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

some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

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

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

one step local quasi likelihood estimation summary local quasi likelihood estimation is a useful extension of local least squares methods but its computational cost and algorithmic convergence problems make the procedure less appealing particularly when it is iteratively used in methods such as the back fitting algorithm cross validation and bootstrapping a one step local quasi likelihood estimator is introduced to overcome the computational drawbacks of the local quasi likelihood method we demonstrate that as long as the initial estimators are reasonably good the one step estimator has the same asymptotic behaviour as the local quasi likelihood method our simulation shows that the one step estimator performs at least as well as the local quasi likelihood method for a wide range of choices of bandwidths a data driven bandwidth selector is proposed for the one step estimator based on the pre asymptotic substitution method of fan and gijbels it is then demonstrated that the datadriven one step local quasi likelihood estimator performs as well as the maximum local quasilikelihood estimator by using the ideal optimal bandwidth

analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near ideal estimation performance in various asymptotic frameworks however image processing practice has shown that universal thresholding methods are outperformed by simple bayesian estimators assuming independent wavelet coefficients and heavy tailed priors such as generalized gaussian distributions ggds in this paper we investigate various connections between shrinkage methods and maximum a posteriori map estimation using such priors in particular we state a simple condition under which map estimates are sparse we also introduce a new family of complexity priors based upon rissanen s universal prior on integers one particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length mdl principle we develop analytical expressions for the shrinkage rules implied by ggd and complexity priors this allows us to show the equivalence between universal hard thresholding map estimation using a very heavy tailed ggd and mdl estimation using one of the new complexity priors theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis specifications of the prior a basic concern in image processing applications

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

estimating and time updating the 2 d coherence spectrum in this correspondence we extend the recently introduced one dimensional capon based magnitude square coherence msc spectral estimator to form two dimensional capon and amplitude and phase estimation apes based msc spectral estimators the resulting estimators are found to yield significantly improved estimates as compared to the typical welch based estimator furthermore we introduce a computationally efficient time updating of the presented msc estimators exploiting their inherent time varying displacement structure the presented updating is found to dramatically lower the computational requirement of reevaluating the msc spectral estimates

understanding waveshrink variance and bias estimation summary donoho johnstone s waveshrink procedure has proved valuable for function estimation and nonparametric regression waveshrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise waveshrink has very broad asymptotic near optimality properties and achieves the optimal risk to within a factor of log n in this paper we derive computationally efficient formulae for computing the exact bias variance and l2 risk of waveshrink estimates in finite sample situations we use these formulae to understand the behaviour of waveshrink estimators construct approximate confidence intervals and bias estimates for waveshrink and compute ideal thresholds for a given function we show that hard shrinkage has smaller bias but larger variance than soft shrinkage and that significantly smaller thresholds should be used for soft shrinkage we also compute minimax thresholds for waveshrink estimators and demonstrate that the minimax thresholds can nearly achieve the ideal rank for a range of functions

interpolation methods for nonlinear wavelet regression with irregularly spaced design we introduce interpolation methods that enable nonlinear wavelet estimators to be employed with stochastic design or nondyadic regular design in problems of nonparametric regression this approach allows relatively rapid computation involving dyadic approximations to wavelet after interpolation techniques new types of interpolation are described enabling first order variance reduction at the expense of second order increases in bias the effect of interpolation on threshold choice is addressed and appropriate thresholds are suggested for error distributions with as few as four finite moments

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

umbral presentations for polynomial sequences abstract using random variables as motivation this paper presents an exposition of formalisms developed in 1 2 for the classical umbral calculus a new construction and a variety of examples are presented culminating in several descriptions of sequences of binomial type in terms of umbral polynomials

series with general exponents abstract we define the artinian and noetherian algebra which consist of formal series involving exponents which are not necessarily integers all of the usual operations are defined here and characterized as an application we compute the algebra of symmetric functions with nonnegative real exponents the applications to logarithmic series and the umbral calculus are deferred to another paper 9

wavelet methods for curve estimation abstract the theory of wavelets is a developing branch of mathematics with a wide range of potential applications compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties they are useful for the detection of edges and singularities in image and sound analysis and for data compression but most of the wavelet based procedures currently available do not explicitly account for the presence of noise in the data a discussion of how this can be done in the setting of some simple nonparametric curve estimation problems is given wavelet analogies of some familiar kernel and orthogonal series estimators are introduced and their finite sample and asymptotic properties are studied we discover that there is a fundamental instability in the asymptotic variance of wavelet estimators caused by the lack of translation invariance of the wavelet transform this is related to the properties of certain lacunary sequences the practi

the classical umbral calculus and the flow of a drinfeld module the notion of a flow appears in many areas of mathematics and physics for example it is one of the fundamental notions in studying ordinary differential equations classically a flow on a set x is a group action of the additive group of the set of real numbers r on the set x one of the classical theorems in analysis the classical taylor theorem can be expressed in terms of a flow more precisely let d dt be the formal derivative acting on the polynomial ring r t for each x r let e r t r t be the linear operator defined by

the algebraic properties of the generalized pascal functional matrices associated with the exponential families abstract in this paper pascal matrices are generalized to functional matrices by using the exponential families and the restricted sequences of polynomials of binomial type some algebraic properties of the pascal functional matrices are presented and proved in addition several important combinatorial identities are obtained

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

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

time variant displacement structure and triangular arrays the authors extend the concept of displacement structure to time variant matrices and use it to efficiently and recursively propagate the cholesky factor of such matrices a natural implementation of the algorithm is via a modular triangular array of processing elements when the algorithm is applied to solve the normal equations that arise in adaptive least squares filtering they get the so called qr algorithm with the extra bonus of a parallelizable procedure for determining the weight vector it is shown that the general algorithm can also be implemented in time variant lattice form a specialization of this result yields a time variant schur algorithm

local strong homogeneity of a regularized estimator this paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones where typically they are constant or linear or more generally satisfy a linear equation the estimate is defined as the minimizer of an objective function combining a quadratic data fidelity term and a regularization prior term the latter term is the sum of the values obtained by applying a potential function pf to each component called a difference of a linear transform of the signal minimizers of functions of this form arise in various settings in statistics and optimization the features exhibited by such an estimate are closely related to the shape of the pf our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data to this end we require that the strongly homogeneous zones recovered by the estimator be insensitive to any variation of th

computationally efficient two dimensional capon spectrum analysis we present a computationally efficient algorithm for computing the 2 d capon 1969 spectral estimator the implementation is based on the fact that the 2 d data covariance matrix will have a toeplitz block toeplitz structure with the result that the inverse covariance matrix can be expressed in closed form by using a special case of the gohberg heinig 1974 formula that is a function of strictly the forward 2 d prediction matrix polynomials furthermore we present a novel method based on a 2 d lattice algorithm to compute the needed forward prediction matrix polynomials and discuss the difference in the so obtained 2 d spectral estimate as compared with the one obtained by using the prediction matrix polynomials given by the whittle wiggins robinson 1963 1965 algorithm numerical simulations illustrate the improved resolution as well as the clear computational gain in comparison to both the well known classical implementation and the method published by liu et al see ieee trans aerosp electron syst vol 34 p 1314 19 1998

computationally efficient capon and apes based coherence spectrum estimation the coherence spectrum is of notable interest as a bivariate spectral measure in a variety of application and the topic has lately attracted interest with the recent formulation of several high resolution data adaptive estimators in this work we further this development with the presentation of computationally efficient implementations of the capon and apes based msc estimators the presented implementations furthers the recent development of exploiting the estimators inherently low displacement rank of the necessary products of toeplitz like matrices to include also the required cross correlation covariance matrices for the mentioned coherence algorithms numerical simulations together with theoretical complexity measures illustrate the performance of the proposed implementations

efficient direction finding methods employing forward backward averaging in this paper we develop a general approach for reducing the computational complexity of any direction finding method implemented with forward backward fb averaging we develop simplified fb data matrices in a manner paralleling previous work related to centrohermitian correlation matrices based on these simplified data matrices efficient construction and updating of the fb correlation matrix is developed in addition efficient fb fft fb beamspace fb evd updating fb svd and fb svd updating methods are derived in most cases fb based direction finding methods can be simplified so that the computational complexity is reduced below that of an analogous forward only implementation thus effectively twice the amount of data is processed with less total computation

better subset regression using the nonnegative garrote a new method called the nonnegative nn garrote is proposed for doing subset regression it both shrinks and zeroes coefficients in tests on real and simulated data it produces lower prediction error than ordinary subset selection it is also compared to ridge regression if the regression equations generated by a procedure do not change drastically with small changes in the data the procedure is called stable subset selection is unstable ridge is very stable and the nn garrote is intermediate simulation results illustrate the effects of instability on prediction error

postmodern logarithmo technia abstract in order to solve ordinary differential equations we use an equation with the so called logarithmic derivative similar equations with a linear operator permit us to define logarithmic and antilogarithmic mappings and obtain some results unknown before for the classical derivation operator this method and its applications are exposed in detail in the author s book 1 and may be treated as an introduction into the 21 st century logarithmo technia according to the original meaning of this word in the first section there are given basic notions and facts of algebraic analysis without proofs the second section consists of the most important definitions and theorems without proofs concerning logarithms and antilogarithms the third section is concerned with properties of multiplicative true shifts in the last section is given a generalization of the binomial theorem for harmonic logarithms

umbral calculus and special functions developpement du calcul symbolique des puissances et de la methode des coalgebres et des comodules associee application a l etude des fonctions speciales classiques hypergeometriques de laguerre sonine d hermite weber de legendre et de bessel

dimension reduction for hypothesis testing in worst case scenarios this paper considers a one among many detection problem where one has to discriminate between pure noise and one among l alternatives that are known up to an amplitude factor two issues linked to high dimensionality arise in this framework first the computational complexity associated to the generalized likelihood ratio glr with the constraint of sparsity one inflates linearly with l which can be an obstacle when multiple data sets have to be tested second standard procedures based on dictionary learning aimed at reducing the dimensionality may suffer from severe power losses for some alternatives thus suggesting a worst case scenario strategy in the case where the learned dictionary has k 1 column we show that the exact solution of the resulting detection problem which can be formulated as a minimax problem can be obtained by quadratic programming because it allows a better sampling of the diversity of the alternatives the case k 1 is expected to improve the detection performances over the case k 1 the worst case analysis of this case which is more involved leads us to propose two minimax learning algorithms numerical results show that these algorithms indeed allow to increase performances over the k 1 case and are in fact comparable to the glr using the full set of alternatives while being computationally simpler

detection tests using sparse models with application to hyperspectral data the problem of finding efficient methods for the detection of unknown sparse signals buried in noise is addressed we present two detection tests adapted to sparse signals based on the maximum a posteriori map estimate of the sparse vector of parameters the first is the posterior density ratio test which computes the ratio of the a posteriori distribution under each hypothesis of the data model the second is a likelihood ratio test in which the map replaces the maximum likelihood ml estimate the behaviors and the relative differences between these tests are investigated through a detailed study of their structural characteristics the proposed approaches are compared to the generalized likelihood ratio test glr showing successful results in the case of a simple model first and then for a model in which sparsity is promoted through the use of a highly redundant dictionary

the iterated logarithmic algebra ii sheffer sequences abstract an extension of the theory of the iterated logarithmic algebra 1 gives the logarithmic analog of a sheffer or appell sequence of polynomials this leads to several examples including stirling s formula and a logarithmic version of the euler maclaurin summation formula

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

nonlinear wavelet shrinkage with bayes rules and bayes factors abstract wavelet shrinkage the method proposed by the seminal work of donoho and johnstone is a disarmingly simple and efficient way of denoising data shrinking wavelet coefficients was proposed from several optimality criteria in this article a wavelet shrinkage by coherent bayesian inference in the wavelet domain is proposed the methods are tested on standard donoho johnstone test functions

wavelet domain image restoration with adaptive edge preserving regularization in this paper we consider a wavelet based edge preserving regularization scheme for use in linear image restoration problems our efforts build on a collection of mathematical results indicating that wavelets are especially useful for representing functions that contain discontinuities i e edges in two dimensions or jumps in one dimension we interpret the resulting theory in a statistical signal processing framework and obtain a highly flexible framework for adapting the degree of regularization to the local structure of the underlying image in particular we are able to adapt quite easily to scale varying and orientation varying features in the image while simultaneously retaining the edge preservation properties of the regularizer we demonstrate a half quadratic algorithm for obtaining the restorations from observed data

