General Moment Research Articles

Estimation of Moment Parameter in Elliptical Distributions

As a typical non-normal case, we consider a family of elliptically symmetric dis- tributions. Then, the moment parameter and its consistent estimator are presented. Also, the asymptotic expectation and the asymptotic variance of the consistent es- timator of the general moment parameter are given. Besides, the numerical results obtained by Monte Carlo simulation for some selected parameters are provided. The general moment parameter includes the important kurtosis parameter in the study of multivariate statistical analysis for elliptical populations. The kurtosis parameter, especially with relation to the estimation problem, has been considered by many authors. Mardia (1970, 1974) defined a measure of multi- variate sample kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Also, the testing normality was considered by using the asymptotic result. The related discussion of the kurtosis parameter under the elliptical distribution has been given by Anderson (1993), and Seo and Toyama (1996). Henze (1994) has discussed the asymptotic variance of the mul- tivariate sample kurtosis for general distributions. Here we deal with the estima- tion of the general moment parameters in elliptical distributions. In particular, we make a generalization of the results of Anderson (1993) and give an extension of asymptotic properties in Mardia (1970, 1974), Seo and Toyama (1996). In general, it is not easy to derive the exact distribution of test statistics or the percentiles for the testing problem under the elliptical populations, and so the asymptotic expansion of the statistics is considered. Especially that given up to the higher order includes not only the kurtosis parameter but the more gen- eral higher order moment parameters as well. Then we have to speculate about the estimation of the moment parameters as a practical problem. The present paper is organized in the following way. First, the probability density function, characteristic function and subclasses of the elliptical distribution are explained. Secondly, we prove the results concerning the moments and define the moment

Dipole and rotational strengths for overtone transitions of a C2-symmetry HCCH molecular fragment using Van Vleck perturbation theory

Contact transformation theory up to second order is employed to treat CH-stretching overtone transitions and to calculate dipole and rotational strengths. A general Hamiltonian describing two interacting CH-stretching oscillators is considered, and the Darling–Dennison resonance is appropriately taken into account. The two CH bonds are supposed to be dissymmetrically disposed, so as to represent a chiral HCCH fragment, endowed with C2 symmetry. Analytical expressions of transition moments and dipole and rotational strengths are given in the hypothesis of general electric and magnetic dipole moments with quadratic dependence on coordinates and momenta. Dipole and rotational strengths are then calculated together with frequencies for the fundamental and first three overtone regions in the simplifying hypothesis of the valence optical approach on the coupled-oscillator framework. Simplified analytical expressions thereof in the relevant parameters are presented.

A novel approach to determining physicochemical and absorption properties of 6-fluoroquinolone derivatives: experimental assessment

The ToSS MoDe approach is used to estimate the n-octanol/buffer partition coefficient, the apparent intestinal absorption rate constant and intestinal permeability from a 6-fluoroquinolone data set. Improved in silico methods for predicting a drug's ability to be transported across biological membranes and other biopharmaceutical properties is highly desirable to optimize new drug development. The physicochemical property (Log P) of 26 6-fluoroquinolone derivatives and the absorption properties (Log K a and Log P eff) of 21 derivatives were well described by the present approach. The models obtained confirm the important role of lipophilicity in the absorption process and its relation with the piperazinyl ring spectral moment and general local spectral moment. The normalized group contributions to each property, at the R 4 and R 5 positions of a 6-fluoroquinolone framework, were calculated. Principal factor analysis between these contributions and the Hammett and Hansch constant, molar refractivity and sterimol parameters was also carried out. Three principal factors explained 78% of the total variance and the correlation coefficients were higher than 0.98. The isocontribution zone analysis for the Log P and Log K a of Sarafloxacin and Sparfloxacin, used as external corroboration compounds, was carried out. The absorption rate constants (in situ rat gut technique) for these drugs were also evaluated, and the results were compared with the values predicted by theoretical models for evaluating predictive performance. The present approach proved to be a good method for studying the oral absorption of drug candidates in drug development studies.

