Boundary Correction Research Articles

Miscellanea Kernel-Type Density Estimation on the Unit Interval

We consider kernel-type methods for the estimation of a density on 0,1 which eschew explicit boundary correction. We propose using kernels that are symmetric in their two arguments; these kernels are conditional densities of bivariate copulas. We give asymptotic theory for the version of the new estimator using Gaussian copula kernels and report on simulation comparisons of it with the beta-kernel density estimator of Chen ([1]). We also provide automatic bandwidth selection in the form of ‘rule-of-thumb’ bandwidths for both estimators. As well as its competitive integrated squared error performance, advantages of the new approach include its greater range of possible values at 0 and 1, the fact that it is a bona fide density and that the individual kernels and resulting estimator are comprehensible in terms of a single simple picture.

Boundaries and the Casimir effect in noncommutative space-time

We calculate modifications to the scalar Casimir force between two parallel plates due to space-time noncommutativity. We devise a heuristic approach to overcome the difficulties of describing boundaries in noncommutative theories and predict that boundary corrections are of the same order as noncommutative volume corrections. Further, both corrections have the form of more conventional finite surface effects.

Free volume of the hard spheres gas

The Enskog factor χ plays a central role in the theory of dense gases, quantifying how the finite size of molecules causes many physical quantities, such as the equation of state, the mean free path, and the diffusion coefficient, to deviate from those of an ideal gas. We suggest an intuitive but rigorous derivation of this fact by showing how all these instances of χ amount to different ways of looking at the derivative of the free volume with respect to the packing density. We show how to compute the free volume explicitly for finitely many molecules in a finite box and demonstrate excellent agreement between its derivative and mean free paths obtained from computer simulations, where the number of molecules N varies from 1000 down to 2, and where the mean free paths vary from many times the molecular diameter at low density down to a small fraction of the molecular diameter at high density. Since the boundary corrections involved are relatively simple and intuitive this strengthens the link between the teaching of large N theory for real physical systems, and the running of small N simulations in undergraduate physics laboratories.

Re-interpreting conventional interval estimates taking into account bias and extra-variation

BackgroundThe study design with the smallest bias for causal inference is a perfect randomized clinical trial. Since this design is often not feasible in epidemiologic studies, an important challenge is to model bias properly and take random and systematic variation properly into account. A value for a target parameter might be said to be "incompatible" with the data (under the model used) if the parameter's confidence interval excludes it. However, this "incompatibility" may be due to bias and/or extra-variation.DiscussionWe propose the following way of re-interpreting conventional results. Given a specified focal value for a target parameter (typically the null value, but possibly a non-null value like that representing a twofold risk), the difference between the focal value and the nearest boundary of the confidence interval for the parameter is calculated. This represents the maximum correction of the interval boundary, for bias and extra-variation, that would still leave the focal value outside the interval, so that the focal value remained "incompatible" with the data. We describe a short example application concerning a meta analysis of air versus pure oxygen resuscitation treatment in newborn infants. Some general guidelines are provided for how to assess the probability that the appropriate correction for a particular study would be greater than this maximum (e.g. using knowledge of the general effects of bias and extra-variation from published bias-adjusted results).SummaryAlthough this approach does not yet provide a method, because the latter probability can not be objectively assessed, this paper aims to stimulate the re-interpretation of conventional confidence intervals, and more and better studies of the effects of different biases.

Smoothing Splines with Boundary Correction

Regular smoothing splines are known to have a type of boundary bias problem that can reduce their estimation efficiency. In this paper, a boundary corrected smoothing spline with general order is designed in a way that the risk will decay at an optimal rate. An O(n) algorithm is also developed to compute the resultant estimator efficiently.

