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

wavelet thresholding via a bayesian approach we discuss a bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression a prior distribution is imposed on the wavelet coefficients of the unknown response function designed to capture the sparseness of wavelet expansion that is common to most applications for the prior specified the posterior median yields a thresholding procedure our prior model for the underlying function can be adjusted to give functions falling in any specific besov space we establish a relationship between the hyperparameters of the prior model and the parameters of those besov spaces within which realizations from the prior will fall such a relationship gives insight into the meaning of the besov space parameters moreover the relationship established makes it possible in principle to incorporate prior knowledge about the function s regularity properties into the prior model for its wavelet coefficients however prior knowledge about a function s regularity properties might be difficult to elicit with this in mind we propose a standard choice of prior hyperparameters that works well in our examples several simulated examples are used to illustrate our method and comparisons are made with other thresholding methods we also present an application to a data set that was collected in an anaesthesiological study

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

spiral vortex solution of birkhoff rott equation abstract evolution of a spiral vortex sheet in an unbounded inviscid fluid is considered and a similarity solution to the birkhoff rott equation an integro differential equation governing vortex sheet dynamics is presented this solution exhibits evolution of a double branched vortex sheet of an exponential spiral it is shown that this is equivalent to the previously known solution obtained by the other method however the present analysis discloses an anomaly of this similarity solution which is associated with the behavior at infinity

functional clustering by bayesian wavelet methods summary we propose a nonparametric bayes wavelet model for clustering of functional data the wavelet based methodology is aimed at the resolution of generic global and local features during clustering and is suitable for clustering high dimensional data based on the dirichlet process the nonparametric bayes model extends the scope of traditional bayes wavelet methods to functional clustering and allows the elicitation of prior belief about the regularity of the functions and the number of clusters by suitably mixing the dirichlet processes posterior inference is carried out by gibbs sampling with conjugate priors which makes the computation straightforward we use simulated as well as real data sets to illustrate the suitability of the approach over other alternatives

the variable selection problem abstract the problem of variable selection is one of the most pervasive model selection problems in statistical applications often referred to as the problem of subset selection it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors but there is uncertainty about which subset to use this vignette reviews some of the key developments that have led to the wide variety of approaches for this problem

separability assignment problem for q dimensional linear discrete time systems separability assignment problem for q dimensional system is formulated sufficient conditions for the existence of a solution to the problem are presented an algorithm for finding the feedback gain matrix is given and illustrated by an example of a 4 d system

frequentist optimality of bayesian wavelet shrinkage rules for gaussian and non gaussian noise the present paper investigates theoretical performance of various bayesian wavelet shrinkage rules in a nonparametric regression model with i i d errors which are not necessarily normally distributed the main purpose is comparison of various bayesian models in terms of their frequentist asymptotic optimality in sobolev and besov spaces we establish a relationship between hyperparameters verify that the majority of bayesian models studied so far achieve theoretical optimality state which bayesian models cannot achieve optimal convergence rate and explain why it happens

note on the pole assignment of q dimensional linear systems the method in chu 1986 for the solution of the pole assignment problem of separable 2 d linear discrete systems with state feedback is improved and extended to q d linear discrete systems and systems with output feedback

bayes and empirical bayes do they merge bayesian inference is attractive for its coherence and good frequentist properties however eliciting a honest prior may be difficult and a common practice is to take an empirical bayes approach using some empirical estimate of the prior hyperparameters despite not rigorous the underlying idea is that for sufficiently large sample size empirical bayes leads to similar inferential answers as a proper bayesian inference however precise mathematical results seem missing in this work we give more rigorous results in terms of merging of bayesian and empirical bayesian posterior distributions we study two notions of merging bayesian weak merging and frequentist merging in total variation we also show that under regularity conditions empirical bayes asymptotically gives an oracle selection of the prior hyperparameters examples include empirical bayes density estimation with dirichlet process mixtures