Moment Equations for a Granular Material in a Thermal Bath

We compute the moment equations for a granular material under the simplifying assumption of pseudo-Maxwellian particles approximating dissipative hard spheres. We obtain the general moment equations of second and third order and the isotropic moment equations of any order. Our equations describe, in the space homogeneous case, the granular system described by a Boltzmann-like collision term and subject to a Brownian motion due to the interaction with a bath, described by a Fokker–Planck term. The trend to equilibrium is studied in detail.

Lower extremity general muscle moment patterns in healthy individuals during recumbent cycling

Objective. The purpose of this study was to compare lower extremity generalized muscle moments across two workloads during recumbent bicycling in younger and older healthy adults. Design. The study design was a comparative investigation of cycling patterns. Background. Biomechanical data regarding muscle activation, kinematic, and kinetic patterns have been presented for upright cycling, but only a few studies have evaluated biomechanical patterns during the alternative configuration of recumbent cycling. Methods. Twenty-four healthy adults, classified by age into two different groups, under 35 and over 50 years of age, rode a recumbent bicycle at a constant cadence (60–65 rpm) and at two different resistances (0.5 and 1.0 kg m) while kinematic and kinetic data were recorded. General muscle moments were calculated using joint kinematic and kinetic data via inverse dynamic equations. Results. The ankle general muscle moment remained plantar flexor throughout the pedaling cycle; the knee general muscle moment remained flexor throughout the cycle, except during the power phase of the higher workload where an extensor general muscle moment was observed; and the hip general muscle moment was extensor with a transient flexor general muscle moment period during the recovery phase. Increased workload led to increases in ankle plantar flexor and knee extensor general muscle moment magnitudes, but no changes at the hip. Age had no effect on general muscle moment magnitudes or patterns. Conclusions. Configurational differences between the upright and recumbent bicycle do not affect patterns, but the total output requirements do affect the magnitudes of the general muscle moments. Relevance Based on previous studies, the recumbent bicycle appears to be a safe rehabilitation tool for post-cerebrovascular accident and cardiorespiratory patients, but in order to more properly and efficiently use the recumbent bicycle as a rehabilitation tool, normative biomechanical data are necessary. The current study is the first such investigation to report normative data of lower extremity general muscle moment patterns during recumbent cycling. Effects of age and workload were also demonstrated.

MOMCO: method of moment components for passive model order reduction of RLCG interconnects

We introduce a new concept called moment components and a new method based on it to obtain passive reduced-order models of interconnect networks. In this method, the impedance matrix moments of the interconnect network are partitioned into their inductive, capacitive and mixed inductive/capacitive moment components. The method of moment components is described in a formal manner using analysis and synthesis equalities. Two significant contributions of the method of moment components are: 1) new decomposition of moments into parts which reflect passivity in all moments of passive networks; and 2) the analysis equalities impart a regular pattern in terms of the moment components, thus simplifying the moment generation for our method. None of these features is observable in conventional complete moment terms. Two new methods for obtaining passive reduced-order models based on the method of moment components are introduced. The passive reduced-order models are obtained by matching their impedance moment components to those of the original interconnect network through the synthesis equalities. Due to nonnegative definiteness of the moment components, the match in the moment components preserves the passivity of the original interconnect in the reduced-order model. The method of moment components does not have the instability problem of general moment matching techniques. The reduced-order model is specifically targeted for fast timing simulators so that interconnect effects can be simulated efficiently.

From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach

The trigonometric moment problem is a classical moment problem with numerous applications in mathematics, physics, and engineering. The rational covariance extension problem is a constrained version of this problem, with the constraints arising from the physical realizability of the corresponding solutions. Although the maximum entropy method gives one well-known solution, in several applications a wider class of solutions is desired. In a seminal paper, Georgiou derived an existence result for a broad class of models. In this paper, we review the history of this problem, going back to Carath{éodory, as well as applications to stochastic systems and signal processing. In particular, we present a convex optimization problem for solving the rational covariance extension problem with degree constraint. Given a partial covariance sequence and the desired zeros of the shaping filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgiou's existence result and his conjecture, a generalized version of which we have recently resolved using geometric methods. We also survey recent related results on constrained Nevanlinna--Pick interpolation in the context of a variational formulation of the general moment problem.

