Above are papers related to the following paper: [Paper_ID] Paper #40583. 	[Title] Wavelet Thresholding via a Bayesian Approach [Abstract] We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in non-parametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be hard to elicit; with this in mind, we propose a standard choise of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set collected in an anaesthesiological study. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #31489. 	[Title] Minimax Estimation via Wavelet Shrinkage  a pleasure to acknowledge friendly conversations with Gerard Kerkyacharian, [Abstract] We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints, and asymptotically minimax over Besov bodies with p q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p &lt; 2, so our method can significantly outperform every linear method (kernel, smoothing spline, sieve, : : : ) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which may vary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper discuss practical implementation, spatial adaptation properties and applications to inverse problems. Acknowledgements. This work was completed while the first author was on leave from U.C. Berkeley, where his research was supported by NSF DMS 88-10192, by NASA Contract NCA2-488, and by a grant from ATT Foundation. The second author was supported in part by NSF grants DMS 84-51750, 86-00235, and NIH PHS grant GM21215-12. Supersedes an earlier version, titled "Wavelets and Optimal Function Estimation", dated November 10, 1990, and issued as Technical reports by the Departments of Statistics at both Stanford and at U.C. Berkeley. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #632935. 	[Title] Density and hazard rate estimation for right censored data using wavelet methods [Abstract] This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators possess pointwise and global mean square consistency, obtain the best possible asymptotic MISE convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #633721. 	[Title] The Stationary Wavelet Transform and some Statistical Applications [Abstract] Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A `stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transform are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #632874. 	[Title] Minimax Bayes, asymptotic minimax and sparse wavelet priors [Abstract] Pinsker(1980) gave a precise asymptotic evaluation of the minimax mean squared error of estimation of a signal in Gaussian noise when the signal is known a priori to lie in a compact ellipsoid in Hilbert space. This `Minimax Bayes' method can be applied to a variety of global non-parametric estimation settings with parameter spaces far from ellipsoidal. For example it leads to a theory of exact asymptotic minimax estimation over norm balls in Besov and Triebel spaces using simple co-ordinatewise estimators and wavelet bases. This paper outlines some features of the method common to several applications. In particular, we derive new results on the exact asymptotic minimax risk over weak ` p balls in R n as n ! 1, and also for a class of `local' estimators on the Triebel scale. By its very nature, the method reveals the structure of asymptotically least favorable distributions. Thus we may simulate `least favorable' sample paths. We illustrate this for estimation of a signal in Gaussian white noise over norm balls in certain Besov spaces. In wavelet bases, when p &lt; 2, the least favorable priors are sparse, and the resulting sample paths strikingly different from those observed in Pinsker's ellipsoidal setting (p = 2). Acknowledgements. I am grateful for many conversations with David Donoho and Carl Taswell, and to a referee for helpful comments. This work was supported in part by NSF grants DMS 84-51750, 9209130, and NIH PHS grant GM21215-12. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #632796. 	[Title] Density estimation by wavelet thresholding [Abstract] Density estimation is a commonly used test case for non-parametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes B s;p;q and for a range of global L 0 p error measures, 1 p 0 &lt; 1. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when p 0 &gt; p, some form of non-linearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of n. A second approach, using an approximation of a Gaussian white noise model in a Mallows metric, is used Acknowledgements: We thank Alexandr Sakhanenko for helpful discussions and references to his work on Berry Esseen theorems used in Section 5. This work was supported in part by NSF DMS 92-09130. The second author would like to thank Universite de Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #9581. 	[Title] DE-NOISING BY reconstruction f n is defined in the wavelet domain by translating all the [Abstract] p n. We prove two results about that estimator. [Smooth]: With high probability ^ f fl n is at least as smooth as f , in any of a wide variety of smoothness measures. [Adapt]: The estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. Our proof of these results develops new facts about abstract statistical inference and its connection with Acknowledgements. These results were described at the Symposium on Wavelet Theory, held in connection with the Shanks Lectures at Van-derbilt University, April 3-4 1992. The author would like to thank Professor L.L. Schumaker for hospitality at the conference, and R.A. DeVore, Iain Johnstone, Gerard Kerkyacharian, Bradley Lucier, A.S. Nemirovskii, Ingram Olkin, and Dominique Picard for interesting discussions and correspondence on related topics. The author is also at the University of California, Berkeley Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #633585. 	[Title] Choice of Thresholds for Wavelet Shrinkage Estimate of the Spectrum fff j g are level-dependent [Abstract] We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t j;n depending on sample size n, wavelet basis and resolution level j. At fine resolution levels (j = 1; 2; :::), we propose The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties. t j;n = ff j log n; Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #227286. 	[Title] Space-Frequency Localized Basis Function Networks for Nonlinear System Estimation and Control [Abstract] Stable neural network control and estimation may be viewed formally as a merging of concepts from nonlinear dynamic systems theory with tools from multivariate approximation theory. This paper extends earlier results on adaptive control and estimation of nonlinear systems using gaussian radial basis functions to the on-line generation of irregularly sampled networks, using tools from multiresolution analysis and wavelet theory. This yields much more compact and efficient system representations while preserving global closed-loop stability. Approximation models employing basis functions that are localized in both space and spatial frequency admit a measure of the approximated function's spatial frequency content that is not directly dependent on reconstruction error. As a result, these models afford a means of adaptively selecting basis functions according to the local spatial frequency content of the approximated function. An algorithm for stable, on-line adaptation of output weights simultaneously with node configuration in a class of non-parametric models with wavelet basis functions is presented. An asymptotic bound on the error in the network's reconstruction is derived and shown to be dependent solely on the minimum approximation error associated with the steady state node configuration. In addition, prior bounds on the temporal bandwidth of the system to be identified or controlled are used to develop a criterion for on-line selection of radial and ridge wavelet basis functions, thus reducing the rate of increase in network's size with the dimension of the state vector. Experimental results obtained by using the network to predict the path of an unknown light bluff object thrown through air, in an active-vision based robotic catching system, are given to illustrate the network's performance in a simple real-time application. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152150. 	[Title]  [Abstract] TECHNICAL REPORT NO. 947 June 5, 1995 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #633081. 	[Title] Minimax Risk over l p -Balls for l q -error Key Words. Minimax Decision Theory. [Abstract] Consider estimating the mean vector from data N n (; 2 I) with l q norm loss, q 1, when is known to lie in an n-dimensional l p ball, p 2 (0; 1). For large n, the ratio of minimax linear risk to minimax risk can be arbitrarily large if p &lt; q. Obvious exceptions aside, the limiting ratio equals 1 only if p = q = 2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. When p &lt; q, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. Our results are basic to a theory of estimation in Besov spaces Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1130780. 	[Title] Wavelet Shrinkage: Asymptopia? [Abstract] Considerable effort has been directed recently to develop asymptotically minimax methods in problems of recovering infinite-dimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly- or exactly- minimax estimators being obtained for a variety of interesting problems. Unfortunately, the results have often not been translated into practice, for a variety of reasons sometimes, similarity to known methods, sometimes, computational intractability, and sometimes, lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coefficients towards the origin by an amount method is different from methods in common use today, is computationally practical, and is spatially adaptive; thus it avoids a number of previous objections to minimax estimators. At the same time, the method is nearly minimax for a wide variety of loss functions - e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives and for a wide range of smoothness classes, including standard Holder classes, Sobolev classes, and Bounded Variation. This is a much broader near-optimality than anything previously proposed in the minimax literature. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity. Acknowledgements: These results have been described at the Oberwolfach meeting `Mathematische Stochastik' December, 1992 and at the AMS Annual meeting, January 1993. This work was supported by NSF DMS 92-09130. The authors would like to thank Paul-Louis Hennequin, who organized the Ecole d' Ete de Probabilites at Saint Flour 1990, where this collaboration began, and to Universite de Paris VII (Jussieu) and Universite de Paris-sud (Orsay) for supporting visits of DLD and IMJ. The authors would like to thank Ildar Ibragimov and Arkady Nemirovskii for personal correspondence cited below. p Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #40605. 	[Title] Nonlinear wavelet shrinkage with Bayes rules and Bayes factors 1 [Abstract] Wavelet shrinkage,the method proposed by seminal work of Donohoand Johnstone is a disarmingly simple and efficient way of de-noising data. Shrinking wavelet coefficients was proposed from several optimality criteria. The most notable are the asymptotic minimax and cross-validation criteria. In this paper a wavelet shrinkage by imposing natural properties of Bayesian models on data is proposed. The performance of methods are tested on standard Donoho-Johnstone test functions. Key Words and Phrases: Wavelets, Discrete Wavelet Transform, Thresholding, Bayes Model. 1991 AMS Subject Classification: 42A06, 62G07. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #884094. 	[Title] ESTIMATING THE SQUARE ROOT OF A DENSITY VIA COMPACTLY SUPPORTED WAVELETS [Abstract] A large body of nonparametric statistical literature is devoted to density estimation. Overviews are given in Silverman (1986) and Izenman (1991). This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by Cencov (1962). The orthonor-mal basis used is a basis of compactly supported wavelets from Daubechies' family. Kerkyacharian and Picard (1992, 1993), Donoho et al. (1996), and Delyon and Judit-sky (1993), among others, applied wavelets in density estimation. The local nature of wavelet functions makes the wavelet estimator superior to projection estimators that use classical orthonormal bases (Fourier, Hermite, etc.) Instead of estimating the unknown density directly, we estimate the square root of the density, which enables us to control the positiveness and the L 1 norm of the density estimate. However, in that approach one needs a pre-estimator of the density to calculate sample wavelet coefficients. We describe VISUSTOP, a data-driven procedure for determining the maximum number of levels in the wavelet density estimator. Coefficients in the selected levels are thresholded to make the estimator parsimonious. Please summarise all the given information.

Paper #40583 has the title `Wavelet Thresholding via a Bayesian Approach  
` and the abstract `We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in non-parametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be hard to elicit; with this in mind, we propose a standard choise of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set collected in an anaesthesiological study. 
`.
Question: Which category should Paper #40583 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #40583 has the title `Wavelet Thresholding via a Bayesian Approach  
` and the abstract `We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in non-parametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be hard to elicit; with this in mind, we propose a standard choise of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set collected in an anaesthesiological study. 
`.
Question: Which category should Paper #40583 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Probabilistic_Methods