on the shape of a set of points in the plane a generalization of the convex hull of a finite set of points in the plane is introduced and analyzed this generalization leads to a family of straight line graphs alpha shapes which seem to capture the intuitive notions of fine shape and crude shape of point sets it is shown that a shapes are subgraphs of the closest point or furthest point delaunay triangulation relying on this result an optimal o n log n algorithm that constructs alpha shapes is developed

surface states and spectra let d 1 let h 0 be the discrete laplacian on the hilbert space l 2 d 1 with a dirichlet boundary condition and let v be a potential supported on the boundary d 1 we introduce the notions of surface states and surface spectrum of the operator h h 0 v and explore their properties our main result is that if the potential v is random and if the disorder is either large or small enough then in dimension two h has no surface spectrum on h 0 with probability one to prove this result we combine aizenman molchanov theory with techniques of scattering theory

anderson localization and lifshits tails for random surface potentials we consider schrodinger operators on l2 rd with a random potential concentrated near the surface rd1 0 rd we prove that the integrated density of states of such operators exhibits lifshits tails near the bottom of the spectrum from this and the multiscale analysis by boutet de monvel and stollmann arch math 80 2003 87 97 we infer anderson localization pure point spectrum and dynamical localization for low energies our proof of lifshits tails relies on spectral properties of schrodinger operators with partially periodic potentials in particular we show that the lowest energy band of such operators is parabolic

on estimation of monotone and concave frontier functions abstract when analyzing the productivity of firms one may want to compare how the firms transform a set of inputs x typically labor energy or capital into an output y typically a quantity of goods produced the economic efficiency of a firm is then defined in terms of its ability to operate close to or on the production frontier the boundary of the production set the frontier function gives the maximal level of output attainable by a firm for a given combination of its inputs the efficiency of a firm may then be estimated via the distance between the attained production level and the optimal level given by the frontier function from a statistical viewpoint the frontier function may be viewed as the upper boundary of the support of the population of firms density in the input and output space it is often reasonable to assume that the production frontier is a concave monotone function then a famous estimator in the univariate input and output case is the data envelopment analysis dea estimato

a nonparametric approach to the estimation of lengths and surface areas fil cuevas antonio universidad de san andres departamento de matematica y ciencias argentina

asymptotical minimax recovery of sets with smooth boundaries in this paper optimal rates of convergence are derived for estimates of sets in n dimensional black and white pictures under smoothness conditions it is assumed that the boundaries of the black regions have a smooth parameterisation that is that the boundaries are given by smooth functions from the sphere s n 1 into r n furthermore classes of convex regions are considered two models are studied edge estimation models motivated by image segmentation problems and density support estimation

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

smooth discrimination analysis ties f and g can be based on the estimation of the set g x fx gx we consider applications where it is appropriate to assume that the region g has a smooth boundary or belongs to another nonparametric class of sets in particular this assumption makes sense if discrimination is used as a data analytic tool decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered their rates of convergence are obtained we show that these rules achieve optimal rates for estimation of g and optimal rates of convergence for bayes risks an interesting conclusion is that the optimal rates for bayes risks can be very fast in particular faster than the parametric root n rate these fast rates cannot be guaranteed for plug in rules

a random approximation of set valued cadlag functions abstract in this paper we study the problem of estimating a cadlag function f whose values are compact convex sets for this purpose a random selection of points in the interval 0 1 is considered and for each selected point x a random sample in f x on the basis of this a sequence of approximants f n m is constructed where n and m are the respective sample sizes under general conditions rates of convergence are obtained for skorokhod s j 1 topology and in case of continuity of the estimated function also for the uniform one

bit error probabilities mdpsk over the nonselective rayleigh fading channel with diversity reception data transmission using m ary differential phase shift keying mdpsk over the nonselective rayleigh fading channel with diversity reception is considered while previous studies on error probability mostly assume no fading fluctuation the author considers exclusively the case in which the fading process fluctuates from one symbol interval to the next exact bit error probability results for 2 4 and 8 dpsk as well as tight upper bounds are derived some applications of the results are discussed