Source Depth and Mechanism Inversion at Teleseismic Distances Using a Neighborhood Algorithm

We performed nonlinear waveform inversion for source depth, time function, and mechanism, by modeling direct P and S waves and corresponding surface reflections at teleseismic distances. This technique was applied to moderate size events, and so we make use of short period or broadband records, and utilize SV waveforms in addition to P and SH . For the inversion we used a direct search method called the neighborhood algorithm (NA), which requires just two control parameters to guide the search in a conceptually simple manner, and is based on the rank of a user-defined misfit measure. We use a simple generalized ray scheme to calculate synthetic seismograms for comparison with observations, and show that the use of a derivative-free method such as the NA allows us to easily substitute more complex synthetics if necessary. The source mechanism is represented in two different ways; the superposition of a double-couple component with an isotropic component, and a general moment tensor with six independent components. Good results are obtained with both synthetic input data and real data. We achieve good depth resolution and obtain useful constraints on the source-time function and source mechanism, including an isotropic component estimate. Such estimates provide important discriminants between man-made events and earthquakes. We illustrate inversion with real data using two earthquakes, and in both cases the source parameter estimates compare well with the corresponding centroid moment tensor solutions. We also apply our technique to a known nuclear explosion and obtain a very shallow depth estimate and a large isotropic component.

An asymptitic analysis of the fokker-planck limit of a general spatial moment method for the transport equation

An asymptotic analysis of the spatially discretized particle transport equation in the Fokker-Planck limit is performed. This asymptotic limit corresponds to particle transport in medium with highly forward-peaked scattering. A general spatial moment (GSM) method is analyzed. It defines a family of spatial discretization methods. The GSM method is based on a set of spatial moments of the transport equation and linear auxiliary equations with coefficients that nonlinearly depend on cross sections. A majority of existing discretization schemes for the transport equation belongs to the group of such methods. The general conditions, under which the considered family of methods satisfies the Fokker-Planck asymptotic limit, are derived. The characteristic methods are studied. The analysis gives an insight into the asymptotic properties of a broad group of transport discretization methods.

Defining the scalar moment of a seismic source with a general moment tensor

Abstract We compare several published definitions of the scalar moment M0, a measure of the size of a seismic disturbance derived from the second-order seismic moment tensor M (with eigenvalues m1 ≥ m3 ≥ m2). While arbitrary, a useful definition is in terms of a total moment, MT0 = MI + MD, where MI = |M|, with M = (m1 + m2 + m3)/3, is the isotropic moment, and MD = max(|mj − M|; j = 1, 2, 3), is the deviatoric moment. This definition is consistent with other definitions of M0 if M is a double couple. This definition also gives physically appealing and simple results for the explosion and crack sources. Furthermore, our definitions of MT0, MI and MD are in accord with the parameterization of the moment tensor into a deviatoric part (represented by T which lies in [−1,1]) and a volumetric part (represented by k which lies in [−1, 1]) proposed by Hudson et al. (1989).

Error reduction in density estimation under shape restrictions

AbstractFor the problems of nonparametric estimation of nonincreasing and symmetric unimodal density functions with bounded supports we determine the projections of estimates onto the convex families of possible parent densities with respect to the weighted integrated squared error. We also describe the method of approximating the analogous projections onto the respective density classes satisfying some general moment conditions. The method of projections reduces the estimation errors for all possible values of observations of a given finite sample size in a uniformly optimal way and provides estimates sharing the properties of the parent densities.

Mechanical response of linear viscoelastic composite laminates incorporating non-isothermal physical aging effects

This paper presents a method of predicting the mechanical response of composite laminates including the effects of linear viscoelasticity and physical aging. Effective-time theory has been used to characterize the physical aging behavior of each linear viscoelastic lamina. In accordance with experimental findings, the aging behavior of each lamina is allowed to differ in the shear and transverse directions. The mechanical loading is restricted to the linear range, which decouples the aging and load behavior. A recursive algorithm has been used to solve the hereditary convolution integral that governs the response of each ply. Classical thin-laminate theory is then used to assemble the individual ply response equations and determine the overall laminate response to general in-plane force and moment loading. The method automatically recovers the ply-level stresses and strains, which are often critical to strength and durability predictions. The model can use either lamina compliance or modulus properties as its basis. Several illustrative examples of long-term laminate response to variable loading are presented and the impact of physical aging is explored. It is shown that for multidirectional laminates, the stiffness of the lamina in the fiber directions can allow simplifications of the model.

