Multivariate Bernstein Polynomials Research Articles

Concerning Multivariate Bernstein Polynomials and Stochastic Logic

Among the applications of the Bernstein polynomials in one variable is their use in solving problems associated with stochastic computing. Taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel-Rosenberg, the aim of this paper is to investigate some necessary and sufficient conditions for guaranteeing whether polynomial operations can be implemented with stochastic logic based on multivariate Bernstein polynomials with coefficients in the unit interval.

Nonparametric Estimation of Multivariate Copula Using Empirical Bayes Methods

In the fields of finance, insurance, system reliability, etc., it is often of interest to measure the dependence among variables by modeling a multivariate distribution using a copula. The copula models with parametric assumptions are easy to estimate but can be highly biased when such assumptions are false, while the empirical copulas are nonsmooth and often not genuine copulas, making the inference about dependence challenging in practice. As a compromise, the empirical Bernstein copula provides a smooth estimator, but the estimation of tuning parameters remains elusive. The proposed empirical checkerboard copula within a hierarchical empirical Bayes model alleviates the aforementioned issues and provides a smooth estimator based on multivariate Bernstein polynomials that itself is shown to be a genuine copula. Additionally, the proposed copula estimator is shown to provide a more accurate estimate of several multivariate dependence measures. Both theoretical asymptotic properties and finite-sample performances of the proposed estimator based on simulated data are presented and compared with some nonparametric estimators. An application to portfolio risk management is included based on stock prices data.

Dependent Bayesian nonparametric modeling of compositional data using random Bernstein polynomials

We discuss Bayesian nonparametric procedures for the regression analysis of compositional responses, that is, data supported on a multivariate simplex. The procedures are based on a modified class of multivariate Bernstein polynomials and on the use of dependent stick-breaking processes. A general model and two simplified versions of the general model are discussed. Appealing theoretical properties such as continuity, association structure, support, and consistency of the posterior distribution are established. Additionally, we exploit the use of spike-and-slab priors for choosing the version of the model that best adapts to the complexity of the underlying true data-generating distribution. The performance of the proposed model is illustrated in a simulation study and in an application to solid waste data from Colombia.

Meta-heuristic algorithms for solving nonlinear differential equations based on multivariate Bernstein polynomials

Meta-heuristic algorithms for solving nonlinear differential equations based on multivariate Bernstein polynomials

An Answer to an Open Problem on the Multivariate Bernstein Polynomials on a Simplex

We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit triangle.

Bernstein series solutions of multidimensional linear and nonlinear Volterra integral equations with fractional order weakly singular kernels

This paper proposes a quadrature method based on multi-variate Bernstein polynomials. The method is used to solve multidimensional Volterra integral equations with weakly singular kernels. Firstly, we use multi-variate Bernstein polynomials to approximate the unknown function of an equation, then a discrete function equation can be obtained by substituting the approximate solution into the equation. Secondly, the discrete function system is transformed into an algebra equation system by using some discrete points. We can perform the integral operations without discrete kernel function, and the weakly singular integrals can be calculated directly by using quadrature method, so the method is easy to implement. Thirdly, we prove the existence and uniqueness of the solution of the approximate equation, as well as the error analysis of the proposed method. Six numerical examples are given to illustrate the efficiency of this method.

On partial derivatives of multivariate Bernstein polynomials

It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus.

Bayesian density estimation for compositional data using random Bernstein polynomials

We propose a Bayesian nonparametric procedure for density estimation for data in a d-dimensional simplex. To this aim, we propose a prior distribution on probability measures based on a modified class of multivariate Bernstein polynomials. The model for the probability distribution corresponds to a mixture of Dirichlet distributions, with random weights and a random number of components. Theoretical properties of the proposal are provided, including posterior consistency and concentration rates of the posterior distribution.

Random sampling scattered data with multivariate Bernstein polynomials

In this paper, the multivariate Bernstein polynomials defined on a simplex are viewed as sampling operators, and a generalization by allowing the sampling operators to take place at scattered sites is studied. Both stochastic and deterministic aspects are applied in the study. On the stochastic aspect, a Chebyshev type estimate for the sampling operators is established. On the deterministic aspect, combining the theory of uniform distribution and the discrepancy method, the rate of approximating continuous function and L p convergence for these operators are studied, respectively.

Formalization of Bernstein Polynomials and Applications to Global Optimization

This paper presents a formalization in higher-order logic of a practical representation of multivariate Bernstein polynomials. Using this representation, an algorithm for finding lower and upper bounds of the minimum and maximum values of a polynomial has been formalized and verified correct in the Prototype Verification System (PVS). The algorithm is used in the definition of proof strategies for formally and automatically solving polynomial global optimization problems.