local likelihood tracking of fault lines and boundaries summary we suggest locally parametric methods for estimating curves such as boundaries of density supports or fault lines in response surfaces in a variety of spatial problems the methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane as a function of that point the local likelihood might be a regular likelihood computed locally with kernel weights e g in the case of support boundary estimation or a local version of a likelihood ratio statistic e g in fault line estimation in either case the local likelihood surface represents a function which is relatively large near the target curve and relatively small elsewhere therefore the curve may be estimated as a ridge line of the surface we require only a numerical algorithm for tracking the projection of a ridge into the plane this approach offers several potential advantages over alternative methods first the local log likelihood surface can be graphed and the degree of ridginess assessed visually to determine how the level of local smoothing should be varied in different spatial locations in order to emphasize the ridge and hence the curve adequately secondly the local likelihood surface does not need to be computed in anything like its entirety once we have a reasonable approximation to a point on the curve we may track it by numerically walking along the ridge line thirdly the method is appropriate without change for many different types of spatial explanatory vanables gridded stochastic or otherwise three examples are explored in detail fault lines in response surfaces and in intensity or density surfaces and boundaries of supports of probability densities

minimax estimation of the volume of a set under the rolling ball condition abstractwe consider the problem of estimating the volume of a compact domain in a euclidean space based on a uniform sample from the domain we assume that the domain has a boundary with positive reach we propose a data splitting approach to correct the bias of the plug in estimator based on the sample convex hull we show that this simple estimator achieves a minimax lower bound that we derive some numerical experiments corroborate our theoretical findings

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

minimax optimal level set estimation this paper describes a new methodology and associated theoretical analysis for rapid and accurate extraction of level sets of a multivariate function from noisy data the identification of the boundaries of such sets is an important theoretical problem with applications for digital elevation maps medical imaging and pattern recognition this problem is significantly different from classical segmentation because level set boundaries may not correspond to singularities or edges in the underlying function as a result segmentation methods which rely upon detecting boundaries would be potentially ineffective in this regime this issue is addressed in this paper through a novel error metric sensitive to both the error in the location of the level set estimate and the deviation of the function from the critical level hoeffding s inequality is used to derive a novel regularization term that is distinctly different from regularization methods used in conventional image denoising settings building upon this foundation it is possible to derive error performance bounds for the proposed estimator and demonstrate that it exhibits near minimax optimal error decay rates for large classes of level set problems the proposed method automatically adapts to the spatially varying regularity of both the boundary of the level set and the underlying function

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

adaptive estimation of convex and polytopal density support we estimate the support of a uniform density when it is assumed to be a convex polytope or more generally a convex body in mathbb r d in the polytopal case we construct an estimator achieving a rate which does not depend on the dimension d unlike the other estimators that have been proposed so far for d ge 3 our estimator has a better risk than the previous ones and it is nearly minimax up to a logarithmic factor we also propose an estimator which is adaptive with respect to the structure of the boundary of the unknown support

new methods for bias correction at endpoints and boundaries we suggest two new translation based methods for estimating and correcting for bias when estimating the edge of a distribution the first uses an empirical translation applied to the argument of the kernel in order to remove the main effects of the asymmetries that are inherent when constructing estimators at boundaries placing the translation inside the kernel is in marked contrast to traditional approaches such as the use of high order kernels which are related to the jackknife and in effect apply the translation outside the kernel our approach has the advantage of producing bias estimators that while enjoying a high order of accuracy are guaranteed to respect the sign of bias our second method is a new bootstrap technique it involves translating an initial boundary estimate toward the body of the dataset constructing repeated boundary estimates from data that lie below the respective translations and employing averages of the resulting empirical bias approximations to estimate the bias of the original estimator the first of the two methods is most appropriate in univariate cases and is studied there the second approach may be used to bias correct estimates of boundaries of multivariate distributions and is explored in the bivariate case

clustering based on pairwise distances when the data is of mixed dimensions in the context of clustering we consider a generative model in a euclidean ambient space with clusters of different shapes dimensions sizes and densities in an asymptotic setting where the number of points becomes large we obtain theoretical guaranties for some emblematic methods based on pairwise distances a simple algorithm based on the extraction of connected components in a neighborhood graph hierarchical clustering with single linkage and the spectral clustering method of ng jordan and weiss the methods are shown to enjoy some near optimal properties in terms of separation between clusters and robustness to outliers the local scaling method of zelnik manor and perona is shown to lead to a near optimal choice for the scale in the first and third methods we also provide a lower bound on the spectral gap to consistently choose the correct number of clusters in the spectral method

