set valued measure and fuzzy set valued measure abstract fuzzy set valued measure and set valued measure are two different subjects in this paper we shall introduce a couple of new results regarding set valued measure and establish the relation between set valued measure and fuzzy set valued measure lastly we offer a theorem which illustrates the application of set valued measure to fuzzy set valued measure

component selection and smoothing in multivariate nonparametric regression we propose a new method for model selection and model fitting in multivariate nonparametric regression models in the framework of smoothing spline anova the cosso is a method of regularization with the penalty functional being the sum of component norms instead of the squared norm employed in the traditional smoothing spline method the cosso provides a unified framework for several recent proposals for model selection in linear models and smoothing spline anova models theoretical properties such as the existence and the rate of convergence of the cosso estimator are studied in the special case of a tensor product design with periodic functions a detailed analysis reveals that the cosso does model selection by applying a novel soft thresholding type operation to the function components we give an equivalent formulation of the cosso estimator which leads naturally to an iterative algorithm we compare the cosso with mars a popular method that builds functional anova models in simulations and real examples the cosso method can be extended to classification problems and we compare its performance with those of a number of machine learning algorithms on real datasets the cosso gives very competitive performance in these studies

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

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

sparse additive models summary we present a new class of methods for high dimensional non parametric regression and classification called sparse additive models our methods combine ideas from sparse linear modelling and additive non parametric regression we derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size sparse additive models are essentially a functional version of the grouped lasso of yuan and lin they are also closely related to the cosso model of lin and zhang but decouple smoothing and sparsity enabling the use of arbitrary non parametric smoothers we give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data showing that they can be effective in fitting sparse non parametric models in high dimensional data

dirichlet laplace priors for optimal shrinkage penalized regression methods such as l1 regularization are routinely used in high dimensional applications and there is a rich literature on optimality properties under sparsity assumptions in the bayesian paradigm sparsity is routinely induced through two component mixture priors having a probability mass at zero but such priors encounter daunting computational problems in high dimensions this has motivated continuous shrinkage priors which can be expressed as global local scale mixtures of gaussians facilitating computation in contrast to the frequentist literature little is known about the properties of such priors and the convergence and concentration of the corresponding posterior distribution in this article we propose a new class of dirichlet laplace priors which possess optimal posterior concentration and lead to efficient posterior computation finite sample performance of dirichlet laplace priors relative to alternatives is assessed in simulated and real data examples

nonlinearity of the fuzzy integral we prove that the fuzzy integral has some linearity properties only for small classes of fuzzy measures we also show that an extension of this concept to bounded functions as proposed by kandel 2 5 is not well defined in general

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

disklikeness of planar self affine tiles we consider the disklikeness of the planar self affine tile t generated by an integral expanding matrix a and a consecutive collinear digit set v 0 v 2v q 1 v z 2 let f x x 2 px q be the characteristic polynomial of a we show that the tile t is disklike if and only if 2 p g 2 moreover t is a hexagonal tile for all the cases except when p 0 in which case t is a square tile the proof depends on certain special devices to count the numbers of nodal points and neighbors of t and a criterion of bandt and wang 2001 on disklikeness

mathematical considerations for the problem of fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent function from the magnitude of its fourier transform this so called phase retrieval problem will first be posed under its different underlying models we will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued sequence x from the magnitude of the output of a linear distortion hx j j 1 cdots n a number of important results will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval number of feasible solutions stability of the essential

bayesian methods and maximum entropy for ill posed inverse problems in this paper we study linear inverse problems where some generalized moments of an unknown positive measure are observed we introduce a new construction called the maximum entropy on the mean method mem which relies on a suitable sequence of finite dimensional discretized inverse problems its advantage is threefold it allows us to interpret all usual deterministic methods as bayesian methods it gives a very convenient way of taking into account prior information it also leads to new criteria for the existence question concerning the linear inverse problem which will be a starting point for the investigation of superresolution phenomena the key tool in this work is the large deviations property of some discrete random measure connected with the reconstruction procedure

fuzzy stochastic fuzzy time series and its models in this paper as an extension of the concept of time series we will present the definition and models of fuzzy stochastic fuzzy time series fsfts both of whose values and probabilities with which the fsfts assumes its values are fuzzy sets and which may not be modeled properly by the concept of time series to investigate fsfts the definition of fuzzy valued probability distributions is considered and discussed when the fsfts is time invariant several preliminary conclusions are derived

fuzzy probability measures abstract in 4 hohle has defined fuzzy measures on g fuzzy sets 2 where g stands for a regular boolean algebra consequently since the unit interval is not complemented fuzzy sets in the sense of zadeh 8 do not fit in this framework in a straightforward manner it is the purpose of this paper to continue the work started in 5 which deals with 0 1 fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space 5 we give necessary and sufficient conditions for such a measure to be a classical integral as in 9 in the case the space is generated a counterexample in the general case is also presented finally it is shown that a fuzzy probability measure is always an integral if the space is generated if we replace the operations and by the t norm to and its dual s0 see 6

high dimensional asymptotics of prediction ridge regression and classification we provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model we work in a high dimensional asymptotic regime where p n to infty and p n to gamma 0 and allow for arbitrary covariance among the features for both methods we provide an explicit and efficiently computable expression for the limiting predictive risk which depends only on the spectrum of the feature covariance matrix the signal strength and the aspect ratio gamma especially in the case of regularized discriminant analysis we find that predictive accuracy has a nuanced dependence on the eigenvalue distribution of the covariance matrix suggesting that analyses based on the operator norm of the covariance matrix may not be sharp our results also uncover an exact inverse relation between the limiting predictive risk and the limiting estimation risk in high dimensional linear models the analysis builds on recent advances in random matrix theory

