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Showing papers in "Journal of the royal statistical society series b-methodological in 1995"


Journal ArticleDOI
TL;DR: In this paper, a different approach to problems of multiple significance testing is presented, which calls for controlling the expected proportion of falsely rejected hypotheses -the false discovery rate, which is equivalent to the FWER when all hypotheses are true but is smaller otherwise.
Abstract: SUMMARY The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses -the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Therefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferronitype procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples.

83,420 citations


Journal ArticleDOI
TL;DR: 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 and draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves.
Abstract: 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

1,639 citations


Journal ArticleDOI
TL;DR: In this article, a Bayesian approach to estimating structural uncertainty about unknown quantities is presented, which can be applied to forecasting the price of oil and the chance of catastrophic failure of the US space shuttle.
Abstract: In most examples of inference and prediction, the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model M that formalizes assumptions about how x and y are related. M will typically have two parts: structural assumptions S, such as the form of the link function and the choice of error distribution in a generalized linear model, and parameters θ whose meaning is specific to a given choice of S. It is common in statistical theory and practice to acknowledge parametric uncertainty about θ given a particular assumed structure S; it is less common to acknowledge structural uncertainty about S itself. A widely used approach involves enlisting the aid of x to specify a plausible single «best» choice S* for S, and then proceeding as if S* were known to be correct. In general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x. When miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y, leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty. In this paper I discuss a Bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible, by virtue of recent computational advances, and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the US space shuttle

1,459 citations


Journal ArticleDOI
TL;DR: This paper presents a framework for Bayesian model choice, along with an MCMC algorithm that does not suffer from convergence difficulties, and applies equally well to problems where only one model is contemplated but its proper size is not known at the outset.
Abstract: SUMMARY Markov chain Monte Carlo (MCMC) integration methods enable the fitting of models of virtually unlimited complexity, and as such have revolutionized the practice of Bayesian data analysis. However, comparison across models may not proceed in a completely analogous fashion, owing to violations of the conditions sufficient to ensure convergence of the Markov chain. In this paper we present a framework for Bayesian model choice, along with an MCMC algorithm that does not suffer from convergence difficulties. Our algorithm applies equally well to problems where only one model is contemplated but its proper size is not known at the outset, such as problems involving integer-valued parameters, multiple changepoints or finite mixture distributions. We illustrate our approach with two published examples.

985 citations


Journal ArticleDOI
TL;DR: In this paper, a new variant of the partial Bayes factor, the fractional Bayes Factor (FBPF), is proposed to deal with weak prior information for model comparison.
Abstract: Bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves. However, there are difficulties with this approach when prior information about the parameters of the various models is weak. Partial Bayes factors offer a resoIution of the problem by setting aside part of the data as a training sampIe. The training sampIe is used to obtain an initiaI informative posterior distribution of the parameters in each model. Model comparison is then based on a Bayes factor calculated from the remaining data. Properties of partial Bayes factors are discussed, particularly in the context of weak prior information, and they are found to have advantages over other proposed methods of model comparison. A new variant of the partial Bayes factor, the fractional Bayes factor, is advocated on grounds of consistency, simplicity, robustness and coherence

693 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a data-driven bandwidth selection procedure, which can be used to select both constant and variable bandwidths, based on a residual squares criterion along with a good approximation of the bias and variance of the estimator.
Abstract: When estimating a mean regression function and its derivatives, locally weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimating curves with a complicated structure, a variable bandwidth is desirable. Furthermore, the bandwidth should be indicated by the data themselves. Recent developments in nonparametric smoothing techniques inspired us to propose such a data-driven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. The idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator. The procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives. The resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves. This is illustrated via a large variety of testing examples, including examples with a large spatial variability. The results are also compared with wavelet thresholding techniques, and it seems that our results are at least comparable, i.e. local polynomial regression using our data-driven variable bandwidth has spatial adaptation properties that are similar to wavelets.