systematic sampling with errors in sample locations systematic sampling of points in continuous space is widely used in microscopy and spatial surveys classical theory provides asymptotic expressions for the variance of estimators based on systematic sampling as the grid spacing decreases however the classical theory assumes that the sample grid is exactly periodic real physical sampling procedures may introduce errors in the placement of the sample points this paper studies the effect of errors in sample positioning on the variance of estimators in the case of one dimensional systematic sampling first we sketch a general approach to variance analysis using point process methods we then analyze three different models for the error process calculate exact expressions for the variances and derive asymptotic variances errors in the placement of sample points can lead to substantial inflation of the variance dampening of zitterbewegung that is fluctuation effects and a slower order of convergence this suggests that the current practice in some areas of microscopy may be based on over optimistic predictions of estimator accuracy

recovering convex boundaries from blurred and noisy observations we consider the problem of estimating convex boundaries from blurred and noisy observations in our model the convolution of an intensity function f is observed with additive gaussian white noise the function f is assumed to have convex support g whose boundary is to be recovered rather than directly estimating the intensity function we develop a procedure which is based on estimating the support function of the set g this approach is closely related to the method of geometric hyperplane probing a well known technique in computer vision applications we establish bounds that reveal how the estimation accuracy depends on the ill posedness of the convolution operator and the behavior of the intensity function near the boundary short title recovering convex boundaries

finite dimensional projection for classification and statistical learning in this paper a new method for the binary classification problem is studied it relies on empirical minimization of the hinge risk over an increasing sequence of finite dimensional spaces a suitable dimension is picked by minimizing the regularized risk where the regularization term is proportional to the dimension an oracle type inequality is established for the excess generalization risk i e regret to bayes of the procedure which ensures adequate convergence properties of the method we suggest to select the considered sequence of subspaces by applying kernel principal components analysis kpca in this case the asymptotical convergence rate of the method can be better than what is known for the support vector machine svm exemplary experiments are presented on benchmark data sets where the practical results of the method are comparable to the svm

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

a plug in approach to support estimation we suggest a new approach based on the use of density estimators for the problem of estimating the compact support of a multivariate density this subject motivated in terms of pattern analysis by grenander has interesting connections with detection and clustering a natural class of density based estimators is defined universal consistency results and convergence rates are established for these estimators with respect to the usual measure based metric d between sets further convergence rates with respect to both d and the hausdorff metric d h are also obtained under some fairly intuitive shape restrictions

likelihood based confidence bands for fault lines in response surfaces we consider the problem of constructing asymptotic confidence bands both pointwise and simultaneous for a smooth fault line in a response surface when the design is represented by a point process either deterministic or stochastic with intensity n diverging to infinity the estimator of the fault line is defined as the ridge line on the likelihood surface which arises from locally fitting a model that employs a linear approximation to the fault line to a kernel smooth of the data the construction is based on analysis of the limiting behaviour of perpendicular distance from a point on the true fault line to the nearest point on the ridge we derive asymptotic properties of bias and the limiting distribution of stochastic error this distribution is given by the location of the maximum of a gaussian process with quadratic drift although the majority of attention is focused on the regression problem the limiting distribution is shown to have wider relevance to local likelihood approaches to fault line estimation for density or intensity surfaces

recovering convex edges of an image from noisy tomographic data we consider the problem of recovering edges of an image from noisy tomographic data the original image is assumed to have a discontinuity jump edge along the boundary of a compact convex set the radon transform of the image is observed with noise and the problem is to estimate the edge we develop an estimation procedure which is based on recovering the support function of the edge it is shown that the proposed estimator is nearly optimal in order in a minimax sense numerical examples illustrate reasonable practical behavior of the estimation procedure

