Regular Fuzzy Neural Networks Research Articles

A fast learning algorithm based on extreme learning machine for regular fuzzy neural network

The regular fuzzy neural network (RFNN) is a kind of fuzzy neural network by fuzzifying the feed-forward neural network. The RFNN can directly deal with the language information and it has the merits of fuzzy system and neural network. It is presented a fast learning algorithm based on the extreme learning machine (ELM) for the RFNN in this paper. The RFNN referred here is a three-layer feed-forward fuzzy neural network and the connected weights in the RFNN are all fuzzy numbers. A simulation example is given to approximately realize the fuzzy if-then rules by the RFNN. The results show that the RFNN trained by the proposed algorithm has good performance and approximation ability.

The functional space in fuzzy analysis

In this paper, we briefly summarize some work on the area of functional spaces in fuzzy analysis and devoted to four directions. The first direction is the development of the fuzzy functional analysis on the setting of topological linear space and several kinds of fuzzy normed linear spaces. Next direction is fuzzy number spaces with different metrics and the embeddings to concrete functional spaces with a linear structure. The third direction is fuzzy continuous function spaces and the approximation by multilayer regular fuzzy neural networks for various fuzzy continuities. The last direction is the space of measurable functions and the space of (Sugeno)-integrable functions based on the theory of monotonic measures. In this paper, we also present several suggestions on future research. Wu Congxin was an undergraduate student of the Mathematics Department of Northeastern Peoples University during 1952-1955, and Professor Hsu guided and supported him so much for long time.

Approximation of fuzzy-valued functions by regular fuzzy neural networks and the accuracy analysis

In this paper, it is shown that four-layer regular fuzzy neural networks can serve as universal approximators for the sendograph-metric-continuous fuzzy-valued functions. The proof is constructive. We propose a principled method to design four-layer regular fuzzy neural neural network to approximate the target functions. In the previous work, a step function is used as the activation function. To improve the approximation accuracy, in the present work, we also consider using a semi-linear sigmoidal function as the activation function. Then it shows how to design the regular fuzzy neural networks (RFNNs) when the activation functions are the semi-linear sigmoidal function and the step function, respectively. After analyze the approximation accuracy of these two classes of RFNNs, it is found that the former has a much better performance than the latter in approximation accuracy. This conclusion also holds when the target functions satisfy other types of continuity. So the results in this paper can also be used to improve the related work. At last, we give a simulation example to validate the theoretical results.

Approximation of fuzzy functions by regular fuzzy neural networks

In this paper, we investigate the ability of regular fuzzy neural networks to provide approximations to fuzzy functions. Since the operation of regular fuzzy neural networks is based on Zadeh's extension principle, we first present a level characterization of the Zadeh's extensions of level-continuous fuzzy-valued functions and consider the continuity of these extensions. On the basis of this, we give characterizations of fuzzy functions which can be approximated by a class of four-layer regular fuzzy neural networks according to supremum-metric and level convergence.

Approximation capabilities of multilayer fuzzy neural networks on the set of fuzzy-valued functions

This paper first introduces a piecewise linear interpolation method for fuzzy-valued functions. Based on this, we present a concrete approximation procedure to show the capability of four-layer regular fuzzy neural networks to perform approximation on the set of all d p continuous fuzzy-valued functions. This approach can also be used to approximate d ∞ continuous fuzzy-valued functions. An example is given to illustrate the approximation procedure.

Approximation of level continuous fuzzy-valued functions by multilayer regular fuzzy neural networks

This paper proves the capability of approximation by four-layer regular fuzzy neural networks on the set of all level continuous fuzzy-valued functions according to level convergence. A numerical example is given to illustrate the results.

Approximation analysis of feedforward regular fuzzy neural network with two hidden layers

The closure fuzzy mapping is employed to analyze approximation capability of feedforward regular fuzzy neural networks (FNNs) with two hidden layers. The bridge to do that is the fuzzy-valued Bernstein polynomial. By the fuzzy polynomial approximation theorem, this paper establishes some equivalent conditions for the fuzzy functions that can be approximated by the regular FNNs to any degree of accuracy. Thus, universal approximation of regular FNNs based on Zadeh's extension principle and fuzzy arithmetic can be studied, systematically. Finally, a simulation example is used to demonstrate the constructive procedure of approximating FNNs.