Generalized Dirichlet distribution in Bayesian analysis

Generalized Dirichlet distribution has a more general covariance structure than Dirichlet distribution. This makes the generalized Dirichlet distribution to be more practical and useful. The concept of complete neutrality will be used to derive the general moment function for the generalized Dirichlet distribution, and then some properties of the generalized Dirichlet distribution will be established. Similar to the Dirichlet distribution, the generalized Dirichlet distribution will be shown to conjugate to multinominal sampling. Two experiments are designed for studying the differences between the Dirichlet and the generalized Dirichlet distributions in Bayesian analysis. A method for generating samples from a generalized Dirichlet in presented. When a prior distribution is either a Dirichlet or a generalized Dirichlet distribution, the way for constructing such a prior is discussed.

On the moment characteristics of distributions

We study different moment characteristics of (mostly discrete) distributions: general, central, binomial, and factorial moments, and also ordinary and factorial semi-invariants. We establish the relations that connect various kinds of “moments” of arbitrary order, and we derive recursion formulas.

The effect of diaphragms on distortion vibration of thin-walled box beams

The present paper considers the problem of static and dynamic distortion behaviour of thin-walled box beams provided with diaphragms. The problem of distortion is formulated in a generalized way using the finite approaches of numerical analysis. The parallel processing FETM method, in combination with the theory of flexure and torsion of thin-walled box beams, is adopted for treatment of the problem. A general distortion moment is defined together with normal and shear distortion stress in the thin-walled box member studied. A discussion on the interaction of flexural, torsion and distortion effects is presented in the formulation suitable for the adoption of the FETM method. Numerical and experimental verifications of the theoretical approach with a discussion of the results obtained are included. © 1998 Published by Elsevier Science Ltd

A multi-level algorithm for the solution of moment problems

We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multi-level algorithms, based on the conjugate gradient method and the Landweber-Richardson method are proposed that determine the "optimal" reconstruction level a posteriori from quantities that arise during the numerical calculations. As an important example we discuss the reconstruction of band-limited signals from irregularly spaced noisy samples, when the actual bandwidth of the signal is not available. Numerical examples show the usefulness of the proposed algorithms

Hermitean Matrix Ensembles and Orthogonal Polynomials

In this article, we investigate orthogonal polynomials associated with complex Hermitean matrix ensembles using the combination of the methods of Coulomb fluid (or potential theory), chain sequences, and Birkhoff–Trjitzinsky theory. We give a general formula for the largest eigenvalue of the N×N Jacobi matrices (which is equivalent to estimating the largest zero of a sequence of orthogonal polynomials) and the two‐level correlation function for the α ensembles (α>0) introduced previously for α>1. In the case of 0<α<1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems.

On a general moment problem on the half axis

We consider the integral representations on the half axis for a pair of nonnegative matrices which satisfy certain matrix equations. The problem of describing all such representations generalizes the classical Stieltjes moment problem.

Eigenelement Statistics of Sample Covariance Matrix in the Correlated Data Case

This paper derives the asymptotic second-order moments of the distinct eigenvalues and the corresponding eigenvectors of a sample covariance matrix in the correlated snapshot case. The general moment formulas obtained herein encompass several previously published results as special cases. The formulas are useful in analysing the statistical performance of eigenstructure-based methods for sinusoidal frequency estimation or for estimating the directions-of-arrival in the case where the emitter signals exhibit temporal correlation.

Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities

The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the p p moment of a sum of independent symmetric random variables to that of the p p and 2 2 moments of the individual variables, are computed in the range 2 > p ≤ 4 2>p\le 4 . This complements the work of Utev who has done the same for p > 4 p>4 . The qualitative nature of the extreme cases turns out to be different for p > 4 p>4 than for p > 4 p>4 . The method developed yields results in some more general and other related moment inequalities.