calibrating the excess mass and dip tests of modality summary nonparametric tests of modality are a distribution free way of assessing evidence about inhomogeneity in a population provided that the potential subpopulations are sufficiently well separated they include the excess mass and dip tests which are equivalent in univariate settings and are alternatives to the bandwidth test only very conservative forms of the excess mass and dip tests are available at present however and for that reason they are generally not competitive with the bandwidth test in the present paper we develop a practical approach to calibrating the excess mass and dip tests to improve their level accuracy and power substantially our method exploits the fact that the limiting distribution of the excess mass statistic under the null hypothesis depends on unknowns only through a constant which may be estimated our calibrated test exploits this fact and is shown to have greater power and level accuracy than the bandwidth test has the latter tends to be quite conservative even in an asymptotic sense moreover the calibrated test avoids difficulties that the bandwidth test has with spurious modes in the tails which often must be discounted through subjective intervention of the experimenter

differentially amplitude and phase encoded qam for the correlated rayleigh fading channel with diversity reception this paper studies the differentially amplitude and phase encoded dape quadrature amplitude modulation qam transmission over correlated rayleigh channels with diversity reception operating over two successive received symbols the optimum and an asymptotic maximum likelihood aml differentially coherent receiver are developed and compared with a conventional switched diversity combining sdc grid receiver it is shown that the aml and sdc grid receivers are much simpler in complexity than the optimum receiver in that no channel side information is required in their realization an exact expression of the bit error probability bep is obtained for the sdc grid receiver based on a union bound argument a bep upper bound for the aml receiver is also derived and verified by simulation numerical results on 16 and 64 point constellations show that the aml receiver exhibits an almost optimum performance and the sdc grid receiver with a small level of diversity is nearly optimum it is also shown by simulation that the conventional equal gain diversity receiver is almost optimal for demodulating a 16 point dape qam signal

a systematic approach to the design and analysis of optimum dpsk receivers for generalized diversity communications over rayleigh fading channels a generalized diversity channel is introduced that models a variety of wireless communication systems that use time frequency multipath and or antenna diversity with various interbranch correlations between signaling waveforms and the fading and additive noise processes in the context of this general model a systematic approach to the design and analysis of optimum noncoherent differential phase shift keying dpsk receivers is introduced in particular it is shown how the minimum error probability mep and the generalized likelihood ratio tests glrt can be applied to obtain optimal noncoherent combining rules a comparative error rate analysis of the glrt and mep detectors and an ad hoc equal gain combiner is provided for binary signaling and the suitability of the three schemes is determined as a function of fading characteristics the asymptotic bit error rate analysis is undertaken for the mep detector for slow and fast fading channels an estimator detector decomposition of the noncoherent mep rule is obtained which allows an insightful comparative study of the fundamental limits of binary phase shift keying and dpsk modulation detection methods for both slow and fast fading the results of this paper are also applicable to postdecorrelative receivers in multiuser channels

distribution estimators and confidence intervals for stereological volumes assessing the precision of volume estimates from systematic samples is a question of great practical importance but statistically a challenging task due to the strong spatial dependence of the data and typically small sample sizes the approach taken in this paper is more ambitious than earlier methodologies the goal of which was estimation of the variance of a volume estimator rather than estimation of the distribution of we shall show that bootstrap methods yield consistent estimators of the distribution of and also suggest a variety of confidence intervals for the true volume our new methodology covers cases where serial sections are exactly periodic as well as instances where the physical slicing procedure introduces errors in the placement of the sampling points measurement errors within sections are also taken into account the performance of the method is illustrated by a simulation study with synthetic data and also applied to real datasets

many body localization in the droplet spectrum of the random xxz quantum spin chain abstract we study many body localization properties of the disordered xxz spin chain in the ising phase disorder is introduced via a random magnetic field in the z direction we prove a strong form of dynamical exponential clustering for eigenstates in the droplet spectrum for any pair of local observables separated by a distance l the sum of the associated correlators over these states decays exponentially in l in expectation this exponential clustering persists under the time evolution in the droplet spectrum our result applies to the large disorder regime as well as to the strong ising phase at fixed disorder with bounds independent of the support of the observables

