408 research outputs found
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
Non-parametric linear time-invariant system identification by discrete wavelet transforms
We describe the use of the discrete wavelet transform (DWT) for non-parametric linear time-invariant system identification. Identification is achieved by using a test excitation to the system under test (SUT) that also acts as the analyzing function for the DWT of the SUT's output, so as to recover the impulse response. The method uses as excitation any signal that gives an orthogonal inner product in the DWT at some step size (that cannot be 1). We favor wavelet scaling coefficients as excitations, with a step size of 2. However, the system impulse or frequency response can then only be estimated at half the available number of points of the sampled output sequence, introducing a multirate problem that means we have to 'oversample' the SUT output. The method has several advantages over existing techniques, e.g., it uses a simple, easy to generate excitation, and avoids the singularity problems and the (unbounded) accumulation of round-off errors that can occur with standard techniques. In extensive simulations, identification of a variety of finite and infinite impulse response systems is shown to be considerably better than with conventional system identification methods.Department of Computin
On the Construction of Wavelets and Multiwavelets for General Dilation Matrices
This thesis is concerned with the construction of (pre-)wavelets and (pre-)multiwavelets. In particular, we identify minimal requirements such that a construction is still possible. To this end, we weaken the assumptions made in the definition of the multiresolution analysis. Based on this generalized multiresolution analysis, we develop construction procedures for compactly supported (pre-)wavelets and for compactly supported (pre-)multiwavelets. These construction procedures involve general dilation matrices which allow us to reduce the number of mother wavelets to a minimum. To illustrate the theory developed in this work, we choose exponential box splines as generators for the generalized multiresolution analysis and construct compactly supported (pre-)wavelets and (pre-)multiwavelets
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
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