707 research outputs found
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
Wavelets: mathematics and applications
The notion of wavelets is defined. It is briefly described {\it what} are
wavelets, {\it how} to use them, {\it when} we do need them, {\it why} they are
preferred and {\it where} they have been applied. Then one proceeds to the
multiresolution analysis and fast wavelet transform as a standard procedure for
dealing with discrete wavelets. It is shown which specific features of signals
(functions) can be revealed by this analysis, but can not be found by other
methods (e.g., by the Fourier expansion). Finally, some examples of practical
application are given (in particular, to analysis of multiparticle production}.
Rigorous proofs of mathematical statements are omitted, and the reader is
referred to the corresponding literature.Comment: 16 pages, 5 figures, Latex, Phys. Atom. Nuc
Wavelet Multiresolution Analysis of High-Frequency Asian FX Rates, Summer 1997
FX pricing processes are nonstationary and their frequency characteristics are time-dependent. Most do not conform to geometric Brownian motion, since they exhibit a scaling law with a Hurst exponent between zero and 0.5 and fractal dimensions between 1.5 and 2. This paper uses wavelet multiresolution analysis, with Haar wavelets, to analyze the nonstationarity (time-dependence) and self-similarity (scale-dependence) of intra-day Asian currency spot exchange rates. These are the ask and bid quotes of the currencies of eight Asian countries (Japan, Hong Kong, Indonesia, Malaysia, Philippines, Singapore, Taiwan, Thailand), and of Germany for comparison, for the crisis period May 1, 1998 - August 31, 1997, provided by Telerate (U.S. dollar is the numeraire). Their time-scale dependent spectra, which are localized in time, are observed in wavelet based scalograms. The FX increments can be characterized by the irregularity of their singularities. This degrees of irregularity are measured by homogeneous Hurst exponents. These critical exponents are used to identify the fractal dimension, relative stability and long term dependence of each Asian FX series. The invariance of each identified Hurst exponent is tested by comparing it at varying time and scale (frequency) resolutions. It appears that almost all FX markets show anti-persistent pricing behavior. The anchor currencies of the D-mark and Japanese Yen are ultra-efficient in the sense of being most anti-persistent. The Taiwanese dollar is the most persistent, and thus unpredictable, most likely due to administrative control. FX markets exhibit these non- linear, non-Gaussian dynamic structures, long term dependence, high kurtosis, and high degrees of non-informational (noise) trading, possibly because of frequent capital flows induced by non-synchronized regional business cycles, rapidly changing political risks, unexpected informational shocks to investment opportunities, and, in particular, investment strategies synthesizing interregional claims using cash swaps with different duration horizons.foreign exchange markets, anti-persistence, long-term dependence, multi-resolution analysis, wavelets, time-scale analysis, scaling laws, irregularity analysis, randomness, Asia
Wavelet Multiresolution Analysis of High-Frequency FX Rates, Summer 1997
FX pricing processes are nonstationary and their frequency characteristics are time-dependent. Most do not conform to geometric Brownian motion, since they exhibit a scaling law with a Hurst exponent between zero and 0.5 and fractal dimensions between 1.5 and 2. This paper uses wavelet multiresolution analysis, with Haar wavelets, to analyze the nonstationarity (time-dependence) and self-similarity (scale-dependence) of intra-day Asian currency spot exchange rates.foreign exchange, anti-persistence, multi-resolution analysis, wavelets, Asia
Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
Left-invariant PDE-evolutions on the roto-translation group (and
their resolvent equations) have been widely studied in the fields of cortical
modeling and image analysis. They include hypo-elliptic diffusion (for contour
enhancement) proposed by Citti & Sarti, and Petitot, and they include the
direction process (for contour completion) proposed by Mumford. This paper
presents a thorough study and comparison of the many numerical approaches,
which, remarkably, is missing in the literature. Existing numerical approaches
can be classified into 3 categories: Finite difference methods, Fourier based
methods (equivalent to -Fourier methods), and stochastic methods (Monte
Carlo simulations). There are also 3 types of exact solutions to the
PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in
previous works by Duits and van Almsick in 2005. Here we provide an overview of
these 3 types of exact solutions and explain how they relate to each of the 3
numerical approaches. We compute relative errors of all numerical approaches to
the exact solutions, and the Fourier based methods show us the best performance
with smallest relative errors. We also provide an improvement of Mathematica
algorithms for evaluating Mathieu-functions, crucial in implementations of the
exact solutions. Furthermore, we include an asymptotical analysis of the
singularities within the kernels and we propose a probabilistic extension of
underlying stochastic processes that overcomes the singular behavior in the
origin of time-integrated kernels. Finally, we show retinal imaging
applications of combining left-invariant PDE-evolutions with invertible
orientation scores.Comment: A final and corrected version of the manuscript is Published in
Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9),
p.1-50, 201
Non-equispaced B-spline wavelets
This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.Comment: 42 pages, 2 figure
Wavelet Analysis and Denoising: New Tools for Economists
This paper surveys the techniques of wavelets analysis and the associated methods of denoising. The Discrete Wavelet Transform and its undecimated version, the Maximum Overlapping Discrete Wavelet Transform, are described. The methods of wavelets analysis can be used to show how the frequency content of the data varies with time. This allows us to pinpoint in time such events as major structural breaks. The sparse nature of the wavelets representation also facilitates the process of noise reduction by nonlinear wavelet shrinkage , which can be used to reveal the underlying trends in economic data. An application of these techniques to the UK real GDP (1873-2001) is described. The purpose of the analysis is to reveal the true structure of the data - including its local irregularities and abrupt changes - and the results are surprising.Wavelets, Denoising, Structural breaks, Trend estimation
Denoising by multiwavelet singularity detection
Wavelet denoising by singularity detection was proposed as an algorithm that combines Mallat and Donoho’s denoising approaches. With wavelet transform modulus sum, we can avoid the error and ambiguities of tracing the modulus maxima across scales and the complicated and computationally demanding reconstruction process. We can also avoid the visual artifacts produced by shrinkage. In this paper, we investigate a multiwavelet denoising algorithm based on a modified singularity detection approach. Improved signal denoising results are obtained in comparison to the single wavelet case
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