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The asymptotic bias and variance of a general class of local polynomial estimators of M-regression functions are studied over the whole compact support of the multivariate covariate under a minimal assumption on the support. The support assumption ensures that the vicinity of the boundary of the support will be visited by the multivariate covariate. The results show that like in the univariate case, multivariate local polynomial estimators have good bias and variance properties near the boundary. For the local polynomial regression estimator, we establish its asymptotic normality near the boundary and the usual optimal uniform convergence rate over the whole support. For local polynomial quantile regression, we establish a uniform linearization result which allows us to obtain similar results to the local polynomial regression. We demonstrate both theoretically and numerically that with our uniform results, the common practice of trimming local polynomial regression or quantile estimators to avoid “the boundary effect” is not needed.

The purpose of this paper is to overcome the application limitations of other multi-variable regression based on polynomials due to the huge computation room and time cost.

In this paper, based on the idea of feature selection and cascaded regression, two strategies including Laguerre polynomials and manifolds optimization are proposed to enhance the accuracy of multi-variable regression. Laguerre polynomials were combined with the genetic algorithm to enhance the capacity of polynomials approximation and the manifolds optimization method was introduced to solve the co-related optimization problem.

Two multi-variable Laguerre polynomials regression methods are designed. Firstly, Laguerre polynomials are combined with feature selection method. Secondly, manifolds component analysis is adopted in cascaded Laguerre polynomials regression method. Two methods are brought to enhance the accuracy of multi-variable regression method.

With the increasing number of variables in regression problem, the stable accuracy performance might not be kept by using manifold-based optimization method. Moreover, the methods mentioned in this paper are not suitable for the classification problem.

Experiments are conducted on three types of datasets to evaluate the performance of the proposed regression methods. The best accuracy was achieved by the combination of cascade, manifold optimization and Chebyshev polynomials, which implies that the manifolds optimization has stronger contribution than the genetic algorithm and Laguerre polynomials.

The purpose of this paper is to propose an improved optimization methodology of information granulation‐based fuzzy radial basis function neural networks (IG‐FRBFNN). In the IG‐FRBFNN, the membership functions of the premise part of fuzzy rules are determined by means of fuzzy c‐means (FCM) clustering. Also, high‐order polynomial is considered as the consequent part of fuzzy rules which represent input‐output relation characteristic of sub‐space and weighted least squares learning is used to estimate the coefficients of polynomial. Since the performance of IG‐RBFNN is affected by some parameters such as a specific subset of input variables, the fuzzification coefficient of FCM, the number of rules and the order of polynomial of consequent part of fuzzy rules, we need the structural as well as parametric optimization of the network. The proposed model is demonstrated with the use of two kinds of examples such as nonlinear function approximation problem and Mackey‐Glass time‐series data.

The type of polynomial of each fuzzy rule is determined by selection algorithm by considering the local error as performance index. In addition, the combined local error is introduced as a performance index considered by two kinds of parameters such as the polynomial type of each rule and the number of polynomial coefficients of each rule. Besides this, other structural and parametric factors of the IG‐FRBFNN are optimized to minimize the global error of model by means of the hierarchical fair competition‐based parallel genetic algorithm.

The performance of the proposed model is illustrated with the aid of two examples. The proposed optimization method leads to an accurate and highly interpretable fuzzy model.

The proposed hybrid optimization methodology is interesting for designing an accurate and highly interpretable fuzzy model. Hybrid optimization algorithm comes in the form of the combination of the combined local error and the global error.

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.

Develops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.

The four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.

This paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

This paper aims to study the sampling methods (or design of experiments) which have a large influence on the performance of the surrogate model. To improve the adaptability of modelling, a new sequential sampling method termed as sequential Chebyshev sampling method (SCSM) is proposed in this study.

The high-order polynomials are used to construct the global surrogated model, which retains the advantages of the traditional low-order polynomial models while overcoming their disadvantage in accuracy. First, the zeros of Chebyshev polynomials with the highest allowable order will be used as sampling candidates to improve the stability and accuracy of the high-order polynomial model. In the second step, some initial sampling points will be selected from the candidates by using a coordinate alternation algorithm, which keeps the initial sampling set uniformly distributed. Third, a fast sequential sampling scheme based on the space-filling principle is developed to collect more samples from the candidates, and the order of polynomial model is also updated in this procedure. The final surrogate model will be determined as the polynomial that has the largest adjusted R-square after the sequential sampling is terminated.

The SCSM has better performance in efficiency, accuracy and stability compared with several popular sequential sampling methods, e.g. LOLA-Voronoi algorithm and global Monte Carlo method from the SED toolbox, and the Halton sequence.

The SCSM has good performance in building the high-order surrogate model, including the high stability and accuracy, which may save a large amount of cost in solving complicated engineering design or optimisation problems.

The main objective of this study is to highlight the relationship between RGB values of digital camera as a device-dependent color space and a device-independent CIE color space. Calibration and testing databases are the colorimetric specifications of colored polyester fabrics. The colored fabrics were dyed with a variety of disperse dyestuffs. Camera characterization was done based on the polynomial regression technique. The methods adopted in the study were non-linear filtering applied to camera RGB values and a polynomial regression function directly applied to the CIELAB space.

The performances of the methods improved by increasing the number of terms in the polynomial. The estimation errors of the linear polynomial regression to CIEXYZ, the nonlinear polynomial regression to CIEXYZ, the linear polynomial regression to CIELAB, and the nonlinear polynomial regression to CIELAB were 3.29, 4.43, 3.05, and 3.04 DE*ab respectively. The generalization capability decreased by increasing the number of terms in the polynomial regression. The best generalization capability was obtained by the linear polynomial regression to CIELAB. The best result was obtained by non-linear filtering while the second-best result was obtained by the polynomial regression to the CIELAB values.

In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.

To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.

Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.

In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

The purpose of this paper is to propose methods to improve the least square error polynomial fitting of multi-input nonlinear sources that suffer from strong correlating inputs.

The polynomial fitting is improved by amplitude normalization, reducing the order of the model, utilizing Chebychev polynomials and finally perturbing the correlating controlling voltage spectra. The fitting process is estimated by the reliability figure and the condition number.

It is shown in the paper that perturbing one of the controlling voltages reduces the correlation to a large extend especially in the cross-terms of the multi-input polynomials. Chebychev polynomials reduce the correlation between the higher-order spectra derived from the same input signal, but cannot break the correlation between correlating input and output voltages.

Optimal perturbations are sought in a separate optimization loop, which slows down the fitting process. This is due to the fact that each nonlinear source that suffers from the correlation needs a different perturbation.

The perturbation, harmonic balance run and refitting of an individual nonlinear source inside a device model is new and original way to characterize and fit polynomial models.

Normal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables.

The new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies.

Comprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems.

This study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

The author studies forms over finite fields obtained as the determinant of Hermitian matrices and use these determinatal forms to define and study the base polynomial of a square matrix over a finite field.

The authors give full proofs for the new results, quoting previous works by other authors in the proofs. In the introduction, the authors quoted related references.

The authors get a few theorems, mainly describing some monic polynomial arising as a base polynomial of a square matrix.

As far as the author knows, all the results are new, and the approach is also new.