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.     

Eigenvector distribution of Wigner matrices

We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν _ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν _ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.     

Generating a Random Collection of Discrete Joint Probability Distributions Subject to Partial Information

In this paper, we develop a practical and flexible methodology for generating a random collection of discrete joint probability distributions, subject to a specified information set, which can be expressed as a set of linear constraints (e.g., marginal assessments, moments, or pairwise correlations). Our approach begins with the construction of a polytope using this set of linear constraints. This polytope defines the set of all joint distributions that match the given information; we refer to this set as the “truth set.” We then implement a Monte Carlo procedure, the Hit-and-Run algorithm, to sample points uniformly from the truth set. Each sampled point is a joint distribution that matches the specified information. We provide guidelines to determine the quality of this sampled collection. The sampled points can be used to solve optimization models and to simulate systems under different uncertainty scenarios.     

An investigation of phase-distribution moment-matching algorithms for use in queueing models

Algorithms for matching moments to phase-type distributions are evaluated on the basis of their performance in their intended application, queueing models. The moment-matching algorithms under consideration match two moments to a hyperexponential distribution with balanced means and three moments to a mixture of two Erlang distributions of common order. These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms. Three performance measures are considered, and attention is focussed on the steady-state mean queue length (number in system) of theGI/M/1 queue. Performance-measure approximations are compared to three-moment bounds and performance-measure values arising from hypothetical approximated distributions.     

Novel algorithm for reconstruction of a distribution by fitting its first-four statistical moments

This paper proposes a novel algorithm to reconstruct an unknown distribution by fitting its first-four moments to a proper parametrized probability distribution (PPD) model. First, a PPD system containing three previously developed PPD models is suggested to approximate the unknown distribution, rather than empirically adopting a single distribution model. Then, a two-step algorithm based on the moments matching criterion and the maximum entropy principle is proposed to specify the appropriate (final) PPD model in the system for the distribution. The proposed algorithm is first verified by approximating several commonly used analytical distributions, along with a set of real dataset, where the existing measures are also employed to demonstrate the effectiveness of the proposed two-step algorithm. Further, the effectiveness of the algorithm is demonstrated through an application to three typical moments-based reliability problems. It is found that the proposed algorithm is a robust tool for selecting an appropriate PPD model in the system for recovering an unknown distribution by fitting its first-four moments.     

An inequality unscented transformation for estimating the statistical moments

Point estimate method (PEM) is convenient for estimating statistical moments. This paper focuses on discussing the existing PEMs and presenting a new PEM for the efficient and accurate estimation of statistical moments. Firstly, a classification method of PEMs is proposed based on the strategy of choosing sigma points. Secondly, the minimum number of sigma points and the error of inverse Nataf transformation are derived corresponding to certain order and dimensionality of PEMs. Then the inequality unscented transformation (IUT) is presented to estimate the statistical moments. The proposed IUT permits the existing of limited errors in the matching of the first several order moments to decrease the number of sigma points, it opens new strategy of PEMs. The proposed method has two advantages. The first advantage is overcoming the growth of the number of sigma points with dimensionality since it parameterizes the number of sigma points and accuracy order. The second advantage is the wide applicability, for it has the ability to handle correlated and asymmetric random input variables and to match cross moments. Numerical and engineering results show the good accuracy and efficiency of the proposed IUT.     

Asymptotic distributions for Random Median Quicksort

The first complete running time analysis of a stochastic divide and conquer algorithm was given for Quicksort, a sorting algorithm invented 1961 by Hoare. We analyse here the variant Random Median Quicksort. The analysis includes the expectation, the asymptotic distribution, the moments and exponential moments. The asymptotic distribution is characterized by a stochastic fixed point equation. The basic technic will be generating functions and the contraction method.     

Approximate unitary 3-designs from transvection Markov chains

Designs, Codes and Cryptography - Unitary k-designs are probabilistic ensembles of unitary matrices whose first k statistical moments match that of the full unitary group endowed with the Haar...     

A model-order reduction method based on Krylov subspaces for mimo bilinear dynamical systems

In this paper, we present a Krylov subspace based projection method for reduced-order modeling of large scale bilinear multi-input multioutput (MIMO) systems. The reduced-order bilinear system is constructed in such a way that it can match a desired number of moments of multivariable transfer functions corresponding to the kernels of Volterra series representation of the original system. Numerical examples report the effectiveness of this method.     

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.     

Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models

We derive an asymptotic expansion for the log-likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal-to-noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation-Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone towards the analysis of EM and maximum likelihood estimation in a wide range of models. A motivating application for the study of low SNR mixture models is cryo-electron microscopy data, which can be modeled as a GMM with algebraic constraints imposed on the mixture centers. We discuss the application of our expansion to algebraically constrained GMMs, among other example models of interest. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Monte Carlo approximate tensor moment simulations

An algorithm to generate samples with approximate first‐order, second‐order, third‐order, and fourth‐order moments is presented by extending the Cholesky matrix decomposition to a Cholesky tensor decomposition of an arbitrary order. The tensor decomposition of the first‐order, second‐order, third‐order, and fourth‐order objective moments generates a non‐linear system of equations. The algorithm solves these equations by numerical methods. The results show that the optimization algorithm delivers samples with an approximate residual error of less than 10¹⁶ between the components of the objective and the sample moments. The algorithm is extended for a n‐th‐order approximate tensor moment version, and simulations of non‐normal samples replicated from distributions with asymmetries and heavy tails are presented. An application for sensitivity analysis of portfolio risk assessment with Value‐at‐Risk (VaR) is provided. A comparison with previous methods available in the literature suggests that the methodology proposed reduces the error of the objective moments in the generated samples. ? ? JEL Classification: C14, C15, G32.
Copyright © 2016 John Wiley & Sons, Ltd.     

The MacLaurin expansion for a G/G/1 queue with Markov-modulated arrivals and services

Consider a Markov-modulated G/G/1 queueing system in which the arrival and the service mechanisms are controlled by an underlying Markov chain. The classical approaches to the waiting time of this type of queueing system have severe computational difficulties. In this paper, we develop a numerical algorithm to calculate the moments of the waiting time based on Gong and Hu's idea. Our numerical results show that the algorithm is powerful. A matrix recursive equation for the moments of the waiting time is also given under certain conditions.     

Moment characterization of matrix exponential and Markovian arrival processes

This paper provides a general framework for establishing the relation between various moments of matrix exponential and Markovian processes. Based on this framework we present an algorithm to compute any finite dimensional moments of these processes based on a set of required (low order) moments. This algorithm does not require the computation of any representation of the given process. We present a series of related results and numerical examples to demonstrate the potential use of the obtained moment relations. This work is partially supported by the Italian-Hungarian bilateral R&D programme, by OTKA grant n. T-34972, by MIUR through PRIN project Famous and by EEC project Crutial.     

The disteribution of a moment estimator for a parameter of the generalized poision distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

Win–win match using a genetic algorithm

As the service industries grow, tasks are not directly assigned to the skills but the knowledge of the worker which is to be valued more in finding the best match. The problem becomes difficult mainly because the match has to be seen with the objectives of both sides. Assignment methods fail to respond to a multi-objective, multi-constraint problem with complicated match; whereas, metaheuristics is preferable based on computational simplicity. A conditional genetic algorithm is developed in this study to propose both global and composite match using different fitness functions. This algorithm kills the infeasibilities to achieve the maximum number of matches. The proposed algorithm is applied on an academic problem of multi-alternative candidates and multi-alternative tasks (m × n problem) in two stages. In the first stage, four different fitness functions are evaluated and in the second stage using one of the fitness functions global and composite matching have been compared. The achievements will contribute both to the academic and business world.     

Moments’ Analysis in Homogeneous Markov Reward Models

We analyze the moments of the accumulated reward over the interval (0,t) in a continuous-time Markov chain. We develop a numerical procedure to compute efficiently the normalized moments using the uniformization technique. Our algorithm involves auxiliary quantities whose convergence is analyzed, and for which we provide a probabilistic interpretation.     

A Heuristic for Moment-Matching Scenario Generation

In stochastic programming models we always face the problem of how to represent the random variables. This is particularly difficult with multidimensional distributions. We present an algorithm that produces a discrete joint distribution consistent with specified values of the first four marginal moments and correlations. The joint distribution is constructed by decomposing the multivariate problem into univariate ones, and using an iterative procedure that combines simulation, Cholesky decomposition and various transformations to achieve the correct correlations without changing the marginal moments.With the algorithm, we can generate 1000 one-period scenarios for 12 random variables in 16 seconds, and for 20 random variables in 48 seconds, on a Pentium III machine.     

On a control problem for a nonlinear second-order system

A control problem for a nonlinear second-order system of differential equations in the presence of uncontrollable effects is investigated. A solution algorithm is proposed in the case when one phase coordinate of the system is measured at discrete moments. The algorithm is stable with respect to information noises and computational errors. Results of a computer experiment are presented.     

Improved probabilistic point estimation schemes for uncertainty analysis

The probabilistic point estimation (PPE) methods replace the probability distribution of the random parameters of a model with a finite number of discrete points in sample space selected in such a way to preserve limit probabilistic information of involved random parameters. Most PPE methods developed thus far match the distribution of random parameters up to the third statistical moment and, in general, could provide reasonable accurate estimation only for the first two statistical moments of model output. This study proposes two optimization-based point selection schemes for the PPE methods to enhance the accuracy of higher-order statistical moments estimation for model output. Several test models of varying degrees of complexity and nonlinearity are used to examine the performance of the proposed point selection schemes. The results indicate that the proposed point selection schemes provide significantly more accurate estimation of model output uncertainty features than the existing schemes.