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

quantitative classical realizability introduced by dal lago and hofmann quantitative realizability is a technique used to define models for logics based on multiplicative linear logic a particularity is that functions are interpreted as bounded time computable functions it has been used to give new and uniform proofs of soundness of several type systems with respect to certain time complexity classes we propose a reformulation of their ideas in the setting of krivine s classical realizability the framework obtained generalizes dal lago and hofmann s realizability and reveals deep connections between quantitative realizability and a linear variant of cohen s forcing

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

phase semantic cut elimination and normalization proofs of first and higher order linear logic abstract we give a natural extension of girard s phase semantic completeness proof of the first order linear logic girard theoret comput sci 1987 to a phase semantic cut elimination proof then we extend this idea to a phase semantic cut elimination proof for higher order linear logic we also extend the phase semantics for provability to a phase semantics like framework for proofs by modifying the phase space of monoid domain to that of proof structures untyped proofs domain in a natural way the resulting phase semantic like framework for proofs provides various versions of proof normalization theorem

hyperanalytic denoising a new threshold rule for the estimation of a deterministic image immersed in noise is proposed the full estimation procedure is based on a separable wavelet decomposition of the observed image and the estimation is improved by introducing the new threshold to estimate the decomposition coefficients the observed wavelet coefficients are thresholded using the magnitudes of wavelet transforms of a small number of replicates of the image the replicates are calculated by extending the image into a vector valued hyperanalytic signal more than one hyperanalytic signal may be chosen and either the hypercomplex or riesz transforms are used to calculate this object the deterministic and stochastic properties of the observed wavelet coefficients of the hyperanalytic signal at a fixed scale and position index are determined a universal threshold is calculated for the proposed procedure an expression for the risk of an individual coefficient is derived the risk is calculated explicitly when the universal threshold is used and is shown to be less than the risk of universal hard thresholding under certain conditions the proposed method is implemented and the derived theoretical risk reductions substantiated

assessing the adequacy of variance function in heteroscedastic regression models heteroscedastic data arise in many applications in heteroscedastic regression analysis the variance is often modeled as a parametric function of the covariates or the regression mean we propose a kernel smoothing type nonparametric test for checking the adequacy of a given parametric variance structure the test does not need to specify a parametric distribution for the random errors it is shown that the test statistic has an asymptotical normal distribution under the null hypothesis and is powerful against a large class of alternatives we suggest a simple bootstrap algorithm to approximate the distribution of the test statistic in finite sample size numerical simulations demonstrate the satisfactory performance of the proposed test we also illustrate the application by the analysis of a radioimmunoassay data set

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

deconvolution by thresholding in mirror wavelet bases the deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients a general criterion is established to choose the orthonormal basis in order to minimize the estimation risk wavelet bases are highly sub optimal to restore signals and images blurred by a low pass filter whose transfer function vanishes at high frequencies a new orthonormal basis called mirror wavelet basis is constructed to minimize the risk for such deconvolutions an application to the restoration of satellite images is shown

pseudoconvex domains with smooth boundary in projective spaces given a pseudoconvex domain u with mathscr c 1 boundary in mathbb p n n geqslant 3 we show that if h 2n 2 mathrm dr u ne 0 then there is a strictly psh function in a neighborhood of partial u we also solve the overline partial equation in x mathbb p n setminus u for data in mathscr c infty 0 1 x we discuss levi flat domains in surfaces if z is a real algebraic hypersurface in mathbb p 2 resp a real analytic hypersurface with a point of strict pseudoconvexity then there is a strictly psh function in a neighborhood of z

a semantic measure of the execution time in linear logic we give a semantic account of the execution time i e the number of cut elimination steps leading to the normal form of an untyped mell net we first prove that 1 a net is head normalizable i e normalizable at depth 0 if and only if its interpretation in the multiset based relational semantics is not empty and 2 a net is normalizable if and only if its exhaustive interpretation a suitable restriction of its interpretation is not empty we then give a semantic measure of execution time we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut link from the interpretations of the two cut free nets these results are inspired by similar ones obtained by the first author for the untyped lambda calculus