nonparametric estimation of surface integrals the estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics with powerful applications to physics and image analysis among other fields provided that one can determine whether random shots hit the body cuevas et al ann statist 35 2007 1031 1051 estimate the boundary measure the boundary length for planar sets and the surface area for 3 dimensional objects via the consideration of shots at a box containing the body the statistics considered by these authors as well as those in subsequent papers are based on the estimation of minkowski content and depend on a smoothing parameter which must be carefully chosen for the same sampling scheme we introduce a new approach which bypasses this issue providing strongly consistent estimators of both the boundary measure and the surface integrals of scalar functions provided one can collect the function values at the sample points examples arise in experiments in which the density of the body can be measured by physical properties of the impacts or in situations where such quantities as temperature and humidity are observed by randomly distributed sensors our method is based on random delaunay triangulations and involves a simple procedure for surface reconstruction from a dense cloud of points inside and outside the body we obtain basic asymptotics of the estimator perform simulations and discuss via google earth s data an application to the image analysis of the aral sea coast and its cliffs

multiple symbol differential detection of mpsk a differential detection technique for mpsk multiple phase shift keying which uses a multiple symbol observation interval is presented and its performance is analyzed and simulated the technique makes use of maximum likelihood sequence estimation of the transmitted phases rather than symbol by symbol detection as in the conventional differential detection thus the performance of this multiple symbol detection scheme fills the gap between conventional two symbol observation differentially coherent detection of mpsk and ideal coherent of mpsk with differential encoding the amount of improvement gained over conventional differential detection depends on the number of phases m and the number of additional symbol intervals added to the observation what is particularly interesting is that substantial performance improvement can be obtained for only one or two additional symbol intervals of observation the analysis and simulation results presented are for uncoded mpsk

on nonparametric estimation of density level sets let x 1 x n be independent identically distributed observations from an unknown probability density f consider the problem of estimating the level set g g f x r 2 f x from the sample x 1 x n under the assumption that the boundary of g has a certain smoothness we propose piecewise polynomial estimators of g based on the maximization of local empirical excess masses we show that the estimators have optimal rates of convergence in the asymptotically minimax sense within the studied classes of densities we find also the optimal convergence rates for estimation of convex level sets a generalization to the n dimensional case where n 2 is given

boundary density and voronoi set estimation for irregular sets in this paper we study the inner and outer boundary densities of some sets with self similar boundary having minkowski dimension s d 1 in mathbb r d these quantities turn out to be crucial in some problems of set estimation theory as we show here for the voronoi approximation of the set with a random input constituted by n iid points in some larger bounded domain we prove that some classes of such sets have positive inner and outer boundary density and therefore satisfy berry essen bounds in n s 2d for kolmogorov distance the von koch flake serves as an example and a set with cantor boundary as a counter example we also give the almost sure rate of convergence of hausdorff distance between the set and its approximation

limit theorems for functionals of convex hulls in 4 a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived this is done by approximating the process of vertices of the convex hull by the process of extreme points of a poisson point process and by considering the latter process of extreme points as a markov process for a particular parametrization we show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull this extents results of renyi and sulanke 1963 and buchta 1984 and shows that the boundary length and the area have a strikingly different probabilistic behavior

localization of surface spectra we study spectral properties of the discrete laplacian h on the half space with random boundary condition the v n are independent random variables on a probability space and is the coupling constant it is known that if the v n have densities then on the interval 2 d 1 2 d 1 h 0 the spectrum of the dirichlet laplacian the spectrum of h is p a s absolutely continuous for all jl1 here we show that if the random potential p satisfies the assumption of aizenman molchanov am then there are constants d and d such that for d the spectrum of h outside h 0 is p a s pure point with exponentially decaying eigenfunctions

spectral structure of anderson type hamiltonians we study self adjoint operators of the form h h 0 n n where the n s are a family of orthonormal vectors and the n s are independently distributed random variables with absolutely continuous probability distributions we prove a general structural theorem saying that for each pair n m if the cyclic subspaces corresponding to the vectors n and m are not completely orthogonal then the restrictions of h to these subspaces are unitarily equivalent with probability one this has some consequences for the spectral theory of such operators in particular we show that well behaved absolutely continuous spectrum of anderson type hamiltonians must be pure and use this to prove the purity of absolutely continuous spectrum in some concrete cases

optimal decoding of coded psk and qam signals in correlated fast fading channels and awgn a combined envelope multiple differential and coherent detection approach the maximum likelihood sequence estimator for the reception of coded digital phase modulated signals with single or multiamplitude constellations transmitted over a multiplicative frequency nonselective i e flat correlated fast fading rayleigh or rician channel and corrupted by additive white gaussian noise awgn is derived due to this correlation the errors caused by fading tend to occur in bursts in the analysis no assumption simplifying the problem is made for fast fading the authors consider the most general case where both phase and amplitude distortion resulting from the fading process could change significantly and thus cannot be assumed to be constant over a number of transmitted symbols it is shown that the estimator s hardware structure consists of a combination of envelope multiple differential and coherent detectors with multiple differential detectors they define a receiver structure consisting of a combination of more than one distinct differential detectors each of them employing a progressively increasing by the symbol duration time delay element the outputs of these detectors are jointly processed by means of an algorithm which is presented in a recursive form the derivation of this new receiver is general enough to accommodate trellis coded phase shift keying psk and quadrature amplitude modulated qam systems differentially encoded signals such as the spl pi 4 shift differential quadrature phase shift keying dqpsk scheme can also be incorporated in order to reduce the overall receiver implementation complexity several reduced complexity near optimal versions of the algorithm are presented these reduced complexity receivers are based on the use of only a few multiple differential detectors performance evaluation results for reduced complexity trellis coded spl pi 4 shift dqpsk spl pi 4 shift 8 dqam differential quadrature amplitude modulation and 8 dpsk differential phase shift keying systems have demonstrated that the proposed receivers significantly reduce the error floors caused by fading

on the area and perimeter of a random convex hull in a bounded convex set suppose k is a compact convex set in 2 and x i 1 i n is a random sample of points in the interior of k under general assumptions on k and the distribution of the x i we study the asymptotic properties of certain statistics of the convex hull of the sample

limit theorems for the typical poisson voronoi cell and the crofton cell with a large inradius in this paper we are interested in the behavior of the typical poisson voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity we prove a law of large numbers for its number of vertices and the area of the cell outside the disk moreover for the latter we establish a central limit theorem as well as moderate deviation type results the proofs deeply rely on precise connections between poisson voronoi tessellations convex hulls of poisson samples and germ grain models in the unit ball besides we derive analogous facts for the crofton cell of a stationary poisson line process in the plane

image denoising pointwise adaptive approach a new method of pointwise adaptation has been proposed and studied in spokoiny 1998 ann statist 26 1356 1378 in the context of estimation of piecewise smooth univariate functions the present paper extends that method to estimation of bivariate grey scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design the proposed estimator hat f x at a point x is simply the average of observations over a window widehat u x selected in a data driven way the theoretical properties of the procedure are studied for the case of piecewise constant images we present a nonasymptotic bound for the accuracy of estimation at a specific grid point x as a function of the number of pixels n of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary it is also shown that the proposed method provides a near optimal rate of estimation near edges and inside homogeneous regions we briefly discuss algorithmic aspects and the complexity of the procedure the numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues an example from satellite sar imaging illustrates the applicability of the method

tracking a smooth fault line in a response surface we suggest a sequential or tracking algorithm for estimating a smooth fault line in a response surface the method starts with an approximation to a point on the line and from there the line is tracked as it meanders through the plane the technique differs from recent approaches in that it does not require a large part of the plane to be searched for evidence of a fault line this offers potential computational savings and produces a method that is invariant under rotations of coordinate axes except insofar as a rotation might affect the estimated starting point and the relative orientation of the grid on which calculations are done that feature is important if design points are not located on a regular grid we investigate properties of the method under very general conditions on the design allowing poisson cluster processes jiggled grid processes and deterministic regular lattices uniform rates of convergence are derived in all these settings for the case of noisy data and shown to be within logarithmic factors of optimal pointwise convergence rates in the no noise setting

rbf nets for approximating an object s boundary by image random sampling abstract our aim is to discuss the problem of approximating the boundary of an object in a binary image in contrast to earlier papers on similar topics we avoid finding contours and then their approximations instead we sample an image in a relatively small number of points 1 3 of pixels at random the collected data is used for imposing constraints on parameters of a radial basis functions rbf neural net it is proved that if the rbf net structure is sufficiently rich to approximate an object boundary then estimates of rbf net parameters tend to their true values as the number of samples approaches to infinity then a seemingly linear algorithm of estimating linear weights is proposed and its consistency is also proved its implementation which is based on an iterative use of the linear programming is briefly discussed and the results of its testing are shortly reported

set estimation from reflected brownian motion we study the problem of estimating a compact set s rd from a trajectory of a reflected brownian motion in s with reflections on the boundary of s this problem has relevant applications in ecology in estimating the home range of an animal based on tracking data

wave operators for the surface maryland model we study scattering properties of the discrete laplacian h on the half space z d 1 zd z with the boundary condition n 1 tan n n 0 where 0 1 d we denote by h0 the dirichlet laplacian on z d 1 khoruzenko and pastur phys rep 288 109 126 1997 have shown that if has typical diophantine properties then the spectrum of h on r h0 is pure point and that corresponding eigenfunctions decay exponentially we demonstrated in an earlier paper lett math phys 45 185 1998 that for every independent over the rationals the spectrum of h on h0 is purely absolutely continuous in this paper we continue the analysis of h on h0 and prove that whenever is independent over the rationals the wave operators h h0 exist and are complete on h0 moreover we show that under the same conditions h has no surface states on h0

measuring mass concentrations and estimating density contour clusters an excess mass approach by using empirical process theory the so called excess mass approach is studied it can be applied to various statistical problems especially in higher dimensions such as testing for multimodality estimating density contour clusters estimating nonlinear functionals of a density density estimation regression problems and spectral analysis we mainly consider the problems of testing for multimodality and estimating density contour clusters but the other problems also are discussed the excess mass over mathbb c is defined as a supremum of a certain functional defined on mathbb c where mathbb c is a class of subsets of the d dimensional euclidean space comparing excess masses over different classes mathbb c yields information about the modality of the underlying probability measure f this can be used to construct tests for multimodality if f has a density f the maximizing sets of the excess mass are level sets or density contour clusters of f provided they lie in mathbb c the excess mass and the density contour clusters can be estimated from the data asymptotic properties of these estimators and of the test statistics are studied for general classes mathbb c including the classes of balls ellipsoids and convex sets

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

change curve estimation via wavelets abstract the recently developed theory of wavelets has a remarkable ability to zoom in on very short lived frequency phenomena such as transients in signals and singularities in functions and hence provides an ideal tool to study localized changes this article proposes a wavelet method for estimating jump and sharp cusp curves of a function in the plane the method involves first computing wavelet transformation of data and then estimating jump and sharp cusp curves by wavelet transformation across fine scales asymptotic theory is established and simulations are carried out to lend some credence to the asymptotic theory the wavelet estimate is nearly optimal and can be computed by fast algorithms the method is applied to a real image

simultaneous adaptation to the margin and to complexity in classification we consider the problem of adaptation to the margin and to complexity in binary classification we suggest a learning method with a numerically easy aggregation step adaptivity both to the margin and complexity in classification usually involves empirical risk minimization or rademacher complexities which lead to numerical difficulties on the other hand there exist classifiers that are easy to compute and that converge with fast rates but are not adaptive combining these classifiers by our aggregation procedure we get numerically realizable adaptive classifiers that converge with fast rates

decision feedback differential detection of differentially encoded 16apsk signals multiple symbol differential detection dd with reference signal estimation based on the feedback of past detected symbols is presented for differentially encoded 16 level amplitude phase shift keying 16dapsk a suboptimal decision is derived the approximate analysis of the bit error rate ber performance taking into account the decision error propagation effect is presented in an additive white gaussian noise awgn channel and the ber performance is compared with those of 16dpsk and 16 quadrature amplitude modulation 16qam

maximum likelihood differential detection of uncoded and trellis coded amplitude phase modulation over awgn and fading channels spl minus metrics and performance this paper derives metrics for maximum likelihood differential detection of uncoded and trellis coded mpsk and qam transmitted over rayleigh and rician fading channels receiver structures based on these metrics are proposed and their error probability performance analyzed and or simulated the results represent a generalization of the notion of multiple symbol differential detection previously introduced by the authors for mpsk over an awgn to the fading channel and other modulations for the coded cases ideal interleaving deinterleaving is assumed and furthermore the presence or absence of channel state information an interesting side result is that for a constant envelope modulation transmitted over a fading channel with unknown but rapidly varying phase error the other extreme to the slowly varying phase error case normally assumed for differential detection under certain practical assumptions it is shown that the optimum receiver is of the limiter discriminator type

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk