Integral Approximation Method Research Articles

Locating branch points in power flow problems using the integral approximation method

In this paper, we present an efficient continuation method that can reconstruct solution branches and locate branch points in power flow problems. For this purpose, we propose the construction of an integral approximation using points from a solution branch interval obtained using conventional power flow software. This is in contrast to single-point Padé approximation methods, where the power flow problem must be embedded in a complex plane and therefore requires a new type of software. The paper presents simulation examples designed to explain how the proposed algorithm works, as well as to demonstrate the accuracy and computational efficiency of the method.

Analysis of Scattering characteristics of PEMC sphere by vortex Bessel beams

Generalized Lorenz-Mie theory (GLMT) is well known to analyze the scattering characteristics of spherical surfaces and metamaterials such as perfect electromagnetic conductor (PEMC) sphere. Also, metamaterials are explored very well when illuminated by electromagnetic waves/beams and their characteristics depend upon propagation and scattering response. So, in this manuscript, vortex Bessel beam (VBB) interaction from PEMC sphere has been investigated. The VBB interaction with material particles is of quite interesting due to association of orbital angular momentum (OAM) with them. By employing the Integral localized approximation (ILA) method, the beam‐shape coefficients (BSCs) regarding VBB are obtained. Computations for the scattering efficiency QSca and extinction efficiency QExt for PEMC sphere by considering the co‐polarized and cross‐polarized scattering components are presented and analyzed. Moreover, the influences of OAM mode number, electromagnetic admittance parameter, and diffraction angle are discussed. Numerical results have applications regarding transfer of OAM of VBB to PEMC sphere and associated applications in light manipulation and scattering, optical tweezers, metamaterial fields, and remote sensing.

Potts-Cox survival regression

A Bayesian semi-parametric survival regression model with latent partitions is introduced. Its goal is to predict survival and to cluster survival patients within the context of building prognosis systems. In order to drive cluster formation on individuals, the Potts random partition model is chosen as a prior on the covariates space. For any given partition, the proposed model assumes an interval-wise exponential distribution for the baseline hazard rate. The number of intervals is unknown. It can be estimated with a fused-lasso type penalty given by a sequential double exponential prior. Estimation and inference are done with the aid of Markov chain Monte Carlo. To simplify the computations, the Laplace's integral approximation method is used to estimate some constants and to propose parameter updates within Markov chain Monte Carlo. The methodology is illustrated with an application to cancer survival.

A Mixed-Effects Model in Which the Parameters of the Autocorrelated Error Structure Can Differ between Individuals

Research in psychology has seen a rapid increase in the usage of experience sampling methods and daily diary methods. The data that result from using these methods are typically analyzed with a mixed-effects or a multilevel model because it allows testing hypotheses about the time course of the longitudinally assessed variable or the influence of time-varying predictors in a simple way. Here, we describe an extension of this model that does not only allow to include random effects for the mean structure but also for the residual variance, for the parameter of an autoregressive process of order 1 and/or the parameter of a moving average process of order 1. After we have introduced this extension, we show how to estimate the parameters with maximum likelihood. Because the likelihood function contains complex integrals, we suggest using adaptive Gauss-Hermite quadrature and Quasi-Monte Carlo integration to approximate it. We illustrate the models using a real data example and also report the results of a small simulation study in which the two integral approximation methods are compared.

An Approximation Method for Variational Inequality with Uncertain Variables

In this paper, a Stieltjes integral approximation method for uncertain variational inequality problem (UVIP) is studied. Firstly, uncertain variables are introduced on the basis of variational inequality. Since the uncertain variables are based on nonadditive measures, there is usually no density function. Secondly, the expected value model of UVIP is established after the expected value is discretized by the Stieltjes integral. Furthermore, a gap function is constructed to transform UVIP into an uncertain constraint optimization problem, and the optimal value of the constraint problem is proved to be the solution of UVIP. Finally, the convergence of solutions of the Stieltjes integral discretization approximation problem is proved.

Frequency-dependent analytic models for scattering off finite half planes

Established theories of wave scattering off points and infinite reflectors have limitations for many problems encountered in wave propagation. Our goal is to extend acoustic scattering theory towards a model that not only describes conventional scattering but also the less idealized case of scattering off finite objects. We develop an analytic object-centric forward frequency domain model of the signal at the receiver. The expression is explicitly parameterized in terms of medium and finite object properties that explicitly incorporates frequency-dependence. Our construction uses perturbation theory and integrates over effective point sources along the object surfaces that acts as finite half planes. From this, we derive closed-form expressions using integral approximation methods; and we also determine frequency, material and geometry conditions under which the conventional theories and associated operators apply. We utilize limiting cases to guide and test our model. The model also provides nontraditional extensions of reflection theory, e.g., for sources with spherical radiation patterns. The unified treatment of the continuum of wave phenomena is not only important theoretically but may also be useful in frequency-based object-centric estimation, array optimization and designing signal processing and acquisition schemes.

