Exact Moments Research Articles

Method for recording ultrasound images of moving objects

The invention relates to a method for recording ultrasound images of moving objects (1), according to which an ultrasound transmitter transmits ultrasound onto the object (1) and an ultrasound receiver receives the ultrasound reflected by the object (1). A data processing system (4) detects and stores the received ultrasound and the corresponding movement state of the object (1). The ultrasound transmitter and ultrasound receiver are moved alongside the moving object (1) and produce images of individual partial image areas (9) of the object (1). At those moments when the movement of the object is greatest, no or only few images are taken. When the object moves only little several images of individual partial image areas (9) of the object (1) are taken and/or analysed. By assigning the individual partial image areas (9) to the corresponding movement states of the object (1) either selective individual movement states or entire movement sequences of the object (1) can be shown in real time. The method provided for by the invention is especially suited for taking images of the blood vessels of an organism, whereby the exact moments when the individual partial image areas (9) of the blood vessel are recorded are controlled by the electrocardiogram of the organism.

Radial Moment Calculations of Coupled Electron-Photon Beams

We consider the steady-state transport of normally incident pencil beams of radiation in slabs of material. A method has been developed for determining the exact radial moments of three-dimensional (3-D) beams of radiation as a function of depth into the slab, by solving systems of one-dimensional (1-D) transport equations. We implement these radial-moment equations in the ONEBFP discrete ordinates code and simulate energy-dependent, coupled electron-photon beams using CEPXS-generated cross sections. Modified PN synthetic acceleration is employed to speed up the iterative convergence of the 1-D charged-particle calculations. For high-energy photon beams, a hybrid Monte Carlo/discrete ordinates method is examined. We demonstrate the efficiency of the calculations and make comparisons with 3-D Monte Carlo calculations. Thus, by solving 1-D transport equations, we obtain realistic multidimensional information concerning the broadening of electron-photon beams. This information is relevant to fields such as industrial radiography, medical imaging, radiation oncology, particle accelerators, and lasers.

X-ray Faraday effect at theL2,3edges of Fe, Co, and Ni: Theory and experiment

The x-ray Faraday effect at the ${L}_{2,3}$ edges of the $3d$ ferromagnets Fe, Co, Ni and of ${\mathrm{Fe}}_{0.5}{\mathrm{Ni}}_{0.5}$ alloy is studied both theoretically and experimentally. We perform ab initio calculations of the x-ray Faraday effect on the basis of the local spin-density approximation and we adopt the linear-response formalism to describe the material's response to the incident light. Experimental x-ray Faraday rotation and ellipticity spectra are measured with linearly polarized soft-x-ray synchrotron radiation at BESSY, Berlin. The measured x-ray Faraday rotations are remarkably large, up to $2.8\ifmmode\times\else\texttimes\fi{}{10}^{5} \mathrm{d}\mathrm{e}\mathrm{g}/\mathrm{m}\mathrm{m},$ which is more than one order of magnitude larger than those observed in the visible range. From the measured Faraday spectra we determine the intrinsic dichroic contributions to the dispersive and absorptive parts of the refractive index, and compare these to ab initio calculated counterparts. The theoretical dichroic spectra are in good qualitative agreement with the experimental data. The inclusion of the spin polarization of the core states leads to a small, yet non-negligible, improvement of the theoretical dichroic spectra. Our results illustrate that the many-particle x-ray excitation spectrum can be sufficiently well approximated by the Kohn-Sham single-particle spectrum. From the computed magneto-x-ray spectra we determine, using the sum rules, the orbital moments, which we compare to the exact orbital moments.

Reliability, Risk, and Uncertainty Analysis Using Generic Expectation Functions

In engineering design and analysis, mathematical models that generally involve a number of uncertain parameters are frequently employed for decision making. Over the years, a number of techniques have been developed to quantify model output uncertainty contributed by uncertain input parameters. Typically, the methods that are easy to apply may give inaccurate estimates of model output uncertainty. Other methods that reliably produce very accurate results are either difficult to apply or require intensive computational effort. This paper describes the development of generic expectation functions as a function of means and coefficients of variation of input random variables. The generic expectation functions are straightforward to develop, and apply to problems related to reliability, risk, and uncertainty analysis. Several expectation functions based on commonly used probability distributions have been developed. Using them, any order of moment can be estimated exactly. It is found that if exact moments of the model output are available, one can find a good estimate of reliability, risk, and uncertainty of a system without knowing its model output distribution exactly. This technique is applicable when an output variable is a function of several independent random variables in multiplicative, additive, or combined (multiplicative and additive) forms. A practical example is presented to demonstrate the application of generic expectation functions.

