Wavelet System Research Articles

Artificial intelligence methods for oil price forecasting: a review and evaluation

Artificial intelligent methods are being extensively used for oil price forecasting as an alternate approach to conventional techniques. There has been a whole spectrum of artificial intelligent techniques to overcome the difficulties of complexity and irregularity in oil price series. The potential of AI as a design tool for oil price forecasting has been reviewed in this study. The following price forecasting techniques have been covered: (i) artificial neural network, (ii) support vector machine, (iii) wavelet, (iv) genetic algorithm, and (v) hybrid systems. In order to investigate the state of artificial intelligent models for oil price forecasting, thirty five research papers (published during 2001 to 2013) had been reviewed in form of table (for ease of comparison) based on the following parameters: (a) input variables, (b) input variables selection method, (c) data characteristics (d) forecasting accuracy and (e) model architecture. This review reveals procedure of AI methods used in complex oil price related studies. The review further extended above overview into discussions regarding specific shortcomings that are associated with feature selection for designing input vector, and then concluded with future insight on improving the current state-of-the-art technology.

DIRECTIONAL TIME–FREQUENCY ANALYSIS VIA CONTINUOUS FRAMES

Grafakos and Sansing [‘Gabor frames and directional time–frequency analysis’, Appl. Comput. Harmon. Anal.25 (2008), 47–67] have shown how to obtain directionally sensitive time–frequency decompositions in $L^{2}(\mathbb{R}^{n})$ based on Gabor systems in $L^{2}(\mathbb{R})$. The key tool is the ‘ridge idea’, which lifts a function of one variable to a function of several variables. We generalise their result in two steps: first by showing that similar results hold starting with general frames for $L^{2}(\mathbb{R}),$ in the settings of both discrete frames and continuous frames, and second by extending the representations to Sobolev spaces. The first step allows us to apply the theory to several other classes of frames, for example wavelet frames and shift-invariant systems, and the second one significantly extends the class of examples and applications. We consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretise the representations using ${\it\epsilon}$-nets.

Vanishing moment conditions for wavelet atoms in higher dimensions

We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L2(?d), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(?d), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Grochenig, and they can be informally described as follows: Given a function ? ? L2(?d) satisfying fairly mild decay, smoothness and vanishing moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation.

Wandering Operators for Unitary Systems of Hilbert Spaces

Inthispaper,weintroducetheconceptofcompletewanderingoperatorsfor as ystemU of unitary operators acting on a Hilbert space, which can be viewed as an abstract mathematical model for g-orthonormal bases of Hilbert spaces and operator- valued wavelets for L 2 (R). The idea comes from Dai and Larson's work (Mem Am Math Soc 134:640 1998), where the wandering vectors for a unitary system are intro- duced as an abstract model for orthogonal wavelets. The topological and algebraical properties of the set W(U) of all complete wandering operators for a unitary system U are studied. In particular, properties of the local commutant of U are established. A parametrization formula for W(U) and some interesting algebraic properties of com- pletewanderingoperatorsforaunitarysystemareobtained.Thespecialcaseofgreatest interest is the wavelet system {UD,T }. We pay certain attention on studying this more structured unitary system and some structural theorems are established. Lots of prop- erties of the wandering vectors for a unitary system are extended to the more general case, i.e. the wandering operators for a unitary system. However, operator-valued case is more complicated. We also give some examples to illustrate our results. Our works show that wavelet theory and frame theory are deeply connected with operator theory.

Linear independence of compactly supported separable shearlet systems

This paper examines linear independence of shearlet systems. This property has already been studied for wavelets and other systems such as, for instance, for Gabor systems. In fact, for Gabor systems this problem is commonly known as the HRT conjecture. In this paper we present a proof of linear independence of compactly supported separable shearlet systems. For this, we employ a sampling strategy to utilize the structure of an implicitly given underlying oversampled wavelet system as well as the shape of the supports of the shearlet elements.

Tight wavelet frames in low dimensions with canonical filters

This paper is to construct tight wavelet frame systems containing a set of canonical filters by applying the unitary extension principle of Ron and Shen (1997). A set of filters are canonical if the filters in this set are generated by flipping, adding a conjugation with a proper sign adjusting from one filter. The simplest way to construct wavelets of s-variables is to use the 2s−1 canonical filters generated by the refinement mask of a box spline. However, almost all wavelets (except Haar or the tensor product of Haar) defined by the canonical filters associated with box splines do not form a tight wavelet frame system. We consider how to build a filter bank by adding filters to a canonical filter set generated from the refinement mask of a box spline in low dimension, so that the wavelet system generated by this filter bank forms a tight frame system. We first prove that for a given low dimension box spline of s-variables, one needs at least additional 2s filters to be added to the canonical filters from the refinement mask (that leads to the total number of highpass filters in the filter bank to be 2s+1−1) to have a tight wavelet frame system. We then provide several methods with many interesting examples of constructing tight wavelet systems with the minimal number of framelets that contain canonical filters generated by the refinement masks of box splines. The supports of the resulting framelets are not bigger than that of the corresponding box spline whose refinement mask is used to generate the first 2s−1 canonical filters in the filter bank. In many of our examples, the tight frame filter bank has the double-canonical property, meaning it is generated by adding another set of canonical filters generated from a highpass filter to the canonical filters generated by the refinement mask to make a tight frame system.

Voice Disorder Signal Classification Using M-Band Wavelets and Support Vector Machine

The aim of this work is to present a noninvasive method to classify normal voice signals and those corresponding to voice disorders. Use of wavelet decomposition is prevalent for feature extraction in this field and provides good frequency resolution in lower-frequency subbands. In this work, to provide better frequency resolution in higher-frequency subbands as well, we use M-band wavelet decomposition for feature extraction, employing a genetic algorithm to determine the parameters of the optimal wavelet. Moreover, a support vector machine is used as the final classifier. By employing a well-known pathological voice database, normal and pathological cases are classified using a five-band wavelet system for feature extraction, showing good performance.

Morphlets: a new class of tree-structured morphological descriptors of image shape

In this paper, we present a new class of tree-structured morphological descriptors of image shape. The notion of a Haar-like function is formalized. It is demonstrated that Pytiev morphology of centered images can be described in terms of a Haar-like basis. The morphology of localized mosaic shapes and corresponding localized Haar-like wavelet systems are described. The formal way for generating a hierarchical structure of localized mosaic shapes and corresponding tree of localized Haar-like wavelets is reported. The notion of “morphlets” as Haar-like wavelets corresponding to the mosaic shape of some image is introduced. Some constructive schemes for the generation of morphlet-based image descriptors are considered.

Wavelet and adaptive neuro-fuzzy inference system conjunction model for groundwater level predicting in a coastal aquifer

Accurately predicting groundwater level (GWL) fluctuations is one of the most important issues for managing groundwater resources. In this study, the feasibility of predicting weekly GWL fluctuatio...

Study of Periodic Frames and Trivariate Tight Wavelet Frames and Applications in Materials Engineering

It is shown that there exists a frame wavelet with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.

Simple proposal for radial 3D needlets

We present here a simple construction of a wavelet system for the three-dimensional ball, which we label \emph{Radial 3D Needlets}. The construction envisages a data collection environment where an observer located at the centre of the ball is surrounded by concentric spheres with the same pixelization at different radial distances, for any given resolution. The system is then obtained by weighting the projector operator built on the corresponding set of eigenfunctions, and performing a discretization step which turns out to be computationally very convenient. The resulting wavelets can be shown to have very good localization properties in the real and harmonic domain; their implementation is computationally very convenient, and they allow for exact reconstruction as they form a tight frame systems. Our theoretical results are supported by an extensive numerical analysis.

Multivariate Periodic Wavelets of de la Vallée Poussin Type

In this paper we present a general approach to multivariate periodic wavelets generated by scaling functions of de la Vall\'ee Poussin type. These scaling functions and their corresponding wavelets are determined by their Fourier coefficients, which are sample values of a function, that can be chosen arbitrarily smooth, even with different smoothness in each direction. This construction generalizes the one-dimensional de la Vall\'ee Poussin means to the multivariate case and enables the construction of wavelet systems, where the set of dilation matrices for the two-scale relation of two spaces of the multiresolution analysis may contain shear and rotation matrices. It further enables the functions contained in each of the function spaces from the corresponding series of scaling spaces to have a certain direction or set of directions as their focus, which is illustrated by detecting jumps of certain directional derivatives of higher order.

River Water Prediction Modeling Using Neural Networks, Fuzzy and Wavelet Coupled Model

In this paper, new prediction model introduced by coupling of neural networks model, fuzzy model and wavelet model for the water resources management. Artificial neural network (ANN), fuzzy, wavelet and adaptive neuro-fuzzy inference system (ANFIS) are found to be a sturdy tool to model many non-linear hydrological processes. Wavelet transformation will improve the ability of a prediction model by capturing valuable information on different resolution levels. The target of this research is to compare our model with other famous data-driven models for monthly forecasting of water quality parameter chemical oxygen demand (COD) level monitored at Nizamuddin station, New Delhi, India of river Yamuna based on the past history. The data has been decomposed into wavelet domain constitutive sub series using Daubechies wavelet at level 8 (Db8). Statistical behavior of wavelet domain constitutive series has been studied. The foretelling performance of the wavelet coupled model has been compared with classical neuro fuzzy, artificial neural network and regression models. The result shows that the wavelet coupled model produces considerably higher leads to comparison to neuro fuzzy, neural network, regression models.

