Wavelet Families Research Articles

Wavelet Entropy Based Analysis and Forecasting of Crude Oil Price Dynamics

For the modeling of complex and nonlinear crude oil price dynamics and movement, wavelet analysis can decompose the time series and produce multiple economically meaningful decomposition structures based on different assumptions of wavelet families and decomposition scale. However, the determination of the optimal model specification will critically affect the forecasting accuracy. In this paper, we propose a new wavelet entropy based approach to identify the optimal model specification and construct the effective wavelet entropy based forecasting models. The wavelet entropy algorithm is introduced to determine the optimal wavelet families and decomposition scale, that will produce the improved forecasting performance. Empirical studies conducted in the crude oil markets show that the proposed algorithm outperforms the benchmark model, in terms of conventional performance evaluation criteria for the model forecasting accuracy.

Biorthogonal Wavelet Construction Using Homotopy Method

The biorthogonal wavelets families are usedwidely because they have compact support, complete symmetry and linear phase. According to Bezout’s theorem,the biorthogonal wavelets available now are only some particular examples of total solutions. The quantity of solutions is decided jointly by the scaling function vanishingmoment N and dual vanishing moment N. The relationship of N, N and solutions’ quantity is discussed in detail.According to the constraint conditions which the compactbiorthogonal wavelets satisfy, a number of biorthogonalwavelets are constructed in which the global convergenthomotopy method is used for different N and N. The filter coefficients and plots of scaling function, dual scalingfunction, wavelet function and dual wavelet function aregiven.

Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

The discrete cosine transform (DCT) is known to be asymptotically equivalent to the Karhunen-Loeve transform (KLT) of Gaussian first-order auto-regressive (AR(1)) processes. Since being uncorrelated under the Gaussian hypothesis is synonymous with independence, it also yields an independent-component analysis (ICA) of such signals. In this paper, we present a constructive non-Gaussian generalization of this result: the characterization of the optimal orthogonal transform (ICA) for the family of symmetric- $\alpha$ -stable AR(1) processes. The degree of sparsity of these processes is controlled by the stability parameter $0 with the only non-sparse member of the family being the classical Gaussian AR(1) process with $\alpha=2$ . Specifically, we prove that, for $\alpha , a fixed family of operator-like wavelet bases systematically outperforms the DCT in terms of compression and denoising ability. The effect is quantified with the help of two performance criteria (one based on the Kullback-Leibler divergence, and the other on Stein’s formula for the minimum estimation error) that can also be viewed as statistical measures of independence. Finally, we observe that, for the sparser kind of processes with $0 , the operator-like wavelet basis, as dictated by linear system theory, is undistinguishable from the ICA solution obtained through numerical optimization. Our framework offers a unified view that encompasses sinusoidal transforms such as the DCT and a family of orthogonal Haar-like wavelets that is linked analytically to the underlying signal model.

Wavelet Transform-Based Classification of Electromyogram Signals Using an Anova Technique

Wavelet analysis of surface electromyogram (sEMG) signals has been investigated. Methods to remove noise before processing and further analysis are rather significant for these signals. The sEMG signals were estimated with the following steps, first, the obtained signal was decomposed using wavelet transform; then, decomposed coefficients were analyzed by threshold methods, and, finally, reconstruction was performed. Comparison of the Daubechies wavelet family for effective removing noise from the recorded sEMGs was executed preciously. As was found, wavelet transform db4 performs denoising best among the aforesaid wavelet family. Results inferred that Daubechies wavelet families (db4) were more suitable for the analysis of sEMG signals related to different upper limb motions, and a classification accuracy of 88.90% was achieved. Then, a statistical technique (one-way repeated factorial analysis) for the experimental coefficient was done to investigate the class separ ability among different motions.

Wavelets as basis functions to represent the coarse-graining potential in multiscale coarse graining approach

In this paper, we apply Multiresolution Analysis (MRA) to develop sparse but accurate representations for the Multiscale Coarse-Graining (MSCG) approximation to the many-body potential of mean force. We rigorously framed the MSCG method into MRA so that all the instruments of this theory become available together with a multitude of new basis functions, namely the wavelets. The coarse-grained (CG) force field is hierarchically decomposed at different resolution levels enabling to choose the most appropriate wavelet family for each physical interaction without requiring an a priori knowledge of the details localization. The representation of the CG potential in this new efficient orthonormal basis leads to a compression of the signal information in few large expansion coefficients. The multiresolution property of the wavelet transform allows to isolate and remove the noise from the CG force-field reconstruction by thresholding the basis function coefficients from each frequency band independently. We discuss the implementation of our wavelet-based MSCG approach and demonstrate its accuracy using two different condensed-phase systems, i.e. liquid water and methanol. Simulations of liquid argon have also been performed using a one-to-one mapping between atomistic and CG sites. The latter model allows to verify the accuracy of the method and to test different choices of wavelet families. Furthermore, the results of the computer simulations show that the efficiency and sparsity of the representation of the CG force field can be traced back to the mathematical properties of the chosen family of wavelets. This result is in agreement with what is known from the theory of multiresolution analysis of signals.

The application of continuous wavelet transform and least squares support vector machine for the simultaneous quantitative spectrophotometric determination of Myricetin, Kaempferol and Quercetin as flavonoids in pharmaceutical plants

Flavonoids are γ-benzopyrone derivatives, which are highly regarded in these researchers for their antioxidant property. In this study, two new signals processing methods been coupled with UV spectroscopy for spectral resolution and simultaneous quantitative determination of Myricetin, Kaempferol and Quercetin as flavonoids in Laurel, St. John’s Wort and Green Tea without the need for any previous separation procedure. The developed methods are continuous wavelet transform (CWT) and least squares support vector machine (LS-SVM) methods integrated with UV spectroscopy individually. Different wavelet families were tested by CWT method and finally the Daubechies wavelet family (Db4) for Myricetin and the Gaussian wavelet families for Kaempferol (Gaus3) and Quercetin (Gaus7) were selected and applied for simultaneous analysis under the optimal conditions. The LS-SVM was applied to build the flavonoids prediction model based on absorption spectra. The root mean square errors for prediction (RMSEP) of Myricetin, Kaempferol and Quercetin were 0.0552, 0.0275 and 0.0374, respectively. The developed methods were validated by the analysis of the various synthetic mixtures associated with a well- known flavonoid contents. Mean recovery values of Myricetin, Kaempferol and Quercetin, in CWT method were 100.123, 100.253, 100.439 and in LS-SVM method were 99.94, 99.81 and 99.682, respectively. The results achieved by analyzing the real samples from the CWT and LS-SVM methods were compared to the HPLC reference method and the results were very close to the reference method. Meanwhile, the obtained results of the one-way ANOVA (analysis of variance) test revealed that there was no significant difference between the suggested methods.

Enhanced B-Wavelets via Mixed, Composite Packets

A modified $B$ -wavelet construction with enhanced filter characteristics is considered. The design comprises a superposition of tessellated, integer dilated, “sister” wavelet functions. We here propose a cascaded filter-bank realization of this wavelet family together with some notable extensions. We prove that modifications of low-order members exist in the multiresolution subspace spanned by the half-translates of the original wavelets and, hence, that the resulting modified wavelet coefficients can be computed as convolutions of the undecimated original wavelet coefficients. Finite impulse response filters are thus designed and incorporated into a $B$ -wavelet packet architecture such that the mainlobe-to-sidelobe ratio of the resulting wavelet filter characteristic is improved. This is achieved by designing the filters so that zeros are introduced near to the maxima of the harmonics. It is shown that the numbers of zeros can be balanced with the length of the corresponding filters by controlling the “modification order”. Several constructions are presented. We prove that two such constructions satisfy the perfect reconstruction property for all orders. The resulting modified wavelets preserve many of the properties of the original $B$ -wavelets, such as differentiability, number of vanishing moments, symmetry, anti-symmetry, finite support, and the existence of a closed form expression.

Runoff forecasting using hybrid Wavelet Gene Expression Programming (WGEP) approach

This study presents a novel approach of using the hybrid Wavelet Gene Expression Programming (WGEP) model to forecast the runoff using rainfall data. The rainfall-runoff data from four different catchments located in the different Hydro-Climatic regions of the world is used in the study. The WGEP model is developed by integrating the discrete wavelet transform (DWT) with the Gene Expression Programming (GEP) models so that individual strengths of each approach can be exploited in a synergistic manner. It is implemented by transforming the input rainfall data using the DWT in order to reveal the temporal and the spectral information contained in the data and subsequently this transformed data is used as input for the GEP models. Ten different mother wavelet functions from different wavelet families are employed in the study in order to transform the data. The mother wavelet functions used in the study include the simple mother wavelet function Haar, the db2, db4, db8 wavelet functions of the most popular wavelet family Daubechies, sym2, sym4, sym8 wavelet functions with sharp peaks, coif2, coif4 wavelet functions and the discrete approximation of meyer (dmey) wavelet. The study further investigates the selection of the optimum input vector to be used in conjunction with the WGEP models by considering nine different input vectors. The first five input vectors contain only one rainfall data series ranging from lagged-1 day to lagged-5 day rainfall data series. The remaining four input vectors are selected using the most common approach in which selection of the input vector comprising of the sequential time series data. The performance of the hybrid WGEP models are compared with the simple GEP models developed without the wavelet transformation for the four selected catchments. The study found that the performance of the WGEP models is superior relative to the simple GEP model for all the nine input vectors considered. However, the WGEP models outperformed their respective simple GEP models only for the first five input vectors. Furthermore, the WGEP models exhibits better results with the dmey wavelet function for all the four catchments during both training and testing.

