A Review of Wavelet Networks, Wavenets, Fuzzy Wavenets and their Applications

The combination of wavelet theory and neural networks has lead to the development of wavelet networks. Wavelet networks are feed-forward neural networks using wavelets as activation function. Wavelet networks have been used in classification and identification problems with some success. The strength of wavelet networks lies in their capabilities of catching essential features in „frequency-rich“ signals. In wavelet networks, both the position and the dilation of the wavelets are optimized besides the weights. Wavenet is another term to describe wavelet networks. Originally, wavenets did refer to neural networks using dyadic wavelets. In wavenets, the position and dilation of the wavelets are fixed and the weights are optimized by the network. We propose to adopt this terminology. The theory of wavenets has been generalized by the author to biorthogonal wavelets. This extension to biorthogonal wavelets has lead to the development of fuzzy wavenets. A serious difficulty with most neurofuzzy methods is that they do often furnish rules without a transparent interpretation. A solution to this problem is furnished by multiresolution techniques. The most appropriate membership functions are chosen from a dictionary of membership functions forming a multiresolution. The dictionary contains a number of membership functions that have the property to be symmetric, everywhere positive and with a single maxima. This family includes among others splines and some radial functions. The main advantage of using a dictionary of membership functions is that each term, such as „small“, „large“ is well defined beforehand and is not modified during learning. The multiresolution properties of the membership functions in the dictionary function permit to fuse or split membership functions quite easily so as to express the rules under a linguistically understandable and intuitive form for the human expert. Different techniques, generally referred by the term „fuzzy-wavelet“, have been developed for data on a regular grid. Fuzzy wavenets extend these techniques to online learning. A major advantage of fuzzy wavenets techniques in comparison to most neurofuzzy methods is that the rules are validated online during learning by using a simple algorithm based on the fast wavelet decomposition algorithm. Significant applications of wavelet networks and fuzzy wavenets are discussed to illustrate the potential of these methods.