covariate information matrix for sufficient dimension reduction building upon recent research on the applications of the density information matrix we develop a tool for sufficient dimension reduction sdr in regression problems called covariate information m

sufficient dimension reduction via squared loss mutual information estimation the goal of sufficient dimension reduction in supervised learning is to find the low dimensional subspace of input features that contains all of the information about the output values that the input features possess in this letter we propose a novel sufficient dimension reduction method using a squared loss variant of mutual information as a dependency measure we apply a density ratio estimator for approximating squared loss mutual information that is formulated as a minimum contrast estimator on parametric or nonparametric models since cross validation is available for choosing an appropriate model our method does not require any prespecified structure on the underlying distributions we elucidate the asymptotic bias of our estimator on parametric models and the asymptotic convergence rate on nonparametric models the convergence analysis utilizes the uniform tail bound of a u process and the convergence rate is characterized by the bracketing entropy of the model we then develop a natural gradient algorithm on the grassmann manifold for sufficient subspace search the analytic formula of our estimator allows us to compute the gradient efficiently numerical experiments show that the proposed method compares favorably with existing dimension reduction approaches on artificial and benchmark data sets

a level set approach to 3d reconstruction from range data this paper presents a method that uses the level sets of volumes to reconstruct the shapes of 3d objects from range data the strategy is to formulate 3d reconstruction as a statistical problem find that surface which is mostly likely given the data and some prior knowledge about the application domain the resulting optimization problem is solved by an incremental process of deformation we represent a deformable surface as the level set of a discretely sampled scalar function of three dimensions i e a volume such level set models have been shown to mimic conventional deformable surface models by encoding surface movements as changes in the greyscale values of the volume the result is a voxel based modeling technology that offers several advantages over conventional parametric models including flexible topology no need for reparameterization concise descriptions of differential structure and a natural scale space for hierarchical representations this paper builds on previous work in both 3d reconstruction and level set modeling it presents a fundamental result in surface estimation from range data an analytical characterization of the surface that maximizes the posterior probability it also presents a novel computational technique for level set modeling called the sparse field algorithm which combines the advantages of a level set approach with the computational efficiency and accuracy of a parametric representation the sparse field algorithm is more efficient than other approaches and because it assigns the level set to a specific set of grid points it positions the level set model more accurately than the grid itself these properties computational efficiency and subcell accuracy are essential when trying to reconstruct the shapes of 3d objects results are shown for the reconstruction objects from sets of noisy and overlapping range maps

contour regression a general approach to dimension reduction we propose a novel approach to sufficient dimension reduction in regression based on estimating contour directions of small variation in the response these directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response leading to simple and general contour regression scr and gcr methodology in comparison with existing sufficient dimension reduction techniques this contour based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions while maintaining sqrt n consistency and computational ease moreover it proves robust to departures from ellipticity we establish population properties for both scr and gcr and asymptotic properties for scr simulations to compare performance with that of standard techniques such as ordinary least squares sliced inverse regression principal hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses we demonstrate the use of contour based methods on a data set concerning soil evaporation

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

svm decision boundary based discriminative subspace induction we study the problem of linear dimension reduction for classification with a focus on sufficient dimension reduction i e finding subspaces without loss of discrimination power first we formulate the concept of sufficient subspace for classification in parallel terms as for regression then we present a new method to estimate the smallest sufficient subspace based on an improvement of decision boundary analysis dba the main idea is to combine dba with support vector machines svm to overcome the inherent difficulty of dba in small sample size situations while keeping dba s estimation simplicity the compact representation of svm boundary results in a significant gain in both speed and accuracy over previous dba implementations alternatively this technique can be viewed as a way to reduce the run time complexity of svm itself comparative experiments on one simulated and four real world benchmark datasets highlight the superior performance of the proposed approach

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

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

efficient and adaptive linear regression in semi supervised settings we consider the linear regression problem under semi supervised settings wherein the available data typically consists of i a small or moderate sized labeled data and ii a much larger sized unlabeled data such data arises naturally from settings where the outcome unlike the covariates is expensive to obtain a frequent scenario in modern studies involving large databases like electronic medical records emr supervised estimators like the ordinary least squares ols estimator utilize only the labeled data it is often of interest to investigate if and when the unlabeled data can be exploited to improve estimation of the regression parameter in the adopted linear model in this paper we propose a class of efficient and adaptive semi supervised estimators ease to improve estimation efficiency the ease are two step estimators adaptive to model mis specification leading to improved optimal in some cases efficiency under model mis specification and equal optimal efficiency under a linear model this adaptive property often unaddressed in the existing literature is crucial for advocating safe use of the unlabeled data the construction of ease primarily involves a flexible semi nonparametric imputation including a smoothing step that works well even when the number of covariates is not small and a follow up refitting step along with a cross validation cv strategy both of which have useful practical as well as theoretical implications towards addressing two important issues under smoothing and over fitting we establish asymptotic results including consistency asymptotic normality and the adaptive properties of ease we also provide influence function expansions and a double cv strategy for inference the results are further validated through extensive simulations followed by application to an emr study on auto immunity

