Interpolation

This paper illustrates the application of a "Sinc-Galerkin" method to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The method is based on approximating functions and their derivatives by use of the Whittaker cardinal function. The DE is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products, the evaluation of which does not require any numerical integration. Using n function evaluations, the error in the final approximation to the solution of the DE is O ( e − c n 1 / 2 d ) O({e^{ - c{n^{1/2d}}}}) , where c is independent of n, and d denotes the dimension of the region on which the DE is defined. This rate of convergence is optimal in the class of n-point methods which assume that the solution is analytic in the interior of the interval, and which ignore possib...

Approximations are given for the universal function ϕ(λ) appearing in Landau's theory on the energy loss distribution of fast electrons by ionization and also for the function P^(−1)(r) inverse to the integral of ϕ(λ). Values of parameters have been determined with the criterion of the best approximation in the Tchebyschev sense. Two results are presented for ϕ(λ); the simpler one is accurate within an absolute error corresponding to 1.0% of the peak value of ϕ(λ) over the interval −2.9 ≤ λ < ∞, and the other, within a relative error of 1.4 × 10−3 over −3.275 ≤ λ ≤ 100. The approximation to P^(−1)(r) is accurate within a relative error of 9 × 10^(−4) over 0.001 ≤ r ≤ 0.999.

Summary form only given. The purpose of this paper is to determine the radial distribution of the emission coefficient from the measured intensity distribution emitted by an extended source of radiation, particularly a plasma source. The source is assumed to be optically thin and axially symmetrical. This problem is solved by inverting Abel&amp;amp;amp;#39;s integral equation. A smoothing procedure is made

This paper presents a study highlighting the predictive performance of a radial basis function (RBF) network in estimating the grade of an offshore placer gold deposit. In applying the radial basis function network to grade estimation of the deposit, several pertinent issues regarding RBF model construction are addressed in this study. One of the issues is the selection of the RBF network along with its center and width parameters. Selection was done by an evolutionary algorithm that utilizes the concept of cooperative coevolutions of the RBFs and the associated network. Furthermore, the problem of data division, which arose during the creation of the training, calibration and validation of data sets for the RBF model development, was resolved with the help of an integrated approach of data segmentation and genetic algorithms (GA). A simulation study conducted showed that nearly 27% of the time, a bad data division would result if random data divisions were adopted in this study. In addition, the efficacy of the RBF network was tested against a feed-forward network and geostatistical techniques. The outcome of this comparative study indicated that the RBF model performed decisively better than the feed-forward network and the ordinary kriging (OK).

In recent years there has been an increased interest in audio steganography and watermarking. This is due primarily to two reasons. First, an acute need to improve our national security capabilities in light of terrorist and criminal activity has driven new ideas and experimentation. Secondly, the explosive proliferation of digital media has forced the music industry to rethink how they

Gridding methods produce a regularly spaced, rectangular array of Z values from irregularly spaced XYZ data. Gridding generates a Z value at each grid node by interpolating or extrapolating the data values. Different gridding methods provide different interpretations of data because each method calculates grid node values by using a different algorithm. In this paper, we apply and compare different gridding methods including Inverse Distance Weighting, Kriging, Minimum Curvature, Modified Shepard's Method, Natural Neighbor, Nearest Neighbor, Polynomial Regression, Radial Basis Function, Triangulation with Linear Interpolation, Moving Average and Local Polynomial for various pilot regions with varied elevations. Grid method parameters which are able to set when producing a grid file, control the interpolation procedures. We also change these parameters to evaluate their effects on the resulted precision. We use a set of checkpoints to compare the precision of these methods. To implement these methods, various elevation data with equal precision is used; therefore, the results are affected only by the interpolation method. In other words, the influences of data and other environmental factors on the resulted precision are reduced by using them similarly in all gridding methods. Eventually, RMSE is computed for each method with its special parameters in different regions using checkpoints. The resulted precision acquired from applying these methods as well as their advantages and disadvantages in various topographic regions are represented in this study.

Verbmobil, a German research project, aims at machine translation of spontaneous speech input. The ultimate goal is the development of a portable machine translator that will allow people to negotiate in their native language. Within this project the University of Karlsruhe has developed a speech recognition engine that has been evaluated on a yearly basis during the project and shows very promising speech recognition word accuracy results on large vocabulary spontaneous speech. We introduce the Janus Speech Recognition Toolkit underlying the speech recognizer. The main new contributions to the acoustic modeling part of our 1996 evaluation system-speaker normalization, channel normalization and polyphonic clustering-are discussed and evaluated. Besides the acoustic models we delineate the different language models used in our evaluation system: word trigram models interpolated with class based models and a separate spelling language model were applied. As a result of using the toolk...

The purpose of this experiment is two-fold: 1. To demonstrate how a common data analysis procedure – interpolation – can be implemented in Azure Machine Learning Studio (Azure MLS) and 2. To show how the use of R can extend capabilities of the Azure MLS.
Botchkarev, A. (2018). Data Analysis and Interpolation Using R. Experiment in Microsoft Azure Machine Learning Studio. Azure AI Gallery. https://gallery.azure.ai/Experiment/Data-Analysis-and-Interpolation-Using-R The purpose of this experiment is to demonstrate how interpolation can be implemented in Azure MLS using R.

Full and reliable rainfall data are required for hydrological modeling, integrated water resource management, and planning. However, these data suffer from record gaps and sparse rain gage distribution, which implies that the use of an imputation method is crucial. This study aims to compare the outputs from ten imputation methods that were used to infill missing rainfall depth data (MRD) in an arid Mediterranean region. Different statistical tests were used to assess the outputs from the imputation methods. The results showed that for MRD ranges between 5% and 20% the stepwise multiple linear regression (MLRsw) method was valid and produced the best results with a root mean square error (RSME) and mean absolute error (MAE) of less than 7 mm and 2 mm, respectively. This was followed by the Monte Carlo Markov chain expectation‐maximization‐based multiple imputation (MI‐MCMC) method, which had an RSME and MAE of 1.01 mm and 0.08 mm, respectively, at 20% MRD. On the other hand, the use of satellite data for imputation (LR_GPCC estimates) was appropriate for MRD ranging between 10% and 15%, while the statistical and spatial method was suitable for MRD of less than 5%. This article is protected by copyright. All rights reserved.