Pseudo-Likelihood or Quadrature? What We Thought We Knew, What We Think We Know, and What We Are Still Trying to Figure Out

Two predominant computing methods for generalized linear mixed models (GLMMs) are linearization, e.g., pseudo-likelihood (PL), and integral approximation, e.g., Gauss–Hermite quadrature. The primary GLMM package in R, LME4, only uses integral approximation. The primary GLMM procedure in SAS®, PROC GLIMMIX, was originally developed using linearization, but integral approximation methods were added in the 2008 release. This presents a dilemma for GLMM users: Which method should one use, and why? Linearization methods are more versatile and able to handle both conditional and marginal GLMMs. Linearization can be implemented with REML-like variance component estimation, whereas quadrature is strictly maximum likelihood. However, GLMM software documentation and the literature on which it is based tend to focus on linearization’s limitations. Stroup (Generalized linear mixed models: modern concepts, methods and applications, CRC Press, Boca Raton, 2013) reiterates this theme in his GLMM textbook. As a result, “conventional wisdom” has arisen that integral approximation—quadrature when possible—is always best. Meanwhile, ongoing experience with GLMMs and research about their small sample behavior suggest that “conventional wisdom” circa 2013 is often not true. Above all, it is clear there is no one-size-fits-all best method. The purpose of this paper is to provide an updated look at what we now know about quadrature and PL and to offer some general operating principles for making an informed choice between the two. A series of simulation studies investigating distributions and designs representative of research in agricultural and related disciplines provide an overview of each method with respect to estimation accuracy, type I error control, and robustness (or lack thereof) to model misspecification. Supplementary materials accompanying this paper appear online.

Restricted maximum likelihood estimation of a censored random effects panel regression model

Panel data sets have been developed in various areas, and many recent studies have analyzed panel, or longitudinal data sets. Maximum likelihood (ML) may be the most common statistical method for analyzing panel data models; however, the inference based on the ML estimate will have an inflated Type I error because the ML method tends to give a downwardly biased estimate of variance components when the sample size is small. The under estimation could be severe when data is incomplete. This paper proposes the restricted maximum likelihood (REML) method for a random effects panel data model with a censored dependent variable. Note that the likelihood function of the model is complex in that it includes a multidimensional integral. Many authors proposed to use integral approximation methods for the computation of likelihood function; however, it is well known that integral approximation methods are inadequate for high dimensional integrals in practice. This paper introduces to use the moments of truncated multivariate normal random vector for the calculation of multidimensional integral. In addition, a proper asymptotic standard error of REML estimate is given.

O точности численных методов решения уравнений Вольтерра I рода в задачах теплопереноса

The article is devoted to the study of the accuracy of methods for solving the measurement challenge that arises when determining the temperature inside an object subjected to the influence of an external control thermal effect. The approach to the construction of a numerical solution of the measurement problem associated with the problem of determining temperature is based on reducing the initial problem to solving an integral equation that are characterized the direct dependence of temperature on the measured values. The integral equation is obtained using the direct and inverse Laplace transforms with the involvement of the regularizing approach and the regularization teqhiqie. The resulting integral equation is the Volterra equations of the first kind of convolution type with a specific kernel. In this paper, we investigate the accuracy of numerical methods for solving an integral equation with a specific kernel from the point of view of the mechanisms for the implementation of machine arithmetic. Computing method schemas are based on the product integration method, squaring the middle rectangles. The article also presents the results of a study of the error of the computational scheme of the order-optimal method based on the application of Fourier transforms and the projection regularization method. The method is used to directly solve the original problem without redusing it to an integral model and allows one to obtain numerical solutions with guaranteed accuracy. In order to obtain experimental estimates of the accuracy of numerical methods and a comparative analysis of the machine accuracy of the integral approximation methods and the order-optimal method, a computational experiment was carry out. The experimental results indicate that it is possible to obtain the numerical solutions of the measurement challenge with a high level of accuracy.