The exact likelihood for a state space model with stochastic inputs

In this work, we derive exact and approximate expressions for the conditional mean and variance of the initial state of a state space model, allowing for unit roots and stochastic inputs. These results provide adequate initial conditions to compute the exact likelihood using the Kalman filter. The exact conditional moments are the best choice when the stochastic structure of the inputs is known. If this is not the case, the approximate expressions are a good alternative, as some simulation results illustrate.

Moment operations on random variables, with applications for probabilistic analysis

A general method is developed for conducting simple operations on random variables, avoiding difficult integrals and singularities, which must be overcome when obtaining exact solutions. For sum, difference and product operations, and combinations thereof, exact moments are first determined from the moments of the constituent variables. The method of orthogonal expansion, developed in the previous paper [Probabilistic Engineering Mechanics 2000;15:371–379], is then used to produce approximate probability density functions (PDFs). The quotient operation is also considered; it requires knowledge of the negative moments of the denominator variable. The quotient and difference operations are used in a first example to establish PDFs for the hazard quotient and excess wind loading on a concrete chimney. A second example demonstrates how the proposed method may be used as an alternative to Monte Carlo simulation for simple probabilistic risk calculations; a PDF for predicted contaminant concentration at a groundwater well compares favorably with a histogram obtained by simulation.

Exact Moments of Curvature Measures in the Boolean Model

Curvature measures are important for the characterization of spatial structures since many physical phenomena depend essentially on the geometry of spatial configurations. Curvature-weighted correlation functions can be defined and calculated explicitly within the Boolean model. This standard model in statistical physics generates random geometries by overlapping grains (spheres, sticks) each with arbitrary location and orientation. A general decomposition relation for characteristic functions of submanifolds based on Chern's differential kinematic formula is used to obtain exact expressions for mean values and second order moments of curvature integrals, i.e., of Minkowski functionals. An exact relation for the canonical and grand-canonical second order moments of the morphological Minkowski measures can be derived based only on thermodynamic arguments. The results are applied on the morphological thermodynamics of complex fluids where specific heats are expressed in terms of curvature moments.

Convergence of the multipole expansion for electrostatic potentials of finite topological atoms

The exact atomic electrostatic potential (AEP) and atomic multipole moments are calculated using the topological partitioning of the electron density. High rank (l⩽20) spherical tensor multipole moments are used to examine the convergence properties of the multipole expansion. We vary independently the maximum multipole rank, lmax, and the radius of the spherical grid around an atom in a molecule where we measure the discrepancy between the exact AEP and the one obtained via multipole expansion. The root mean square values are between 0.1 and 1.6 kJ/mol for four atoms (C, N, O, S) on a spherical grid with the ρ=0.001 a.u. convergence radius and for lmax=4. Our calculations demonstrate that this fast convergence is due to the decay of the electron density. We show that multipole moments generated by finite atoms are adequate for use in the multipole expansion of the electrostatic potential, contrary to some claims made in the literature. Moreover they can be used to model intermolecular and in principle intramolecular interactions as well.

Approximating the distribution of the maximum partial sum of normal deviates

Abstract The largest partial sum of deviations from the mean is a statistic of importance in several areas of application, including hydrology and in testing for a change-point. Approximations to its distribution for the simple normal case have appeared in the literature, based either on functionals of Brownian motion asymptotics or on a methodology developed for boundary crossing problems in sequential analysis. The former approximation is inaccurate except for very large samples, while the latter is based on a rather difficult theory. In this paper, we first review some early findings about exact moments and extend them somewhat. We then use these moments to fit simple chi-squared and beta approximations and show that they work very well.

Theory & Methods: A Weighted Least Squares Approach to Levene's Test of Homogeneity of Variance

In 1960 Levene suggested a potentially robust test of homogeneity of variance based on an ordinary least squares analysis of variance of the absolute values of mean‐based residuals. Levene's test has since been shown to have inflated levels of significance when based on the F‐distribution, and tests a hypothesis other than homogeneity of variance when treatments are unequally replicated, but the incorrect formulation is now standard output in several statistical packages. This paper develops a weighted least squares analysis of variance of the absolute values of both mean‐based and median‐based residuals. It shows how to adjust the residuals so that tests using the F‐statistic focus on homogeneity of variance for both balanced and unbalanced designs. It shows how to modify the F‐statistics currently produced by statistical packages so that the distribution of the resultant test statistic is closer to an F‐distribution than is currently the case. The weighted least squares approach also produces component mean squares that are unbiased irrespective of which variable is used in Levene's test. To complete this aspect of the investigation the paper derives exact second‐order moments of the component sums of squares used in the calculation of the mean‐based test statistic. It shows that, for large samples, both ordinary and weighted least squares test statistics are equivalent; however they are over‐dispersed compared to an F variable.