577 citations


Journal ArticleDOI
TL;DR: A unified theory of sequential changepoint detection is introduced which leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on-line implementation and yet are nearly optimal under several performance criteria.
Abstract: After a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing, we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection. This theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on-line implementation and yet are nearly optimal under several performance criteria.

563 citations


Journal ArticleDOI
TL;DR: This EM gradient algorithm approximately solves the M-step of the EM algorithm by one iteration of Newton's method, and the proof of global convergence applies and improves existing theory for the EM algorithms.
Abstract: In many problems of maximum likelihood estimation, it is impossible to carry out either the E-step or the M-step of the EM algorithm. The present paper introduces a gradient algorithm that is closely related to the EM algorithm. This EM gradient algorithm approximately solves the M-step of the EM algorithm by one iteration of Newton's method. Since Newton's method converges quickly, the local properties of the EM gradient algorithm are almost identical with those of the EM algorithm. Any strict local maximum point of the observed likelihood locally attracts the EM and EM gradient algorithm at the same rate of convergence, and near the maximum point the EM gradient algorithm always produces an increase in the likelihood. With proper modification the EM gradient algorithm also exhibits global convergence properties that are similar to those of the EM algorithm. Our proof of global convergence applies and improves existing theory for the EM algorithm. These theoretical points are reinforced by a discussion of three realistic examples illustrating how the EM gradient algorithm can succeed where the EM algorithm is intractable

371 citations


Journal ArticleDOI
TL;DR: In this article, a predictive Bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model, and using criteria derived from a certain predictive density and a prior specification that emphasizes the observables, they implement the proposed methodology for three common problems arising in normal linear models: variable subset selection, selection of a transformation of predictor variables and estimation of a parametric variance function.
Abstract: We consider the problem of selecting one model from a large class of plausible models. A predictive Bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model. Using criteria derived from a certain predictive density and a prior specification that emphasizes the observables, we implement the proposed methodology for three common problems arising in normal linear models: variable subset selection, selection of a transformation of predictor variables and estimation of a parametric variance function. Interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria. Relationships between the proposed criteria and other well-known criteria are examined

337 citations


Journal ArticleDOI
TL;DR: In this article, a unified approach to fitting traditional Box-Jenkins ARIMA processes as well as stationary and non-stationary fractional ARIMAs is presented, and a simple algorithm for calculating the estimate of d and the ARMA parameters is given.
Abstract: In practical applications of Box-Jenkins autoregressive integrated moving average (ARIMA) models, the number of times that the observed time series must be differenced to achieve approximate stationarity is usually determined by careful, but mostly informal, analysis of the differenced series. For many time series, some differencing seems appropriate, but taking the first or the second difference may be too strong. As an alternative, Hosking, and Granger and Joyeux proposed the use of fractional differences. For -1/2 -1/2 can be estimated by an approximate maximum likelihood method. We thus obtain a unified approach to fitting traditional Box-Jenkins ARIMA processes as well as stationary and non-stationary fractional ARIMA processes. A confidence interval for d can be given. Tests, such as for unit roots in the autoregressive parameter or for stationarity, follow immediately. The resulting confidence intervals for the ARMA parameters take into account the additional uncertainty due to estimation of d. A simple algorithm for calculating the estimate of d and the ARMA parameters is given. Simulations and two data examples illustrate the results.

303 citations


Journal ArticleDOI
TL;DR: In this article, the multivariate logistic transform introduced by McCullagh and Nelder can be used to define a class of regression models that is, in many applications, particularly suitable for relating the joint distribution of the responses to predictors.
Abstract: SUMMARY When data composed of several categorical responses together with categorical or continuous predictors are observed, the multivariate logistic transform introduced by McCullagh and Nelder can be used to define a class of regression models that is, in many applications, particularly suitable for relating the joint distribution of the responses to predictors. In this paper we give a general definition of this class of models and study their properties. A computational scheme for performing maximum likelihood estimation for data sets of moderate size is described and a system of model formulae that succinctly define particular models is introduced. Applications of these models to longitudinal problems are illustrated by numerical examples.

Journal ArticleDOI
TL;DR: In this article, it was shown that the usual Laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter n.
Abstract: SUMMARY It is shown that the usual Laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter n. The formal Laplace expansion for multidimensional integrals is given and used to construct asymptotic approximations for high dimensional integrals. One example is considered in which the dimension of the integral is O(n 1/2) and the relative error of the unmodified Laplace approximation is 0(1). Nevertheless, it is possible to construct a valid asymptotic expansion by regrouping terms in the formal expansion according to asymptotic order in n.

Journal ArticleDOI
TL;DR: In this article, the Gibbs sampling scheme converges geometrically in terms of Pearson χ 2 -distance for both systematic and random scans under conditions that guarantee the compactness of the Markov forward operator and irreducibility of the corresponding chain.
Abstract: This paper presents results on covariance structure and convergence for the Gibbs sampler with both systematic and random scans. It is shown that, under conditions that guarantee the compactness of the Markov forward operator and irreducibility of the corresponding chain, the Gibbs sampling scheme converges geometrically in terms of Pearson χ 2 -distance. In particular, for the random scan, the autocovariance can be expressed as variances of iterative conditional expectations. As a consequence, the autocorrelations are all positive and decrease monotonically

Journal ArticleDOI
TL;DR: In this paper, both likelihood-based and non-likelihood (generalized estimating equations) regression models for longitudinal binary responses when there are drop-outs are reviewed. But the performance of the methods is compared, in terms of asymptotic bias, under mis-specification of the association between the responses and the missing data mechanism or drop-out process.
Abstract: This paper reviews both likelihood-based and non-likelihood (generalized estimating equations) regression models for longitudinal binary responses when there are drop-outs. Throughout, it is assumed that the regression parameters for the marginal expectations of the binary responses are of primary scientific interest. The association or time dependence between the responses is largely regarded as a nuisance characteristic of the data. The performance of the methods is compared, in terms of asymptotic bias, under mis-specification of the association between the responses and the missing data mechanism or drop-out process.

Journal ArticleDOI
TL;DR: In this article, a general method of adjusting any conveniently defined initial estimates to result in estimates which are asymptotically unbiased and consistent is proposed, motivated by iterative bias correction and can be applied to any parametric model.
Abstract: SUMMARY Obtaining estimates that are nearly unbiased has proven to be difficult when random effects are incorporated into a generalized linear model. In this paper, we propose a general method of adjusting any conveniently defined initial estimates to result in estimates which are asymptotically unbiased and consistent. The method is motivated by iterative bias correction and can be applied in principle to any parametric model. A simulation-based approach of implementing the method is described and the relationship of the method proposed with other sampling-based methods is discussed. Results from a small scale simulation study show that the method proposed can lead to estimates which are nearly unbiased even for the variance components while the standard errors are only slightly inflated. A new analysis of the famous salamander mating data is described which reveals previously undetected between-animal variation among the male salamanders and results in better prediction of mating outcomes.

Journal ArticleDOI
TL;DR: In this article, the ideas of functional principal component analysis are extended to deal with data that are hybrids of 'functional' and 'parametric' effects, and several possible ways of treating the estimated parameter values are discussed.
Abstract: SUMMARY The ideas of functional principal component analysis are extended to deal with data that are hybrids of 'functional' and 'parametric' effects. The parametric effects may be more general than just the addition of a multiple of a given function. A detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval, and some remarks are made for the extension to a far more general case. Given data, a Procrustes fitting method can be used to estimate the parametric effects. Several possible ways of treating the estimated parameter values are discussed. The methods are illustrated by reference to temperature data at 35 Canadian weather-stations.

