Traditional Approximation Research Articles

Modified empirical likelihood-based confidence intervals for data containing many zero observations

Data containing many zeroes is popular in statistical applications, such as survey data. A confidence interval based on the traditional normal approximation may lead to poor coverage probabilities, especially when the nonzero values are highly skewed and the sample size is small or moderately large. The empirical likelihood (EL), a powerful nonparametric method, was proposed to construct confidence intervals under such a scenario. However, the traditional empirical likelihood experiences the issue of under-coverage problem which causes the coverage probability of the EL-based confidence intervals to be lower than the nominal level, especially in small sample sizes. In this paper, we investigate the numerical performance of three modified versions of the EL: the adjusted empirical likelihood, the transformed empirical likelihood, and the transformed adjusted empirical likelihood for data with various sample sizes and various proportions of zero values. Asymptotic distributions of the likelihood-type statistics have been established as the standard chi-square distribution. Simulations are conducted to compare coverage probabilities with other existing methods under different distributions. Real data has been given to illustrate the procedure of constructing confidence intervals.

Period spacings of gravity modes in rapidly rotating magnetic stars

Context.Stellar internal magnetic fields have recently been shown to leave a detectable signature on period spacing patterns of gravity modes.Aims.We aim to investigate the effect of the obliquity of a mixed (poloidal and toroidal) dipolar internal fossil magnetic field with respect to the rotation axis on the frequency of gravity modes in rapidly rotating stars.Methods.We used the traditional approximation of rotation to compute non-magnetic modes, and a perturbative treatment of the magnetic field to compute the corresponding frequency shifts. We applied the new formalism to HD 43317, a magnetic, rapidly rotating, slowly pulsating B-type star, whose field has an obliquity angle of about 80°.Results.We find that frequency shifts induced by the magnetic field on high-radial-order gravity modes are larger with increasing obliquity angle, when the magnetic axis is closer to the equatorial region, where these modes are trapped. The maximum value is reached for an obliquity angle of 90°. This trend is observed for all mode geometries.Conclusions.Our results predict that the signature of an internal oblique dipolar magnetic field is detectable using asteroseismology of gravity modes.

High-dimensional Edgeworth expansion of the determinant of sample correlation matrix and its error bound

The paper considers the asymptotic distribution on the determinant of the high-dimensional sample correlation matrix of the Gaussian population with independent components. In particular, when the dimension p and the sample size N satisfy , and , the asymptotic expansion and a uniform error bound of the distribution function of the logarithmic determinant of the sample correlation matrix are obtained by the Edgeworth expansion method. An application of the result to high-dimensional independence test is also proposed, some numerical simulations reveal that the proposed method outperforms the traditional chi-square approximation method and performs as efficient as the method introduced by Jiang and Yang [Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions, Ann. Statist. 41(4) (2013), pp. 2029–2074].

Efficient and robust Schur complement approximations in the augmented Lagrangian preconditioner for the incompressible laminar flows

This paper introduces three new Schur complement approximations for the augmented Lagrangian preconditioner. The incompressible Navier-Stokes equations discretized by a stabilized finite element method are utilized to evaluate these new approximations of the Schur complement. A wide range of numerical experiments in the laminar context determines the most efficient Schur complement approximation and investigates the effect of the Reynolds number, mesh anisotropy and refinement on the optimal choice. Furthermore, the advantage over the traditional Schur complement approximation is exhibited.

Accurate Low-Frequency Approximation for Wires Within a Conducting Half-Space

The most popular approximate models for the analysis of vertical and horizontal wires within a conducting half-space are based on image theory. However, their accuracy is often confined within some low-frequency (LF) range. Such models are often based on the integral equation technique with the so-called “traditional” choice of potentials. In this letter, a new LF approximation is derived from an alternative choice of potentials, i.e., “formulation B.” The accuracy of the proposed approximation and the traditional image approximations is evaluated by comparison with the electromagnetic simulation software FEKO for different wire geometries and different values of earth resistivity in a frequency range up to 10 MHz. It is shown that the new LF approximation is consistently more accurate in all analyzed cases, which widens its applicable frequency range in relation to the image approximations based on the traditional choice of potentials.