The mean consistency of wavelet density estimators

The wavelet estimations have made great progress when an unknown density function belongs to a certain Besov space. However, in many practical applications, one does not know whether the density function is smooth or not. It makes sense to consider the mean $L_{p}$ -consistency of the wavelet estimators for $f\in L_{p}$ ( $1\leq p\leq\infty$ ). In this paper, the authors will construct wavelet estimators and analyze their $L_{p}(\mathbb{R})$ performance. They prove that, under mild conditions on the family of wavelets, the estimators are shown to be $L_{p} $ ( $1\leq p\leq\infty$ )-consistent for both noiseless and additive noise models.

MEO based secured, robust, high capacity and perceptual quality image watermarking in DWT-SVD domain.

The aim of this paper is to present multiobjective evolutionary optimizer (MEO) based highly secured and strongly robust image watermarking technique using discrete wavelet transform (DWT) and singular value decomposition (SVD). Many researchers have failed to achieve optimization of perceptual quality and robustness with high capacity watermark embedding. Here, we achieved optimized peak signal to noise ratio (PSNR) and normalized correlation (NC) using MEO. Strong security is implemented through eight different security levels including watermark scrambling by Fibonacci-Lucas transformation (FLT). Haar wavelet is selected for DWT decomposition to compare practical performance of wavelets from different wavelet families. The technique is non-blind and tested with cover images of size 512x512 and grey scale watermark of size 256x256. The achieved perceptual quality in terms of PSNR is 79.8611dBs for Lena, 87.8446 dBs for peppers and 93.2853 dBs for lake images by varying scale factor K1 from 1 to 5. All candidate images used for testing namely Lena, peppers and lake images show exact recovery of watermark giving NC equals to 1. The robustness is tested against variety of attacks on watermarked image. The experimental demonstration proved that proposed method gives NC more than 0.96 for majority of attacks under consideration. The performance evaluation of this technique is found superior to all existing hybrid image watermarking techniques under consideration.

Numerical approximation of 2D Fredholm integral eigenvalue problems by orthogonal wavelets

We investigate the numerical approximation of two-dimensional, second kind Fredholm integral eigenvalue problems by the Galerkin method with the Cohen–Daubechies–Vial (CDV) wavelet family. This choice provides us orthogonal bases for bounded domains, avoiding the need of periodization or domain truncation. The CDV family is indexed by the number of vanishing moments, which drives the regularity of the basis. We generate the Galerkin basis from tensorized scaling functions and employ weighted Gaussian quadratures derived from refinement equations. Numerical experiments address the relative computational cost of this approach with respect to the Haar basis and the relationship between convergence rate and number of vanishing moments.

Signal Design for Context Aware Distributed Radar Sensing Networks Based on Wavelets

In the field of radar sensing, a paradigm shift is taking place from the traditional radar system to distributed sensing/radar networks. Comparing with the single radar system distributed radar networks have a superior performance in detection and estimation of parameters of target. They also enhance the imaging and classification performance of sensing systems. As in distributed radar networks several radars operate at the same frequency band in a network and share the spectrum they may interfere each other, and accordingly the radar-to-radar interference becomes a major problem to be addressed. Therefore, the context aware design of radar signals for radar sensing networks to cope with the interference is pronounced. On the other hand Wavelet packet transform has recently emerged as a novel signal design technique with attractions such as good time–frequency resolution, low sidelobes, and the reconfigurability capability. The wavelet approach is advantageous for the signal design of cognitive radar networks mainly because of its flexibility, lower sensitivity to distortion and interference, and better utilization of spectrum. For minimum interference to the adjacent radar bands, the wavelets should be maximally frequency selective. Commonly known wavelets are not frequency selective in nature and hence result in poor spectrum sharing performance. To alleviate this problem, we design a family of wavelets that are maximally frequency selective in nature. To this end, the design constraints are first enlisted. Then the problem, originally non-convex, is reformulated into a convex optimization problem and solved using Semi Definite Programming tools. Through simulation studies the benefits of the newly designed wavelets for a distributed radar sensing network are demonstrated.

