Types Of Wavelet Functions Research Articles

Numerical solution of different kinds of fractional‐order optimal control problems using generalized Lucas wavelets and the least squares method

AbstractGeneralized Lucas wavelets (GLWs) have two more parameters ( and ), comparing with some existing classical wavelet functions. In this manner, we have different types of wavelet functions (orthogonal and non‐orthogonal) by choosing various values of parameters and . Due to the impressive feature of the GLWs, we design a new computational method for the solution of fractional optimal control problems and fractional pantograph optimal control problems. This technique uses the GLWs and least squares method. The scheme includes expanding the required functions using GLW elements. We present new Riemann–Liouville and pantograph operational matrices for GLWs. Applying the operational matrices and least squares method, the considered problems lead to systems of algebraic equations, which can be solved numerically. A brief discussion of the error of the estimate used is investigated. Finally, some numerical experiments are exhibited to demonstrate the validity and applicability of the suggested scheme. The proposed algorithm is easy to implement and presents very accurate results.

Accuracy improvement of quantitative analysis in VIS-NIR spectroscopy using the GKF-WTEF algorithm.

The extraction of effective information in visible-near-infrared (VIS-NIR) spectroscopy is crucial and difficult for spectral analysis. In this research, an algorithm of wavelet feature extraction based on the Gaussian kernel function (GKF-WTEF) was developed to suppress the influence of external interference on VIS-NIR spectroscopy and improve the accuracy of quantitative analysis. This algorithm takes the root-mean-square error of the prediction set (RMSEP) of the model, which is established by partial least-squares regression, as the optimization criteria. First, the optimal type of wavelet function, the decomposition level, and the Gauss kernel function central frequency band are determined according to the RMSEP. Second, the Gauss kernel function bandwidth is determined by Newton's method. Then, the Hadamard product of the Gaussian kernel function and the wavelet coefficient is obtained. Finally, the wavelet coefficients after the Hadamard product can be reconstructed to obtain the spectral data after feature extraction. In order to verify the effectiveness of this algorithm, the difference in the optical parameters of the polyvinyl chloride material container was used as an external interference source. And the spectrum of Intra-lipid and India-ink mixed solution with different concentrations was collected therein. The volume fraction of India-ink in complex mixed solution was quantitatively analyzed by using the RMSEP and the average relative error of the prediction set as the evaluation criteria. The research results demonstrated that the Gaussian-wavelet transform feature extraction algorithm is an effective pretreatment method, it can satisfactorily suppress the influence of external interference on the spectrum, and it can improve the analytical accuracy of VIS-NIR spectroscopy.

Damage Identification in Beam Structure using Spatial Continuous Wavelet Transform

In this paper the applicability of spatial continuous wavelet transform (CWT) technique for damage identification in the beam structure is analyzed by application of different types of wavelet functions and scaling factors. The proposed method uses exclusively mode shape data from the damaged structure. To examine limitations of the method and to ascertain its sensitivity to noisy experimental data, several sets of simulated data are analyzed. Simulated test cases include numerical mode shapes corrupted by different levels of random noise as well as mode shapes with different number of measurement points used for wavelet transform. A broad comparison of ability of different wavelet functions to detect and locate damage in beam structure is given. Effectiveness and robustness of the proposed algorithms are demonstrated experimentally on two aluminum beams containing single mill-cut damage. The modal frequencies and the corresponding mode shapes are obtained via finite element models for numerical simulations and by using a scanning laser vibrometer with PZT actuator as vibration excitation source for the experimental study.

Reservoir Inflow Forecasting Using Ensemble Models Based on Neural Networks, Wavelet Analysis and Bootstrap Method

Accurate and reliable forecasting of reservoir inflow is necessary for efficient and effective water resources planning and management. The aim of this study is to develop an ensemble modeling approach based on wavelet analysis, bootstrap resampling and neural networks (BWANN) for reservoir inflow forecasting. In this study, performance of BWANN model is also compared with wavelet based ANN (WANN), wavelet based MLR (WMLR), bootstrap and wavelet analysis based multiple linear regression models (BWMLR), standard ANN, and standard multiple linear regression (MLR) models for inflow forecasting. Robust ANN and WANN models are ensured considering state of the art methodologies in the field. For development of WANN models, initially original time series data is decomposed using wavelet transformation, and wavelet sub-time series are considered to develop WANN models instead of standard data used for development of ANN model. To ensure a robust WANN model different types of wavelet functions are utilized. Further, a comparative analysis is carried out among different approaches of WANN model development using wavelet sub time series. Seven years of reservoir inflow data along with outflow data from two upstream reservoirs in the Damodar catchment along with rainfall data of 5 upstream rain gauge stations are considered in this study. Out of 7 years daily data, 5 years data are used for training the model, 1 year data are used for cross-validation and remaining 1 year data are used to evaluate the performance of the developed models. Different performance indices indicated better performance of WANN model in comparison with WMLR, ANN and MLR models for inflow forecasting. This study demonstrated the effectiveness of proper selection of wavelet functions and appropriate methodology for wavelet based model development. Moreover, performance of BWANN models is found better than BWMLR model for uncertainty assessment, and is found that instead of point predictions, range of forecast will be more reliable, accurate and can be very helpful for operational inflow forecasting.

Multiple crack identification in Euler beams by means of B-spline wavelet

In this paper, an effective method for identification of multiple cracks is presented based on discrete wavelet transform in a cracked beam. First, a compactly supported semi-orthogonal B-spline wavelet on interval (BSWI) is employed to construct Euler beam-bending element for free vibration analysis of cracked beams. Next, the construction of general order one-dimensional B-spline wavelets is presented and applied for damage identification in a cantilever beam modeled by wavelet-based elements. Also, principles of an appropriate wavelet selection are presented. Natural vibration modes of a cantilever beam with three cracks are analyzed using one-dimensional fourth-order B-spline wavelet. The results illustrate that BSWI elements can be used in determining the un-cracked and cracked beam natural frequencies with a high accuracy and efficiency. Moreover, the applicability of the presented method in crack identification is studied by numerical examples under several situations, such as in the presence of random noises, and the efficiency of B-spline wavelets in damage prognosis is compared with other types of wavelet functions. The obtained results show the effectiveness of B-spline wavelets in modeling of the damaged beam and identifying multiple crack locations in a free baseline scheme.

Intermittent Streamflow Forecasting and Extreme Event Modelling using Wavelet based Artificial Neural Networks

Forecasting of intermittent stream flow is necessary for water resource planning and management at catchment scale. Forecasting of extreme events and events outside the range of training data used for artificial neural network (ANN) model development has been a major bottleneck in their generalization capabilities till date. Despite of several studies using wavelet analysis in water resource modelling, no study has yet been conducted to explore capabilities of hybrid ANN modelling techniques for extreme events outside the training range. In this study a wavelet based ANN model (WANN) is proposed for intermittent streamflow forecasting and extreme event modelling. This study is carried out in a watershed in semi arid middle region of Gujarat, India. 6 years of hydro-climatic data are used in this study. 4 years of data are used for model training, 1 year for cross-validation and remaining 1 year data are used to evaluate the effectiveness of the WANN model. Two different approaches of data arrangement are considered in this study, in one approach testing data are within the range of training dataset, whereas in another approach testing data are outside the training dataset range. Performance of four different training algorithms and different types of wavelet functions are also evaluated for WANN model development. In this study it is found that WANN model performed significantly better than standard ANN models. It is observed in this study that different wavelet functions have different role in modelling complexities of normal and extreme events. WANN model simulated peak values very well and it shows that WANN model has the potential to be applied successfully for intermittent streamflow forecasting even for the data outside the training range and for extreme events.

Wavelet network meta-models for the analysis of slender offshore structures

Mooring lines and risers are crucial components of offshore oil and gas production platforms. The usual design practice requires the use of complex Finite-Element (FE) time-domain simulation tools, to represent their severe nonlinear dynamic behavior. However, such tools may require excessively high computational times; this fact motivates the study of expeditious methods – the so-called surrogate models or meta-models. In this context, this work presents an approach based on Wavelet Networks (WN) – a combination of the feed-forward neural network architecture with the wavelet transform. The goal is to obtain dramatic reductions in processing time, while providing results nearly as good as those from nonlinear dynamic FE methods.Case studies are presented to evaluate the performance of the model, in terms of accuracy and computational time. Extensive parametric studies are performed to fine-tune the model (in terms of several parameters such as type of wavelet function; number of nodes in the hidden layer; size of the training/validation sets), to find the configuration most suited for the problem at hand. It is shown that the WN-based models proposed in this work are more efficient than ANN models; also, the fine-tuned configuration performs substantially better than the standard configuration with typical values for the parameters.

EEG signal classification for epilepsy diagnosis via optimum path forest – A systematic assessment

Epilepsy refers to a set of chronic neurological syndromes characterized by transient and unexpected electrical disturbances of the brain. The detailed analysis of the electroencephalogram (EEG) is one of the most influential steps for the proper diagnosis of this disorder. This work presents a systematic performance evaluation of the recently introduced optimum path forest (OPF) classifier when coping with the task of epilepsy diagnosis directly through EEG signal analysis. For this purpose, we have made extensive use of a benchmark dataset composed of five classes, whose full discrimination is very hard to achieve. Four types of wavelet functions and three well-known filter methods were considered for the tasks of feature extraction and selection, respectively. Moreover, support vector machines configured with radial basis function (SVM-RBF) kernel, multilayer perceptron neural networks (ANN-MLP), and Bayesian classifiers were used for comparison in terms of effectiveness and efficiency. Overall, the results evidence the outperformance of the OPF classifier in both types of criteria. Indeed, the OPF classifier was usually extremely fast, with average training/testing times much lower than those required by SVM-RBF and ANN-MLP. Moreover, when configured with Coiflets as feature extractors, the performance scores achieved by the OPF classifier include 89.2% as average accuracy and sensitivity/specificity values higher than 80% for all five classes.

A wavelet-based approach for vibration analysis of framed structures

In this paper, an explicit time integration scheme is proposed for structural vibration analysis by using wavelet functions. Initially, the differential equation of vibration governing SDOF (single-degree of freedom) systems has been solved by wavelet operators, and later the proposed approach has been generalized for MDOF (multi-degrees of freedom) systems. For this purpose, two different types of wavelet functions have been exemplified including, complex Chebyshev wavelet functions and simple Haar wavelet functions. In the proposed approach, a straightforward formulation has been derived from the numerical approximation of response through the wavelet definition. Emphasizing on frequency-domain approximation, a simple step-by-step algorithm has been implemented and improved to calculate the response of MDOF systems. Moreover, stability and accuracy of results have been evaluated. The effectiveness of the proposed approach is demonstrated using three examples compared with some of the existing numerical integration schemes such as family of Newmark-β, Wilson-θ and central difference method. In all the procedures, computation time involved has also been considered. Finally, it is concluded that the vibration analysis of structures is improved by lesser computation time and high accuracy of proposed approach, particularly, in large-scaled systems.

Crack identification in composite elements with non-linear geometry using spatial wavelet transform

A great popularity of constructional polymer composites in industrial applications requires the development of appropriate diagnostic methods for the elements made of these materials. These methods should provide damage detection and identification in possible early stages of its development. A large group of such methods is based on vibration testing and modal analysis. One of the promising techniques of processing vibration data for damage identification is a wavelet transform. Considering the practical needs the method should be applicable for plane structures with non-linear geometry. Results presented in this paper consider numerous analyses including the application of different types of wavelet transforms, different types of wavelet functions and other factors, which have an influence on the accuracy of damage identification. The proposed method shows good effectiveness in damage identification problems and could be applied in industrial conditions as well.

Sound quality estimation for nonstationary vehicle noises based on discrete wavelet transform

A new method based on the discrete wavelet transform (DWT) for sound quality estimation (SQE) of nonstationary vehicle noises is presented in this paper. Sample interior and exterior vehicle noises are first measured and denoised using the wavelet threshold method to initially lower the noise level with negligible signal distortion. A multirate filter bank combined by a set of low-pass and band-pass filters and a DWT-based filter bank are then developed for sound octave-band analysis (OBA). To build the DWT octave-band filters, the Mallat pyramidal algorithm is introduced, and five types of wavelet function with different filter lengths are investigated and compared. Finally, the Daubechies wavelet with a filter order of 35 is determined and applied to estimate sound pressure levels (SPLs) of both the interior and exterior vehicle noises. Verification results show that the newly proposed multirate-filter-based method (MF-OBA) and DWT-based method (DWT-OBA) are accurate and effective for SQE of nonstationary vehicle noises. Due to outstanding time-frequency characteristics of the wavelet analysis, the DWT-OBA can be suggested to deal with not only SQE of nonstationary vehicle noises, but also any other sound-related signal processing in engineering.

Artificial wavelet neural network and its application in neuro-fuzzy models

In the proposed work, two types of artificial neural networks are proposed by using well-known advantages and valuable features of wavelets and sigmoidal activation functions. Two neurons are derived by adding and multiplying the outputs of the wavelet and the sigmoidal activation functions. These neurons in a feed-forward single hidden layer network result summation wavelet neural network (SWNN) and multiplication wavelet neural network (MWNN). An algorithm is introduced for structure determination of the proposed networks. Approximation properties of SWNN and MWNN have been evaluated with different wavelet functions. The above networks in the consequent part of the neuro-fuzzy model result summation wavelet neuro-fuzzy (SWNF) and multiplication wavelet neuro-fuzzy (MWNF) models. Different types of wavelet function are tested with the proposed networks and fuzzy models on four different dynamical examples. Convergence of the learning process is also guaranteed by adaptive learning rate and performing stability analysis using Lyapunov function.

Artificial wavelet neuro-fuzzy model based on parallel wavelet network and neural network

From the well-known advantages and valuable features of wavelets when used in neural network, two type of networks (i.e., SWNN and MWNN) have been proposed. These networks are single hidden layer network. Each neuron in the hidden layer is comprised of wavelet and sigmoidal activation functions. First model is derived from adding the outputs of wavelet and sigmoidal activation functions, while in the second model outputs of wavelet and sigmoidal activation function are multiplied together. Using these proposed networks in consequent part of the neuro-fuzzy model, which result summation wavelet neuro-fuzzy and multiplication wavelet neuro-fuzzy models, are also proposed. Different types of wavelet function are tested with proposed networks and fuzzy models on four different types of examples. Convergence of the learning process is also guaranteed by performing stability analysis using Lyapunov function.

Analysis and localization of epileptic events using wavelet packets

This article compares results obtained in previous studies using time–frequency representations (Wigner–Ville, Choi–Williams and Parametric) and the wavelet transform with those obtained with wavelet packet functions to show new findings about their quality in the analysis of ECoG recordings in human intractable epilepsy: data from 21 patients have been analyzed and processed with four types of wavelet functions, including Orthogonal, Biorthogonal and Non-Orthogonal basis. These functions were compared in order to test their quality to represent spikes in the ECoG. The energy based on the wavelet coefficients to different scales was also calculated. The best results were found with the biorthogonal-6.8 wavelet on 5–7 scales, which gave 0.92 sensitivity, but with a high percentage of false positives; this representation was highly correlated with spike events on time and duration. To improve these results we have studied the wavelet packet coefficients energy. We found that reconstruction wavelet packet coefficients at 4 and 9 nodes contain significant information to characterize the spike event. These nodes' reconstruction coefficients were multiplied and this product was highly correlated with spikes events on time and duration. With this procedure we improved the sensitivity up to 0.96 with the same biorthogonal-6.8 wavelet at four levels. With this technique we do not sacrifice computation time: 896 samples are processed at only 0.16 s, so that it is possible to show the spike scattering path on line, because 896 samples (7 s)/16 channels are processed at 3.13 s.

