adaptive bayesian wavelet shrinkage abstract when fitting wavelet based models shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data this article outlines a bayesian approach to shrinkage obtained by placing priors on the wavelet coefficients the prior for each coefficient consists of a mixture of two normal distributions with different standard deviations the simple and intuitive form of prior allows us to propose automatic choices of prior parameters these parameters are chosen adaptively according to the resolution level of the coefficients typically shrinking high resolution frequency coefficients more heavily assuming a good estimate of the background noise level we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients the latter may be used to assess uncertainty in the reconstruction several examples are used to illustrate the method and comparisons are made with other shrinkage methods

wavelet thresholding via a bayesian approach we discuss a bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression a prior distribution is imposed on the wavelet coefficients of the unknown response function designed to capture the sparseness of wavelet expansion that is 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 relationship between the hyperparameters of the prior model and the parameters of those besov spaces within which realizations from the prior will fall such a relationship gives insight into the meaning of the besov space parameters moreover the relationship established 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 difficult to elicit with this in mind we propose a standard choice 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 that was collected in an anaesthesiological study

on condition numbers of the spectral projections associated with periodic eigenproblems abstract in this paper we study and analyze absolute and relative condition numbers of the spectral projections for regular periodic eigenproblems the main contribution is to derive explicit expressions of the condition numbers of the j th left and right spectral projections numerical examples are given to illustrate the proposed condition numbers for the spectral projections associated with periodic eigenproblems

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

complex valued wavelet lifting and applications abstractsignals with irregular sampling structures arise naturally in many fields in applications such as spectral decomposition and nonparametric regression classical methods often assume a regular sampling pattern thus cannot be applied without prior data processing this work proposes new complex valued analysis techniques based on the wavelet lifting scheme that removes one coefficient at a time our proposed lifting transform can be applied directly to irregularly sampled data and is able to adapt to the signal s characteristics as our new lifting scheme produces complex valued wavelet coefficients it provides an alternative to the fourier transform for irregular designs allowing phase or directional information to be represented we discuss applications in bivariate time series analysis where the complex valued lifting construction allows for coherence and phase quantification we also demonstrate the potential of this flexible methodology over real valued analysis in the nonparametric re

the discrete multiple wavelet transform and thresholding methods we propose thresholding for multiwavelets considering the coefficient vectors as a whole rather than thresholding individual elements a multivariate universal threshold is obtained using the spl chi sup 2 distribution simulations indicate that using the ghm multiwavelet with appropriate preprocessing our method outperforms univariate thresholding of both ghm and daubechies wavelet decompositions

analytic wavelet thresholding summary we introduce so called analytic stationary wavelet transform thresholding where using the discrete hilbert transform we create a complex valued analytic vector from which an amplitude vector is defined thresholding of a real valued wavelet coefficient at some transform level is carried out according to the corresponding value in this amplitude vector relevant statistical results follow from properties of the discrete hilbert transform analytic stationary wavelet transform thresholding is found to produce consistently a reduced mean squared error compared to using standard stationary wavelet transform or cycle spinning thresholding for signals with extensive oscillations at some transform levels this improvement is very marked furthermore we show that our thresholding test is invariant to phase shifts in the data whereas if complex wavelet filters are being used the filters must be analytic or anti analytic at each level of the wavelet transform

perturbation of eigenvalues for periodic matrix pairs via the bauer fike theorems abstract in earlier papers the bauer fike technique was applied to the ordinary eigenvalue problem a x x the generalized eigenvalue problem a x b x and the matrix polynomial eigenvalue problem k 0 m k a k x 0 general multiple eigenvalues were dealt with and condition numbers were obtained for individual as well as clusters of eigenvalues in this paper we shall generalize the technique to the eigenvalue problem for regular periodic matrix pairs perturbation to eigenvalues simple or multiple finite or infinite will be considered for perturbations of any size for small perturbations clusters of eigenvalues will be considered

minimax threshold for denoising complex signals with waveshrink for the problem of signal extraction from noisy data waveshrink has proven to be a powerful tool both from an empirical and an asymptotic point of view waveshrink is especially efficient at estimating spatially inhomogeneous signals a key step of the procedure is the selection of the threshold parameter donoho and johnstone 1994 propose a selection of the threshold based on a minimax principle their derivation is specifically for real signals and real wavelet transforms in this paper we propose to extend the use of waveshrink to denoising complex signals with complex wavelet transforms we illustrate the problem of denoising complex signals with an electronic surveillance application

real nonparametric regression using complex wavelets summary wavelet shrinkage is an effective nonparametric regression technique especially when the underlying curve has irregular features such as spikes or discontinuities the basic idea is simple take the discrete wavelet transform dwt of data consisting of a signal corrupted by noise shrink or remove the wavelet coefficients to remove the noise and then invert the dwt to form an estimate of the true underlying curve various authors have proposed increasingly sophisticated methods of doing this using real valued wavelets complex valued wavelets exist but are rarely used we propose two new shrinkage techniques which use complex valued wavelets extensive simulations show that our methods almost always give significantly more accurate estimates when compared with methods based on real valued wavelets further one of our methods is both simpler and dramatically faster than its competitors in an attempt to understand the excellent performance of this latter method we present a new risk bound on its hard thresholded coefficients

directed random walks in continuous space the investigation on diffusion with directed motion in a two dimensional continuous space is completed by using the model of the continuous directed random walks the average square end to end distance approximately t 2nu is calculated the results show that this type of walks belongs asymptotically to the same class nu 1 0 as the ballistic motions for short time we observe a crossover from purely random walks nu 0 5 to ballistic motions nu 1 0 the dependence of the crossover on the direction parameter theta is studied there exists a scaling relation of the form approximately tf t theta 2 the return probability p00 t is also investigated and the scaling form similar to is obtained

on bayesian testimation and its application to wavelet thresholding summary we consider the problem of estimating the unknown response function in the gaussian white noise model we first utilize the recently developed bayesian maximum a posteriori testimation procedure of abramovich et al 2007 for recovering an unknown high dimensional gaussian mean vector the existing results for its upper error bounds over various sparse lp balls are extended to more general cases we show that for a properly chosen prior on the number of nonzero entries of the mean vector the corresponding adaptive estimator is asymptotically minimax in a wide range of sparse and dense lp balls the proposed procedure is then applied in a wavelet context to derive adaptive global and level wise wavelet estimators of the unknown response function in the gaussian white noise model these estimators are then proven to be respectively asymptotically near minimax and minimax in a wide range of besov balls these results are also extended to the estimation of derivatives of the response function simulated examples are conducted to illustrate the performance of the proposed level wise wavelet estimator in finite sample situations and to compare it with several existing counterparts

wavelet shrinkage using cross validation summary wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics this paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage a modified form of twofold cross validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2 the cross validation algorithm is then extended to data sets of any length and to multidimensional data sets the algorithms are compared with established threshold choosers by using simulation an application to a real data set arising from anaesthesia is presented

applications of complex valued wavelet transforms to subband decomposition a method for constructing complex valued linear phase fir conjugate quadrature filters and associated wavelet bases is described each filter is derived by replating certain zeros of a real valued fir conjugate quadrature filter by their reciprocal conjugates the derived filters have the same frequency response magnitudes as the original filters and their linear phase property permits the use of symmetrization in subband decomposition to avoid border discontinuities that result from signal periodization subband decomposition and reconstruction using both a length 6 filter associated with a daubechies 1988 wavelet bases and a related length 6 complex valued linear phase filter are compared to illustrate the reduced border effects

a note on the approximation of two sided rayleigh quotient to eigenvalues abstract we consider the two sided rayleigh quotient introduced by ostrowski as convergence acceleration tool for eigenvalue solvers an interesting property of the two sided rayleigh quotient is established its approximation to eigenvalues depends on the eigenvalue condition number this fact is valid not only for matrix but also for matrix polynomial multiparameter system and periodic matrix pairs

dicussion on the meeting on statistical approaches to inverse problems johnstone kerkyacharian picard and raimondo johnstone kerkyacharian picard and raimondo are interested in the inverse problem of estimating f where f has been convolved with g and then contaminated with white noise this popular problem has been tackled by a wide variety of procedures and wavelet methods have recently generated considerable interest donoho s 1995 seminal wavelet vaguelette paper introduced the notion that wavelets would be a good choice for the representation of f since real life objects such as images are more likely to be efficiently represented using wavelets when compared with for example fourier representations johnstone and his colleagues have moved the field on significantly in particular their procedure is more direct than wavelet vaguelette or abramovich and silverman s 1998 vaguelette wavelet method it can handle boxcar blur theoretically and practically they have rates of convergence forp 2 p defines the type of loss and the paper innovates through use of the new maxiset approach for me the most appealing of these innovations is that of enabling the treatment of boxcar blur which is one of the most common types of inverse problem however is it really really the case that for rational a nothing can be done formula 4 compels us to say no nothing can but naively it still feels wrong formula 19 is the popular signal plus noise model but here it is a little different from what normally appears in the literature because the quantities are complex valued random variables more specifically the zl are zero mean gaussian variables which are complex valued and satisfy e zlzk lk one question is why threshold the k and not the yl directly the covariance of the k is given by cov k l n m km l m

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

the monogenic wavelet transform this paper extends the 1 d analytic wavelet transform to the 2 d monogenic wavelet transform the transformation requires care in its specification to ensure suitable transform coefficients are calculated and it is constructed so that the wavelet transform may be considered as both local and monogenic this is consistent with defining the transform as a real wavelet transform of a monogenic signal in analogy with the analytic wavelet transform classes of monogenic wavelets are proposed with suitable local properties it is shown that the monogenic wavelet annihilates anti monogenic signals that the monogenic wavelet transform is phase shift covariant and that the transform magnitude is phase shift invariant a simple form for the magnitude and orientation of the isotropic transform coefficients of a unidirectional signal when observed in a rotated frame of reference is derived the monogenic wavelet ridges of local plane waves are given

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

the rayleigh ritz method refinement and arnoldi process for periodic matrix pairs we extend the rayleigh ritz method to the eigen problem of periodic matrix pairs assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero we show that there exist periodic ritz values that converge to the desired periodic eigenvalues unconditionally yet the periodic ritz vectors may fail to converge to overcome this potential problem we minimize residuals formed with periodic ritz values to produce the refined periodic ritz vectors which converge under the same assumption these results generalize the corresponding well known ones for rayleigh ritz approximations and their refinement for non periodic eigen problems in addition we consider a periodic arnoldi process which is particularly efficient when coupled with the rayleigh ritz method with refinement the numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors

the difference periodic ricati equation for the periodic prediction problem gives a comprehensive treatment of several important aspects of the discrete time periodic riccati equation dpre arising from the prediction problem for linear discrete time periodic systems the authors analyze the symmetric periodic positive semidefinite spps solution of the dpre under appropriate assumptions of stabilizability and detectability of the periodic system among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the spps solution and the stability of the resulting closed loop system some of these results can be seen as extensions of the corresponding results for the time invariant case however a number of them contain contributions to the time invariant case as well the paper also gives a numerical algorithm based on an iterative linearization procedure for computing the spps solution the algorithm is a periodic version of kleinman s algorithm for the time invariant case

noise reduction of cdna microarray images using complex wavelets noise reduction is an essential step of cdna microarray image analysis for obtaining better quality gene expression measurements wavelet based denoising methods have shown significant success in traditional image processing the complex wavelet transform cwt is preferred to the classical discrete wavelet transform for denoising of microarray images due to its improved directional selectivity for better representation of the circular edges of spots and near shift invariance property existing cwt based denoising methods are not efficient for microarray image processing because they fail to take into account the signal as well as noise correlations that exist between red and green channel images in this paper two bivariate estimators are developed for the cwt based denoising of microarray images using the standard maximum a posteriori and linear minimum mean squared error estimation criteria the proposed denoising methods are capable of taking into account both the interchannel signal and noise correlations significance of the proposed denoising methods is assessed by examining the effect of noise reduction on the estimation of the log intensity ratio extensive experimentations are carried out to show that the proposed methods provide better noise reduction of microarray images leading to more accurate estimation of the log intensity ratios as compared to the other cwt based denoising methods

perturbation analysis for the eigenproblem of periodic matrix pairs this paper is devoted to perturbation analysis for the eigenproblem of periodic matrix pairs a j e j j 1 k we first study perturbation expansions of periodic deflating subspaces and eigenv

periodic descriptor systems solvability and conditionability the authors consider discrete time linear periodic descriptor systems and study the concepts of solvability and conditionability introduced by luenberger they prove that solvability is equivalent to conditionability just as in the time invariant case we give a characterization of solvability conditionability in terms of a cyclic matrix pencil and furthermore propose a simple test via the periodic schur decomposition to check for either property this could lead to further systematic study of these systems

generalized fiducial inference for wavelet regression we apply fisher s fiducial idea to conduct statistical inference for wavelet regression we first develop a general methodology for handling model selection problems within the fiducial framework with this new methodology we then propose fiduical based methods for performing wavelet curve estimation as well as constructing both pointwise and curvewise confidence intervals it is shown that under some mild regularity conditions both the new fiducial based pointwise and curvewise confidence intervals have asymptotically correct coverage furthermore simulation results show that these new fiducial based methods especially for constructing pointwise confidence intervals also possess promising empirical properties to the best of our knowledge this is the first time that the fiducial idea has been applied to a nonparametric estimation problem

flexible empirical bayes estimation for wavelets summary wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression although bayes estimators can provide excellent mean squared error mse properties the selection of an effective prior is a difficult task to address this problem we propose empirical bayes eb prior selection methods for various error distributions including the normal and the heavier tailed student t distributions under such eb prior distributions we obtain threshold shrinkage estimators based on model selection and multiple shrinkage estimators based on model averaging these eb estimators are seen to be computationally competitive with standard classical thresholding methods and to be robust to outliers in both the data and wavelet domains simulated and real examples are used to illustrate the flexibility and improved mse performance of these methods in a wide variety of settings

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed