Beta Basis Function Research Articles

Rate of Approximation for Modified Lupaş-Jain-Beta Operators

The main intent of this paper is to innovate a new construction of modified Lupaş-Jain operators with weights of some Beta basis functions whose construction depends onσsuch thatσ0=0andinfx∈0,∞σ′x≥1. Primarily, for the sequence of operators, the convergence is discussed for functions belong to weighted spaces. Further, to prove pointwise convergence Voronovskaya type theorem is taken into consideration. Finally, quantitative estimates for the local approximation are discussed.

Convergence of Generalized Lupaş-Durrmeyer Operators

The main aim of this paper is to establish summation-integral type generalized Lupaş operators with weights of Beta basis functions which depends on μ having some properties. Primarily, for these new operators, we calculate moments and central moments, weighted approximation is discussed. Further, Voronovskaya type asymptotic theorem is proved. Finally, quantitative estimates for the local approximation is taken into consideration.

The Integral Type Modification of Jain Operators and its Approximation Properties

In the present paper, we discuss the approximation properties of Durrmeyer-Stancu type variant of Jain operators with the modified forms of the Beta basis functions. We establish some direct results, which include the asymptotic formula, the error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct a King modification of these operators which preserves the test functions e0 and e1.

A Beta basis function Interval Type-2 Fuzzy Neural Network for time series applications

The huge complexity and uncertainty in real life requires the use of advanced automatic learning methods to find out better approximators and suitable relationship in real data behavior. Neuro fuzzy systems have been proved to be excellent universal approximators. In this paper we propose a new based function Interval Type-2 Fuzzy Neural Network denoted ”Beta basis function Interval Type-2 Fuzzy Neural Network”, the BIT2FNN. The main idea is to involve type-2 beta fuzzy sets in the design process of fuzzy networks. The proposed architecture is based on beta type-2 fuzzy sets in the antecedent part, while the consequent part achieves the TSK (Takagi–Sugeno–Kang) fuzzy output strategy. Thanks to the beta function flexibility, the network achieve a good performance and shows a good resistance to noisy data. First order derivatives of type-1 and type-2 Beta functions were developed for the first time for designing fuzzy logic systems based on given input–output pairs. The backpropagation algorithm was used for the learning process of antecedent fuzzy beta parameters and the consequent part. The performance of the proposed model of Beta fuzzy logic system is evaluated with mainly two problems of time series applications : the Mackey Glass Chaotic Time-Series prediction problem with different setting of parameters and levels of noise and the ECG heart-rate Time Series monitoring problem.

Multi-agent architecture for Multi‐objective optimization of Flexible Neural Tree

In this paper, a multi-agent system is introduced to parallelize the Flexible Beta Basis Function Neural Network (FBBFNT)’ training as a response to the time cost challenge. Different agents are formed; a Structure Agent is designed for the FBBFNT structure optimization and a variable set of Parameter Agents is used for the FBBFNT parameter optimization. The main objectives of the FBBFNT learning process were the accuracy and the structure complexity. With the proposed multi-agent system, the main purpose is to reach a good balance between these objectives. For that, a multi-objective context was adopted which based on Pareto dominance. The agents use two algorithms: the Pareto dominance Extended Genetic Programming (PEGP) and the Pareto Multi-Dimensional Particle Swarm Optimization (PMD_PSO) algorithms for the structure and parameter optimization, respectively. The proposed system is called Pareto Multi-Agent Flexible Neural Tree (PMA_FNT).To assess the effectiveness of PMA_FNT, four benchmark real datasets of classification are tested. The results compared with some classifiers published in the literature.

Evolving flexible beta basis function neural tree using extended genetic programming & Hybrid Artificial Bee Colony

In this paper, a new hybrid learning algorithm is introduced to evolve the flexible beta basis function neural tree (FBBFNT). The structure is developed using the Extended Genetic Programming (EGP) and the Beta parameters and connected weights are optimized by the Hybrid Artificial Bee Colony algorithm. This hybridization is essentially based on replacing the random Artificial Bee Colony (ABC) position with the guided Opposite-based Particle Swarm Optimization (OPSO) position. Such modification can minimize the delay which might be lead by the random position, in reaching the global solution. The performance of the proposed model is evaluated for benchmark problems drawn from time series prediction area and is compared with those of related methods.