Application of wavelet transform in reflection seismic data analysis

A seismic signal is characterized by its travel time, frequency and phase information, as well as noises, especially coherent noises. The wavelet decomposition of a seismic signal involves simultaneous representation of its time and frequency characteristics. The key advantage of the wavelet transform (WT) over the conventional Fourier transform is that it can not only provide insight on the combined temporal and spectral character- istics of signals, but it can also localize the target information in the time-frequency domain simultaneously. The wavelet trans- form distinguished itself from the short time Fourier transform for time-frequency analysis in that it has a zoom-in and zoom-out capability. Thus the WT approach is suitable for time-frequency analysis of seismic signals. Application capability of the wavelet transform depends on the selection of the wavelet functions from which a basis function can be constructed for signal decomposition. There are two types of wavelet functions: orthogonal and non-orthogonal wavelet func- lions, thus the algorithms of the wavelet transform vary. The corn- moldy used wavelet functions are orthonormal and compactly supported, but do not have a finite impulse response and linear phase. These features are, however, undesirable for applications in exploration seismology, especially when further subsequent pro- ccssing is required in complex domain. Compactly supported non- orthogonal wavelets do not have phase distortion problem and provide a better choice for seismic data processing. Reflection and refraction events, coherent noises such as ground roll, air-wave, and ringing in seismic data have characteristic fea- tures and can be distinguished in the time-frequency space. The Morlet wavelet, a well-known example of non-orthogonal wave- lets, is tested in this study for effectively suppressing coherent noise. It was also demonstrated that the reconstructed signals after the weighted wavelet transform show significant improvements in the S/N ratio.

A wavelet-based algorithm for detecting clustered microcalcifications in digital mammograms.

A computerized scheme to detect clustered microcalcifications in digital mammograms has been developed. Detection of individual microcalcifications in regions of interest (ROIs) was also performed. The mammograms were previously classified into fatty and dense, according to their breast tissue. The most appropriate wavelet basis and reconstruction levels were selected. To select the wavelet basis, 40 profiles of microcalcifications were decomposed and reconstructed using different types of wavelet functions and different combinations of wavelet coefficients. The symlets with a basis of length 8 were chosen for fatty tissue. For dense tissue, the Daubechies' wavelets with a four-element basis were employed. Two methods to detect individual microcalcifications were evaluated: (a) two-dimensional wavelet transform, and (b) one-dimensional wavelet transform. The second technique yielded the best results, and was used to detect clustered microcalcifications in the complete mammogram. When detecting individual microcalcifications by using two-dimensional wavelet transform we have obtained, for fatty ROIs, a sensitivity of 71.11% at a false positive rate of 7.13 per image. For dense ROIs the sensitivity was 60.76% and the false positive rate, 7.33. The areas (A1) under the AFROC curves were 0.33+/-0.04 and 0.28+/-0.02, respectively. The one-dimensional wavelet transform method yielded 80.44% of sensitivity and 6.43 false positives per image (A1=0.39+/-0.03) for fatty ROIs, and 62.17% and 5.82 false positives per image (A1=0.37+/-0.02) for dense ROIs. For the detection of clusters of microcalcifications in the entire mammogram, the sensitivity was 80.00% with 0.94 false positives per image (A1=0.77+/-0.09) for fatty mammograms, and 72.85% of sensitivity at a false positive detection rate of 2.21 per image (A1=0.64+/-0.07) for dense mammograms. Globally, a sensitivity of 76.43% at a false positive detection rate of 1.57 per image was obtained.