An intelligent sales forecasting system through integration of artificial neural networks and fuzzy neural networks with fuzzy weight elimination

Sales forecasting plays a very prominent role in business strategy. Numerous investigations addressing this problem have generally employed statistical methods, such as regression or autoregressive and moving average (ARMA). However, sales forecasting is very complicated owing to influence by internal and external environments. Recently, artificial neural networks (ANNs) have also been applied in sales forecasting since their promising performances in the areas of control and pattern recognition. However, further improvement is still necessary since unique circumstances, e.g. promotion, cause a sudden change in the sales pattern. Thus, this study utilizes a proposed fuzzy neural network (FNN), which is able to eliminate the unimportant weights, for the sake of learning fuzzy IF–THEN rules obtained from the marketing experts with respect to promotion. The result from FNN is further integrated with the time series data through an ANN. Both the simulated and real-world problem results show that FNN with weight elimination can have lower training error compared with the regular FNN. Besides, real-world problem results also indicate that the proposed estimation system outperforms the conventional statistical method and single ANN in accuracy.

Universal approximations of continuous fuzzy-valued functions by multi-layer regular fuzzy neural networks

The fact that four-layer feedforward regular fuzzy neural networks with sigmoid function in the first hidden layer are capable of approximately representing continuous fuzzy valued functions on any compact set of R is shown. At first, Bernstein polynomials associated with fuzzy valued functions are employed to approximate continuous fuzzy valued function defined on a compact set. Secondly, by the conclusions related to standard feedforward networks, universal approximations of continuous fuzzy valued functions by regular fuzzy neural networks are obtained.

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.

Analyses of regular fuzzy neural networks for approximation capabilities

By introducing a trapezoidal fuzzy function and using the properties of the function, we set up a continuously increasing fuzzy function that cannot be arbitrarily closely approximated on a compact set of F 0(R) by the regular fuzzy neural network (RFNN). Thus, the conclusions in Buckley and Hayashi (Fuzzy Sets and Systems 61 (1994) 43–51) are improved and the problem if the RFNN is the universal approximator to the class of continuously increasing fuzzy functions is solved. Finally, we obtain the universal approximation to the extended fuzzy functions by RFNN.

Approximation analyses for fuzzy valued functions inL 1(μ)-norm by regular fuzzy neural networks

By defining fuzzy valued simple functions and givingL1(μ) approximations of fuzzy valued integrably bounded functions by such simple functions, the paper analyses byL1(μ)-norm the approximation capability of four-layer feedforward regular fuzzy neural networks to the fuzzy valued integrably bounded functionF : Rn →Fc0(R). That is, if the transfer functionσ :R →R is non-polynomial and integrable function on each finite interval,F may be inL1(μ)-norm approximated by fuzzy valued functions defined as\(\sum\limits_{k = 1}^q {\tilde W_k } \cdot \sum\limits_{j = 1}^p {\nu _{kj} \cdot \sigma \left( {\left\langle {x,u\left( j \right)} \right\rangle + \theta _j } \right)} \) to any degree of accuracy. Finally some real examples demonstrate the conclusions.

Approximation capability of regular fuzzy neural networks to continuous fuzzy functions

The approximation capability of regular fuzzy neural networks to fuzzy functions is studied. When σ is a nonconstant, bounded and continuous function of\(\mathbb{R}\), some equivalent conditions are obtained, with which continuous fuzzy functions can be approximated to any degree of accuracy by the four-layer feedforward regular fuzzy neural networks\(\sum\limits_{k = 1}^q {\tilde W_k } \cdot \left( {\sum\limits_{j = 1}^p {\tilde V_{kj} \cdot \sigma (\tilde X \cdot \tilde U_j + \tilde \Theta _j )} } \right)\). Finally a few examples of such fuzzy functions are given.