Fryzlewicz, Piotr · London School of Economics and Political Science Research Online (London School of Economics and Political Science) · 2010
This overview article motivates the use of wavelets in statistics, and introduces the basic mathematics behind the construction of wavelets. Topics covered include the continuous and discrete wavelet transforms, multiresolution analysis and the non-decimated wavelet transform. We describe the basic mechanics of nonparametric function estimation via wavelets, emphasising the concepts of sparsity and thresholding. A simple proof of the mean-square consistency of the wavelet estimator is also included. The article ends with two special topics: function estimation with Unbalanced Haar wavelets, and variance stabilisation via the Haar-Fisz transformation. Wavelets aremathematical functions which, when plotted, resemble “little waves”: that is, they are compactly or almost-compactly supported, and they integrate to zero. This is in contrast to “big waves” – sines and cosines in Fourier analysis, which also oscillate, but the amplitude of their oscillation never changes. Wavelets are useful for decomposing data into “wavelet coefficients”, which can then be processed in a way which depends on the aim of the analysis. One possibly advantageous feature of this decomposition is that in some set-ups, the decomposition will be sparse, i.e. most of the coefficients will be close to zero, with only a few coefficients carrying most of the information about the data. One can imagine obvious uses of this fact, e.g. in image compression. The decomposition is particularly informative, fast and easy to invert if it is performed using wavelets at a range of scales and locations. The role of scale is similar to the role of frequency in Fourier analysis. However, the concept of location is unique to wavelets: as mentioned above, they are localised around a particular point of the domain, unlike Fourier functions. This article provides a self-contained introduction to the applications of wavelets in statistics and attempts to justify the extreme popularity which they have enjoyed in the literature over the past 15 years.
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David Salomon, Giovanni Motta · 2009
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Santanu Saha Ray, Srikanta Behera · 2025
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Mani Mehra · Forum for interdisciplinary mathematics · 2018
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Tao Li, Sheng Ma, Mitsunori Ogihara · 2010
Recently there has been significant development in the use of wavelet methods in various Data Mining processes. This article presents general overview of their applications in Data Mining. It first presents a high-level data-mining framework in which the overall process is divided into smaller components. It reviews a p plications of wavelets for each component. It discusses the impact of wavelets on Data Mining research and outlines potential future research directions and applications.
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Peter Schröder · 1996
Introduction In this chapter we will explain how wavelets can be used to solve integral equations. The example we use is an important integral equation in graphics, the radiosity equation. The radiosity equation governs the transport of light between surfaces under the assumption that all reflection occurs isotropically. The resulting integral equation is linear and can be analyzed as a linear operator. Since wavelets can be used as bases for function spaces, linear operators can be expressed in them. If these operators satisfy certain smoothness conditions---as radiosity does--- the resulting matrices are approximately sparse and can be solved asymptotically faster if only finite precision is required of the answer. We develop this subject by first introducing the Galerkin method which is used to solve integral equations. Applying the method results in a linear system whose solution approximates the solution of the original integral equation. This discussion is kept very genera
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Mathematisches Forschungsinstitut Oberwolfach · Mathematisches Forschungsinstitut Oberwolfach · 2004
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Silvia Bacchelli, Serena Papi · Journal of Computational and Applied Mathematics · 2004
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Peter Schröder · 2007
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K)not Cables Braids, Sergei Sobolev, Hans Triebel et al. · 2008
Society Oxford University Press is pleased to announce that all EMS members can benefit from a 20 % discount on a large range of our Mathematics books.
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Benjamin Lillard · arXiv (Cornell University) · 2023
A new integration method drastically improves the efficiency of the dark matter direct detection calculation. In this work I introduce a complete, orthogonal basis of spherical wavelet-harmonic functions, designed for the new vector space integration method. This factorizes the numeric calculation into a ``vector'' that depends only on the astrophysical velocity distribution; a second vector, depending only on the detector form factor; and a scattering matrix defined on the basis functions, which depends on the details of the dark matter (DM) particle model (e.g.~its mass). For common spin-independent DM--Standard Model interactions, this scattering matrix can be evaluated analytically in the wavelet-harmonic basis. This factorization is particularly helpful for the more complicated analyses that have become necessary in recent years, especially those involving anisotropic detector materials or more realistic models of the local DM velocity distribution. With the new method, analyses studying large numbers of detector orientations and DM particle models can be performed more than 10~million times faster. This paper derives several analytic results for the spherical wavelets, including an extrapolation in the space of wavelet coefficients, and a generalization of the vector space method to a much broader class of linear functional integrals. Both results are highly relevant outside the field of DM direct detection.
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Albert Cohen, Wolfgang Dahmen, Ronald DeVore et al. · Oberwolfach Reports · 2011
Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers. These have advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts.This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines. A special emphasis was placed on high dimensional problems since these amplify even further the need for novel theory and computation.
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(:unkn) unknown · TIB Repositorium · 2004
[no abstract available]
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Mathematisches Forschungsinstitut Oberwolfach · Mathematisches Forschungsinstitut Oberwolfach · 2007
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(:unkn) unknown · TIB Repositorium · 2010
Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers. These have advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines. A special emphasis was placed on high dimensional problems since these amplify even further the need for novel theory and computation.
Cited by 0Open access
(:unkn) unknown · TIB Repositorium · 2007
Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines.
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Benjamin Lillard · Physical review. D/Physical review. D. · 2025
A new integration method drastically improves the efficiency of the dark matter direct detection calculation. In this work I introduce a complete, orthogonal basis of spherical wavelet-harmonic functions, designed for the new vector space integration method. This factorizes the numeric calculation into a “vector” that depends only on the astrophysical velocity distribution; a second vector, depending only on the detector form factor; and a scattering matrix defined on the basis functions, which depends on the details of the dark matter (DM) particle model (e.g. its mass). For common spin-independent DM–Standard Model interactions, this scattering matrix can be evaluated analytically in the wavelet-harmonic basis. This factorization is particularly helpful for the more complicated analyses that have become necessary in recent years, especially those involving anisotropic detector materials or more realistic models of the local DM velocity distribution. With the new method, analyses studying large numbers of detector orientations and DM particle models can be performed more than 10 million times faster. This paper derives several analytic results for the spherical wavelets, including an extrapolation in the space of wavelet coefficients, and a generalization of the vector space method to a much broader class of linear functional integrals. Both results are highly relevant outside the field of DM direct detection.
Cited by 2Open access
Manoj Kumar, Sapna Pandit · 2012
Wavelet transformation is a new development in the area of applied mathematics. Wavelets are mathematical tools that cut data or functions or operators into different frequency components, and then study each component with a resolution matching to its scale. In this article, we have made a brief discussion on historical development of wavelets, basic definitions, formulations of wavelets and different numerical meth- ods based on Haar and Daubechies wavelets for the numerical solution of differential equations, integral and integro-differential equations.
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Tao Li, Sheng Ma, Mitsunori Ogihara · 2006
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Hassan M. Aljohani, Aykroyd, Robert G · White Rose eTheses Online (University of Leeds, The University of Sheffield, University of York) · 2016
Archaeological investigations are designed to acquire information without damaging the archaeological site. Magnetometry is one of the important techniques for producing a surface grid of readings, which can be used to infer underground features. The inversion of this data, to give a fitted model, is an inverse problem. This type of problem can be ill-posed or ill-conditioned, making the estimation of model parameters less stable or even impossible. More precisely, the relationship between archaeological data and parameters is expressed by a likelihood. It is not possible to use the standard regression estimate obtained through the likelihood, which means that no maximum likelihood estimate exists. \nInstead, various constraints can be added through a prior distribution with an estimate produced using the posterior distribution. Current approaches incorporate prior information describing smoothness, which is not always appropriate. The biggest challenge is that the reconstruction of an archaeological site as a single layer requires various physical features such as depth and extent to be assumed. By applying a smoothing prior in the analysis of stratigraphy data, however, these features are not easily estimated. Wavelet analysis has proved to be highly efficient at eliciting information from noisy data. Additionally, complicated signals can be explained by interpreting only a small number of wavelet coefficients. It is possible that a modelling approach, which attempts to describe an underlying function in terms of a multi-level wavelet representation will be an improvement on standard techniques. Further, a new method proposed uses an elastic-net based distribution as the prior. Two methods are used to solve the problem, one is based on one-stage estimation and the other is based on two stages. The one-stage considers two approaches a single prior for all wavelet resolution levels and a level-dependent prior, with separate priors at each resolution level. In a simulation study and a real data analysis, all these techniques are compared to several existing methods. It is shown that the methodology using a single prior provides good reconstruction, comparable even to several established wavelet methods that use mixture priors.
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