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

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

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

bayes model averaging with selection of regressors when a number of distinct models contend for use in prediction the choice of a single model can offer rather unstable predictions in regression stochastic search variable selection with bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors here we look at bayes model averaging incorporating variable selection for prediction this offers similar mean square errors of prediction but with a vastly reduced predictor space this can greatly aid the interpretation of the model it also reduces the cost if measured variables have costs the development here uses decision theory in the context of the multivariate general linear model in passing this reduced predictor space bayes model averaging is contrasted with single model approximations a fast algorithm for updating regressions in the markov chain monte carlo searches for posterior inference is developed allowing many more variables than observations to be contemplated we discuss the merits of absolute rather than proportionate shrinkage in regression especially when there are more variables than observations the methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution

semiparametric regression using student t processes in this paper we propose a latent factor regression model in which priors are assigned to both the latent regression vector and the error term by using reproducing kernels the resulting regression function follows a stochastic process known as a student process the model is attractive because its implementation is based on a tractable posterior predictive distribution and a simple expectation maximization em estimation algorithm in addition treating the transductive inference as a missing data problem we devise the em algorithm to deal with the parameter estimation as well as the response prediction in a single paradigm the model is also elaborated for multivariate response regression problems for this purpose we present a generalization of multivariate models and some of its properties experimental results show our approaches to be effective

hardy spaces and the two dimensional euler equations with nonnegative vorticity we prove that certain quadratic expressions involving the gradient of a weakly superharmonic function in r belong to a local hardy space as an application we provide a new proof of j m delort s convergence theorem for solutions of the two dimensional euler equations with vorticities of one sign department of mathematics university of california berkeley california 94720 sfb 256 institut fur angewandte mathematik beringstrasse 4 d5300 bonn 1 ger many this content downloaded from 207 46 13 101 on sat 08 oct 2016 05 44 35 utc all use subject to http about jstor org terms

wavelet based sequential monte carlo blind receivers in fading channels with unknown channel statistics recently an adaptive bayesian receiver for blind detection in flat fading channels was developed by the present authors based on the sequential monte carlo methodology that work is built on a parametric modeling of the fading process in the form of a state space model and assumes the knowledge of the second order statistics of the fading channel in this paper we develop a nonparametric approach to the problem of blind detection in fading channels without assuming any knowledge of the channel statistics the basic idea is to decompose the fading process using a wavelet basis and to use the sequential monte carlo technique to track both the wavelet coefficients and the transmitted symbols a novel resampling based wavelet shrinkage technique is proposed to dynamically choose the number of wavelet coefficients to best fit the fading process under such a framework blind detectors for both flat fading channels and frequency selective fading channels are developed simulation results are provided to demonstrate the excellent performance of the proposed blind adaptive receivers

efficient functional anova through wavelet domain markov groves we introduce a wavelet domain method for functional analysis of variance fanova it is based on a bayesian hierarchical model that employs a graphical hyperprior in the form of a markov grove mg

long time existence for a slightly perturbed vortex sheet consider a flat two dimensional vortex sheet perturbed initially by a small analytic disturbance by a formal perturbation analysis moore derived an approximate differential equation for the evolution of the vortex sheet we present a simplified derivation of moore s approximate equation and analyze errors in the approximation the result is used to prove existence of smooth solutions for long time if the initial perturbation is of size and is analytic in a strip m existence of a smooth solution of birkhoff s equation is shown for time t k2p if is sufficiently small with 1 as 0 for the particular case of sinusoidal data of wave length and amplitude e moore s analysis and independent numerical results show singularity development at time tc log o log log our results prove existence for t log if is sufficiently small with k 1 as 0 thus our existence results are nearly optimal

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

pole assignment problem in two dimensional linear systems sufficient conditions for existence of a solution to the pole assignment problem in a class of 2 d linear systems using state feedback are given an algorithm for finding the feedback gain matrix for a class of 2 d linear systems is presented and illustrated by simple numerical example

a criterion for the equivalence of the birkhoff rott and euler descriptions of vortex sheet evolution in this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible euler equations and as a weak solution of the birkhoff rott equations we begin by discussing the classical birkhoff rott equations with respect to arbitrary parametrizations of the sheet we introduce a notion of weak solution to the birkhoff rott system and we prove consistency of this notion with the classical formulation of the equations our main purpose in this paper is to present a sharp criterion for the equivalence of the weak euler and weak birkhoff rott descriptions of vortex sheet dynamics

