Continuous Multivariate Functions Research Articles

Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

In this paper, we provide tight linear lower bounding functions for multivariate polynomials given over boxes. These functions are obtained by the expansion of polynomials into Bernstein basis and using the linear least squares function. Convergence properties for the absolute difference between the given polynomials and their lower bounds are shown with respect to raising the degree and the width of boxes and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate continuous rational function based on the Bernstein control points, the convex hull of a non-positive polynomial s, and degree elevation. Numerical comparisons with the well-known Bernstein constant lower bounding function are given. Finally, with these affine functions, the positivity of polynomials and rational functions can be certified by computing the Bernstein coefficients of their linear lower bounds.

(1903-5072) Continuity of super- and sub-additive transformations of continuous functions

We prove a continuity inheritance property for super- and sub-additive transformations of non-negative continuousmultivariate functions defined on the domain of all non-negative points and vanishing at the origin. As a corollaryof this result we obtain that super- and sub-additive transformations of continuous aggregation functions are againcontinuous aggregation functions.

A New Nonparametric Test for Two Sample Multivariate Location Problem with Application to Astronomy

This paper provides a nonparametric test for the identity of two multivariate continuous distribution functions when they differ in locations. The test uses Wilcoxon rank-sum statistics on distances between observations for each of the components and is unaffected by outliers. It is numerically compared with two existing procedures in terms of power. The simulation study shows that its power is strictly increasing in the sample sizes and/or in the number of components. The applicability of this test is demonstrated by use of two astronomical data sets on early-type galaxies.

Korovkin Type Results for Multivariate Continuous Periodic Functions

Let k be a natural number and $$C_{2\pi }\left( \mathbb {R}^{k}\right) $$ the real Banach space of the all continuous $$2\pi $$ -periodic functions $$f: \mathbb {R}^{k}\rightarrow \mathbb {R}$$ . We prove the following Korovkin-type result: Let $$\gamma :\mathbb {R}^{k}\times \mathbb {R}^{k}\rightarrow \left[ 0,\infty \right) $$ be a continuous trigonometric separating function which is $$2\pi $$ -periodic with respect to every variable and such that $$\gamma \left( t,t\right) =0$$ for all $$t\in \mathbb {R}^{k}$$ . If $$V_{n}:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow C_{2\pi }\left( \mathbb {R} ^{k}\right) $$ is a sequence of positive linear operators such that $$ \lim \nolimits _{n\rightarrow \infty }V_{n}\left( \gamma \left( \cdot ,t\right) \right) \left( t\right) =0$$ uniformly with respect to $$t\in \mathbb {R}^{k}$$ and $$\lim \nolimits _{n\rightarrow \infty }V_{n}\left( \mathbf {1}\right) = \mathbf {1}$$ uniformly on $$\mathbb {R}^{k}$$ then, $$\lim \nolimits _{n\rightarrow \infty }V_{n}\left( f\right) =f$$ uniformly on $$\mathbb {R}^{k}$$ for every $$ f\in C_{2\pi }\left( \mathbb {R}^{k}\right) $$ . As an application we obtain the Korovkin theorem for positive linear operators on the space $$C_{2\pi }\left( \mathbb {R}^{k}\right) $$ . Various applications are given.

A note on continuous sums of ridge functions

In this paper we prove that if a continuous multivariate function is represented by a sum of arbitrarily behaved ridge functions, then, under a suitable condition, it can be represented by a sum of continuous ridge functions.

On the approximation by single hidden layer feedforward neural networks with fixed weights

Single hidden layer feedforward neural networks (SLFNs) with fixed weights possess the universal approximation property provided that approximated functions are univariate. But this phenomenon does not lay any restrictions on the number of neurons in the hidden layer. The more this number, the more the probability of the considered network to give precise results. In this note, we constructively prove that SLFNs with the fixed weight 1 and two neurons in the hidden layer can approximate any continuous function on a compact subset of the real line. The proof is implemented by a step by step construction of a universal sigmoidal activation function. This function has nice properties such as computability, smoothness and weak monotonicity. The applicability of the obtained result is demonstrated in various numerical examples. Finally, we show that SLFNs with fixed weights cannot approximate all continuous multivariate functions.

On the error of approximation by ridge functions with two fixed directions

We consider the problem of approximation of a continuous multivariate function by sums of two ridge functions in the uniform norm. We obtain a formula for the approximation error in terms functionals generated by closed paths.

A sufficient condition on a general class of interval type-2 Takagi-Sugeno fuzzy systems with linear rule consequent as universal approximators

Whether a rule-based interval type-2 fuzzy system has the ability to approximate any continuous multivariate function arbitrarily well is a fundamentally important question for fuzzy control and modeling. The only approximation results available are the preliminary ones that we previously obtained. They state that two general classes of the interval T2 fuzzy systems, one for the Mamdani type and the other for the TS type, are universal approximators. We now further our investigation on the same T2 TS fuzzy systems to establish a quantitative sufficient approximation condition. These TS fuzzy systems use the linear rule consequent. Their input fuzzy sets are interval T2 and can be in any shape (e.g., Gaussian or trapezoidal) and the fuzzy AND operators in the rules can be any one type or mixed types. We first proved that the fuzzy systems could uniformly approximate any multivariate polynomials arbitrarily accurately and then utilized the Weierstrass approximation theorem to produce the sufficient approximation condition. The condition is a formula calculating the number of the input fuzzy sets needed to achieve any given approximation accuracy (the number of fuzzy rules needed can be easily computed afterward). A numerical example is provided for illustration.

Application of Ant Colony Algorithm in Optimal Preform Design of Forging Process

For the forging with complex shape, the preform design is a difficult problem in the process and die design. It has an enormous influence on the quality of the forging. In this paper, combining with the FEM, the Ant Colony Algorithm was used for the preform optimization design. The general Ant Colony Algorithm was improved to fit for the multivariate continuous function optimization. The preform die shape was represented by B-spline and the coordinates of the control point of the B-spline were taken as the optimization design variables. The optimization program was developed. Finally, aimed to decrease the material cost of the forging, the preform optimization of a typical H-section forging was obtained using the self-developed program. The optimization results show that the improved Ant Colony Algorithm is suitable for the preform optimization design of forging.

Experiments with an adaptive multicut-HDMR map generation for slowly varying continuous multivariate functions

In many areas of science an technology there is a need for efficient methods of approximating multivariate functions. Methods of this kind often suffer from the exponential growth (known as the curse of dimensionality) of the computational costs, with the increasing number of independent variables. The approximation called cut-HDMR (High Dimensional Model Representation), elaborated in recent years, avoids this problem, but is valid only locally. Its extensions, called multicut-HDMR approximations, are applicable to larger function domains, but are less developed. Numerical experiments performed in this study reveal advantages and weaknesses of three multicut-HDMR variants known from the literature, and two new variants defined by us. The experiments focus on an adaptive, error-driven generation of approximants to slowly varying continuous functions. Accuracy, convergence rate, computational times, and amounts of data needed, are compared. Three most satisfactory variants are identified. Two of them average multiple cut-HDMR maps using weights dependent on distances to cut subspaces. In the third one the grid of cut points splits the function domain into hyperrectangles, and a single cut-HDMR map having the smallest errors represents the approximant in every hyperrectangle.

Approximation by ridge functions and neural networks with a bounded number of neurons

In this paper, we consider the problem of approximation of continuous multivariate functions by neural networks with a bounded number of neurons in hidden layers. We prove the existence of single-hidden-layer networks with bounded number of neurons, which have approximation capabilities not worse than those of networks with arbitrarily many neurons. Our analysis is based on the properties of ridge functions.

Multivariate Jackson-type inequality for a new type neural network approximation

In this paper, we introduce a new type neural networks by superpositions of a sigmoidal function and study its approximation capability. We investigate the multivariate quantitative constructive approximation of real continuous multivariate functions on a cube by such type neural networks. This approximation is derived by establishing multivariate Jackson-type inequalities involving the multivariate modulus of smoothness of the target function. Our networks require no training in the traditional sense.

On the approximation by neural networks with bounded number of neurons in hidden layers

In the current note, we show that a two hidden layer neural network with d inputs, d neurons in the first hidden layer, 2d+2 neurons in the second hidden layer and with a specifically constructed sigmoidal and infinitely differentiable activation function can approximate any continuous multivariate function with arbitrary accuracy.

Multivariate error function based neural network approximations

Here we present multivariate quantitative approximations of real and complex valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), by the multivariate quasi-interpolation, Baskakov type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last three types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the Gaussian error special function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Interpolation and superpositions of multivariate continuous functions

A new approach to proving the unrepresentability of multivariate continuous functions by superpositions of continuous functions of fewer variables is suggested. The approach is based on the idea of interpolating functions with the use of sufficiently sparse interpolation nodes but with relatively high accuracy. The approach is illustrated by Vitushkin’s well-known results on superpositions of functions that depend on a given number of variables and have derivatives of a given order.

On the Chebyshev rank of multivariate splines in [formula omitted]-approximation

Weighted L1-approximation of continuous multivariate real-valued functions by finite-dimensional subspaces of multivariate splines of degree m≥1 is studied. Lower and upper bounds for the Chebyshev rank, i.e., for the largest dimension of the sets of best L1-approximations, are established. The exact value of the Chebyshev rank of linear splines is determined. Our investigations extend known results for bivariate splines to the multivariate case.

Inequalities for a class of multivariate operators

This paper introduces and studies a class of generalized multivariate Bernstein operators defined on the simplex. By means of the modulus of continuity and so-called Ditzian-Totik’s modulus of function, the direct and inverse inequalities for the operators approximating multivariate continuous functions are simultaneously established. From these inequalities, the characterization of approximation of the operators follows. The obtained results include the corresponding ones of the classical Bernstein operators. MSC:41A25, 41A36, 41A60, 41A63.

High-order expansion of the solution of preliminary orbit determination problem

A method for high-order treatment of uncertainties in preliminary orbit determination is presented. The observations consist in three couples of topocentric right ascensions and declinations at three observation epochs. The goal of preliminary orbit determination is to compute a trajectory that fits with the observations in two-body dynamics. The uncertainties of the observations are usually mapped to the phase space only when additional observations are available and a least squares fitting problem is set up. A method based on Taylor differential algebra for the analytical treatment of observation uncertainties is implemented. Taylor differential algebra allows for the efficient computation of the arbitrary order Taylor expansion of a sufficiently continuous multivariate function. This enables the mapping of the uncertainties from the observation space to the phase space as high-order multivariate Taylor polynomials. These maps can then be propagated forward in time to predict the observable set at successive epochs. This method can be suitably used to recover newly discovered objects when a scarce number of measurements is available. Simulated topocentric observations of asteroids on realistic orbits are used to assess the performances of the method.

Multivariate numerical approximation using constructive $$ L^{2} (\mathbb{R}) $$ RBF neural network

For the multivariate continuous function, using constructive feedforward $$ L^{2} (\mathbb{R}) $$ radial basis function (RBF) neural network, we prove that a $$ L^{2} (\mathbb{R}) $$ RBF neural network with n + 1 hidden neurons can interpolate n + 1 multivariate samples with zero error. Then, we prove that the $$ L^{2} (\mathbb{R}) $$ RBF neural network can uniformly approximate any continuous multivariate function with arbitrary precision. The correctness and effectiveness are verified through eight numeric experiments.

Multivariate sigmoidal neural network approximation

Here we study the multivariate quantitative constructive approximation of real and complex valued continuous multivariate functions on a box or RN, N∈N, by the multivariate quasi-interpolation sigmoidal neural network operators. The "right" operators for our goal are fully and precisely described. This approximation is derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the logarithmic sigmoidal function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.