Nonlinear Approximation Rates and Besov Regularity for Elliptic PDEs on Polyhedral Domains

We investigate the Besov regularity for solutions of elliptic PDEs. This is based on regularity results in Babuska–Kondratiev spaces. Following the argument of Dahlke and DeVore, we first prove an embedding of these spaces into the scale $$B^r_{\tau ,\tau }(D)$$ of Besov spaces with $$\frac{1}{\tau }=\frac{r}{d}+\frac{1}{p}$$ . This scale is known to be closely related to $$n$$ -term approximation w.r.t. wavelet systems, and also adaptive finite element approximation. Ultimately, this yields the rate $$n^{-r/d}$$ for $$u\in {\mathcal {K}}^m_{p,a}(D)\cap H^s_p(D)$$ for $$r<r^*\le m$$ . In order to improve this rate to $$n^{-m/d}$$ , we leave the scale $$B^r_{\tau ,\tau }(D)$$ and instead consider the spaces $$B^m_{\tau ,\infty }(D)$$ . We determine conditions under which the space $${\mathcal {K}}^m_{p,a}(D)\cap H^s_p(D)$$ is embedded into some space $$B^m_{\tau ,\infty }(D)$$ for some $$\frac{m}{d}+\frac{1}{p}>\frac{1}{\tau }\ge \frac{1}{p}$$ , which in turn indeed yields the desired $$n$$ -term rate. As an intermediate step, we also prove an extension theorem for Kondratiev spaces.

Multiresolution representation for curves based on ternary subdivision schemes

Subdivision offers a way to increase the resolution of models, while reverse subdivision possesses the opposite ability. Combining the two theories could realize the multiresolution (MR) representation of models. Based on two ternary subdivision schemes, we present the trial and refined filters and an algorithm to realize MR representation for curves, which has some difference compared with the work relating to binary schemes. And the filters yield biorthogonal wavelet systems which are the underlying theory fundament of curves MR. By experiments and numerical calculations, we demonstrate that by using the ternary methods one can accomplish the MR representation for curves and the low-resolution results obtained by reverse subdivision can approximate the original curves well. Besides, ternary methods need smaller number of decomposition times than binary methods to get low-resolution results at similar levels of resolution for the same original curve.

CHARACTERIZATIONS OF THE MULTIVARIATE WAVE PACKET SYSTEMS

Some characterizations of the multivariate wave packet systems in terms of the Fourier transforms of the wave packet systems' generating functions are discussed in this paper. First, the characterizations of the orthogonal wave packet systems are provided. Secondly, a sufficient condition of the completeness of wave packet system in some special cases is established. Finally, the necessary conditions and sufficient conditions for the wave packet systems to be wave packet Parseval frames with the very general lattices are obtained. Thus, the corresponding known results in Gabor systems and wavelet systems are obtained as some corollaries.

Reconstruction of Wavelet Coefficients

A comparative study of different methods of reconstruction of wavelet coefficients is presented. The following are the different techniques for the reconstruction of wavelet coefficients. To start with, we show how to design and construct Daubechies four coefficient wavelet system which are orthogonal and compactly supported wavelets. Then we outline the multi resolution analysis of wavelets using Mallat transform. Multi resolution analysis can be illustrated by the decomposition and reconstruction of wavelet system using Laplacian Pyramid. To construct wavelet systems with finite support and regularity using orthonormal and interpolariting units, only multicomponent wavelets are possible. When image function is expressed in terms of scaling functions and wavelet functions of higher resolution, we need to consider only few wavelet coefficients and wavelet coefficients are dominant only near edges. The wavelet coefficient near edges can be estimated using wavelet transform maxima and statistical inference of coefficients using Markov tree model. An alternate method is the reconstruction of wavelet coefficients using total variation minimisation. If we use thresholding, so that when we neglect wavelet coefficients having values less than a threshold value there is ringing across edges. Hence we propose an improved algorithm of reconstruction of wavelet coefficients using zero padding and cycle spinning. PSNR of images with wavelet based interpolation and denoising by cycle spinning is moderately high.

Multiresolution Analysis Through Low-Pass Filter on Local Fields of Positive Characteristic

The concept of wavelet basis on the integers can be generalized to a countable subset of a local field having positive characteristic by using a prime element of such a field. In this paper, we provide a characterization of first-stage discrete wavelet system on a countable subset of a local field of positive characteristic. Further, we obtain some results on refinement equation and refinement coefficients which provide sufficient conditions for a function to be a solution of the refinement equation and generate a multiresolution analysis on the local fields.

On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

AbstractThe unitary extension principle (UEP) by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

A New Algorithm for Computationally Efficient Modified Dual Tree Complex Wavelet Transform

We introduce a new generation functionally distinct redundant free Modified Dual Tree Complex Wavelet structure with improved orthogonality and symmetry properties. Traditional Dual Tree Complex Wavelets Transform (DTCWT), which incorporates two operationally similar, procedurally different Discrete Wavelet Transform (DWT) trees, is inherently redundant and computationally complex. In this paper, we propose Symmetrically Modified DTCWT (SMDTCWT) to explore the close relationships between the wavelet coefficients from the real and imaginary tree of the dualtree CWT with an advent of a Quadrature Filter. This exploitation can reduce the level of redundancy that currently exists in a dual-tree wavelet system and decrease the computational complexity .Some of the primary constraints include that the designed algorithm should be satisfying the Hilbert transform pair condition and should have high coding gain, good directional sensitivity, and sufficient degree of regularity.