A revisit to testing the equality of means for several lognormal distributions

In this paper, we propose a new test statistic for testing the equality of several lognormal means based on Score statistic. Besides the traditional chi-square approximation to Score statistic, we use a parametric bootstrap method called computational approach test (CAT). We compare the proposed tests in terms of type I error rate and power under different number of groups and sample sizes to evaluate efficiency of these tests against the existing tests in the literature. Simulation results indicate that powers of these tests are outstandingly higher compared to their competitors under all the considered cases. Although the performance of chi-square approximation of Score statistic is insufficient in small sample cases for many problems, this approximation performs quite well for our problem even in some small sample sizes. Besides, we illustrate the considered methods using both numerical example and a real data example.

Terahertz-Based Porosity Measurement of Pharmaceutical Tablets: a Tutorial

Porosity, one of the important quality attributes of pharmaceutical tablets, directly affects the mechanical properties, the mass transport and hence tablet disintegration, dissolution and ultimately the bioavailability of an orally administered drug. The ability to accurately and quickly monitor the porosity of tablets during manufacture or during the manufacturing process will enable a greater assurance of product quality. This tutorial systematically outlines the steps involved in the terahertz-based measurement method that can be used to quantify the porosity of a tablet within seconds in a non-destructive and non-invasive manner. The terahertz-based porosity measurement can be performed using one of the three main methods, which are (i) the zero-porosity approximation (ZPA); (ii) the traditional Bruggeman effective medium approximation (TB-EMA); and (iii) the anisotropic Bruggeman effective medium approximation (AB-EMA). By using a set of batches of flat-faced and biconvex tablets as a case study, the three main methods are compared and contrasted. Overall, frequency-domain signal processing coupled with the AB-EMA method was found to be most suitable approach in terms of accuracy and robustness when predicting the porosity of tablets over a range of complexities and geometries. This tutorial aims to concisely outline all the necessary steps, precautions and unique advantages associated with the terahertz-based porosity measurement method.

Reflection and transmission approximations for weak contrast orthorhombic media

Subsurface media are in general anisotropic, and this fact should be taken into account for analyzing reflection and transmission (R/T) coefficients. Orthorhombic (ORT) media are commonly regarded as a practical symmetry system to account for polar anisotropy and azimuthal anisotropy. We have focused on the model made up of two welded ORT half-spaces to analyze the R/T coefficients normalized by vertical energy flux. The two half-spaces have azimuthally nonaligned vertical symmetry planes and are parameterized in two local 3D Cartesian coordinate frames, respectively. The vertical coordinate planes in each local frame coincide with the vertical symmetry planes for the corresponding ORT half-spaces. Under the weak contrast assumption for the two half-spaces, this model is taken as the perturbed model in R/T approximations with the perturbation theory. The unperturbed model is also composed of two unperturbed ORT half-spaces with their different vertical symmetry plane orientations inheriting the counterparts for the perturbed half-spaces above and below the interface, respectively. The unperturbed ORT half-spaces above and below the interface have the same model parameters defined in the two local coordinate frames, respectively. With the perturbations respectively evaluated in the local coordinate frames above and below the interface, the azimuth angle that indicates the local frames’ azimuthal difference is decoupled from the model parameter contrasts. Compared with the traditional approximation method with the perturbation theory in a global coordinate frame, the proposed R/T approximations depend on fewer model parameter discontinuities. We also consider the isotropic background medium under the weak anisotropy assumption. Influences of S-wave singularity points are mitigated by introducing pseudowaves for approximations. Numerical tests are implemented to demonstrate the accuracy.

Asymptotic normality and moderate deviation principle for high-dimensional likelihood ratio statistic on block compound symmetry covariance structure

ABSTRACTThe paper considers a high-dimensional likelihood ratio (LR) test on the block compound symmetric (BCS) covariance structure of a multivariate Gaussian population. When the dimension of each block p, the number of blocks u and the sample size n satisfy that and pu<n−1 as , the asymptotic normality and the moderate deviation principle of the logarithmic LR statistic are obtained. Some numerical simulations demonstrate that the proposed method in high-dimensional BCS test outperforms the traditional Chi-square approximation method, and it is as efficient as the Edgeworth expansion method by Mitsui et al. [Likelihood ratio test statistic for block compound symmetry covariance structure and its asymptoic expansion. Technical Report No.15-03, Statistical Research Group, Hiroshima University, Japan; 2015]. In addition, the proposed method is more applicable because the asymptotic distribution of the test statistic is more concise and the restriction on parameters p, u is milder.

Verification of the current coupling collision probability method with orthogonal flux expansion for the case of single cell

Within this article, current coupling collision probability method with expansion of flux by orthogonal polynomials is verified and validated via comparison with analytical and Monte-Carlo solutions for the single cell with reflective boundary conditions. The results of the calculations for a single unit cell demonstrate good agreement with the results of Monte Carlo calculations. The comparison of the new method with the state-of-the-art flat flux approximation demonstrates either an improved quality of the result for identical cell discretisation or significantly increased computational efficiency to achieve an identical solution. Application of the numerical methods for evaluation of the coefficients of orthogonal polynomials increase the flexibility of the method significantly since higher orders of the polynomials for the neutron source/flux expansion can be applied now for much wider range of the 2D problems. Application of the orthogonal polynomials for the flux expansion allows a significant reduction of the number of calculation regions, while maintaining or improving the accuracy of the results. It also improves efficiency (in terms of the accuracy/run time ratio) of the calculations. Finally, it has a potential to relieve the user from determining the correct discretisation of the calculation regions, which is required in the case of the traditional flat flux approximation and can be quite challenging for the complex geometries.

RETRACTED ARTICLE: Automatic epilepsy detection using hybrid decomposition with multi class support vector method

The epilepsy has been detected from the electroencephalogram (EEG) by utilizing the complex wavelet transform with support vector machine. These methods successfully examine each and every frequency in the EEG signal for detecting the epilepsy with effective manner because epilepsy is one of the most important brain abnormalities which affect the entire brain function. The epilepsy is occurring due to the brain stroke, lack of blood flow, brain fever and so on, these lead to create the number of human deaths. So, the brain epilepsy needs to be analyzed and it has to be detected for improving the epilepsy recognition rate. But the major problem with the epilepsy recognition is the accuracy and efficiency of the classifier because of the traditional approximation entropy only extracts the minimum number of features which is difficult to detect the epilepsy with effective manner. These problems increase the false classification rate while analyzing the brain features. So, brain abnormalities has been automatically recognized by using the various machine learning steps like preprocessing, signal decomposition, feature extraction, feature selection and classification. In this research, the brain Epilepsy is recognized by applying the Hybrid Multi Class Support Vector Machine (HMCSVM). Then the performance of the system is analyzed using the experimental results and discussions.

Rank minimization on tensor ring: an efficient approach for tensor decomposition and completion

In recent studies, tensor ring decomposition (TRD) has become a promising model for tensor completion. However, TRD suffers from the rank selection problem due to the undetermined multilinear rank. For tensor decomposition with missing entries, the sub-optimal rank selection of traditional methods leads to the overfitting/underfitting problem. In this paper, we first explore the latent space of the TRD and theoretically prove the relationship between the TR-rank and the rank of the tensor unfoldings. Then, we propose two tensor completion models by imposing the different low-rank regularizations on the TR-factors, by which the TR-rank of the underlying tensor is minimized and the low-rank structures of the underlying tensor are exploited. By employing the alternating direction method of multipliers scheme, our algorithms obtain the TR factors and the underlying tensor simultaneously. In experiments of tensor completion tasks, our algorithms show robustness to rank selection and high computation efficiency, in comparison to traditional low-rank approximation algorithms.

Gravity-mode period spacings and near-core rotation rates of 611 γ Doradus stars with Kepler

Abstract We report our survey of γ Dor stars from the 4-yr Kepler mission. These stars pulsate mainly in g modes and r modes, showing period-spacing patterns in the amplitude spectra. The period-spacing patterns are sensitive to the chemical composition gradients and the near-core rotation, hence they are essential for understanding the stellar interior. We identified period-spacing patterns in 611 γ Dor stars. Almost every star pulsates in dipole g modes, while about 30% of stars also show clear patterns for quadrupole g modes and 16% of stars present r mode patterns. We measure periods, period spacings, and the gradient of the period spacings. These three observables guide the mode identifications and can be used to estimate the near-core rotation rate. We find many stars are hotter and show longer period-spacing patterns than theory. Using the Traditional Approximation of Rotation (TAR), we inferred the asymptotic spacings, the near-core rotation rates, and the radial orders of the g and r modes. Most stars have a near-core rotation rate around 1 d−1and an asymptotic spacing around 4000 s. We also find that many stars rotate more slowly than predicted by theory for unclear reasons. 11 stars show rotational splittings with fast rotation rates. We compared the observed slope–rotation relation with the theory and find a large spread. We detected rotational modulations in 58 stars and used them to derive the core-to-surface rotation ratios. The interiors rotate faster than the cores in most stars, but by no more than 5%.