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

deterministic edge preserving regularization in computed imaging many image processing problems are ill posed and must be regularized usually a roughness penalty is imposed on the solution the difficulty is to avoid the smoothing of edges which are very important attributes of the image in this paper we first give conditions for the design of such an edge preserving regularization under these conditions we show that it is possible to introduce an auxiliary variable whose role is twofold first it marks the discontinuities and ensures their preservation from smoothing second it makes the criterion half quadratic the optimization is then easier we propose a deterministic strategy based on alternate minimizations on the image and the auxiliary variable this leads to the definition of an original reconstruction algorithm called artur some theoretical properties of artur are discussed experimental results illustrate the behavior of the algorithm these results are shown in the field of 2d single photon emission tomography but this method can be applied in a large number of applications in image processing

a unitary transformation method for angle of arrival estimation eigenstructure methods for estimating angles of arrival of radiation sources generally require complex computations in computing eigencomponents of the covariance matrix and calculating the search function a unitary transformation method that transforms the complex covariance matrix of an equally spaced linear array which is hermitian persymmetric and the complex search vector into a real symmetric matrix and a real vector respectively is presented both tasks can be accomplished by real computations the sampled covariance matrix available is not persymmetric to suit the unitary transformation method a persymmetrized estimator of the sampled covariance matrix which is optimal in the sense of euclidean distance is proposed

test of significance based on wavelet thresholding and neyman s truncation abstract traditional nonparametric tests such as the kolmogorov smirnov test and the cramer von mises test are based on the empirical distribution functions although these tests possess root n consistency they effectively use only information contained in the low frequencies this leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high frequency alternations the drawback can be repaired via smoothing based test statistics in this article we propose two such kind of test statistics based on the wavelet thresholding and the neyman truncation we provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations while maintaining the same capability in detecting smooth alternative densities as the traditional tests similar conclusions can be made for two sample nonparametric tests of distribution functions in that case the traditional linear rank tests such as th

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

two identities in the theory of polynomials of binomial type demonstration de deux identites generalisant celles etablies par hurwitz 1902 application aux cycles de permutations

recursive matrices and umbral calculus two major objectives can be seen to guide much recent work in enumeration 1 to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide enough type 2 to recover from empirical data an underlying set theoretic structure which would reveal the source of the given recursion we are here concerned with the first of these objectives though the eventual understanding of the second is tacitly present if only as a goal we noticed the coincidence of several computations which similar as they are in retrospect had failed to realize their kinship on leafing through the unique assembly of recursively solvable combinatorial problems in comtet s and sloane s invaluable collections one is struck by the repeated occurrence of one and the same kind of double recursion more strikingly the same recursion is seen to occur in the polynomial sequences of the umbra1 calculus of roman and rota see 25 everywhere the lagrange inversion formula for power series plays a pivotal role much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs andrews foata garsia gessel joni raney reiner schtitzenberger to name but a few whatever their origins the identities abutting lagrange inversion are expressed by integers alone this suggests not only a hidden set theoretic layer but a characteristic free generalization as well this generalization is the central theme of our work we define a monoid of infinite matrices recursive matrices for short the entries of these matrices give the sought out recursion for example that for coefficients of binomial and sheffer polynomials and factor sequences as well as that of the special sequences recently introduced by roman in 26 546 002 l 8693 82 040546 28 02 00 0

more on the umbral calculus with emphasis on the q umbral calculus abstract the interrelationship between distinct umbral calculi is studied these ideas are applied in particular to q umbral calculus which shows how andrews q theory relates to the q theory discussed in the previous paper by the author the q hermite polynomials and basic hypergeometric series are briefly discussed

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

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

multiple shrinkage and subset selection in wavelets this paper discusses bayesian methods for multiple shrinkage estimation in wavelets wavelets are used in applications for data denoising via shrinkage of the coefficients towards zero and for data compression by shrinkage and setting small coefficients to zero we approach wavelet shrinkage by using bayesian hierarchical models assigning a positive prior probability to the wavelet coefficients being zero the resulting estimator for the wavelet coefficients is a multiple shrinkage estimator that exhibits a wide variety of nonlinear patterns we discuss fast computational implementations with a focus on easy to compute analytic approximations as well as importance sampling and markov chain monte carlo methods multiple shrinkage estimators prove to have excellent mean squared error performance in reconstructing standard test functions we demonstrate this in simulated test examples comparing various implementations of multiple shrinkage to commonly used shrinkage rules finally we illustrate our approach with an application to the so called glint data

sets with a negative number of elements we will define an analog of a set which can contain either a positive or negative number of elements we will allow sums to be calculated over an arbitrary hybrid set this will lead us to a generalization of symmetric functions to a negative number of variable which agrees with all previous known generalizations of symmetric functions in this direction from these new symmetric functions we calculate generalized binomial coefficients then these coefficients will be given a combinatorial interpretation in terms of a new partial order on the hybrid sets next we will generalize linear partitions using this partial order and enumerate them finally we define sum from one to an arbitrary integer and derive formulas involving them including a generalization of the stirling numbers

linear regression with a sparse parameter vector we consider linear regression under a model where the parameter vector is known to be sparse using a bayesian framework we derive the minimum mean square error mmse estimate of the parameter vector and a computationally efficient approximation of it we also derive an empirical bayesian version of the estimator which does not need any a priori information nor does it need the selection of any user parameters as a byproduct we obtain a powerful model basis selection tool for sparse models the performance and robustness of our new estimators are illustrated via numerical examples

wavelet shrinkage for nonequispaced samples standard wavelet shrinkage procedures for nonparametric regression are restricted to equispaced samples there data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients the estimators are obtained via the inverse transform of the denoised wavelet coefficients in many applications however the samples are nonequispaced it can be shown that these procedures would produce suboptimal estimators if they were applied directly to nonequispaced samples we propose a wavelet shrinkage procedure for nonequispaced samples we show that the estimate is adaptive and near optimal for global estimation the estimate is within a logarithmic factor of the minimax risk over a wide range of piecewise holder classes indeed with a number of discontinuities that grows polynomially fast with the sample size for estimating a target function at a point the estimate is optimally adaptive to unknown degree of smoothness within a constant in addition the estimate enjoys a smoothness property if the target function is the zero function then with probability tending to 1 the estimate is also the zero function

the stochastic difference between econometric statistics dans un contexte general et dans divers cas particuliers on calcule l ordre en probabilite de la difference entre des racines consistantes d equations d estimation rivales

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

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

the interated logarithmic algebra abstract we generalize the umbral calculus of g c rota adv in math 27 1978 95 188 by studying not only sequences of polynomials and inverse power series or even the logarithms studied by d loeb and g c rota adv in math 75 1989 1 118 but instead we study sequences of formal expressions involving the iterated logarithms and x to an arbitrary real power using a theory of formal power series with real exponents and a more general definition of factorial binomial coefficient and stirling numbers to all the real numbers we define the iterated logarithmic algebra j its elements are the formal representations of the asymptotic expansions of a large class of real functions and we define the harmonic logarithm basis of j which will be interpreted as a generalization of the powers xn since it behaves nicely with respect to the derivative we classify all operators over j which commute with the derivative classically there are known as shift invariant operators and formulate several equivalent definitions of a sequence of binomial type we then derive many formulas useful towards the calculation of these sequences including the recurrence formula the transfer formula and the lagrange inversion formula in the sequel 22 we study sheffer sequences and give many examples

distributions and power of optimal signal detection statistics in finite case for detecting weak and sparse signals by a set of n input p values the higher criticism hc type statistics the berk jones b j type statistics and the phi divergence statistics have the equivalent asymptotic optimality as n goes to infinity however they can have significantly different performance in practical data analysis where n is always finite and even very small to address this problem in a broader context this paper introduces a general family of goodness of fit statistics called the ggof which unifies a broad signal detection statistics including these optimal ones efficient and accurate analytical calculations for the distributions of the ggof statistics are provided under arbitrary i i d continuous models of the null and the alternative hypotheses based on that a systematic power study reveals that in finite case the number of signals is often more relevant than the signal proportion the hc and the reverse hc have advantages for relatively sparser and denser signals respectively while the b j is more robust a general framework is given to apply the ggof into data analysis based on the generalized linear models an application to the snp set based genome wide association study gwas for crohn s disease shows that these optimal statistics have a good potential for detecting novel disease genes with weak snp effects the calculations have been implemented into an r package settest and published on the cran

data compression and harmonic analysis in this paper we review some recent interactions between harmonic analysis and data compression the story goes back of course to shannon s r d theory in the case of gaussian stationary processes which says that transforming into a fourier basis followed by block coding gives an optimal lossy compression technique practical developments like transform based image compression have been inspired by this result in this paper we also discuss connections perhaps less familiar to the information theory community growing out of the field of harmonic analysis recent harmonic analysis constructions such as wavelet transforms and gabor transforms are essentially optimal transforms for transform coding in certain settings some of these transforms are under consideration for future compression standards we discuss some of the lessons of harmonic analysis in this century typically the problems and achievements of this field have involved goals that were not obviously related to practical data compression and have used a language not immediately accessible to outsiders nevertheless through an extensive generalization of what shannon called the sampling theorem harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations we explain why harmonic analysis has interacted with data compression and we describe some interesting recent ideas in the field that may affect data compression in the future

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

formulae for mean integrated squared error of nonlinear wavelet based density estimators we provide an asymptotic formula for the mean integrated squared error mise of nonlinear wavelet based density estimators we show that unlike the analogous situation for kernel density estimators this mise formula is relatively unaffected by assumptions of continuity in particular it is available for densities which are smooth in only a piecewise sense another difference is that in the wavelet case the classical mise formula is valid only for sufficiently small values of the bandwidth for larger bandwidths mise assumes a very different form and hardly varies at all with changing bandwidth this remarkable property guarantees a high level of robustness against oversmoothing not encountered in the context of kernel methods we also use the mise formula to describe an asymptotically optimal empirical bandwidth selection rule

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 decomposition approaches to statistical inverse problems summary a wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise primary interest is in some function f t but data are accessible only about some linear transform corrupted by noise the usual linear methods for such inverse problems do not perform satisfactorily when f t is spatially inhomogeneous one existing nonlinear alternative is the wavelet vaguelette decomposition method based on the expansion of the unknown f t in wavelet series in the vaguelette wavelet decomposition method proposed here the observed data are expanded directly in wavelet series the performances of various methods are compared through exact risk calculations in the context of the estimation of the derivative of a function observed subject to noise a result is proved demonstrating that with a suitable universal threshold somewhat larger than that used for standard denoising problems both the wavelet based approaches have an ideal spatial adaptivity property

markovian reconstruction using a gnc approach this paper is concerned with the reconstruction of images or signals from incomplete noisy data obtained at the output of an observation system the solution is defined in maximum a posteriori map sense and it appears as the global minimum of an energy function joining a convex data fidelity term and a markovian prior energy the sought images are composed of nearly homogeneous zones separated by edges and the prior term accounts for this knowledge this term combines general nonconvex potential functions pfs which are applied to the differences between neighboring pixels the resultant map energy generally exhibits numerous local minima calculating its local minimum placed in the vicinity of the maximum likelihood estimate is inexpensive but the resultant estimate is usually disappointing optimization using simulated annealing is practical only in restricted situations several deterministic suboptimal techniques approach the global minimum of special map energies employed in the field of image denoising at a reasonable numerical cost the latter techniques are not directly applicable to general observation systems nor to general markovian prior energies this work is devoted to the generalization of one of them the graduated nonconvexity gnc algorithm in order to calculate nearly optimal map solutions in a wide range of situations in fact gnc provides a solution by tracking a set of minima along a sequence of approximate energies starting from a convex energy and progressing toward the original energy in this paper we develop a common method to derive efficient gnc algorithms for the minimization of map energies which arise in the context of any observation system giving rise to a convex data fidelity term and of markov random field mrf energies involving any nonconvex and or nonsmooth pfs as a side result we propose how to construct pertinent initializations which allow us to obtain meaningful solutions using local minimization of these map energies two numerical experiments an image deblurring and an emission tomography reconstruction illustrate the performance of the proposed technique

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the