intuitionistic differential nets and lambda calculus we define pure intuitionistic differential proof nets extending ehrhard and regnier s differential interaction nets with the exponential box of linear logic normalization of the exponential reduction and confluence of the full one is proved these results are directed and adjusted to give a translation of boudol s untyped calculus with resources extended with a linear nonlinear reduction a la ehrhard and regnier s differential calculus such reduction comes in two flavours baby step and giant step s reduction the translation based on girard s encoding a b a b and as such extending the usual one for calculus into proof nets enjoys bisimulation for giant step s reduction from this result we also derive confluence of both reductions

score test for the first order autoregressive model with heteroscedasticity summary a score test is proposed for simultaneous testing of independence and homoscedasticity in the first order autoregressive model with nonconstant variance the relationships between the score test statistic and the local influence of minor perturbations on a statistical model are examined

wavelet processes and adaptive estimation of the evolutionary wavelet spectrum this article defines and studies a new class of non stationary random processes constructed from discrete non decimated wavelets which generalizes the cramer fourier representation of stationary time series we define an evolutionary wavelet spectrum ews which quantifies how process power varies locally over time and scale we show how the ews may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time localized autocovariance we illustrate our theory with a pedagogical example based on discrete non decimated haar wavelets and also a real medical time series example

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

wavelet thresholding techniques for power spectrum estimation estimation of the power spectrum s f of a stationary random process can be viewed as a nonparametric statistical estimation problem we introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown s f this approach offers the ability to capture statistically significant components of ln s f at different resolution levels and guarantees nonnegativity of the spectrum estimator the spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non gaussian noise we propose a wavelet thresholding technique to solve this problem under specified noise resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables the thresholds are computed using a saddle point approximation to the distribution of the noise coefficients

score tests for heteroscedasticity in wavelet regression summary we consider two score tests for heteroscedasticity in the errors of a signal plus noise model where the signal is estimated by wavelet thresholding methods the error variances are assumed to depend on observed covariates through a parametric relationship of known form the tests are based on the approaches of breusch pagan 1979 and koenker 1981 we establish the asymptotic validity of the tests and examine their performance in a simulation study the koenker test is found to perform well in terms of both size and power

stochastic expansions in an overcomplete wavelet dictionary we consider random functions defined in terms of members of an overcomplete wavelet dictionary the function is modelled as a sum of wavelet components at arbitrary positions and scales where the locations of the wavelet components and the magnitudes of their coefficients are chosen with respect to a marked poisson process model the relationships between the parameters of the model and the parameters of those besov spaces within which realizations will fall are investigated the models allow functions with specified regularity properties to be generated they can potentially be used as priors in a bayesian approach to curve estimation extending current standard wavelet methods to be free from the dyadic positions and scales of the basis functions

pfaff systems currents and hulls let s be a pfaff system of dimension 1 on a compact complex manifold m we prove that there is a positive partial overline partial closed current t of bidimension 1 1 and of mass 1 directed by the pfaff system s there is no integrability assumption we also show that local singular solutions always exist under a transversality assumption of s on the boundary of an open set u we prove the existence in u of positive partial overline partial closed currents directed by s in u using i partial overline partial negative currents we discuss jensen measures local maximum principle and hulls with respect to a cone mathcal p of smooth functions in the euclidean complex space subharmonic in some directions the case where mathcal p is the cone of plurisubharmonic functions is classical we use the results to describe the harmonicity properties of the solutions of equations of homogeneous monge ampere type we also discuss extension problems of positive directed currents

automatic noise removal signal removal based on general cross validation thresholding in synchrosqueezed domain and its application on earthquake data abstractrecorded seismic signals are often corrupted by noise we have developed an automatic noise attenuation method for single channel seismic data based upon high resolution time frequency analysis synchrosqueezing is a time frequency reassignment method aimed at sharpening a time frequency picture noise can be distinguished from the signal and attenuated more easily in this reassigned domain the threshold level is estimated using a general cross validation approach that does not rely on any prior knowledge about the noise level the efficiency of the thresholding has been improved by adding a preprocessing step based on kurtosis measurement and a postprocessing step based on adaptive hard thresholding the proposed algorithm can either attenuate the noise either white or colored and keep the signal or remove the signal and keep the noise hence it can be used in either normal denoising applications or preprocessing in ambient noise studies we tested the performance of the proposed method on s

denoising functional mr images a comparison of wavelet denoising and gaussian smoothing we present a general wavelet based denoising scheme for functional magnetic resonance imaging fmri data and compare it to gaussian smoothing the traditional denoising method used in fmri analysis one dimensional wavelab thresholding routines were adapted to two dimensional 2 d images and applied to 2 d wavelet coefficients to test the effect of these methods on the signal to noise ratio snr we compared the snr of 2 d fmri images before and after denoising using both gaussian smoothing and wavelet based methods we simulated a fmri series with a time signal in an active spot and tested the methods on noisy copies of it the denoising methods were evaluated in two ways by the average temporal snr inside the original activated spot and by the shape of the spot detected by thresholding the temporal snr maps denoising methods that introduce much smoothness are better suited for low snrs but for images of reasonable quality they are not preferable because they introduce heavy deformations wavelet based denoising methods that introduce less smoothing preserve the sharpness of the images and retain the original shapes of active regions we also performed statistical parametric mapping on the denoised simulated time series as well as on a real fmri data set false discovery rate control was used to correct for multiple comparisons the results show that the methods that produce smooth images introduce more false positives the less smoothing wavelet based methods although generating more false negatives produce a smaller total number of errors than gaussian smoothing or wavelet based methods with a large smoothing effect

strong normalization from weak normalization in typedλ calculi for some typed calculi it is easier to prove weak normalization than strong normalization techniques to infer the latter from the former have been invented over the last twenty years by nederpelt klop khasidashvili karr de groote and kfoury and wells however these techniques infer strong normalization of one notion of reduction from weak normalization of amore complicatednotion of reduction this paper presents a new technique to infer strong normalization of a notion of reduction in a typed calculus from weak normalization of thesamenotion of reduction the technique is demonstrated to work on some well known systems including second order calculus and the system of positive recursive types it gives hope for a positive answer to the barendregt geuvers conjecture stating that every pure type system which is weakly normalizing is also strongly normalizing the paper also analyzes the relationship between the techniques mentioned above and reviews in less detail other techniques for proving strong normalization

a semantic account of strong normalization in linear logic abstract we prove that given two cut free nets of linear logic by means of their relational interpretations one can determine 1 whether or not the net obtained by cutting the two nets is strongly normalizable 2 in case it is strongly normalizable the maximum length of the reduction sequences starting from that net as a by product of our semantic approach we obtain a new proof of the conservation theorem for multiplicative exponential linear logic mell which does not rely on confluence this yields an alternative proof of strong normalization for mell

a note on studentizing a test for heteroscedasticity abstract breusch and pagan 1979 have recently proposed a convenient test for heteroscedasticity in general linear models this note derives the asymptotic distribution of their test under sequences of contiguous alternatives to the null hypothesis of homoscedasticity the test is shown to possess asymptotically incorrect size nominal significance level except in the case of strictly gaussian disturbances a slight modification of the test is proposed which corrects this defect

strong normalization property for second order linear logic the paper contains the first complete proof of strong normalization sn for full second order linear logic ll girard s original proof uses a standardization theorem which is not proven we introduce sliced pure structures sps a very general version of girard s proof nets and we apply to sps gandy s method to infer sn from weak normalization wn we prove a standardization theorem for sps if wn without erasing steps holds for an sps then it enjoys sn a key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness namely acyclicity slice by slice we conclude by showing how standardization for sps allows to prove sn of ll using as usual girard s reducibility candidates

additives of linear logic and normalization part i a restricted church rosser property we define a generalized cut elimination procedure for proof nets of fall linear logic without constants for which we prove a restricted church rosser property

a study of mexican free tailed bat chirp syllables bayesian functional mixed models for nonstationary acoustic time series we describe a new approach to analyze chirp syllables of free tailed bats from two regions of texas in which they are predominant austin and college station our goal is to characterize any systematic regional differences in the mating chirps and assess whether individual bats have signature chirps the data are analyzed by modeling spectrograms of the chirps as responses in a bayesian functional mixed model given the variable chirp lengths we compute the spectrograms on a relative time scale interpretable as the relative chirp position using a variable window overlap based on chirp length we use two dimensional wavelet transforms to capture correlation within the spectrogram in our modeling and obtain adaptive regularization of the estimates and inference for the regions specific spectrograms our model includes random effect spectrograms at the bat level to account for correlation among chirps from the same bat and to assess relative variability in chirp spectrograms within and between bats the mo

spatially adaptive wavelet thresholding with context modeling for image denoising the method of wavelet thresholding for removing noise or denoising has been researched extensively due to its effectiveness and simplicity much of the literature has focused on developing the best uniform threshold or best basis selection however not much has been done to make the threshold values adaptive to the spatially changing statistics of images such adaptivity can improve the wavelet thresholding performance because it allows additional local information of the image such as the identification of smooth or edge regions to be incorporated into the algorithm this work proposes a spatially adaptive wavelet thresholding method based on context modeling a common technique used in image compression to adapt the coder to changing image characteristics each wavelet coefficient is modeled as a random variable of a generalized gaussian distribution with an unknown parameter context modeling is used to estimate the parameter for each coefficient which is then used to adapt the thresholding strategy this spatially adaptive thresholding is extended to the overcomplete wavelet expansion which yields better results than the orthogonal transform experimental results show that spatially adaptive wavelet thresholding yields significantly superior image quality and lower mse than the best uniform thresholding with the original image assumed known

wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra we derive minimax rates for estimation in anisotropic smoothness classes this rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension it is shown that this basis is superior to its one scale multiresolution analog if different degrees of smoothness in different directions are present as an important application we introduce a new adaptive wavelet estimator of the time dependent spectrum of a locally stationary time series using this model which was resently developed by dahlhaus we show that the resulting estimator attains nearly the rate which is optimal in gaussian white noise simultaneously over a wide range of smoothness classes moreover by our new approach we overcome the difficulty of how to choose the right amount of smoothing i e how to adapt to the appropriate resolution for reconstructing the local structure of the evolutionary spectrum in the time frequency plane

testing for smooth structural changes in time series models via nonparametric regression checking parameter stability of econometric models is a long standing problem almost all existing structural change tests in econometrics are designed to detect abrupt breaks little attention has been paid to smooth structural changes which may be more realistic in economics we propose a consistent test for smooth structural changes as well as abrupt structural breaks with known or unknown change points the idea is to estimate smooth time varying parameters by local smoothing and compare the fitted values of the restricted constant parameter model and the unrestricted time varying parameter model the test is asymptotically pivotal and does not require prior information about the alternative a simulation study highlights the merits of the proposed test relative to a variety of popular tests for structural changes in an application we strongly reject the stability of univariate and multivariate stock return prediction models in the postwar and post oil shocks periods

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

minimax estimation via wavelet shrinkage 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 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators 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 which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

levi problem in complex manifolds let u be a pseudoconvex open set in a complex manifold m when is u a stein manifold there are classical counter examples due to grauert even when u has real analytic boundary or has strictly pseudoconvex points we give new criteria for the steinness of u and we analyze the obstructions the main tool is the notion of levi currents they are positive partial overline partial closed currents t of bidimension 1 1 and of mass 1 directed by the directions where all continuous psh functions in u have vanishing levi form the extremal ones are supported on the sets where all continuous psh functions are constant we also construct under geometric conditions bounded strictly psh exhaustion functions and hence we obtain donnelly fefferman weights to any infinitesimally homogeneous manifold we associate a foliation the dynamics of the foliation determines the solution of the levi problem some of the results can be extended to the context of pseudoconvexity with respect to a pfaff system

adaptive wavelet thresholding for image denoising and compression the first part of this paper proposes an adaptive data driven threshold for image denoising via wavelet soft thresholding the threshold is derived in a bayesian framework and the prior used on the wavelet coefficients is the generalized gaussian distribution ggd widely used in image processing applications the proposed threshold is simple and closed form and it is adaptive to each subband because it depends on data driven estimates of the parameters experimental results show that the proposed method called bayesshrink is typically within 5 of the mse of the best soft thresholding benchmark with the image assumed known it also outperforms sureshrink donoho and johnstone 1994 1995 donoho 1995 most of the time the second part of the paper attempts to further validate claims that lossy compression can be used for denoising the bayesshrink threshold can aid in the parameter selection of a coder designed with the intention of denoising and thus achieving simultaneous denoising and compression specifically the zero zone in the quantization step of compression is analogous to the threshold value in the thresholding function the remaining coder design parameters are chosen based on a criterion derived from rissanen s minimum description length mdl principle experiments show that this compression method does indeed remove noise significantly especially for large noise power however it introduces quantization noise and should be used only if bitrate were an additional concern to denoising

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions 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 and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves 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

fitting time series models to nonstationary processes a general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation the asymptotic properties of the estimate are derived under the assumption of possible model misspecification for autoregressive processes with time varying coefficients the estimate is compared to the least squares estimate furthermore the behavior of estimates is explained when a stationary model is fitted to a nonstationary process

the equation on a positive current we study the induced equation on a positive current in a complex manifold we extend the l 2 estimates for the equation to harmonic currents of bidimension 1 1 satisfying a frobenius type condition we also show that the l 2 estimates are satisfied for the equation on a positive closed current of bidegree 1 1 on a pseudoconvex domain in n

adaptive covariance estimation of locally stationary processes it is shown that the covariance operator of a locally stationary process has approximate eigenvectors that are local cosine functions we model locally stationary processes with pseudo differential operators that are time varying convolutions an adaptive covariance estimation is calculated by searching first for a best local cosine basis which approximates the covariance by a band or a diagonal matrix the estimation is obtained from regularized versions of the diagonal coefficients in the best basis

extending the scope of wavelet regression methods by coefficient dependent thresholding abstract various aspects of the wavelet approach to nonparametric regression are considered with the overall aim of extending the scope of wavelet techniques to irregularly spaced data to regularly spaced datasets of arbitrary size to heteroscedastic and correlated data and to data that contain outliers the core of the methodology is an algorithm for finding ail of the variances and within level covariances in the wavelet table of a sequence with given covariance structure if the original covariance matrix is band limited then the algorithm is linear in the length of the sequence the variance calculation algorithm allows data on any set of independent variable values to be treated by first interpolating to a fine regular grid of suitable length and then constructing a wavelet expansion of the gridded data various thresholding methods are discussed and investigated exact risk formulas for the mean square error of the methodology for given design are derived good performance is obtained by nois

function estimation via wavelet shrinkage for long memory data in this article we study function estimation via wavelet shrinkage for data with long range dependence we propose a fractional gaussian noise model to approximate nonparametric regression with long range dependence and establish asymptotics for minimax risks because of long range dependence the minimax risk and the minimax linear risk converge to 0 at rates that differ from those for data with independence or short range dependence wavelet estimates with best selection of resolution leveldependent threshold achieve minimax rates over a wide range of spaces cross validation for dependent data is proposed to select the optimal threshold the wavelet estimates significantly outperform linear estimates the key to proving the asymptotic results is a wavelet vaguelette decomposition which decorrelates fractional gaussian noise such wavelet vaguelette decomposition is also very useful in fractal signal processing

wavelet based image denoising using a markov random field a priori model this paper describes a new method for the suppression of noise in images via the wavelet transform the method relies on two measures the first is a classic measure of smoothness of the image and is based on an approximation of the local holder exponent via the wavelet coefficients the second novel measure takes into account geometrical constraints which are generally valid for natural images the smoothness measure and the constraints are combined in a bayesian probabilistic formulation and are implemented as a markov random field mrf image model the manipulation of the wavelet coefficients is consequently based on the obtained probabilities a comparison of quantitative and qualitative results for test images demonstrates the improved noise suppression performance with respect to previous wavelet based image denoising methods

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