The traditional approximation of rotation, including the centrifugal acceleration for slightly deformed stars

Context. The traditional approximation of rotation (TAR) is a treatment of the dynamical equations of rotating and stably stratified fluids in which the action of the Coriolis acceleration along the direction of the entropy (and chemicals) stratification is neglected, while assuming that the fluid motions are mostly horizontal because of their inhibition in the vertical direction by the buoyancy force. This leads to the neglect of the horizontal projection of the rotation vector in the equations for the dynamics of gravito-inertial waves (GIWs) that become separable, such as in the non-rotating case, while they are not separable in the case in which the full Coriolis acceleration is taken into account. This approximation, first introduced in geophysical fluid dynamics for thin atmospheres and oceans, has been broadly applied in stellar (and planetary) astrophysics to study low-frequency GIWs that have short vertical wavelengths. The appoximation is now being tested thanks to direct 2D oscillation codes, which constrain its domain of validity. The mathematical flexibility of this treatment allows us to explore broad parameter spaces and to perform detailed seismic modelling of stars. Aims. The TAR treatment is built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating while an adiabatic treatment of the dynamics of the waves is adopted. In addition, their induced gravitational potential fluctuations is neglected. However, it has been recently generalised with including the effects of a differential rotation. We aim to carry out a new generalisation that takes into account the centrifugal acceleration in the case of deformed stars that are moderately and uniformly rotating. Methods. We construct an analytical expansion of the equations for the dynamics of GIWs in a spheroidal coordinates system by assuming the hierarchies of frequencies and amplitudes of the velocity components adopted within TAR in the spherical case. Results. We derive the complete set of equations that generalises TAR by taking the centrifugal acceleration into account. As in the case of a differentially rotating spherical star, the problem becomes 2D but can be treated analytically if we assume the anelastic and JWKB approximations, which are relevant for low-frequency GIWs. This allows us to derive a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods, which can be used to probe the structure and dynamics of rotating deformed stars thanks to asteroseismology. A first numerical exploration of its eigenvalues and horizontal eigenfunctions shows their variation as a function of the pseudo-radius for different rotation rates and frequencies and the development of avoided crossings.

Numerical enhancements of the microwave resonant cavity method for plasma diagnostics

Microwave resonator method is a well established and sensitive technique of plasma density measurements, especially in low pressure plasmas. The traditional Slater approximation of very small electromagnetic field perturbation is extended and corrected by a more accurate, yet relatively simple numerical model, also addressing further phenomena such as the resonator eigenmodes and the stability of the dominant mode. The presence of a discharge tube is found to be a main source of deviation from the Slater approximation. The pre-calculated numerical correction factors are directly applicable for a range of cavity and discharge tube geometries.

Spectroscopy of Diatomic Molecules in an Adiabatic Approximation

Modern molecular spectroscopy of diatomic molecules is the precision study of the structure and dynamics of electronically excited states of isolated molecules in the gas phase. At the same time, the energy, radiation, magnetic, and electrical characteristics of excited molecules, which are of interest for different physicochemical applications, must be determined and predicted at the experimental (spectral) level of accuracy in a broad range of electron vibrational and rotational excitations. This problem cannot be solved within spectroscopy’s traditional adiabatic approximation, since a large number of intramolecular interactions (so-called perturbations) inevitably distorts not only the regular energy structure of different levels but he nodal structure of the corresponding wave functions as well. The most accurate solution to both direct and inverse spectroscopic problems with respect to perturbed molecular states is based on constructing a reduced system of the coupled radial Schrodinger equations encountered in the quantum-mechanical modeling of the nonadiabatic interaction of electronic states. The nonadiabatic approach can be employed successfully only through the joint use of precision spectroscopic data and highly precise calculations of electronic structure.

Scattering off molecules far from equilibrium.

Pump-probe gas phase X-ray scattering experiments, enabled by the development of X-ray free electron lasers, have advanced to reveal scattering patterns of molecules far from their equilibrium geometry. While dynamic displacements reflecting the motion of wavepackets can probe deeply into the reaction dynamics, in many systems, the thermal excitation embedded in the molecules upon optical excitation and energy randomization can create systems that encompass structures far from the ground state geometry. For polyatomic molecular systems, large amplitude vibrational motions are associated with anharmonicity and shifts of interatomic distances, making analytical solutions using traditional harmonic approximations inapplicable. More generally, the interatomic distances in a polyatomic molecule are not independent and the traditional equations commonly used to interpret the data may give unphysical results. Here, we introduce a novel method based on molecular dynamic trajectories and illustrate it on two examples of hot, vibrating molecules at thermal equilibrium. When excited at 200 nm, 1,3-cyclohexadiene (CHD) relaxes on a subpicosecond time scale back to the reactant molecule, the dominant pathway, and to various forms of 1,3,5-hexatriene (HT). With internal energies of about 6 eV, the energy thermalizes quickly, leading to structure distributions that deviate significantly from their vibrationless equilibrium. The experimental and theoretical results are in excellent agreement and reveal that a significant contribution to the scattering signal arises from transition state structures near the inversion barrier of CHD. In HT, our analysis clarifies that previous inconsistent structural parameters determined by electron diffraction were artifacts that might have resulted from the use of inapplicable analytical equations.

Unknown-but-bounded uncertainty propagation in spacecraft structural system: Interval reduced basis method and its integrated framework

Uncertainty propagation analysis is a key issue for the safety and optimal design of spacecraft structures. This paper proposes an interval reduced basis method (IRBM) for predicting static response of spacecraft structures with unknown-but-bounded uncertainties. Meanwhile, an integrated framework on the basis of IRBM is established by coupling the current finite element codes, which lays the foundation for development of an easy-to-use interval finite element software. In this paper, the novel idea is to approximate the static response using a linear combination of interval basis vectors with undetermined coefficients. The reduced order equations for computing the undetermined coefficients are derived based on Galerkin scheme. For the second-order approximation in IRBM, the static response is easily converted into a family of quadratic programming problems subject to box constraints. The difference of convex functions algorithm is employed to solve these quadratic programming problems. The proposed method can improve the accuracy of the traditional first-order approximation and perform the efficient computation of the second-order approximation of the structural response. Two numerical examples are presented to show the good performance of the proposed method. And an engineering example is performed for its application.

Network equivalent by vector fitting‐based rational approximation in wind power integrated power systems

Considering the electromagnetic transient (EMT) simulation for a large power system is complicated and impractical, a feasible solution is to model the area not of interest (external system) as a frequency-dependent network equivalent circuit by vector fitting (VF)-based rational approximation methods. However, when the external system is wind power integrated, errors may be generated from the equivalent network by traditional approximation methods. A proposed algorithm in which the wind turbines or dynamic loads in the external system are reduced at its working points of steady-state firstly and the results from VF method are passivity enforced can improve the accuracy of network equivalent both in frequency and time domain. A case study is considered to evaluate the performance of the proposed algorithm and the simulation results prove the validity of the algorithm for the good accuracy and less time consuming in simulation.

Period spacings of gravity modes in rapidly rotating magnetic stars

Context.Stellar magnetic fields are often invoked to explain the missing transport of angular momentum observed in models of stellar interiors. However, the properties of an internal magnetic field and the consequences of its presence on stellar evolution are largely unknown.Aims.We study the effect of an axisymmetric internal magnetic field on the frequency of gravity modes in rapidly rotating stars to check whether gravity modes can be used to detect and probe such a field.Methods.Rotation is taken into account using the traditional approximation of rotation and the effect of the magnetic field is computed using a perturbative approach. As a proof of concept, we compute frequency shifts due to a mixed (i.e. with both poloidal and toroidal components) fossil magnetic field for a representative model of a known magnetic, rapidly rotating, slowly pulsating B-type star: HD 43317.Results.We find that frequency shifts induced by the magnetic field scale with the square of its amplitude. A magnetic field with a near-core strength of the order of 150 kG (which is consistent with the observed surface field strength of the order of 1 kG) leads to signatures that are detectable in period spacings for high-radial-order gravity modes.Conclusions.The predicted frequency shifts can be used to constrain internal magnetic fields and offer the potential for a significant step forward in our interpretation of the observed structure of gravity-mode period spacing patterns in rapidly rotating stars.