Exact smoothing for stationary and non-stationary time series

In this work we derive an analytical relationship between exact fixed-interval smoothed moments and those obtained from an arbitrarily initialized smoother. Combining this result with a conventional smoother we obtain an exact algorithm that can be applied to stationary, non-stationary or partially non-stationary systems. Other advantages of our method are its computational efficiency and numerical stability. Its extension to forecasting, filtering, fixed-point and fixed-lag smoothing is immediate, as it only requires modification of a conditioning information set. Three examples illustrate the adverse effect of an inadequate initialization on smoothed estimates.

On estimating equations for parameters in generalized linear mixed models with application to binary data

In view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models and other nonlinear variance component models. For example, we refer to the penalized quasi-likelihood (PQL) approach of Breslow and Clayton (1993), the Stein-type estimating function based approach of Waclawiw and Liang (1993), and the corrected PQL approach of Breslow and Lin (1995). Recently, Sutradhar and Godambe (1998) provided a semiparametric solution to the estimation problem dealt with by Waclawiw and Liang. In the present article, we provide a semiparametric solution to the estimation problem investigated by Breslow and Lin, and Breslow and Clayton. More specifically, we propose a two-step joint estimating equations approach to estimate the model parameters. In the first step, we use an estimating function based approach to obtain the estimates of the random effects. In the second step, following Prentice and Zhao (1991), we construct the first two moment based joint estimating equations for the regression parameters and the variance component of the random effects. As the exact first and second order moments of the generalized linear mixed models are not available, these moments are obtained first by expanding the conditional moments for given random effects about their estimating function based estimates and then taking the expectation of the conditional moments over the true distribution of the random effects. Since binary mixed models arise in many biomedical application areas, for example, computations are provided in detail for this special case. We also discuss a second approach, namely, a three-step ad hoc estimating equations approach, in the context of the special binary mixed models. The performance of the proposed estimators are examined through a simulation study. The estimation procedure is also illustrated through an analysis of the child health and development study (CHDS) data from Yerushalmy (1970).

Basis-set convergence of the molecular electric dipole moment

The electric dipole moments (μ) of BH and HF are computed in conventional calculations employing different correlation-consistent basis sets at the levels of Hartree–Fock theory, second-order perturbation theory, and coupled cluster theory with single and double excitations, and single and double excitations with a perturbative triples correction. The basis-set convergence of μ is examined by comparison with results obtained with explicitly correlated wave function models. Inclusion of diffuse functions in the basis set is essential for accurate calculations of μ. They speed up the convergence at the Hartree–Fock level significantly and make the convergence at the correlated levels systematic. Once the outer valence regions important for μ are described accurately via the diffuse functions, the convergence at the correlated levels is governed by the interelectronic Coulomb singularity. For the aug-cc-pVXZ basis sets, the correlation contribution to μ follows μXcorr=μlimcorr+aX−3, which is similar to the form for the correlation energy, and extrapolated values based on this form represent a significant improvement on the ordinary basis-set results. Our estimates of the exact dipole moments μe(HF)=1.8037±0.0007 D and μ0(BH)=1.3586±0.0007 D are in excellent agreement with the experimental values μe(HF)=1.803±0.002 D and μ0(BH)=1.27±0.21 D.

Random p-content of a p-parallelotope in Euclidean n-space

Techniques currently available in the literature in dealing with problems in geometric probabilities seem to rely heavily on results from differential and integral geometry. This paper provides a radical departure in this respect. By using purely algebraic procedures and making use of some properties of Jacobians of matrix transformations and functions of matrix argument, the distributional aspects of the random p-content of a p-parallelotope in Euclidean n-space are studied. The common assumptions of independence and rotational invariance of the random points are relaxed and the exact distributions and arbitrary moments, not just integer moments, are derived in this article. General real matrix-variate families of distributions, whose special cases include the mulivariate Gaussian, a multivariate type-1 beta, a multivariate type-2 beta and spherically symmetric distributions, are considered.

