Despite the remarkable success of deep learning in pattern recognition, deep network models face the problem of training a large number of parameters. In this paper, we propose and evaluate a novel multi-path wavelet neural network architecture for image classification with far less number of trainable parameters. The model architecture consists of a multi-path layout with several levels of wavelet decompositions performed in parallel followed by fully connected layers. These decomposition operations comprise wavelet neurons with learnable parameters, which are updated during the training phase using the back-propagation algorithm. We evaluate the performance of the introduced network using common image datasets without data augmentation except for SVHN and compare the results with influential deep learning models. Our findings support the possibility of reducing the number of parameters significantly in deep neural networks without compromising its accuracy.
The application of deep learning approaches to finance has received a great deal of attention from both investors and researchers. This study presents a novel deep learning framework where wavelet transforms (WT), stacked autoencoders (SAEs) and long-short term memory (LSTM) are combined for stock price forecasting. The SAEs for hierarchically extracted deep features is introduced into stock price forecasting for the first time. The deep learning framework comprises three stages. First, the stock price time series is decomposed by WT to eliminate noise. Second, SAEs is applied to generate deep high-level features for predicting the stock price. Third, high-level denoising features are fed into LSTM to forecast the next day’s closing price. Six market indices and their corresponding index futures are chosen to examine the performance of the proposed model. Results show that the proposed model outperforms other similar models in both predictive accuracy and profitability performance. Wavelet networks (WNs) are a new class of networks which have been used with great success in a wide range of applications. However a general accepted framework for applying WNs is missing from the literature. In this study, we present a complete statistical model identification framework in order to apply WNs in various applications. The following subjects were thoroughly examined: the structure of a WN, training methods, initialization algorithms, variable significance and variable selection algorithms, model selection methods and finally methods to construct confidence and prediction intervals. In addition the complexity of each algorithm is discussed. Our proposed framework was tested in two simulated cases, in one chaotic time series described by the Mackey-Glass equation and in three real datasets described by daily temperatures in Berlin, daily wind speeds in New York and breast cancer classification. Our results have shown that the proposed algorithms produce stable and robust results indicating that our proposed framework can be applied in various applications. Since EEG is one of the most important sources of information in therapy of epilepsy, several researchers tried to address the issue of decision support for such a data. In this paper, we introduce two fundamentally different approaches for designing classification models (classifiers); the traditional statistical method based on logistic regression and the emerging computationally powerful techniques based on artificial neural networks (ANNs). Logistic regression as well as feedforward error backpropagation artificial neural networks (FEBANN) and wavelet neural networks (WNN) based classifiers were developed and compared in relation to their accuracy in classification of EEG signals. In these methods we used FFT and autoregressive (AR) model by using maximum likelihood estimation (MLE) of EEG signals as an input to classification system with two discrete outputs: epileptic seizure or nonepileptic seizure. By identifying features in the signal we want to provide an automatic system that will support a physician in the diagnosing process. By applying AR with MLE in connection with WNN, we obtained novel and reliable classifier architecture. The network is constructed by the error backpropagation neural network using Morlet mother wavelet basic function as node activation function. The comparisons between the developed classifiers were primarily based on analysis of the receiver operating characteristic (ROC) curves as well as a number of scalar performance measures pertaining to the classification. The WNN-based classifier outperformed the FEBANN and logistic regression based counterpart. Within the same group, the WNN-based classifier was more accurate than the FEBANN-based classifier, and the logistic regression-based classifier.  It is known that the force and vibration sensor signals in a turning process are sensitive to the gradually increasing flank wear. Based on this fact, this paper investigates a flank wear assessment technique in turning through force and vibration signals. Mainly to reduce the computational burden associated with the existing sensor-based methods for flank wear assessment, a so-called wavelet network is investigated. The basic idea in this new method is to optimize simultaneously the wavelet parameters (that represent signal features) and the signal-interpretation parameters (that are equivalent to neural network weights) to eliminate the feature extraction phase without increasing the computational complexity of the neural network. A neural network architecture similar to a standard one-hidden-layer feedforward neural network is used to relate sensor signal measurements to flank wear classes. A novel training algorithm for such a network is developed. The performance of this n ew method is compared with a previously developed flank wear assessment method which uses a separate feature extraction step. The proposed wavelet network can also be useful for developing signal interpretation schemes for manufacturing process monitoring, critical component monitoring, and product quality monitoring. We describe a new approach to real time learning of unknown functions based on an interpolating wavelet estimation. We choose a subfamily of a wavelet basis relying on nested hierarchical allocation and update in real time our estimate of the unknown function. Such an interpolation process can be used for real time applications like neural network adaptive control, where learning an unknown function very fast is critical. A new technique, wavelet network, is introduced to predict chaotic time series. By using this technique, firstly, we make accurate short-term predictions of the time series from chaotic attractors. Secondly, we make accurate predictions of the values and bifurcation structures of the time series from dynamical systems whose parameter values are changing with time. Finally we predict chaotic attractors by making long-term predictions based on remarkably few data points, where the correlation dimensions of predicted attractors are calculated and are found to be almost identical to those of actual attractors.  A representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed. It is shown that by appropriate grouping of terms, feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings. It is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks. For the purpose of analysis, the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class. Spatio-spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task. Two synthesis procedures based on spatio-spectral localization that reduce the training problem to one of convex optimization are outlined. > A local linear wavelet neural network (LLWNN) is presented in this paper. The difference of the network with conventional wavelet neural network (WNN) is that the connection weights between the hidden layer and output layer of conventional WNN are replaced by a local linear model. A hybrid training algorithm of particle swarm optimization (PSO) with diversity learning and gradient descent method is introduced for training the LLWNN. Simulation results for the prediction of time-series show the feasibility and effectiveness of the proposed method. We propose a wavelet multiscale decomposition-based autoregressive approach for the prediction of 1-h ahead load based on historical electricity load data. This approach is based on a multiple resolution decomposition of the signal using the non-decimated or redundant Haar a trous wavelet transform whose advantage is taking into account the asymmetric nature of the time-varying data. There is an additional computational advantage in that there is no need to recompute the wavelet transform (wavelet coefficients) of the full signal if the electricity data (time series) is regularly updated. We assess results produced by this multiscale autoregressive (MAR) method, in both linear and non-linear variants, with single resolution autoregression (AR), multilayer perceptron (MLP), Elman recurrent neural network (ERN) and the general regression neural network (GRNN) models. Results are based on the New South Wales (Australia) electricity load data that is provided by the National Electricity Market Management Company (NEMMCO). We consider a wavelet neural network approach for electricity load prediction. The wavelet transform is used to decompose the load into different frequency components that are predicted separately using neural networks. We firstly propose a new approach for signal extension which minimizes the border distortion when decomposing the data, outperforming three standard methods. We also compare the performance of the standard wavelet transform, which is shift variant, with a non-decimated transform, which is shift invariant. Our results show that the use of shift invariant transform considerably improves the prediction accuracy. In addition to wavelet neural network, we also present the results of wavelet linear regression, wavelet model trees and a number of baselines. Our evaluation uses two years of Australian electricity data. In this paper, a prototype wavelet-based neural-network classifier for recognizing power-quality disturbances is implemented and tested under various transient events. The discrete wavelet transform (DWT) technique is integrated with the probabilistic neural-network (PNN) model to construct the classifier. First, the multiresolution-analysis technique of DWT and the Parseval's theorem are employed to extract the energy distribution features of the distorted signal at different resolution levels. Then, the PNN classifies these extracted features to identify the disturbance type according to the transient duration and the energy features. Since the proposed methodology can reduce a great quantity of the distorted signal features without losing its original property, less memory space and computing time are required. Various transient events tested, such as momentary interruption, capacitor switching, voltage sag swell, harmonic distortion, and flicker show that the classifier can detect and classify different power disturbance types efficiently. Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes. Abstract This paper discusses the concept of adaptive wavelets and their application to signal compression and classification. When wavelet non-linear functions are used in the neurons of a neural network we can estimate the wavelet parameters—scale and translation and the relative importance (weight) of each basis function optimally with respect to approximating a given function (signal) in the minimum mean square error sense or with respect to classifying a function (signal) with minimum mean classification error. This estimation can be considered as adaptive (signal or function dependent) sampling. In this paper, we give theoretical proof to show that such an adaptive sampling scheme constitutes a frame which implies that the neural network-based estimation technique allows us to reconstruct the input signal from the adaptive wavelets such that the reconstruction is numerically stable. We apply the proposed representation and classification architectures for coding and classifying biological signals such as electrocardiogram (ECG). Experimental details of ECG coder, and classification of ECG waves such as P, QRS and T are provided. The experimental results reported in this paper are obtained by applying the proposed methodologies to standard ECG (American Heart Association) data base. The results indicate that the compression ratio of about twice better than the current techniques can be obtained and average classification accuracy of 94 can be obtained for abnormal P, QRS and T classification.