger moment problem determinacy conditions by a geometric meaning. Then the resorting to fractional moments is considered; numerical aspects are investigated and a stable algorithm for calculating fractional moments from integer moments is proposed.

Monitoring evolution of laser atmospheric scintillation statistics by use of fractional moments

The time evolution of experimental intensity fluctuation distributions of a laser beam propagating through atmospheric turbulence is discussed. Experimental data, collected under conditions ranging from low to strong scintillation, are compared with four theoretical probability-density functions by means of fractional moments.

Valuation of exotic options using moments

In this paper we discuss the problem of recovering a density from its moments. For theoretical reasons, we propose the use of fractional moments combined with the Maximum Entropy density. We then discuss the application to the pricing of exotic options.

Fractional moments for processing data from low-dynamics acquisition systems in laser atmospheric scintillation

Use of fractional moments of low order, recently proposed by Consortini and Rigal for investigating probability density functions in la- ser scintillation statistics through atmospheric turbulence, is here pro- posed for overcoming the saturation of the electronics due to low- dynamics acquisition systems. Paper 200438. © 2001 Society of Photo-

The Use of Fractional Moments for Estimating the Parameters of a Mixed Exponential Distribution

In this paper the use of fractional moments for estimation purposes is discussed. These ideas are illustrated by means of the mixed exponential distribution. The estimation of the three parameters of the above distribution by the method of moments and by maximum likelihood is investigated numerically in detail. As anticipated, the efficiency of the former method can be greatly increased by using approximately optimal combinations of moments. It is found that the moment method requires only a small amount of calculation when compared with the maximum likelihood method, although charts are presented to greatly ease the computational burden of the latter method.

The Use of Fractional Moments for Estimating the Parameters of a Mixed Exponential Distribution

In this paper the use of fractional moments for estimation purposes is discussed. These ideas are illustrated by means of the mixed exponential distribution. The estimation of the three parameters of the above distribution by the method of moments and by maximum likelihood is investigated numerically in detail. As anticipated, the efficiency of the former method can be greatly increased by using approximately optimal combinations of moments. It is found that the moment method requires only a small amount of calculation when compared with the maximum likelihood method, although charts are presented to greatly ease the computational burden of the latter method.