Random p-content of a p-parallelotope in Euclidean n-space

Techniques currently available in the literature in dealing with problems in geometric probabilities seem to rely heavily on results from differential and integral geometry. This paper provides a radical departure in this respect. By using purely algebraic procedures and making use of some properties of Jacobians of matrix transformations and functions of matrix argument, the distributional aspects of the random p-content of a p-parallelotope in Euclidean n-space are studied. The common assumptions of independence and rotational invariance of the random points are relaxed and the exact distributions and arbitrary moments, not just integer moments, are derived in this article. General real matrix-variate families of distributions, whose special cases include the mulivariate Gaussian, a multivariate type-1 beta, a multivariate type-2 beta and spherically symmetric distributions, are considered.

Exact moments of the product limit estimator

In this article we derive the exact kth moment of the product limit estimator without any assumption on the underlying survival distribution model. An approximate formula is also derived which is more amenable to estimation. Under the assumption of the proportional hazard model these results specialize to those obtained by Chen et al. (1982, J. Am. Statist. Assoc. 77, 141–144). Biases and comparative performances of commonly used asymptotic variance formulas with the exact and approximate formulas are discussed numerically for some specific models. In addition, the first two exact moments of the two nonparametric estimators of the cumulative hazard function commonly used, are also derived.

Uniform Positive-Weight Quadratures for Discrete Ordinate Transport Calculations

Mechanical quadratures that allow systematic improvement and solution convergence are derived for application of the discrete ordinates method to the Boltzmann transport equation. The quadrature directions are arranged on n latitudinal levels, are uniformly distributed over the unit sphere, and have positive weights. Both a uniform and equal-weight quadrature set UEn and a uniform and Gauss-weight quadrature set UGn are derived. These quadratures have the advantage over the standard level-symmetric LQn quadrature sets in that the weights are positive for all orders, and the solution may be systematically converged by increasing the order of the quadrature set. As the order of the quadrature is increased the points approach a uniform continuous distribution on the unit sphere and the quadrature is invariant with respect to spatial rotations. The numerical integrals converge for continuous functions as the order of the quadrature is increased.Numerical calculations were performed to evaluate the application of the UEn quadrature set. Comparisons of the exact moments and those calculated using the UEn quadrature set demonstrate that the moment integrals are performed accurately except for distributions that are very sharply peaked along the direction of the polar axis. A series of DORT transport calculations of the >1-MeV neutron flux for a typical reactor core/pressure vessel geometry were also carried out. These calculations employed the UEn (n = 6, 10, 12, 18, and 24) quadratures and indicate that the UEn solutions have converged to within ~0.5%. The UE24 solutions were also found to be more accurate than the calculations performed with the S16 level-symmetric quadratures.

Full configuration interaction benchmark calculations of first-order one-electron properties of BH and HF

Full configuration interaction benchmark calculations have been carried out for the electric dipole and quadrupole moments and the electric field gradient at the nuclei of BH and HF. The accuracy of perturbation theory from second to fourth order and coupled cluster theory with up to triple excitations has been investigated. For all the properties the coupled cluster models outperform the perturbation series. The convergence of the series of coupled cluster models is significantly faster and more systematic than the one of the perturbation series, and only the coupled cluster series defines a hierarchy of models with well defined levels of accuracy. The CCSD(T) model is a good approximation to the full coupled cluster singles, doubles, and triples model. It recovers 80%–90% of the full effect of triple excitations, and the small error due to the approximate description of triple excitations is comparable in size to the error due to neglect of higher-order excitations. For accurate calculations, the CCSD(T) model is the preferred model for all the properties. Our estimates of the exact dipole moments μe(HF)=1.802±0.003 D and μ0(BH)=1.356±0.004 D are in perfect agreement with the experimental values of μe(HF)=1.803±0.002 D and μ0(BH)=1.27±0.21 D.

Moments and generating functions for the absortion distribution and its negative binomial analogue

The moments of the absorption are difficult to obtain. The generating functions are basic hypergeometric functions. This paper shows how to define two shift operators that allow elementary arguments to be used to develop recursions for the expected values of general functions. The exact moments of the distribution follow. The generating function for the negative binomial analogue gives the moments directly.

Exact Distribution of the Regression Esti Mator in Double Sampling

Asymptotic results about the distribution of the regression estimator in a double sampling scheme when using an auxiliary covariate have been established by Engel and Walstra (1991). Conniffe (1985) studied in the finite sample case the closely related model of simultaneous regression equations in econometrics with unequal numbers of observations. The aim of this paper is to provide a normal approximation of the distribution of the regression estimator. First, the exact distribution is derived in the Gaussian framework and the exact moments of the regression estimator are deduced. An Edgeworth expansion of the exact distribution is then calculated in order to propose a normal approximation. The exact distribution, the Edgeworth expansion and the normal approximation are compared throughout an example.