Journal ArticleDOI
TL;DR: In this paper, the authors reviewed and discussed four non-regular estimation problems and compared modified likelihood and spacings methods with the Box-Cox shifted power transform (BCPT).
Abstract: Four non-regular estimation problems are reviewed and discussed. One (the unbounded likelihood problem) involves distributions with infinite spikes, for which maximum likelihood can fail to give consistent estimators. A comparison is made with modified likelihood and spacings methods which do give efficient estimators in this case. An application to the Box-Cox shifted power transform is given. The other three problems occur when the true parameter lies in some special subregion. In one (the constrained parameter problem) the subregion is a boundary. The other two (the embedded model and the indeterminate parameters problems) occur when the model takes on a special form in the subregion. These last two problems have previously been investigated separately. We show that they are equivalent in some situations. Both often arise in non-linear models and we give a directed graph approach which allows for their occurrence in nested model building. It is argued that many non-regular problems can be handled systematically without having to resort to elaborate technical assumptions. Relatively uncomplicated methods may be used provided that the underlying nature of the non-regularity is understood

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a new class of estimators that remains consistent and asymptotically normal even when the probability that X is missing depends on the observed V and Y.
Abstract: SUMMARY Pepe and Fleming, and Carroll and Wand have recently proposed estimators in a parametric model for the density of a random variable Y conditional on a vector of covariates (X, V) when data on one of the regressors X is missing for some study subjects. We propose a new class of estimators that remains consistent and asymptotically normal even when the probability that X is missing depends on the observed V and Y, includes an estimator whose asymptotic variance attains the semiparametric variance bound for the model and, when the data are missing completely at random, includes an estimator that is asymptotically equivalent to the inefficient estimators proposed by Pepe and Fleming and by Carroll and Wand. The optimal estimator in our class depends on the unknown probability law generating the data. When the vector V of non-missing regressors has at most two continuous components, we propose an adaptive semiparametric efficient estimator and compare the performance of the proposed semiparametric efficient estimator with the estimators proposed by Pepe and Fleming and Carroll and Wand in a small simulation study. When V has many continuous components, we propose an alternative class of adaptive estimators that should have high efficiency.

Journal ArticleDOI
TL;DR: In this article, a new method to identify influential subsets in linear regression problems is presented, which uses the eigenstructure of an influence matrix, defined as the matrix of uncentred covariances of the effect on the whole data set of deleting each observation, normalized to include the univariate Cook statistics on the diagonal.
Abstract: This paper presents a new method to identify influential subsets in linear regression problems. The procedure uses the eigenstructure of an influence matrix which is defined as the matrix of uncentred covariances of the effect on the whole data set of deleting each observation, normalized to include the univariate Cook statistics on the diagonal. It is shown that the eigenstructure of the influence matrix is useful to identify influential subsets and a procedure for detecting influential sets is proposed. The method is illustrated with two examples

Journal ArticleDOI
TL;DR: It is shown how to discriminate between different linear Gaussian state space models for a given time series by means of a Bayesian approach which chooses the model that minimizes the expected loss.
Abstract: SUMMARY It is shown how to discriminate between different linear Gaussian state space models for a given time series by means of a Bayesian approach which chooses the model that minimizes the expected loss. A practical implementation of this procedure requires a fully Bayesian analysis for both the state vector and the unknown hyperparameters and is carried out by Markov chain Monte Carlo methods. An application to some non-standard situations such as testing hypotheses on the boundary of the parameter space, discriminating non-nested models and discrimination of more than two models is discussed in detail.

Journal ArticleDOI
TL;DR: The paper is concerned with masking of Cook's distance, the popular deletion measure of influence of individual data cases in linear regression, and the noteworthy effects possible in joint influence are described as reducing, enhancing and swamping.
Abstract: By A. J. LAWRANCEt University of Birmingham, UK [Received May 1991. Final revision October 1993] SUMMARY The paper is concerned with masking of Cook's distance, the popular deletion measure of influence of individual data cases in linear regression. Masking is broadly concerned with the limitations imposed by the use of individual cases. Two specific views are drawn from quotations: the first is associated with the established idea of joint influence, i.e. the deletion of two or more cases simultaneously, and the second is conditional influence, i.e. the calculation of Cook's distance before and after the deletion of other cases. Attention is mainly focused on pairs of already identified cases. Intuitive understanding is obtained via explicit general forms for joint and conditional Cook's distance influence with pairs of cases; especially tractable results are obtained for replicate and opposite explanatory pairs, and for multiples of individual cases of opposite pairs. The noteworthy effects possible in joint influence are described as reducing, enhancing and swamping; in conditional influence, interesting effects are described as masking and boosting. Thus only in connection with one effect in conditional influence is the term masking used. The work is illustrated on one constructed and one reported data set. Keywords: BOOSTING; CONDITIONAL INFLUENCE; COOK'S DISTANCE; DIAGNOSTICS; ENHANCING; INFLUENCE; JOINT INFLUENCE; LEVERAGE; LINEAR REGRESSION; MASKING; OPPOSITE PAIRS; OUTLIERS; REDUCING; REPLICATE PAIRS; SWAMPING 1. INTRODUCTION

Journal ArticleDOI
TL;DR: In this paper, a robust method for fitting a model from a parametric family is presented, which reduces the influence of information that is not compatible with the model family, while maintaining the basic structure of a familiar model fitting process.
Abstract: A robust method for fitting a model from a parametric family is fundamental to effective statistical analysis. A procedure is given for robustifying any model fitting process by using weights from the family from which the model is to be chosen. The weighting reduces the influence of information that is not compatible with the model family, while maintaining the basic structure of a familiar model fitting process. The procedure produces a parametric family of model fitting functions. The value of the parameter determines the degree to which the weighting influences the robustified model fit. A mechanism for determining an appropriate value for the parameter is described.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed two asymptotically normal statistics to test whether the frequency of a given word is concordant with the first-order Markov chain model or not.
Abstract: Considering a Markov chain model for deoxyribonucleic acid sequences, this paper proposes two asymptotically normal statistics to test whether the frequency of a given word is concordant with the first-order Markov chain model or not The problem is to choose estimates μ(W) of the expectation of the frequency M W of a word W in the observed sequence such that the asymptotic variance of M W −μ(W) is easily computable The first estimator is derived from the frequency of W [−1] , which is W with its last letter deleted The second, following an idea of Cowan, is the conditional expectation M W given the observed frequencies of all two-letter words Two examples on phage lambda and phage T7 are shown

Journal ArticleDOI
TL;DR: In this paper, a general method for simulating infinitely divisible random variates when their Fourier or Laplace transforms are available in the Levy representation is proposed and proved by proving a convergence theorem that is a sampling form of a classical theorem in probability.
Abstract: SUMMARY Stochastic processes with independent increments play a central role in Bayesian nonparametric inference. The distributions of the increments of these processes, aside from fixed points of discontinuity, are infinitely divisible and their Laplace and/or Fourier transforms in the Levy representation are usually known. Conventional Bayesian inference in this context has been limited largely to providing point estimates of the random quantities of interest, although Markov chain Monte Carlo methods have been used to obtain a fuller analysis in the context of Dirichlet process priors. In this paper, we propose and implement a general method for simulating infinitely divisible random variates when their Fourier or Laplace transforms are available in the Levy representation. Theoretical justification is established by proving a convergence theorem that is a 'sampling form' of a classical theorem in probability. The results provide a method for implementing Bayesian nonparametric inference by using a wide range of stochastic processes as priors.

Journal ArticleDOI
TL;DR: In this article, the authors discuss the collapsibility of logistic regression coefficients over a background variable and present necessary and sufficient conditions for such a condition to be true for logistic regressions.
Abstract: SUMMARY In this paper we discuss collapsibility of logistic regression coefficients over a background variable and present necessary and sufficient conditions for collapsibility.

Journal ArticleDOI
TL;DR: In this paper, a local likelihood function is proposed to give more weight to observations near a region of interest in the sample space, which can be used for assessing local departures from a parametric model, and for semiparametric density estimation.
Abstract: By drawing an analogy with likelihood for censored data, a local likelihood function is proposed which gives more weight to observations near a region of interest in the sample space. Resulting methods can be used for assessing local departures from a parametric model, and for semiparametric density estimation. Some theory, and three examples, is given

Journal ArticleDOI
TL;DR: A geometric framework for constructing optimal Bayesian designs and maximin designs for non-linear models with a single unknown parameter and a prior distribution on that parameter, which is restricted in that it comprises exactly two points of support, is presented in this article.
Abstract: SUMMARY A geometric framework for constructing optimal Bayesian designs and maximin designs for non-linear models with a single unknown parameter and a prior distribution on that parameter, which is restricted in that it comprises exactly two points of support, is presented. The approach is illustrated by means of selected examples involving logistic regression and the simple exponential model, and its applicability to the construction of optimal designs for models with uncontrolled variation and to model robust designs is also demonstrated. In addition, the method is shown to provide some valuable insights into the general properties of optimal Bayesian designs for non-linear models.

Journal ArticleDOI
TL;DR: In this article, the authors define a response surface bandit as the sequential design problem that maximizes an expected bandit utility but where the outcomes y n are continuous and can be related through a responsesurface to a set of controllable variables x n = (x 1n, x 2n,..., x kn ).
Abstract: In this paper we define a response surface bandit as the sequential design problem that maximizes an expected bandit utility but where the outcomes y n are continuous and can be related through a response surface to a set of controllable variables x n = (x 1n , x 2n ,..., x kn ). We link this problem to other traditional optimization problems from industrial engineering and to the traditional bandit problem. We consider two approaches to the problem. The first is based on a myopic sequential design. The second approach uses the best design out of a family of designs related to upper bounds for the predicted surface; the family includes myopic and sequential versions of D-optimal designs. These approaches can be generalized to more broadly defined sequential problems.

Journal ArticleDOI
TL;DR: A new algorithm for the approximation of the maximum a posteriori (MAP) restoration of noisy images, considered in a Bayesian setting, which runs in polynomial time and is based on the coding of the colours.
Abstract: We propose a new algorithm for the approximation of the maximum a posteriori (MAP) restoration of noisy images. The image restoration problem is considered in a Bayesian setting. We assume as prior distribution multicolour Markov random fields on a graph whose main restriction is the presence of only pairwise site interactions. The noise is modelled as a Bernoulli field. Computing the mode of the posterior distribution is NP complete, i.e. can (very likely) be done only in a time exponential in the number of sites of the underlying graph. Our algorithm runs in polynomial time and is based on the coding of the colours. It produces an image with the following property: either a pixel is coloured with one of the possible colours or it is left blank. In the first case we prove that this is the colour of the site in the exact MAP restoration. The quality of the approximation is then measured by the number of sites being left blank. We assess the performance of the new algorithm by numerical experiments on the simple three-colour Potts model. More rigorously, we present a probabilistic analysis of the algorithm. The results indicate that the approximation is quite often sufficiently good for the interpretation of the image.

Journal ArticleDOI
TL;DR: This paper proposes a selector based on an iterative plug-in approach for bivariate kernel regression that is shown to give satisfactory results and can be quickly computed.
Abstract: SUMMARY For two and higher dimensional kernel regression, currently available bandwidth selection procedures are based on cross-validation or related penalizing ideas. However, these techniques have been shown to suffer from high sample variability and, in addition, can sometimes be difficult to implement when a vector of bandwidths needs to be selected. In this paper we propose a selector based on an iterative plug-in approach for bivariate kernel regression. It is shown to give satisfactory results and can be quickly computed. Our ideas can be extended to higher dimensions.