perfect sampling for the wavelet reconstruction of signals the coupling from the past cftp procedure is a protocol for finite state markov chain monte carlo mcmc methods whereby the algorithm itself can determine the necessary runtime to convergence in this paper we demonstrate how this protocol can be applied to the problem of signal reconstruction using bayesian wavelet analysis where the dimensionality of the wavelet basis set is unknown and the observations are distorted by gaussian white noise of unknown variance mcmc simulation is used to account for model uncertainty by drawing samples of wavelet bases for approximating integrals or summations on the model space that are either too complex or too computationally demanding to perform analytically we extend the cftp protocol by making use of the central limit theorem to show how the algorithm can also monitor its own approximation error induced by mcmc in this way we can assess the number of mcmc samples needed to approximate the integral to within a user specified tolerance level hence the method automatically ensures convergence and determines the necessary number of iterations needed to meet the error criteria

outlier models and prior distributions in bayesian linear regression summary bayesian inference in regression models is considered using heavy tailed error distri butions to accommodate outliers the particular class of distributions that can be con structed as scale mixtures of normal distributions are examined and use is made of them as both error models and prior distributions in bayesian linear modelling includ ing simple regression and more complex hierarchical models with structured priors depending on unknown hyperprior parameters the modelling of outliers in nominally normal linear regression models using alternative error distributions which are heavy tailed relative to the normal provides an automatic means of both detecting and accommodating possibly aberrant observations such realistic models do however often lead to analytically intractable analyses with complex posterior distributions in several dimensions that are difficult to summarize and understand in this paper we consider a special yet rather wide class of heavy tailed unimodal and symmetric error distributions for which the analyses though apparently intractable can be examined in some depth by exploiting certain properties of the assumed error form the distributions concerned are those that can be con structed as scale mixtures of normal distributions in his paper concerning location parameters de finetti 1961 discusses such distributions and suggests the hypothetical interpretation that each observation is taken using an instrument with normal error but each time chosen at random from a collection of instruments of different precisions the distribution of the

bayesian wavelet regression on curves with application to a spectroscopic calibration problem motivated by calibration problems in near infrared nir spectroscopy we consider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced points and there may be multiple predictands we tackle this regression problem by calculating the wavelet transforms of the discretized curves then applying a bayesian variable selection method using mixture priors to the multivariate regression of predictands on wavelet coefficients for prediction purposes we average over a set of likely models applied to a particular problem in nir spectroscopy this approach was able to find subsets of the wavelet coefficients with overall better predictive performance than the more usual approaches in the application the available predictors are measurements of the nir reflectance spectrum of biscuit dough pieces at 256 equally spaced wavelengths the aim is to predict the composition i e the fat flour sugar and water content of the doug

wavelet based functional mixed models increasingly scientific studies yield functional data in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid we present new methodology that generalizes the linear mixed model to the functional mixed model framework with model fitting done by using a bayesian wavelet based approach this method is flexible allowing functions of arbitrary form and the full range of fixed effects structures and between curve covariance structures that are available in the mixed model framework it yields nonparametric estimates of the fixed and random effects functions as well as the various between curve and within curve covariance matrices the functional fixed effects are adaptively regularized as a result of the non linear shrinkage prior that is imposed on the fixed effects wavelet coefficients and the random effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient because we have posterior samples for all model quantities we can perform pointwise or joint bayesian inference or prediction on the quantities of the model the adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks

bayesian correlation estimation we propose prior probability models for variance covariance matrices in order to address two important issues first the models allow a researcher to represent substantive prior information about the strength of correlations among a set of variables secondly even in the absence of such information the increased flexibility of the models mitigates dependence on strict parametric assumptions in standard prior models for example the model allows a posteriori different levels of uncertainty about correlations among different subsets of variables we achieve this by including a clustering mechanism in the prior probability model clustering is with respect to variables and pairs of variables our approach leads to shrinkage towards a mixture structure implied by the clustering we discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models including the evaluation of normalising constants that are functions of parameters of interest the normalising constants result from the restriction that the correlation matrix be positive definite we discuss examples based on simulated data a stock return dataset and a population genetics dataset

existence of vortex sheets with reflection symmetry in two space dimensions the main purpose of this work is to establish the existence of a weak solution to the incompressible 2d euler equations with initial vorticity consisting of a radon measure with distinguished sign in h 1 compactly supported in the closed right half plane superimposed on its odd reflection in the left half plane we make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis we prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2d euler equations we also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half plane finally we extend our existence result to odd l 1 perturbations without distinguished sign of our original initial vorticity

reduced hausdorff dimension and concentration cancellation for two dimensional incompressible flow where v t v1 v2 is the fluid velocity p is the scalar pressure dv dt av at v v v and vo is an initial incompressible velocity field i e div vo 0 in this paper we study the detailed limiting behavior of approximate solution sequences for 2 d euler with vortex sheet initial data a sequence of smooth velocity fields v8 x t is an approximate solution sequence for 2 d euler provided that v is incompressible i e div v 0 and satisfies the following properties 1 the velocity fields v6 have uniformly bounded local kinetic energy i e

bayesian wavelet based image denoising using the gauss hermite expansion the probability density functions pdfs of the wavelet coefficients play a key role in many wavelet based image processing algorithms such as denoising the conventional pdfs usually have a limited number of parameters that are calculated from the first few moments only consequently such pdfs cannot be made to fit very well with the empirical pdf of the wavelet coefficients of an image as a result the shrinkage function utilizing any of these density functions provides a substandard denoising performance in order for the probabilistic model of the image wavelet coefficients to be able to incorporate an appropriate number of parameters that are dependent on the higher order moments a pdf using a series expansion in terms of the hermite polynomials that are orthogonal with respect to the standard gaussian weight function is introduced a modification in the series function is introduced so that only a finite number of terms can be used to model the image wavelet coefficients ensuring at the same time the resulting pdf to be non negative it is shown that the proposed pdf matches the empirical one better than some of the standard ones such as the generalized gaussian or bessel k form pdf a bayesian image denoising technique is then proposed wherein the new pdf is exploited to statistically model the subband as well as the local neighboring image wavelet coefficients experimental results on several test images demonstrate that the proposed denoising method both in the subband adaptive and locally adaptive conditions provides a performance better than that of most of the methods that use pdfs with limited number of parameters

a bayesian transformation model for wavelet shrinkage wavelet shrinkage estimators in general make the additive normal noise assumption and disregard the nonlinear nature of contamination we develop bayesian wavelet shrinkage estimators based on the power transformations in the linear model to accommodate a broad class of noise models in image processing applications we intend to admit under one roof the widespread additive model the product models common in imaging such as in synthetic aperture radar sar imagery as well as noise that may exist amid these two extremes tactful prior elicitation in this model such as the simultaneous assignment of mixture priors for wavelet coefficients and the transformation imparts flexibility and ample insight into the underlying structure the model permits estimation with unknown noise structure for reasonably unimodal and well behaved on the tails distributions wherein it can outperform common shrinkage estimators extensions with multiple transformations and markov random field priors are also considered for adaptation to local variations in contamination modern markov chain monte carlo mcmc bayesian computation has been used for simulations and several examples are reported

robust nonparametric methods the terms nonparametric statistics and distributionfree methods have historically referred to a collection of statistical tests whose null distributions do not depend on the underlying distribution of the data the area has been expanded to include estimates and confidence intervals derived from the tests and includes asymptotically distribution free tests for complex models as well it is now generally recognized that these statistical methods are highly efficient over large sets of possible models and are robust as well this was not always the case the earliest work was by hotelling and pabst in 1936 references to works in this paragraph can be found in the books cited in the next paragraph on rank correlation and by friedman in 1937 on rank tests in a two way design followed in 1945 with the introduction of the signed rank and rank sum tests by wilcoxon in 1947 mann and whitney extended the ideas of wilcoxon all of these rank tests were considered quick and dirty but not competitive with the presumably more efficient t tests this perspective was about to change in 1948 pitman developed efficiency concepts in a set of lecture notes that although never published were widely disseminated during the 1950s and 1960s lehmann working with hodges and with his students at berkeley showed that rank tests are surprisingly efficient and robust in fact the wilcoxon rank tests are essentially efficient at the normal model and can be much more efficient than least squares methods when the underlying distribution of the data has heavy tails in the early 1960s hodges and lehmann derived estimates and confidence intervals from rank test statistics and also introduced aligned rank tests for use in the regression model in 1967 hajek and sid published their seminal book on the theory of rank tests it is impossible to mention all of the contributions to this early development hence we cite several key mono

denoising based on multivariate stochastic volatility modeling of multiwavelet coefficients in this paper signal denoising using multiwavelet based on multivariate garch model is presented where the multivariate garch modeling captures in addition to the correlations among the multiwavelet streams the dependencies among the multiwavelet coefficients after time decimation at each level of multiwavelet decomposition then a maximum a posteriori map estimator based on multivariate garch model is proposed for the purpose of the multiwavelet coefficients this map estimator separates a heteroscedastic signal from a non heteroscedastic noise then in order to demonstrate that heteroscadisticity assumption for real signals is plausible we show analytically that existence of errors in time varying coefficients of the time varying autoregression tvar for natural signal modeling causes conditional heteroscedasticity a statistical validation estimations at each step of this new denoising approach is provided via bootstrapping in experimental results synthetic and real signals are used for comparison of denoising methods

calibration and empirical bayes variable selection for the problem of variable selection for the normal linear model selection criteria such as aic c p bic and ric have fixed dimensionality penalties such criteria are shown to correspond to selection of maximum posterior models under implicit hyperparameter choices for a particular hierarchical bayes formulation based on this calibration we propose empirical bayes selection criteria that use hyperparameter estimates instead of fixed choices for obtaining these estimates both marginal and conditional maximum likelihood methods are considered as opposed to traditional fixed penalty criteria these empirical bayes criteria have dimensionality penalties that depend on the data their performance is seen to approximate adaptively the performance of the best fixed penalty criterion across a variety of orthogonal and nonorthogonal set ups including wavelet regression empirical bayes shrinkage estimators of the selected coefficients are also proposed

finite time analyticity for the two and three dimensional kelvin helmholtz instability the well posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by birkhoff in two dimensions and is shown here to hold both in two and three dimensions incompressible inviscid and irrotational flow with a velocity jump across an interface is assumed in two dimensions global existence of a weak solution to the euler equation with such initial conditions is established in three dimensions a lagrangian representation of the vortex sheet analogous to the birkhoff equation in two dimensions is presented

mixtures of g priors for bayesian variable selection zellner s g prior remains a popular conventional prior for use in bayesian variable selection despite several undesirable consistency issues in this article we study mixtures of g priors as an alternative to default g priors that resolve many of the problems with the original formulation while maintaining the computational tractability that has made the g prior so popular we present theoretical properties of the mixture g priors and provide real and simulated examples to compare the mixture formulation with fixed g priors empirical bayes approaches and other default procedures please see arnold zellner s letter and the author s response

using simulated annealing to optimize the feature selection problem in marketing applications the feature selection also specification problem is concerned with finding the most influential subset of predictors in predictive modeling from a much larger set of potential predictors that can contain hundreds of predictors the problem belongs to the realm of combinatorial optimization where the objective is to find the subset of variables that optimize the value of some goodness of fit function due to the dimensionality of the problem the feature selection problem belongs to the group of np hard problems most of the available predictors are noisy or redundant and add very little if any to the prediction power of the model using all the predictors in the model often results in strong over fitting and very poor predictions constructing a prediction model by checking out all possible subsets is impractical due to computational volume looking on the contribution of each predictor separately is not accurate because it ignores the inter correlations between predictors as a result no analytic solution is available for the feature selection problem requiring that one resorts to heuristics in this paper we employ the simulated annealing sa approach which is one of the leading stochastic search methods for specifying a large scale linear regression model the sa results are compared to the results of the more common stepwise regression swr approach for model specification the models are applied on realistic data sets in database marketing we also use simulated data sets to investigate what data characteristics make the swr approach equivalent to the supposedly more superior sa approach

concentration cancellation for the velocity fields in two dimensional incompressible fluid flows we show that the weak l2 limit of a sequence of solutions of the two dimensional incompressible euler equation is still a solution provided that a strong concentration set for the reduced defect measure has locally finite one dimensional hausdorff measure in space and time

an algorithm for the computation of 2 d eigenvalues in this paper a definition for the eigenvalues of two dimensional 2 d general discrete state space models is proposed this definition is consistent with that known for 2 d roesser models also a programmable algorithm is developed to compute the 2 d eigenvalues