Beam shape coefficients calculation for an elliptical Gaussian beam with 1-dimensional quadrature and localized approximation methods

The use of a shaped beam for applications relying on light scattering depends much on the ability to evaluate the beam shape coefficients (BSC) effectively. Numerical techniques for evaluating the BSCs of a shaped beam, such as the quadrature, the localized approximation (LA), the integral localized approximation (ILA) methods, have been developed within the framework of generalized Lorenz-Mie theory (GLMT). The quadrature methods usually employ the 2-/3-dimensional integrations. In this work, the expressions of the BSCs for an elliptical Gaussian beam (EGB) are simplified into the 1-dimensional integral so as to speed up the numerical computation. Numerical results of BSCs are used to reconstruct the beam field and the fidelity of the reconstructed field to the given beam field is estimated. It is demonstrated that the proposed method is much faster than the 2-dimensional integrations and it can acquire more accurate results than the LA method. Limitations of the quadrature method and also the LA method in the numerical calculation are analyzed in detail.

Thermal degradation kinetics and density functional study of a macromonomer containing dual-lactone

A macromonomer containing dual-lactone, MMDL, was characterized by thermal degradation kinetics and density functional theory (DFT) calculations. The frontier molecular orbitals, molecular structural geometry, molecular electrostatic potential (MEP) and electrostatic potential (ESP) maps were determined with the help of structure optimizations based on the DFT method with standard 3–21G* as a basis set that has polarization functions on the second row atoms only. The 3–21G* basis set comprises the same number of primitive Gaussian functions. The electronic properties, such as electron affinity, HOMO–LUMO energies, ionization energy, electronegativity, chemical potential, global hardness and softness, global electrophilicity were computed with the help of the DFT method. The MEP and ESP maps were determined to predict the reactive sites of the macromonomer. Finally, the activation energy and thermal degradation mechanism for the initial part of the decomposition process under non-isothermal conditions were determined from the thermogravimetric analysis by integral approximation methods. The decomposition activation energies of the macromonomer were computed with the help of Flynn–Wall–Ozawa, Coats–Redfern and Tang methods. The kinetic equations showed that the reaction mechanism was an R1 mechanism, which is a phase boundary-controlled reaction (one-dimensional movement) solid-state mechanism.

On Some Mixture Models for Over-dispersed Binary Data

In this paper, we consider several binomial mixture models for fitting over-dispersed binary data. The models range from the binomial itself, to the beta-binomial (BB), the Kumaraswamy distributions I and II (KPI \& KPII) as well as the McDonald generalized beta-binomial mixed model (McGBB). The models are applied to five data sets that have received attention in various literature. Because of convergence issues, several optimization methods ranging from the Newton-Raphson to the quasi-Newton optimization algorithms were employed with SAS PROC NLMIXED using the Adaptive Gaussian Quadrature as the integral approximation method within PROC NLMIXED. Our results differ from those presented in Li, Huang and Zhao (2011) for the example data sets in that paper but agree with those presented in Manoj, Wijekoon and Yapa (2013). We also applied these models to the case where we have a $k$ vector of covariates $(x_1, x_2, \ldots, x_k)^{'}$. Our results here suggest that the McGBB performs better than the other models in the GLM framework. All computations in this paper employed PROC NLMIXED in SAS. We present in the appendix a sample of the SAS program employed for implementing the McGBB model for one of the examples.

Molecular structure, kinetics and mechanism of thermal decomposition, molecular electrostatic potential, thermodynamic parameters and HOMO–LUMO analysis of coumarin-containing graft copolymer

This paper contains a combined theoretical and experimental study of thermodynamic, thermogravimetric and electronic properties of coumarin-containing graft copolymer. All the theoretical calculations were performed using the DFT (B3LYP) method with 6-311 + G (d, p) basis set in the ground state. From the optimized geometry of the coumarin-containing graft copolymer, atomic charge distributions, molecular electrostatic potential (MEP) surfaces, frontier molecular orbitals (FMOs) and thermodynamic properties such as entropy, enthalpy and specific heat capacity have been calculated theoretically. Finally, the activation energy and thermal degradation mechanism for graft copolymer were obtained by integral approximation methods under non-isothermal conditions. The activation energies (Ea) for thermal degradation of graft copolymer were determined by Flynn–Wall–Ozawa (FWO), Tang and Coats–Redfern (CR) methods. The study of kinetic equations showed that the reaction mechanism for graft copolymer was R3 mechanism, phase boundary-controlled reaction (contracting volume) of solid-state mechanism.

Blending of Poly(ethyl methacrylate) with Poly(2-hydroxy-3-phenoxypropyl methacrylate): Thermal and Optical Properties

The miscibility behavior of poly(ethyl methacrylate) [poly(EMA)] with poly(2-hydroxy-3-phenoxypropyl methacrylate) [poly(PPMA)] was studied by DSC technique. Thermogravimetric analysis was used to investigate the thermal degradation mechanism and activation energies for initial decomposition process under non-isothermal conditions by integral approximation methods. The reaction mechanism of decomposition process using the kinetic equations of Coats–Redfern method was described by kinetic equations of phase-boundary-controlled reaction, R 3 mechanism. The activation energies for R 3 mechanism are very close to the values obtained in both Kissinger (115.6 kJ/mol) and Flyn–Wall–Ozawa (121.1 kJ/mol) methods. The optical and dispersion parameters such as refractive index, extinction coefficient, steepness parameter, single-oscillator energy and dispersion energy of blend polymer were determined in the visible range as a function of wavelength. The energy of Urbach corresponding to the width of the band tails of localized states and the optical band gap constant were calculated to be 1.08 and 3.33 eV, respectively.

ON THE SMALL SAMPLE BEHAVIOR OF GENERALIZED LINEAR MIXED MODELS WITH COMPLEX EXPERIMENTS

Generalized linear mixed models (GLMMs), regardless of the software used to implement them (R, SAS, etc.), can be formulated as conditional or marginal models and can be computed using pseudo-likelihood, penalized quasi-likelihood, or integral approximation methods. While information exists about the small sample behavior of GLMMs for some cases- notably RCBDs with Binomial or count data- little is known about GLMMs for continuous proportions (e.g. Beta) or time-to-event (e.g. Gamma) data or for more complex designs such as the split-plot. In this presentation we review the major model formulation and estimation options and compare their small sample performance for cases listed above.

Perturbed Integral Collocation Approximation Method For The Solution of a Class of Second Order Differential Equations

This paper concerns the numerical approximation of a class of second order linear ordinary differential equations by Perturbed Integral Collocation Method. The application of the method resulted to system of linear algebraic equations which are then solved by Gaussian elimination method. The results obtained for some illustrative examples showed that the Perturbed Integral Collocation Method is efficient, accurate and reliable. Examples are given to illustrate the method. y p x y x p p x y x p x p p 

Integral Collocation Approximation Methods for the Numerical Solution of Linear Integro Differential Equations.

Integral Collocation Approximation Methods for the Numerical Solution of Linear Integro Differential Equations.

Thermal Degradation Behavior of Methyl Methacrylate Derived Copolymer

A novel copolymer of 2-hydroxy-3-menthyloxy-1-(4-methoxyphenyl)-3-oxopropyl methacrylate (HMOPMA) and methyl methacrylate (MMA), [poly(HMOPMA-co-MMA)], was synthesized by free radical polymerization. The percentages of HMOPMA and MMA units obtained in the copolymer composition were 19% and 81%, respectively. The solubility parameter of poly(HMOPMA-co-MMA) was found to be 10.3 cal(1/2) cm(-3/2) by turbidimetric titration method. Thermal degradation mechanism and activation energies for initial decomposition process under non-isothermal conditions were determined by integral approximation methods from the thermogravimetric study. The decomposition activation energies of poly(HMOPMA-co-MMA) using Kissinger and Flyn-Wall-Ozawa methods were calculated as 119.99 kJ/mol and 125.34 kJ/mol, respectively. The study of kinetic equations showed that the reaction mechanism of decomposition process in the conversion range studied progressed with D1 mechanism, one-dimensional diffusion of deceleration type of solid state mechanism.

Kinetics of pyrolysis and combustion of oil shale sample from thermogravimetric data

Thermogravimetric (TG) data of oil shale obtained at MI (Waste to Energy laboratory) were studied to evaluate the kinetic parameters for El-Lujjun oil shale samples. Different heating rates were employed simulating pyrolysis reaction using Nitrogen as purging gas up to ∼800°C. The extent of char combustion was found out by relating TG data for pyrolysis and combustion with the ultimate analysis. Due to distinct behavior of oil shale during pyrolysis, TG curves were divided into three separate events: moisture release; devolatization; and evolution of fixed carbon/char, where for each event, kinetic parameters, based on Arrhenius theory, were calculated. Three methods were used and compared: integral method; direct Arrhenius plot method; and temperature integral approximation method. Results showed that integral method is closer to the experiment, while no relationship was observed between activation energy and the heating rate.

An Approximate Solution of Energy Partition in Grind-Hardening Process

In order to estimate the energy partition within the workpiece, an integral approximation solution of energy partition is found in the grind-hardening process. Heat transfer models of the abrasive grain, workpiece, chips and fluid were analyzed by using the integral approximation method. The present model can calculate the energy partition with and without film boiling in the grinding zone. The temperature underneath the surface of the workpiece was measured using semi--natural thermocouple method. The workpiece background temperature calculated by the present model agreed very well with the experimental results. Energy partition model can be used to calculate the grinding temperature for controlling thermal damage and predicting the harden layer depth in grinding.