Rate of Convergence of Modified Baskakov-Durrmeyer Type Operators for Functions of Bounded Variation

We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function. We establish rate of convergence for these operators for functions having derivative of bounded variation. Also, we discuss Stancu type generalization of these operators.

On approximation of certain integral operators

In the present paper we introduce the summation-integral type modified Lupas operators with weights of Beta basis functions. We define the operators in terms of hypergeometric series and, using such an approach, establish moments. Our main results are an asymptotic formula and an error estimate in terms of modulus of continuity and weighted approximation and rate of convergence for functions having bounded derivatives.

Some Approximation Properties of Modified Jain-Beta Operators

Generalization of Szász-Mirakyan operators has been considered by Jain, 1972. Using these generalized operators, we introduce new sequences of positive linear operators which are the integral modification of the Jain operators having weight functions of some Beta basis function. Approximation properties, the rate of convergence, weighted approximation theorem, and better approximation are investigated for these new operators. At the end, we generalize Jain-Beta operator with three parameters α, β, and γ and discuss Voronovskaja asymptotic formula.

A hybrid learning algorithm for evolving Flexible Beta Basis Function Neural Tree Model

In this paper, a tree-based encoding method is introduced to represent the Beta basis function neural network. The proposed model called Flexible Beta Basis Function Neural Tree (FBBFNT) can be created and optimized based on the predefined Beta operator sets. A hybrid learning algorithm is used to evolving FBBFNT Model: the structure is developed using the Extended Genetic Programming (EGP) and the Beta parameters and connected weights are optimized by the Opposite-based Particle Swarm Optimization algorithm (OPSO). The performance of the proposed method is evaluated for benchmark problems drawn from control system and time series prediction area and is compared with those of related methods.

Hierarchical multi-dimensional differential evolution for the design of beta basis function neural network

This paper proposes a hierarchical multi-dimensional differential evolution (HMDDE) algorithm, which is an automatic computational frame work for the optimization of beta basis function neural network (BBFNN) wherein the neural network architecture, weights connection, learning algorithm and its parameters are adapted according to the problem. In the HMDDE-designed neural network, the number of individuals of the population multi-dimensions is the number of beta neural networks. The population of HMDDE forms multiple beta networks with different structures at the higher level and each individual of the previous population is optimized at a lower hierarchical level to improve the performance of each individual. For the beta neural network consisting of m neurons, n individuals (different lengths) are formed in the upper level to optimize the structure of the beta neural network. In the lower level, the population within the same length is to optimize the free parameters of the beta neural network. To evaluate the comparative performance, we used benchmark problems drawn from identification system and time series prediction area. Empirical results illustrate that the HMDDE produces a better generalization performance.

Bézier variant of Baskakov-Beta-Stancu operators

In the year 1994, Gupta (Approx Theory Appl (N.S.) 10(3):74–78, 1994) introduced the integral modification of well known Baskakov operators with weights of Beta basis functions and obtained better approximation over the usual Baskakov Durrmeyer operators. The rate of convergence for Bezier variant of these operators for functions of bounded variations were discussed in Gupta (Int J Math Math Sci 32(8):471–479, 2002). The present paper is the extension of the previous work, here we consider the Bezier variant of Baskakov-Beta-Stancu operators. We estimate the rate of convergence of these operators for the bounded functions. In the end of the paper we suggest an open problem.

Rate of convergence for generalized Baskakov operators

In the present paper, we consider the generalized Baskakov operators having the weight functions of Beta basis functions. We study the rate of convergence for functions having derivatives of bounded variation.

A note on improved estimations for integrated Szász-Mirakyan operators

Abstract In the present paper, we propose a modification of the integrated Szász-Mirakyan operators having the weight function of general Beta basis function. We study some direct results on the modified operators. It is also observed that our modified operators have better estimates over the original operators.