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

on 2d discrete schrodinger operators associated with multiple orthogonal polynomials a class of cross shaped difference operators on a two dimensional 2d lattice is introduced the main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials in other words this scheme generalizes the classical connection between jacobi matrices and orthogonal polynomials to the case of operators on lattices furthermore we also show how to obtain 2d discrete schrodinger operators out of this construction and give a number of explicit examples based on known families of multiple orthogonal polynomials

testing predictor contributions in sufficient dimension reduction we develop tests of the hypothesis of no effect for selected predictors in regression without assuming a model for the conditional distribution of the response given the predictors predictor effects need not be limited to the mean function and smoothing is not required the general approach is based on sufficient dimension reduction the idea being to replace the predictor vector with a lower dimensional version without loss of information on the regression methodology using sliced inverse regression is developed in detail

a note on shrinkage sliced inverse regression summary we employ lasso shrinkage within the context of sufficient dimension reduction to obtain a shrinkage sliced inverse regression estimator which provides easier interpretations and better pre diction accuracy without assuming a parametric model the shrinkage sliced inverse regression approach can be employed for both single index and multiple index models simulation studies suggest that the new estimator performs well when its tuning parameter is selected by either the bayesian information criterion or the residual information criterion

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

smart nonlinear diffusion a probabilistic approach in this paper a stochastic interpretation of nonlinear diffusion equations used for image filtering is proposed this is achieved by relating the problem of evolving smoothing images to that of tracking the transition probability density functions of an underlying random process we show that such an interpretation of e g perona malik equation in turn allows additional insight and sufficient flexibility to further investigate some outstanding problems of nonlinear diffusion techniques in particular upon unraveling the limitations as well as the advantages of such an equation we are able to propose a new approach which is demonstrated to improve performance over existing approaches and more importantly to lift the longstanding problem of a stopping criterion for a nonlinear evolution equation with no data term constraint substantiating examples in image enhancement and segmentation are provided

mean field annealing a formalism for constructing gnc like algorithms optimization problems are approached using mean field annealing mfa which is a deterministic approximation using mean field theory and based on peierls s inequality to simulated annealing the mfa mathematics are applied to three different objective function examples in each case mfa produces a minimization algorithm that is a type of graduated nonconvexity when applied to the weak membrane objective mfa results in an algorithm qualitatively identical to the published gnc algorithm one of the examples mfa applied to a piecewise constant objective function is then compared experimentally with the corresponding gnc weak membrane algorithm the mathematics of mfa are shown to provide a powerful and general tool for deriving optimization algorithms

dimension reduction for the conditional mean in regressions with categorical predictors consider the regression of a response y on a vector of quantitative predictors x and a categorical predictor w in this article we describe a first method for reducing the dimension of x without loss of information on the conditional mean mathrm e y x w and without requiring a prespecified parametric model the method which allows for but does not require parametric versions of the subpopulation mean functions mathrm e y x w w includes a procedure for inference about the dimension of x after reduction this work integrates previous studies on dimension reduction for the conditional mean mathrm e y x in the absence of categorical predictors and dimension reduction for the full conditional distribution of y x w the methodology we describe may be particularly useful for constructing low dimensional summary plots to aid in model building at the outset of an analysis our proposals provide an often parsimonious alternative to the standard technique of modeling with interaction terms to adapt a mean function for different subpopulations determined by the levels of w examples illustrating this and other aspects of the development are presented

constrained inverse regression for incorporating prior information inverse regression methods facilitate dimension reduction analyses of high dimensional data by extracting a small number of factors that are linear combinations of the original predictor variables but the estimated factors may not lend themselves readily to interpretation consistent with prior information our approach to solving this problem is to first incorporate prior information via theory or data driven constraints on model parameters and then apply the proposed method constrained inverse regression cir to extract factors that satisfy the constraints we provide chi squared and t tests to assess the significance of each factor and its estimated coefficients and we also generalize cir to other inverse regression methods in situations where both dimension reduction and factor interpretation are important finally we investigate cir s small sample performance test data driven constraints and present a marketing example to illustrate its use in discovering meaningful factors that influence th

identifying regression outliers and mixtures graphically regressions in practice can include outliers and other unknown subpopulation structures for example mixtures of regressions occur if there is an omitted categorical predictor like gender or location and different regressions occur within each category the theory of regression graphics based on central subspaces can be used to construct graphical solutions to long standing problems of this type it is argued that in practice the central subspace automatically expands to incorporate outliers and regression mixtures thus methods of estimating the central subspace can be used to identify these phenomena without specifying a model examples illustrating the power of the theory are presented

a general theory for nonlinear sufficient dimension reduction formulation and estimation in this paper we introduce a general theory for nonlinear sufficient dimension reduction and explore its ramifications and scope this theory subsumes recent work employing reproducing kernel hilbert spaces and reveals many parallels between linear and nonlinear sufficient dimension reduction using these parallels we analyze the properties of existing methods and develop new ones we begin by characterizing dimension reduction at the general level of sigma fields and proceed to that of classes of functions leading to the notions of sufficient complete and central dimension reduction classes we show that when it exists the complete and sufficient class coincides with the central class and can be unbiasedly and exhaustively estimated by a generalized sliced inverse regression estimator gsir when completeness does not hold this estimator captures only part of the central class however in these cases we show that a generalized sliced average variance estimator gsave can capture a larger portion of the class both estimators require no numerical optimization because they can be computed by spectral decomposition of linear operators finally we compare our estimators with existing methods by simulation and on actual data sets

estimating central subspaces via inverse third moments summary modern graphical tools have enhanced our ability to learn many things from data directly in recent years dimension reduction has proven to be an effective tool for generating low dimensional summary plots without appreciable loss of information some well known inverse regression methods for dimension reduction such as sliced inverse regression li 1991 and sliced average variance estimation cook weisberg 1991 have been developed to estimate summary plots for regression and discriminant analysis in this paper we suggest a new method that makes use of inverse third moments this method can find structure beyond that found by sliced inverse regression and sliced average variance estimation particularly regression mixtures illustrative examples are presented

on wielandt s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix a relatively obscure eigenvalue inequality due to wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix the asymptotic distributions are obtained under a fairly general setting an application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given 1 introduction and summary the derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis although the main idea behind most of the derivations is quite basic i e the expansion of the sample roots about the population roots the derivations themselves are often quite involved these complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem one of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices the method is based upon a relatively obscure eigenvalue inequality

a generalized em algorithm for 3 d bayesian reconstruction from poisson data using gibbs priors a generalized expectation maximization gem algorithm is developed for bayesian reconstruction based on locally correlated markov random field priors in the form of gibbs functions and on the poisson data model for the m step of the algorithm a form of coordinate gradient ascent is derived the algorithm reduces to the em maximum likelihood algorithm as the markov random field prior tends towards a uniform distribution three different gibbs function priors are examined reconstructions of 3 d images obtained from the poisson model of single photon emission computed tomography are presented

on divergences and informations in statistics and information theory the paper deals with the f divergences of csiszar generalizing the discrimination information of kullback the total variation distance the hellinger divergence and the pearson divergence all basic properties of f divergences including relations to the decision errors are proved in a new manner replacing the classical jensen inequality by a new generalized taylor expansion of convex functions some new properties are proved too e g relations to the statistical sufficiency and deficiency the generalized taylor expansion also shows very easily that all f divergences are average statistical informations differences between prior and posterior bayes errors mutually differing only in the weights imposed on various prior distributions the statistical information introduced by de groot and the classical information of shannon are shown to be extremal cases corresponding to alpha 0 and alpha 1 in the class of the so called arimoto alpha informations introduced in this paper for 0 alpha 1 by means of the arimoto alpha entropies some new examples of f divergences are introduced as well namely the shannon divergences and the arimoto alpha divergences leading for alphauarr1 to the shannon divergences square roots of all these divergences are shown to be metrics satisfying the triangle inequality the last section introduces statistical tests and estimators based on the minimal f divergence with the empirical distribution achieved in the families of hypothetic distributions for the kullback divergence this leads to the classical likelihood ratio test and estimator

thresholding implied by truncated quadratic regularization we address the problem of the estimation of an unknown signal that is known to involve sharp edges from noisy data given at the output of a linear system the sought solution is defined to be the global minimizer of an objective function combining a quadratic data fidelity term and a regularization term the latter term is a sum whose entries are obtained by applying a truncated quadratic potential function to every difference between adjacent samples such objective functions are naturally formulated either in a statistical framework or in a variational framework and they are customarily used in signal and image reconstruction however these objective functions are nonsmooth and highly nonconvex and many questions related to their minimization as well as to the features of the resulting solutions remain open we present some new facts characterizing the features exhibited by the minimizers of such objective functions our main result states that the magnitude of the differences between adjacent samples of a global minimizer are either smaller than a first threshold or larger than a second strictly larger threshold conversely no difference corresponding to a global minimizer of the objective function can be placed among these thresholds for any data this explains how edges are recovered in a signal and estimated using truncated quadratic regularization these thresholds are independent of the data but are related to the observation system and to the regularization parameters they can be used to derive necessary conditions for the choice of the regularization parameters we also show that the chance to get data for which the objective function has two or more global minimizers is null numerical experiments corroborate the obtained theoretical results

dimension reduction based on the hellinger integral sufficient dimension reduction is a useful tool for studying the dependence between a response and a multi dimensional predictor in this article a new formulation is proposed that is based on the hellinger integral of order two introduced as a natural measure of the regression information contained in the predictor subspace the response may be either continuous or discrete we establish links between local and global central subspaces and propose an efficient local estimation algorithm simulations and an application show that our method compares favourably with existing approaches

a constructive approach to the estimation of dimension reduction directions in this paper we propose two new methods to estimate the dimension reduction directions of the central subspace cs by constructing a regression model such that the directions are all captured in the regression mean compared with the inverse regression estimation methods e g j amer statist assoc 86 1991 328 332 j amer statist assoc 86 1991 316 342 j amer statist assoc 87 1992 1025 1039 the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable and have better performance than the inverse regression estimation methods for finite samples compared with the direct regression estimation methods e g j amer statist assoc 84 1989 986 995 ann statist 29 2001 1537 1566 j r stat soc ser b stat methodol 64 2002 363 410 which can only estimate the directions of cs in the regression mean the new methods can detect the directions of cs exhaustively consistency of the estimators and the convergence of corresponding algorithms are proved

fourier methods for estimating the central subspace and the central mean subspace in regression in regression with a high dimensional predictor vector it is important to estimate the central and central mean subspaces that preserve sufficient information about the response and the mean response using the fourier transform we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively under the normality assumption of the predictors explicit estimates of the central and central mean subspaces are derived bootstrap procedures are used for determining dimensionality and choosing tuning parameters simulation results and an application to a real data are reported our methods demonstrate competitive performance compared with sir save and other existing methods the approach proposed in the article provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future

dimension reduction for the conditional kth moment in regression the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction here we introduce the central kth moment subspace to capture information from the mean variance and so on up to the kth conditional moment of the regression new methods are studied for estimating these subspaces connections with sliced inverse regression are established and examples illustrating the theory are presented

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

essential spectra of difference operators on periodic graphs let x be a discrete metric space we suppose that the group acts freely on x and that the number of orbits of x with respect to this action is finite then we call x a periodic discrete metric space we examine the fredholm property and essential spectra of band dominated operators on lp x when 1 p our approach is based on the theory of band dominated operators on and their limit operators in the case where x is the set of vertices of a combinatorial graph the graph structure defines a schrodinger operator on lp x in a natural way we illustrate our approach by determining the essential spectra of schrodinger operators with slowly oscillating potential both on zig zag and on hexagonal graphs the latter being related to nano structures

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

principal hessian directions for regression with measurement error summary we consider a nonlinear regression problem with predictors with measurement error we assume that the response is related to unknown linear combinations of a p dimensional predictor vector through an unknown link function instead of observing the predictors we observe a surrogate vector with the property that its expectation is linearly related to the predictor vector with constant variance we use an important linear transformation of the surrogates based on the transformed variables we develop the modified principal hessian directions method for estimating the subspace of the effective dimension reduction space we derive the asymptotic variances of the modified principal hessian directions estimators several examples are reported and comparisons are made with the sliced inverse regression method of carroll li 1992

image relaxation restoration and feature extraction the techniques of a posteriori image restoration and iterative image feature extraction are described and compared image feature extraction methods known as graduated nonconvexity gnc variable conductance diffusion vcd anisotropic diffusion and biased anisotropic diffusion bad which extract edges from noisy images are compared with a restoration feature extraction method known as mean field annealing mfa all are shown to be performing the same basic operation image relaxation this equivalence shows the relationship between energy minimization methods and spatial analysis methods and between their respective parameters of temperature and scale as a result of the equivalence vcd is demonstrated to minimize a cost function and that cost is specified explicitly furthermore operations over scale space are shown to be a method of avoiding local minima

