The ability of nonlinear approximation of wavelets [1,2] , neural network [3,4] and neuro-fuzzy models [5–8] has been shown by many researchers. Combining the ability of wavelet transformation for revealing the property of function in localize region with learning ability and general approximation properties of neural network; recently, different types of wavelet neural network (WNN) have been proposed [9–17] . Boubez and Peskin [9], used orthonormal set of wavelet functions as basis functions. Yamakawa [10] and Wang[11,12] applied non-orthogonal wavelet function as an activation function in the single layer feed-forward neural network. They have used a simple cosine wavelet activation function. Neural network (NN) with sigmoidal activation function has already been shown to carry out large dimensional problem very well [18] . WNN instigate a superior system model for complex and seismic application in comparison to the NN with sigmoidal activation function. Majority of the application of wavelet is limited to small dimension [19] , though WNN can handle large dimension problem [18] .

1 引言

小波的[1,2]的非线性逼近能力，神经网络[ 3,4 ] [ 5，8 ]–模糊模型已被证明是由许多研究人员和神经。为揭示函数的性质定位区在学习能力和神经网络的通用逼近性质，结合小波变换的能力；最近，不同类型的小波神经网络（WNN）已经提出了9–[ 17 ]。boubez和佩[ 9 ]，使用正交的小波函数作为基函数。山川[ 10 ]王[11,12]应用非正交小波函数在单层前馈神经网络的激活函数。他们用一个简单的余弦小波激励函数。神经网络（NN）和S形激活功能已被证明进行大尺寸的问题很好[ 18 ]。WNN提出复杂的地震与S型激活函数神经网络的应用比较优越的系统模型。小波的应用多数是有限的小尺寸[ 19 ]，但小波神经网络可以处理大尺寸的问题[ 18 ]。

In this paper two types of wavelet neural network (WNN) namely summation wavelet neural network (SWNN) and multiplication wavelet neural network (MWNN), are proposed. These two proposed WNN, lead to propose two types of wavelet neuro-fuzzy model (WNF) namely summation wavelet neuro-fuzzy model (SWNF) and multiplication wavelet neuro-fuzzy model (MWNF). From the literature survey, it has been found that all studies show the efficacy of wavelets when used in wavelet network or/and in wavelet neuro-fuzzy model. But none of the reported work caters a comparative study for different types of the wavelets. The presented work is an attempt to brought a comparative study for three types of wavelet used in WNN or/and WNF, namely, Mexican hat, Morlet and Sinc wavelet function.

在本文中，两种小波神经网络（WNN）即总和小波神经网络（SWNN）和乘法小波神经网络（MWNN），提出了。这两个建议的小波神经网络，导致了两种小波模糊神经网络模型（WNF）即总和小波神经模糊模型（SWNF）和乘法小波模糊神经网络模型（MWNF）。从文献调查，发现所有的研究表明，小波的疗效时，用小波网络或/和小波模糊神经网络模型。但报告工作的不满足不同类型的小波的比较研究。本文的工作是一个试图把用于WNN或/和WNF，小波三种类型，即，墨西哥帽的比较研究，Morlet小波函数和正弦。

The idea of this work is to use approximation of inputs by sigmoidal function and wavelet functions separately and then combine them. The sigmoidal activation function in NN can modulate low frequency section of the signal and the wavelet activation function in WNN can modulate high frequency section especially sharp section of signal. The idea of proposing SWNN and MWNN is to combine the localize approximation property of wavelets with functional approximation properties of neural network. The temporal change in dynamic system, particularly when the changes are sharp, can be accumulated in wavelets. The output of every neuron in SWNN is summation of the sigmoidal and wavelet activation functions and the output of each neuron in MWNN is the product of these two. With the proving capability of the ANFIS as a powerful approximation method [20] , that has the both ability of the learning parameter in the neural networks and the localized approximation of the TSK fuzzy model[5], different types of networks based on neuro-fuzzy model has been proposed. In TSK fuzzy model the consequent part of each rule is approximated by a linear function of the inputs. Essentially, neuro-fuzzy models based on TSK model are nonlinear, but conceptually, it is an aggregation of the linear models. If the system under consideration is chaos some forecasting able information in the system may not be predicted well, by the aggregation of these linear models. In nonlinear dynamic systems and time series application, the linear local models in TSK are adequate to predict the behavior of the system, but nonlinear local models in TSK are better to predict the nonlinear dynamic behavior of the system under consideration. For example, Wavelet neuro-fuzzy (WNF) model can be used as a good general approximation [21,22] . In these models, the premise part of each fuzzy rule, represents a localized region of the input space in which a wavelet network is used as local model in the consequent part of fuzzy rules. In the present paper, proposed wavelet neural networks, i.e., SWNN and MWNN are used as a local model in the consequent part of fuzzy rules that leads to the proposition of summation wavelet neuro-fuzzy (SWNF) and multiplication wavelet neuro-fuzzy (MWNF), respectively. By joining the localized region transformation of wavelet activation function with the localized approximation of each fuzzy rule; an increase in precision of models has been experienced.

In both wavelet network and wavelet neuro-fuzzy models three types of non-orthogonal wavelet function, namely Mexican hat, Morlet and Sinc, are used. Ability of proposed model is examined with four examples of time series. The rest of the paper is organized as follow: In Section2 ,a brief discussion of the wavelet function and wavelet transform is presented. Section 3 proposes SWNN and MWNN models. Approximation properties and convergence analysis of the proposed networks also describes in this section. Wavelet neuro-fuzzy (WNF) model are proposed in Section 4 . This section also dealt with the convergence analysis of SWNF and MWNF. Experimental results are revealed in Section5 and, finally conclusions are relegated to Section 6 .

2. Wavelet function

The wavelet transform (WT) in its continuous form provides a flexible time-frequency window, which narrows when observing high frequency phenomena and widens when analyzing low frequency behavior. Thus, time resolution becomes arbitrarily good at high frequencies, while the frequency resolution becomes arbitrarily good at low frequen-cies. This kind of analysis is suitable for signals composed of high frequency components with short duration and low frequency components with long duration, which is often the case in practical situations. Here, a brief review from the theory of wavelets is described that gives basic idea about the wavelets and the related work.

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