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

a pole assignment problem for 2 dimensional linear discrete systems in kaczorek 1983 the pole assignment problem of a class of 2 dimensional linear discrete systems is solved in this paper an algorithm for a larger class of systems is presented if degrees of freedom permit the most robust feedback gain matrix can be found numerical examples are included

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

rao blackwellization for bayesian variable selection and model averaging in linear and binary regression a novel data augmentation approach choosing the subset of covariates to use in regression or generalized linear models is a ubiquitous problem the bayesian paradigm addresses the problem of model uncertainty by considering models corresponding to all possible subsets of the covariates where the posterior distribution over models is used to select models or combine them via bayesian model averaging bma although conceptually straightforward bma is often difficult to implement in practice since either the number of covariates is too large for enumeration of all subsets calculations cannot be done analytically or both for orthogonal designs with the appropriate choice of prior the posterior probability of any model can be calculated without having to enumerate the entire model space and scales linearly with the number of predictors p in this article we extend this idea to a much broader class of nonorthogonal design matrices we propose a novel method which augments the observed nonorthogonal design by at most p new rows to obtain

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

bayesian variable selection for latent class models in this article we develop a latent class model with class probabilities that depend on subject specific covariates one of our major goals is to identify important predictors of latent classes we consider methodology that allows estimation of latent classes while allowing for variable selection uncertainty we propose a bayesian variable selection approach and implement a stochastic search gibbs sampler for posterior computation to obtain model averaged estimates of quantities of interest such as marginal inclusion probabilities of predictors our methods are illustrated through simulation studies and application to data on weight gain during pregnancy where it is of interest to identify important predictors of latent weight gain classes

minimax multiple shrinkage estimation on presente des nouveaux estimateurs qui incorporent l information partielle a priori en utilisant les donnees comme emulation du comportement et du risque des estimateurs consideres les plus efficaces

global vortex sheet solutions of euler equations in the plane abstract for any curve close enough to a straight line there is a global solution of euler equations having for all positive time a vortex sheet and admitting the given curve as the initial sheet

on scale mixtures of normal distributions summary the exponential power family of distributions of box tiao 1973 is shown to be a subset of the class of scale mixtures of normals the corresponding mixing distributions are explicitly obtained identifying a close relationship between the exponential power family and a further class of normal scale mixtures namely the stable distributions

a comparative evaluation of wavelet based methods for hypothesis testing of brain activation maps wavelet based methods for hypothesis testing are described and their potential for activation mapping of human functional magnetic resonance imaging fmri data is investigated in this approach we emphasise convergence between methods of wavelet thresholding or shrinkage and the problem of hypothesis testing in both classical and bayesian contexts specifically our interest will be focused on the trade off between type i probability error control and power dissipation estimated by the area under the roc curve we describe a technique for controlling the false discovery rate at an arbitrary level of error in testing multiple wavelet coefficients generated by a 2d discrete wavelet transform dwt of spatial maps of fmri time series statistics we also describe and apply change point detection with recursive hypothesis testing methods that can be used to define a threshold unique to each level and orientation of the 2d dwt and bayesian methods incorporating a formal model for the anticipated sparseness of wavelet coefficients representing the signal or true image the sensitivity and type i error control of these algorithms are comparatively evaluated by analysis of null images acquired with the subject at rest and an experimental data set acquired from five normal volunteers during an event related finger movement task we show that all three wavelet based algorithms have good type i error control the fdr method being most conservative and generate plausible brain activation maps the bayesian method being most powerful we also generalise the formal connection between wavelet based methods for simultaneous multiresolution denoising hypothesis testing and methods based on monoresolution gaussian smoothing followed by statistical testing of brain activation maps

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

posterior probability intervals for wavelet thresholding summary we use cumulants to derive bayesian credible intervals for wavelet regression estimates the first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions these powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale hence a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set johnson transformations then yield the credible intervals themselves simulations show that these intervals have good coverage rates even when the underlying function is inhomogeneous where standard methods fail in the case where the curve is smooth the performance of our intervals remains competitive with established nonparametric regression methods

assessing robustness of intrinsic tests of independence in two way contingency tables for testing nested hypotheses from a bayesian standpoint a desirable condition is that the prior for the alternative model concentrates mass around the smaller or null model for testing independence in contingency tables the intrinsic priors satisfy this requirement furthermore the degree of concentration of the priors is controlled by a discrete parameter t the training sample size which plays an important role in the resulting answer in this article we report on the robustness of the tests of independence for small or moderate sample sizes in contingency tables with respect to intrinsic priors with different degrees of concentration around the null we compare these tests to frequentist tests and other robust bayes tests for large sample sizes robustness is achieved because the intrinsic bayesian tests are consistent examples using real and simulated data are given supplemental materials technical details and data sets are available online

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

optimal predictive model selection often the goal of model selection is to choose a model for future prediction and it is natural to measure the accuracy of a future prediction by squared error loss under the bayesian approach it is commonly perceived that the optimal predictive model is the model with highest posterior probability but this is not necessarily the case in this paper we show that for selection among normal linear models the optimal predictive model is often the median probability model which is defined as the model consisting of those variables which have overall posterior probability greater than or equal to 1 2 of being in a model the median probability model often differs from the highest probability model

pole assignment of two dimensional systems for separable characteristic equations abstract this paper considers the problem of pole assignment of linear time invariant multivariable two dimensional systems the desirable closed loop characteristic function is considered to be separable an algorithm is presented for the solution of the problem using state feedback the necessary condition for the problem to have a solution is established and an illustrative example is given

the focused information criterion a variety of model selection criteria have been developed of general and specific types most of these aim at selecting a single model with good overall properties for example formulated via average prediction quality or shortest estimated overall distance to the true model the akaike the bayesian and the deviance information criteria along with many suitable variations are examples of such methods these methods are not concerned however with the actual use of the selected model which varies with context and application the present article takes the view that the model selector should instead focus on the parameter singled out for interest in particular a model that gives good precision for one estimand may be worse when used for inference for another estimand we develop a method that for a given focus parameter estimates the precision of any submodel based estimator the framework is that of large sample likelihood inference using an unbiased estimate of limiting risk we propose a foc

bayesian curve classification using wavelets we propose classification models for binary and multicategory data where the predictor is a random function we use bayesian modeling with wavelet basis functions that have nice approximation properties over a large class of functional spaces and can accommodate a wide variety of functional forms observed in real life applications we develop an unified hierarchical model to encompass both the adaptive wavelet based function estimation model and the logistic classification model we couple together these two models are to borrow strengths from each other in a unified hierarchical framework the use of gibbs sampling with conjugate priors for posterior inference makes the method computationally feasible we compare the performance of the proposed model with other classification methods such as the existing naive plug in methods by analyzing simulated and real data sets

analytical form for a bayesian wavelet estimator of images using the bessel k form densities a novel bayesian nonparametric estimator in the wavelet domain is presented in this approach a prior model is imposed on the wavelet coefficients designed to capture the sparseness of the wavelet expansion seeking probability models for the marginal densities of the wavelet coefficients the new family of bessel k forms bkf densities are shown to fit very well to the observed histograms exploiting this prior we designed a bayesian nonlinear denoiser and we derived a closed form for its expression we then compared it to other priors that have been introduced in the literature such as the generalized gaussian density ggd or the spl alpha stable models where no analytical form is available for the corresponding bayesian denoisers specifically the bkf model turns out to be a good compromise between these two extreme cases hyperbolic tails for the spl alpha stable and exponential tails for the ggd moreover we demonstrate a high degree of match between observed and estimated prior densities using the bkf model finally a comparative study is carried out to show the effectiveness of our denoiser which clearly outperforms the classical shrinkage or thresholding wavelet based techniques