dimension reduction in binary response regression abstract the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model focusing on the central subspace we investigate such sufficient dimension reduction in regressions with a binary response three existing methods sliced inverse regression principal hessian direction and sliced average variance estimation and one new method difference of covariances are studied for their ability to estimate the central subspace and produce sufficient summary plots combining these numerical methods with the graphical methods proposed earlier by cook leads to a novel paradigm for the analysis of binary response regressions

difference equations and pseudo differential operators on zn abstract in this paper we develop the calculus of pseudo differential operators on the lattice z n which we can call pseudo difference operators an interesting feature of this calculus is that the global frequency space t n is compact so the symbol classes are defined in terms of the behaviour with respect to the lattice variable we establish formulae for composition adjoint transpose and for parametrix for the elliptic operators we also give conditions for the l 2 weighted l 2 and l p boundedness of operators and for their compactness on l p we describe a link to the toroidal quantization on the torus t n and apply it to give conditions for the membership in schatten classes on l 2 z n furthermore we discuss a version of fourier integral operators on the lattice and give conditions for their l 2 boundedness the results are applied to give estimates for solutions to difference equations on the lattice z n moreover we establish garding and sharp garding inequalities with an application to the unique solvability of parabolic equations on the lattice z n

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

crossed products of c algebras and spectral analysis of quantum hamiltonians we study spectral properties of a hamiltonian by analyzing the structure of certain c algebras to which it is affiliated the main tool we use for the construction of these algebras is the crossed product of abelian c algebras generated by the classical potentials by actions of groups we show how to compute the quotient of such a crossed product with respect to the ideal of compact operators and how to use the resulting information in order to get spectral properties of the hamiltonians this scheme provides a unified approach to the study of hamiltonians of anisotropic and many body systems including quantum fields

on the interpretation of regression plots abstract a framework is developed for the interpretation of regression plots including plots of the response against selected covariates residual plots added variable plots and detrended added variable plots it is shown that many of the common interpretations associated with these plots can be misleading the framework also allows for the generalization of standard plots and the development of new plotting paradigms a paradigm for graphical exploration of regression problems is sketched

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

dimension reduction for conditional mean in regression in many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors and less concerned with the other aspects of the conditional distribution in this paper we develop dimension reduction methods that incorporate this consideration we introduce the notion of the central mean subspace cms a natural inferential object for dimension reduction when the mean function is of interest we study properties of the cms and develop methods to estimate it these methods include a new class of estimators which requires fewer conditions than phd and which displays a clear advantage when one of the conditions for phd is violated cms also reveals a transparent distinction among the existing methods for dimension reduction ols phd sir and save we apply the new methods to a data set involving recumbent cows

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

sliced regression for dimension reduction a new dimension reduction method involving slicing the region of the response and applying local kernel regression to each slice is proposed compared with the traditional inverse regression methods e g sliced inverse regression sir the new method is free of the linearity condition and has much better estimation accuracy compared with the direct estimation methods e g mave the new method is much more robust against extreme values and can capture the entire central subspace cs exhaustively to determine the cs dimension a consistent cross validation criterion is developed extensive numerical studies including a real example confirm our theoretical findings

direction estimation in single index regressions we propose a general dimension reduction method that combines the ideas of likelihood correlation inverse regression and information theory we do not require that the dependence be confined to particular conditional moments nor do we place restrictions on the predictors or on the regression that are necessary for methods like ordinary least squares and sliced inverse regression although we focus on single index regressions the underlying idea is applicable more generally illustrative examples are presented

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

inversion of large support ill posed linear operators using a piecewise gaussian mrf we propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support e g a fourier transform on a sparse set this inverse problem is ill posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions this feature is modeled by a piecewise gaussian pg markov random field mrf known also as the weak string in one dimension and the weak membrane in two dimensions the reconstruction is defined as the maximum a posteriori estimate the prerequisite for the use of such a prior is the success of the optimization stage the posterior energy corresponding to a pg mrf is generally multimodal and its minimization is particularly problematic in this context general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support global optimization is dealt with by extending the graduated nonconvexity gnc algorithm to ill posed linear inverse problems gnc has been pioneered by blake and zisserman 1987 in the field of image segmentation the resulting algorithm is mathematically suboptimal but it is seen to be very efficient in practice we show that the original gnc does not correctly apply to ill posed problems our extension is based on a proper theoretical analysis which provides further insight into the gnc the performance of the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography

sufficient dimension reduction via inverse regression a family of dimension reduction methods the inverse regression ir family is developed by minimizing a quadratic objective function an optimal member of this family the inverse regression estimator ire is proposed along with inference methods and a computational algorithm the ire has at least three desirable properties 1 its estimated basis of the central dimension reduction subspace is asymptotically efficient 2 its test statistic for dimension has an asymptotic chi squared distribution and 3 it provides a chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors current methods like sliced inverse regression belong to a suboptimal class of the ir family comparisons of these methods are reported through simulation studies the approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in slic

the essential spectrum of schrodinger operators on lattices the paper is devoted to the study of the essential spectrum of discrete schr dinger operators on the lattice by means of the limit operators method this method has been applied by one of the authors to describe the essential spectrum of continuous electromagnetic schr dinger operators square root klein gordon operators and dirac operators under quite weak assumptions on the behaviour of the magnetic and electric potential at infinity the present paper aims at illustrating the applicability and efficiency of the limit operators method to discrete problems as well we consider the following classes of the discrete schr dinger operators 1 operators with slowly oscillating at infinity potentials 2 operators with periodic and semi periodic potentials 3 schr dinger operators which are discrete quantum analogues of the acoustic propagators for waveguides 4 operators with potentials having an infinite set of discontinuities and 5 three particle schr dinger operators which describe the motion of two particles around a heavy nuclei on the lattice

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

higher order nonlinear priors for surface reconstruction for surface reconstruction problems with noisy and incomplete range data a bayesian estimation approach can improve the overall quality of the surfaces the bayesian approach to surface estimation relies on a likelihood term which ties the surface estimate to the input data and the prior which ensures surface smoothness or continuity this paper introduces a new high order nonlinear prior for surface reconstruction the proposed prior can smooth complex noisy surfaces while preserving sharp geometric features and it is a natural generalization of edge preserving methods in image processing such as anisotropic diffusion an exact solution would require solving a fourth order partial differential equation pde which can be difficult with conventional numerical techniques our approach is to solve a cascade system of two second order pdes which resembles the original fourth order system this strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals we solve one pde for processing the normals and one for refitting the surface to the normals furthermore we implement the associated surface deformations using level sets hence the algorithm can accommodate very complex shapes with arbitrary and changing topologies this paper gives the mathematical formulation and describes the numerical algorithms we also show results using range and medical data

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

an estimating equation approach to dimension reduction for longitudinal data sufficient dimension reduction has been extensively explored in the context of independent and identically distributed data in this article we generalize sufficient dimension reduction to longitudinal data and propose an estimating equation approach to estimating the central mean subspace the proposed method accounts for the covariance structure within each subject and improves estimation efficiency when the covariance structure is correctly specified even if the covariance structure is misspecified our estimator remains consistent in addition our method relaxes distributional assumptions on the covariates and is doubly robust to determine the structural dimension of the central mean subspace we propose a bayesian type information criterion we show that the estimated structural dimension is consistent and that the estimated basis directions are root formula see text consistent asymptotically normal and locally efficient simulations and an analysis of the framingham heart study data confirm the effectiveness of our approach

extending sliced inverse regression sliced inverse regression sir and an associated chi squared test for dimension have been introduced as a method for reducing the dimension of regression problems whose predictor variables are normal in this article the assumptions on the predictor distribution under which the chi squared test was proved to apply are relaxed and the result is extended a general weighted chi squared test that does not require normal regressors for the dimension of a regression is given simulations show that the weighted chi squared test is more reliable than the chi squared test when the regressor distribution digresses from normality significantly and that it compares well with the chi squared test when the regressors are normal

structure adaptive approach for dimension reduction we propose a new method of effective dimension reduction for a multiindex model which is based on iterative improvement of the family of average derivative estimates the procedure is computationally straightforward and does not require any prior information about the structure of the underlying model we show that in the case when the effective dimension m of the index space does not exceed 3 this space can be estimated with the rate n 1 2 under rather mild assumptions on the model

sparse non gaussian component analysis non gaussian component analysis ngca introduced in offered a method for high dimensional data analysis allowing for identifying a low dimensional non gaussian component of the whole distribution in an iterative and structure adaptive way an important step of the ngca procedure is identification of the non gaussian subspace using principle component analysis pca method this article proposes a new approach to ngca called sparse ngca which replaces the pca based procedure with a new the algorithm we refer to as convex projection