A NOTE ON THE GENERALIZED BERNSTEIN POLYNOMIALS

We prove two identities for multivariate Bernstein polynomials on simplex, which are considered on a pointwise. In this paper, we study good approximations of Bernstein polynomials for every continuous functions on simplex and the higher dimensional q-analogues of Bernstein polynomials on simplex.

Exchangeability and sets of desirable gambles

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti’s Finite and Infinite Representation Theorems. We show that the finite representation in terms of count vectors has a very nice geometrical interpretation, and that the representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability and the extension of exchangeable sequences.

Formula omitted]-functionals and multivariate Bernstein polynomials

This paper estimates upper and lower bounds for the approximation rates of iterated Boolean sums of multivariate Bernstein polynomials. Both direct and inverse inequalities for the approximation rate are established in terms of a certain K -functional. From these estimates, one can also determine the class of functions yielding optimal approximations to the iterated Boolean sums.

Division algorithms for Bernstein polynomials

Three division algorithms are presented for univariate Bernstein polynomials: an algorithm for finding the quotient and remainder of two univariate polynomials, an algorithm for calculating the GCD of an arbitrary collection of univariate polynomials, and an algorithm for computing a μ-basis for the syzygy module of an arbitrary collection of univariate polynomials. Division algorithms for multivariate Bernstein polynomials and analogues in the multivariate Bernstein setting of Gröbner bases are also discussed. All these algorithms are based on a simple ring isomorphism that converts each of these problems from the Bernstein basis to an equivalent problem in the monomial basis. This isomorphism allows all the computations to be performed using only the original Bernstein coefficients; no conversion to monomial coefficients is required.

Direct and converse results for multivariate generalized Bernstein polynomials

In this paper, the following generalization of multivariate Bernstein polynomials has been studied: B n ( f ; x 1 , x 2 , … , x m ) = ∑ k 1 , k 2 , … , k m = 0 n f k 1 b n , k 2 b n , … , k m b n ∏ j = 1 m n k j x j k j ( 1 - x j ) n - k j , where ( b n ) is a sequence of positive numbers such that lim n → ∞ n / b n = 1 and direct and inverse theorems for this polynomials have been presented.

A new proof of an identity of Jetter and Stöckler for multivariate Bernstein polynomials

A simple proof of an elegant identity for multivariate Bernstein polynomials discovered recently by K. Jetter and J. Stöckler is given. This identity shows a pointwise orthogonality relation of multivariate Bernstein polynomials and implies a new representation for the dual basis of multivariate Bernstein polynomials. Our method is based on “generating functions”, which reveals a general structure of the identity.

Multivariate approximation by a combination of modified Taylor polynomials

We study an approximation of a multivariate function f by an operator of the form ∑ i = 1 N T ˜ r [ f , x i ] ( x ) φ i ( x ) , where φ 1 , … , φ N are certain basis functions and T ˜ r [ f , x i ] ( x ) are modified Taylor polynomials of degree r expanded at x i . The modification is such that the operator has highest degree of algebraic precision. In the univariate case, this operator was investigated by Xuli [Multi-node higher order expansions of a function, J. Approx. Theory 124 (2003) 242–253]. Special attention is given to the case where the basis functions are a partition of unity of linear precision. For this setting, we establish two types of sharp error estimates. In the two-dimensional case, we show that this operator gives access to certain classical interpolation operators of the finite element method. In the case where φ 1 , … , φ N are multivariate Bernstein polynomials, we establish an asymptotic representation for the error as N → ∞ .

An identity for multivariate Bernstein polynomials

We prove an identity for multivariate Bernstein polynomials on simplices, which may be considered a pointwise orthogonality relation. Its integrated version provides a new representation for the polynomial dual basis of Bernstein polynomials. An identity for the reproducing kernel is used to define quasi-interpolants of arbitrary order.

Asymptotic constants for multivariate Bernstein polynomials

We study the rate of convergence of multivariate Bernstein polynomials on the class of Hölder continuous functions. Using a proper probabilistic representation we are able to derive the asymptotic constant.

Approximation capabilities of multilayer feedforward regular fuzzy neural networks

Four-layer feedforward regular fuzzy neural networks are constructed. Universal approximations to some continuous fuzzy functions defined on ℱ0 (R)n by the four-layer fuzzy neural networks are shown. At first, multivariate Bernstein polynomials associated with fuzzy valued functions are empolyed to approximate continuous fuzzy valued functions defined on each compact set of Rn. Secondly, by introducing cut-preserving fuzzy mapping, the equivalent conditions for continuous fuzzy functions that can be arbitrarily closely approximated by regular fuzzy neural networks are shown. Finally a few of sufficient and necessary conditions for characterizing approximation capabilities of regular fuzzy neural networks are obtained. And some concrete fuzzy functions demonstrate our conclusions.