Performance Analysis of Different Wavelet Families Over Fading Environments for Mobile WiMAX System

Orthogonal Frequency Division Multiplexing (OFDM) has made tremendous improvement possible in wireless technology. Wavelet analysis is used as an alternative to Fourier analysis in OFDM system to transform and map data onto orthogonal subcarriers, as it is a time-frequency domain operation thus it offers the advantages of optimal flexibility with less complexity. This paper presents performance analysis in terms of Bit Error rate (BER) of different wavelet families in OFDM System under the joint effect of path loss, multipath fading and noisy environment for Mobile WiMAX. The results obtained show that the performance of the mobile-multipath environments for 16QAM modulation schemes is enhanced as Daubechies and Biorthogonal wavelet families offers lower BER with different Path loss model. The performance measurements and analysis was done in simulation developed in MATLAB.

Chemometrics-assisted spectrophotometric method for simultaneous determination of Pb2+ and Cu2+ ions in different foodstuffs, soil and water samples using 2-benzylspiro [isoindoline-1,5′-oxazolidine]-2′,3,4′-trione using continuous wavelet transformation and partial least squares – Calculation of pKf of complexes with rank annihilation factor analysis

A new multi-component analysis method based on zero-crossing point-continuous wavelet transformation (CWT) was developed for simultaneous spectrophotometric determination of Cu2+ and Pb2+ ions based on the complex formation with 2-benzyl espiro[isoindoline-1,5oxasolidine]-2,3,4 trione (BSIIOT). The absorption spectra were evaluated with respect to synthetic ligand concentration, time of complexation and pH. Therefore according the absorbance values, 0.015mmolL−1 BSIIOT, 10min after mixing and pH 8.0 were used as optimum values. The complex formation between BSIIOT ligand and the cations Cu2+ and Pb2+ by application of rank annihilation factor analysis (RAFA) were investigated. Daubechies-4 (db4), discrete Meyer (dmey), Morlet (morl) and Symlet-8 (sym8) continuous wavelet transforms for signal treatments were found to be suitable among the wavelet families. The applicability of new synthetic ligand and selected mother wavelets were used for the simultaneous determination of strongly overlapped spectra of species without using any pre-chemical treatment. Therefore, CWT signals together with zero crossing technique were directly applied to the overlapping absorption spectra of Cu2+ and Pb2+. The calibration graphs for estimation of Pb2+ and Cu 2+were obtained by measuring the CWT amplitudes at zero crossing points for Cu2+ and Pb2+ at the wavelet domain, respectively. The proposed method was validated by simultaneous determination of Cu2+ and Pb2+ ions in red beans, walnut, rice, tea and soil samples. The obtained results of samples with proposed method have been compared with those predicted by partial least squares (PLS) and flame atomic absorption spectrophotometry (FAAS).

Threshold Analysis of Wavelet Based Fingerprint Feature Extraction Methods on Multiple Impression Dataset

In recent years, fingerprint recognition has been moving through series of evolutions with the intent to decrease the False Acceptance Rate (FAR) and the False Rejection Rate (FRR) in order to achieve minimum Equal Error Rate (EER) while increasing recognition rate. In practical cases, fingerprint images stored in fingerprint databases may have come from scanners with different specifications under variant environmental conditions which may produce different or multiple impressions and backgrounds. The choice of what single and acceptable threshold value to use in order to characterize fingerprint features in images for recognition is therefore crucial in establishing a minimal EER. In this paper, we investigate and analyze the effect of several threshold values on EER when several families of wavelets based methods for feature extraction are applied on multiple impression datasets (Fingerprint Verification Competition-FVC2004). After conducting several threshold analysis on extracted features from multiple impression dataset, the results show

Multiscale analysis of surface morphologies by curvelet and contourlet transforms

The surface topographies of precision components are critical to their functionalities. However, it is challenging to characterize the topographies of complex surfaces, especially for structured surfaces. The wavelet families are promising for the multiscale geometry analysis of nonstochastic surfaces. The second-generation curvelet transform provides a sparse representation and good multiscale decomposition for curve singularities. However, the contourlet expansion, composed of bases oriented along various directions in multiple scales with smaller redundancy rates, has a remarkable capability of representing borderlines. In this paper they are both adopted for the characterization of surface topographies. Different components can be extracted according to their scales and morphological characteristics; as a result, the corresponding manufacturing processes and functionalities can be analyzed specifically. Numerical experiments are given to demonstrate the capabilities of these methods in sparse representation and effective extraction of geometry features of different nonstochastic surfaces.

Wavelet Threshold Denoising Experiments of Air-coupled Ultrasonic Signal

In air-coupled ultrasonic testing, the received ultrasonic signal has poor signal to noise ratio(SNR) due to low transmission ratio, noises introduced by transmitting and receiving system, and scattering noises during the propagation of ultrasound in the media. The use of wavelet threshold denoising technique to deal with handicaps above is faced with problems of parameter selection including wavelet base, decomposition level, and threshold function. The wavelet threshold denoising experiments are carried out using the single factor analysis method based on the basic principles of wavelet theory. Different wavelet bases from different wavelet family(Daubechies, Symlet, and Coiflet), decomposition level(4-8 levels), and threshold function(soft threshold function or improved threshold function based on hard and soft threshold) are used to wavelet threshold denoising process for the realistic noisy ultrasonic signal. The impacts of different parameters on noise filtering are analyzed from the comparison of SNR and frequency spectrum of the denoised signal. From the results, denoised signal possesses higher SNR and signal amplitude using wavelet base from coiflet family. The higher the decomposition level, the higher the SNR of the denoised signal, but growth trend gradually stabilized. The impact of threshold function on the performance of denoising is not very significant, and acceptable filtering results can be acquired from soft threshold or improved threshold function.

Efficient De-noising Performance of a Combined Algorithm of Translation Invariant (TI) Wavelets and Independent Component Analysis over TI Wavelets for Speech-Auditory Brainstem Responses

In this paper we have presented a research for de-noising the EEG collected Brainstem Speech Evoked Potentials data collected in an audiology lab in University of Ottawa, from 10 different human subjects. Here the de-noising techniques we have considered are Yule-Walker Multiband Filter, Cascaded Yule-Walker-Comb Filter, Conventional Wavelet Transform estimation filters: Daubechies, Symlet, Coiflet Wavelet families, Translation Invariant (TI) Wavelet Transform estimation filter, FAST Independent Component Analysis (FASTICA) De-noising Technique, Combined algorithm of “Translation Invariant (TI) Wavelets and Independent Component Analysis” De-noising technique. The performance measures we have considered are Mean Square Error (MSE) and Signal-to-Noise-Ratio (SNR) values. Out of these techniques we found that cascading of Yule-Walker filter and Comb-Peak filter gave better De-noising performance than Yule-Walker Multiband Filter. Then conventional Wavelets performed far better than the cascaded filter, in those Daubechies family of wavelets worked better than all. Then FASTICA Algorithm worked near to the performance of Conventional Wavelets but far better than cascaded filter. Then we have utilized Translation Invariant (TI) wavelet algorithm which provided the excellent performance than above all. Then we have utilized combined Algorithm of “Translation Invariant (TI) Wavelets and Independent Component Analysis – CSTIICA” algorithm which found to be, it may perform better than TI wavelets algorithm. Ultimately TI and CSTIICA algorithms are found to be may be the best auditory artifact removal techniques and can be highly useful in auditory EEG data analysis to the best.

Denoising Acoustic Emission Signal using Wavelet Transforms for Determining the Micro Crack Location Inside of Concrete

Acoustic emission (AE) technique is developed to locate source of damage inside of concrete. However, the AE signal is interfered by much noise, which makes the determination of first time amplitude of AE signal is hard to be carried out. In fact, the determination of this parameter is a significant part for locating the source of damage in concrete. Therefore, one of the denoising methods called wavelet based denoising is proposed. In this case, some wavelet bases function are investigated to find out the proper wavelet bases function to perform the denoising of AE Signal. From the experimental data, the best wavelet basis function for this case is Coiflet, which is shown by providing the best SNR than the other wavelet families. In addition, the determining cracks locations on concrete can be performed easier on denoised AE signal than on noisy AE signal.

Wavelets Centered on a Knot Sequence: Theory, Construction, and Applications

We develop a general notion of orthogonal wavelets ‘centered’ on an irregular knot sequence. We present two families of orthogonal wavelets that are continuous and piecewise polynomial. We develop efficient algorithms to implement these schemes and apply them to a data set extracted from an ocelot image. As another application, we construct continuous, piecewise quadratic, orthogonal wavelet bases on the quasi-crystal lattice consisting of the $$\tau $$ -integers where $$\tau $$ is the golden ratio. The resulting spaces then generate a multiresolution analysis of $$L^2(\mathbf {R})$$ with scaling factor $$\tau $$ .