Direct estimates in simultaneous approximation for Durrmeyer type operators

In the present paper, we study a Durrmeyer type integral modification of the wellknown Baskakov operators with the weight function of Beta basis function. Some approximation properties of these operators were recently studied by Finta [2]. Here we study simultaneous approximation properties for these operators. We estimate local direct result in terms of modulus of continuity. The operators considered in this paper reproduce not only the constant functions but also the linear ones, due to this property we can improve the order of approximation for these operators by applying the iterative combinations, which were first studied by Micchelli [7]. We establish an asymptotic formula and error estimation in terms of higher order modulus of continuity in simultaneous approximation for the Micchelli combinations of these operators. Mathematics subject classification (2000): 41A30, 41A36.

Some results on modified Szász–Mirakjan operators

In this paper we study the mixed summation-integral type operators having Szász and Beta basis functions. We extend the study of Gupta and Noor [V. Gupta, M.A. Noor, Convergence of derivatives for certain mixed Szász–Beta operators, J. Math. Anal. Appl. 321 (1) (2006) 1–9] and obtain some direct results in local approximation without and with iterative combinations. In the last section are established direct global approximation theorems.

Convergence of derivatives for certain mixed Szasz–Beta operators

In this paper we study the mixed summation–integral type operators having Szasz and Beta basis functions in summation and integration, respectively, we obtain the rate of point-wise convergence, a Voronovskaja type asymptotic formula and an error estimate in simultaneous approximation.

The design of beta basis function neural network and beta fuzzy systems by a hierarchical genetic algorithm

We propose an evolutionary method for the design of beta basis function neural networks (BBFNN) and of beta fuzzy systems (BFS). Classical training algorithms start with a predetermined network structure for neural networks and with a predetermined number of fuzzy rules for fuzzy systems. Generally speaking both the neural network and the fuzzy systems are either insufficient or overcomplicated. This paper describes a hierarchical genetic learning model of the BBFNN and the BFS. In order to examine the performance of the proposed algorithm, it is used for functional approximation problem for the case of BBFNN and for the identification of an induction machine fuzzy plant model for the case of BFS. The results obtained have been encouraging.

A Genetic-Designed Beta Basis Function Neural Network for Multi-Variable Functions Approximation

We propose two evolutionary neural network-training algorithms for Beta basis function neural networks (BBFNN). Classic training algorithms for neural networks start with a predetermined network structure, and so the quality of the response of the BBFNN depends strongly on its structure. Generally, the network resulting from learning applied to a predetermined architecture is either insufficient or overcomplicated.This paper describes two genetic learning models of the BBFNN. The first continuous genetic model changes the number of neurons in the hidden layer through the application of specific genetic operators. Each network is coded as a variable length string and some new genetic operators are proposed to evolve a population of individuals. A function is proposed to evaluate the fitness of individual networks. Applications to function approximation problems are considered to demonstrate the performance of the BBFNN and of the evolutionary algorithm. For the second discrete genetic model, each network is coded as a matrix for which number of rows is equal to the number of parameters in the function that will be approximated. The genetic algorithm operators change the number of neurons in the hidden layer. Some applications to functions with one and two parameters are considered to demonstrate the performance of the genetic model and the ability of genetic algorithm to be used for the design of BBFNN.

A Beta basis function neural network in CMOS subthreshold mode

Beta basis function neural networks (BBFNNs) are powerful systems for learning and universal approximation. In this paper, we present a hardware implementation of the Beta neuron using the CMOS subthreshold mode. We describe the low power–low voltage analogue Beta neuron circuit. Three main modules are used to realize the electronic Beta function: a logarithmic currentto-voltage converter, a multiplier and an exponential voltage-to-current converter. Simulation results show the validity of our neural hardware implementation. The parameters of the electronic Beta function are controlled independently by current sources. This analogue implementation could be used easily to realize analogue BBFNNs.