Wind Tunnel Boundary Corrections for a Full Scale 1903 Wright Flyer Replica

A panel code is developed in order to assess and correct wind-tunnel boundary interferences during a test of a full-scale reproduction of the 1903 Wright Flyer in the Langley Full-Scale Tunnel (LFST). The LFST is a very large, low-speed wind tunnel with a 3 4-open test section. The principal concerns relating to boundary corrections are the relatively large size of the replica relative to the test-section dimensions, coupled with the lack of modern correction techniques for aeronautical testing in open-jet test sections. The Flyer is represented either by simple lifting surfaces, with the accuracy of the representation validated by direct calculations using the panel code CMARC, or by an array of simple horseshoe vortex singularities. Vortex ring panels are used in the boundary correction code. The solution starts by solving the flow in a closed test section in order to obtain the flow properties on the boundaries. An iterative method, based on the one-step method employed in adaptive wall wind-tunnel work, is then used to adapt the boundary geometry to a constant-pressure condition. Finally the boundary interference is obtained by calculating the velocities induced in the test region by the boundary singularities. Results are presented for different test-section boundary configurations: fully closed, fully open, and 3 4 open. Corrections are compared to calculations using the classical method of images.

On a Rational Linear Approximation of Fourier Series for Smooth Functions

The Fourier series of a smooth function on a compact interval usually has slow convergence due to the Gibbs phenomena. A class of Fourier-Pade approximations is introduced and studied for performing a boundary correction. Additional acceleration is achieved by applying Fourier---Bernoulli scheme.

A generalized reflection method of boundary correction in kernel density estimation

The kernel method of estimation of curves is now popular and widely used in statistical applications. Kernel estimators suffer from boundary effects, however, when the support of the function to be estimated has finite endpoints. Several solutions to this problem have already been proposed. Here the authors develop a new method of boundary correction for kernel density estimation. Their technique is a kind of generalized reflection involving transformed data. It generates a class of boundary corrected estimators having desirable properties such as local smoothness and nonnegativity. Simulations show that the proposed method performs quite well when compared with the existing methods for almost all shapes of densities. The authors present the theory behind this new methodology, and they determine the bias and variance of their estimators.

Simple and efficient improvements of multivariate local linear regression

This paper studies improvements of multivariate local linear regression. Two intuitively appealing variance reduction techniques are proposed. They both yield estimators that retain the same asymptotic conditional bias as the multivariate local linear estimator and have smaller asymptotic conditional variances. The estimators are further examined in aspects of bandwidth selection, asymptotic relative efficiency and implementation. Their asymptotic relative efficiencies with respect to the multivariate local linear estimator are very attractive and increase exponentially as the number of covariates increases. Data-driven bandwidth selection procedures for the new estimators are straightforward given those for local linear regression. Since the proposed estimators each has a simple form, implementation is easy and requires much less or about the same amount of effort. In addition, boundary corrections are automatic as in the usual multivariate local linear regression.

The Chapman–Enskog procedure for an age-structured population model: initial, boundary and corner layer corrections

We consider a mathematical model of an age-structured population of some fisheries (for example, anchovies, sardines or soles). Two time scales are involved in the problem: the fast time scale for the migration dynamics and the slow time scale for the demographic process. At a first step, we study the so called ‘aggregated’ system by means of the semigroups theory. Then, we study the asymptotic behaviour of the model by using the Chapman–Enskog procedure. In particular, we study initial, boundary and corner layer effects in order to obtain the exact initial and boundary conditions the approximated solution has to satisfy.

Monte Carlo simulation of secondary electron and backscattered electron images for a nanoparticle–matrix system

Contrast simulation of scanning electron microscope (SEM) images for particle–matrix systems is important for understanding the contrast formation mechanism in such systems. Most of the previous works have been limited to backscattered electron (BSE) signals because of the difficulties involved in the simulation of the secondary electron (SE) signals. This limitation prevents some important information from being extracted from experimental SEM images. A comprehensive and accurate Monte Carlo model has been developed for simulation of electron–solid interactions, which enables the simulation of production and emission of cascade SEs and is then quite suitable for investigating both SE and BSE signals. In this paper, we have modified the simulation model for a particle–matrix system and performed a simulation of SE and BSE images for nano-size Pt particles in a carbon matrix. Since the size of the particles is comparable with the electron scattering mean free path, modification of the conventional Monte Carlo procedure for a solid is necessary in order to include the boundary correction due to different scattering mean free paths. SE and BSE line scans and two-dimensional images under various conditions (particle size, depth and beam incident angle) are obtained and discussed in detail. The results have shown that: (1) In the formation of Pt–C contrast, the material factor plays a dominant role. (2) Only those particles located on, or quite close to, the matrix surface can be clearly revealed in a SE image. The maximum visible depth in SE observation for a Pt particle embedded in carbon matrix is about three times the particle size, while it is twice deeper than that with BSE signals. (3) The maximum SE emission from Pt particles appears at the centre of, instead of the sides of, the particle when the size falls down to the nanometre region; (4) Both the SE and BSE contrast have maximum values at a certain primary energy for particles of a specific size. (5) For oblique incidence of the primary beam, the SE contrast is quite different for the particles placed on the surface from that for those embedded in the matrix. These simulation results on image contrast will be helpful in evaluating the fraction of catalyst particles on the matrix surface and the optimum operating condition for imaging.

On boundary correction in kernel density estimation

It is well known now that kernel density estimators are not consistent when estimating a density near the finite end points of the support of the density to be estimated. This is due to boundary effects that occur in nonparametric curve estimation problems. A number of proposals have been made in the kernel density estimation context with some success. As of yet there appears to be no single dominating solution that corrects the boundary problem for all shapes of densities. In this paper, we propose a new general method of boundary correction for univariate kernel density estimation. The proposed method generates a class of boundary corrected estimators. They all possess desirable properties such as local adaptivity and non-negativity. In simulation, it is observed that the proposed method perform quite well when compared with other existing methods available in the literature for most shapes of densities, showing a very important robustness property of the method. The theory behind the new approach and the bias and variance of the proposed estimators are given. Results of a data analysis are also given.

The Index of b-Pseudodifferential Operators on Manifolds with Corners

On compact manifolds with corners of arbitrary codimension, we characterize the ‘multi-cylindrical end’ (or b-type) pseudodifferential operators that are Fredholm on weighted Sobolev spaces and we compute their indices. The index formula contains the usual interior term manufactured from the local symbols of the operator and also contains boundary correction terms corresponding to eta-type invariants of the induced operators on the boundary faces.

A locally adaptive transformation method of boundary correction in kernel density estimation

Kernel smoothing methods are widely used in many research areas in statistics. However, kernel estimators suffer from boundary effects when the support of the function to be estimated has finite endpoints. Boundary effects seriously affect the overall performance of the estimator. In this article, we propose a new method of boundary correction for univariate kernel density estimation. Our technique is based on a data transformation that depends on the point of estimation. The proposed method possesses desirable properties such as local adaptivity and non-negativity. Furthermore, unlike many other transformation methods available, the proposed estimator is easy to implement. In a Monte Carlo study, the accuracy of the proposed estimator is numerically analyzed and compared with the existing methods of boundary correction. We find that it performs well for most shapes of densities. The theory behind the new methodology, along with the bias and variance of the proposed estimator, are presented. Results of a data analysis are also given.

Curved boundary corrections to nodal line statistics in chaotic billiards

A Gaussian random wavefunction that satisfies Dirichlet and Neumann conditions locally on a convex circular boundary is introduced. The average of the square of the wavefunction and its derivatives are computed and their asymptotics studied in the semi-classical limit. The mean nodal line length (L) is calculated and the first order boundary effect shown to be of order log k, where k is the wavenumber. In the limit of vanishing boundary curvature (large boundary radius) these results are shown to approach those for a straight wall boundary.

Inflation as a probe of trans-Planckian physics: a brief review and progress report1

We provide a prescription for parametrizing the vacuum choice ambiguity in cosmological settings. We introduce an arbitrary boundary action representing the initial conditions. A Lagrangian description is moreover the natural setting to study decoupling of high-energy physics. RG flow affects the boundary interactions. As a consequence the boundary conditions are sensitive to high-energy physics through irrelevant terms in the boundary action. Using scalar field theory as an example, we derive the leading dimension four irrelevant boundary operators. We discuss flow the known vacuum choices, e.g. the Bunch-Davies vacuum, appear in the Lagrangian description and square with decoupling. For all choices of boundary conditions encoded by relevant boundary operators, of which the known ones are a subset, backreaction is under control. All, moreover, will generically feel the influence of high-energy physics through irrelevant (dimension four) boundary corrections. Having established a coherent effective field theory framework including the vacuum choice ambiguity, we derive an explicit expression for the flower spectrum of inflationary density perturbations including the leading high energy corrections. In accordance with the dimensionality of the leading irrelevant operators, the effect of high energy physics is linearly proportional to the Hubble radius H and the scale of new physics l = 1/M. Effects of such strength are potentially observable in future measurements of the cosmic micflowave background.

Monte Carlo dosimetric study of Best Industries and Alpha Omega Ir‐192 brachytherapy seeds

Ir-192 seeds are widely used in the USA for low dose rate interstitial brachytherapy. There are two commercially available models: those manufactured by Best Industries filtered with stainless steel, and those manufactured by Alpha-Omega seeds filtered with Pt. Newly developed 3D correction algorithms for brachytherapy are based on dosimetry data obtained on unbounded phantom size, allowing corrections for heterogeneities and actual tissue boundaries. Published dosimetric datasets for both seeds have been obtained under bounded conditions. The aim of the present study is to obtain dosimetric datasets for these seeds under full scatter conditions. The Monte Carlo GEANT4 code has been used to estimate air-kerma strength and dose rate in water around the Ir-192 seeds. Functions and parameters following the TG43 formalism are obtained and presented in tabular forms: the dose rate constant, the radial dose function, and the anisotropy function. Tables for the anisotropy factor have been obtained in order to apply punctual approximation. Differences between dose rate distributions for both seeds show that specific dataset must be used for each type of seed in clinical dosimetry. The data in the present study improve on published data in the following aspects: (i) dosimetric data were obtained under full scatter conditions, which affect dose values at distances greater than 4-5 cm from the source; (ii) the dose rate tables are given at greater distances from the source; and (iii) the spatial resolution in high dose gradient areas, such as those near the longitudinal source axis, has been improved.

Decoupling in an expanding universe: boundary RG-flow affects initial conditions for inflation

We study decoupling in FRW spacetimes, emphasizing a Lagrangian description throughout. To account for the vacuum choice ambiguity in cosmological settings, we introduce an arbitrary boundary action representing the initial conditions. RG flow in these spacetimes naturally affects the boundary interactions. As a consequence the boundary conditions are sensitive to high-energy physics through irrelevant terms in the boundary action. Using scalar field theory as an example, we derive the leading dimension four irrelevant boundary operators. We discuss how the known vacuum choices, e.g. the Bunch-Davies vacuum, appear in the Lagrangian description and square with decoupling. For all choices of boundary conditions encoded by relevant boundary operators, of which the known ones are a subset, backreaction is under control. All, moreover, will generically feel the influence of high-energy physics through irrelevant (dimension four) boundary corrections. Having established a coherent effective field theory framework including the vacuum choice ambiguity, we derive an explicit expression for the power spectrum of inflationary density perturbations including the leading high energy corrections. In accordance with the dimensionality of the leading irrelevant operators, the effect of high energy physics is linearly proportional to the Hubble radius H and the scale of new physics L= 1/M.

Structure and Dynamics of a Benzenedithiol Monolayer on a Au(111) Surface

We use the universal force field (UFF) developed by Rappe et al. (Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A.; Skiff, W. M. J. Am. Chem. Soc. 1992, 114, 10024) and the specific classical potentials developed from ab initio calculations for Au−benzenedithiol (BDT) molecule interaction to perform molecular dynamics (MD) simulations of a BDT monolayer on an extended Au(111) surface. The simulation system consists of 100 BDT molecules and three rigid Au layers in a simulation box that is rhombic in the plane of the Au surface. A multiple time scale algorithm, the double-reversible reference system propagator algorithm (double RESPA) based on the Nose−Hoover dynamics scheme, and the Ewald summation with a boundary correction term for the treatment of long-range electrostatic interactions in a 2-D slab have been incorporated into the simulation technique. We investigate the local bonding properties of Au−BDT contacts and molecular orientation distributions of BDT molecules. These results show t...

Experimental study of a laminar air boundary layer over a water surface

The formation of a laminar boundary layer over a water surface has been experimentally studied for the case of a low wind speed of 3.77 m/s without pressure gradient. Velocity profiles were measured using a small total head tube with an external diameter D=0.7 mm in order to detect the velocity as close to the air/water interface as possible. Precision data were obtained by close attention being paid to near boundary correction procedures arising from the effects of viscous, velocity gradient and wall proximity on the pressure tube. Measurements for the first time support Lock's theory.