Linear Dynamical Identification: An Integral Transform seen as a Complex Wavelet Transform

This paper presents a technique that allows the direct linearidentification of frequency response functions from the computation ofa weighted integral transform. This transform allows toemphasize the influence of the poles and zeros of the frequencyresponse functions its formation is based on the Cauchy--Weierstrass theorem. It isthen shown that this transform is directly linked to a complex wavelettransform. This representation with a wavelet transform provides abetter understanding of the amplification effects of the weightedintegral transform and allows the singularities analysis.     

Orthogonal Wavelets Analysis of Electroanalytical Signals

ABSTRACT

A signal processing technique based on orthogonal wavelet analysis is applied to process various simulated electroanalytical signals. The results indicate that if the scale parameters are selected, the orthogonal wavelet processing method (OWPM) can remove high-frequency noise. Experimental signal was recorded by computer and used to test the OWPM procedure. After processing with OWPM, the processed data was used to analyze the mechanism of the electrode reactions. Processed results of the experimental data are also satisfactory.     

Compactly supported shearlets are optimally sparse

Cartoon-like images, i.e., C² functions which are smooth apart from a C² discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications.In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.     

Solving diffusion equation using wavelet method

A wavelet-based numerical method based on wavelets of Hermite cubic splines is presented for computing singularly perturbed convection-dominated diffusion equation. The advantages of the method are explained. To improve the accuracy of singular areas, wavelets are configured hierarchically for solving algebraic equations. Numerical examples show that the proposed method has good efficiency and precision.     

Long-range dependence in the volatility of commodity futures prices: Wavelet-based evidence

Commodity futures have long been used to facilitate risk management and inventory stabilization. The study of commodity futures prices has attracted much attention in the literature because they are highly volatile and because commodities represent a large proportion of the export value in many developing countries. Previous research has found apparently contradictory findings about the presence of long memory or more generally, long-range dependence. This note investigates the nature of long-range dependence in the volatility of 14 energy and agricultural commodity futures price series using the improved Hurst coefficient (H) estimator of Abry, Teyssière and Veitch. This estimator is motivated by the ability of wavelets to detect self-similarity and also enables a test for the stability of H. The results show evidence of long-range dependence for all 14 commodities and of a non-stationary H for 9 of 14 commodities.     

Solving Multi-dimensional Evolution Problems with Localized Structures using Second Generation Wavelets

A dynamically adaptive numerical method for solving multi-dimensional evolution problems with localized structures is developed. The method is based on the general class of multi-dimensional second-generation wavelets and is an extension of the second-generation wavelet collocation method of Vasilyev and Bowman to two and higher dimensions and irregular sampling intervals. Wavelet decomposition is used for grid adaptation and interpolation, while O ( N ) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The prowess and computational efficiency of the method are demonstrated for the solution of a number of two-dimensional test problems.     

A new wavelet multigrid method

The standard multigrid procedure performs poorly or may break down when used to solve certain problems, such as elliptic problems with discontinuous or highly oscillatory coefficients. The method discussed in this paper solves this problem by using a wavelet transform and Schur complements to obtain the necessary coarse grid, interpolation, and restriction operators. A factorized sparse approximate inverse is used to improve the efficiency of the resulting method. Numerical examples are presented to demonstrate the versatility of the method.     

Inferring mixed-culture growth from total biomass data in a wavelet approach

It is shown that the presence of mixed-culture growth in batch fermentation processes can be very accurately inferred from total biomass data by means of the wavelet analysis for singularity detection. This is accomplished by considering simple phenomenological models for the mixed growth and the more complicated case of mixed growth on a mixture of substrates. The main quantity provided by the wavelet analysis is the Hölder exponent of the singularity that we determine for our illustrative examples. The numerical results point to the possibility that Hölder exponents can be used to characterize the nature of the mixed-culture growth in batch fermentation processes with potential industrial applications. Moreover, the analysis of the same data affected by the common additive Gaussian noise still lead to the wavelet detection of the singularities although the Hölder exponent is no longer a useful parameter.     

Self-organized criticality and stock market dynamics: an empirical study

The stock market is a complex self-interacting system, characterized by intermittent behaviour. Periods of high activity alternate with periods of relative calm. In the present work we investigate empirically the possibility that the market is in a self-organized critical state (SOC). A wavelet transform method is used in order to separate high activity periods, related to the avalanches found in sandpile models, from quiescent. A statistical analysis of the filtered data shows a power law behaviour in the avalanche size, duration and laminar times. The memory process, implied by the power law distribution of the laminar times, is not consistent with classical conservative models for self-organized criticality. We argue that a “near-SOC” state or a time dependence in the driver, which may be chaotic, can explain this behaviour.     

A wavelet adaptive Newton method for the solution of nonlinear equations

For the solution of nonlinear equations, we present an adaptive wavelet scheme, which couples an inexact Newton method and the idea of nonlinear wavelet approximation. In particular, we obtain a result of quadratic convergence.