the role of pseudo data for robust smoothing with application to wavelet regression summary we propose a robust curve and surface estimator based on m type estimators and penalty based smoothing this approach also includes an application to wavelet regression the concept of pseudo data a transformation of the robust additive model to the one with bounded errors is used to derive some theoretical properties and also motivate a computational algorithm the resulting algorithm termed the es algorithm is computationally fast and provides a simple way of choosing the amount of smoothing moreover it is easily described straightforwardly implemented and can be extended to other wavelet regression settings such as irregularly spaced data and image denoising results from a simulation study and real data examples demonstrate the promising empirical properties of the proposed approach

some remarks on a paper by r r yager abstract we show that slight technical changes in the definition transform the probability of fuzzy events introduced by r r yager 16 into a new concept of such probabilities having nice properties both from an intuitive and from a mathematical point of view monotonicity additivity and continuity

consistent inference of probabilities for reproducible experiments the need for inducing a probability distribution from partial data and the complementary problem of the analysis of an observed distribution in terms of fewer relevant variables occur in many branches of physics for reproducible experiments consistency conditions which must be satisfied by any algorithm for inferring a discrete probability distribution with given averages are formulated the only consistent algorithm is the one leading to the distribution of maximal entropy subject to the given constraints

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

variational inference of the fiber orientation density using diffusion mr imaging diffusion mr imaging has enabled the in vivo exploration of the connectional architecture in human brain this method particularly reveals the complex system of long range nerve fibers that integrate the functionally distinct areas of the cerebral cortex since the fibers are not directly observed but the diffusion process of water molecules in the underlying material a forward model is established that maps the microgeometry of nervous tissue onto the diffusion weighted signals this article proposes the spherical deconvolution of the fiber orientation density in a reproducing kernel hilbert space thereby generalizing previous approaches that perform a truncated fourier analysis on the sphere the specified inverse problem is solved within a smoothing spline framework which preserves the characteristic properties of a density function namely its normalization and non negativity a gaussian process model allows the specification of confidence bands for the estimated fiber orientation density and the rigorous selection of the hyperparameters here the high frequency content in the density function and the noise variance of the mr observations in addition we weaken the constant diffusivity assumption frequently made in the spherical convolution methodology the novel approach which uncovers the fiber orientation field of white matter is demonstrated with diffusion weighted data sets featuring high angular resolution

estimation of a sparse group of sparse vectors summary we consider estimating a sparse group of sparse normal mean vectors based on penalized likelihood estimation with complexity penalties on the number of nonzero mean vectors and the numbers of their significant components which can be performed by a fast algorithm the resulting estimators are developed within a bayesian framework and can be viewed as maximum a posteriori estimators we establish their adaptive minimaxity over a wide range of sparse and dense settings a simulation study demonstrates the efficiency of the proposed approach which successfully competes with the sparse group lasso estimator

generalized information theory abstract generalized information theory is viewed in this paper as an information theory that is liberated from the boundaries of probability theory after overviewing classical probabilistic information theory the paper examines recent developments regarding nonprobabilistic measures and principles of uncertainty based information which form a nucleus of the emerging generalized information theory

connectedness of a class of two dimensional self affine tiles associated with triangular matrices abstract we study the connectedness of planar self affine sets t a d generated by a matrix of the form a p 0 a q together with nonconsecutive and noncollinear digit sets of the form d l 0 l 1 l p 1 m 0 m 1 m q 1 where l 0 l 1 l p 1 and m 0 m 1 m q 1 are residue systems for p and q respectively we give a necessary and sufficient condition for t a d to be connected and extend some results by deng and lau 2011 5 to nonconsecutive digit sets

sparse additive regression on a regular lattice summary we consider estimation in a sparse additive regression model with the design points on a regular lattice we establish the minimax convergence rates over sobolev classes and propose a fourier based rate optimal estimator which is adaptive to the unknown sparsity and smoothness of the response function the estimator is derived within a bayesian formalism but can be naturally viewed as a penalized maximum likelihood estimator with the complexity penalties on the number of non zero univariate additive components of the response and on the numbers of the non zero coefficients of their fourer expansions we compare it with several existing counterparts and perform a short simulation study to demonstrate its performance

connectedness of planar self affine sets associated with non consecutive collinear digit sets abstract in the paper we focus on the connectedness of planar self affine sets t a d generated by an integer expanding matrix a with det a 3 and a collinear digit set d 0 1 b v where b 1 and v r 2 such that v a v is linearly independent we discuss the domain of the digit b to determine the connectedness of t a d especially a complete characterization is obtained when we restrict b to be an integer some results on the general case of det a 3 are obtained as well

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

parametric deconvolution of positive spike trains this paper describes a parametric deconvolution method pdps appropriate for a particular class of signals which we call spike convolution models these models arise when a sparse spike train dirac deltas according to our mathematical treatment is convolved with a fixed point spread function and additive noise or measurement error is superimposed we view deconvolution as an estimation problem regarding the locations and heights of the underlying spikes as well as the baseline and the measurement error variance as unknown parameters our estimation scheme consists of two parts model fitting and model selection to fit a spike convolution model of a specific order we estimate peak locations by trigonometric moments and heights and the baseline by least squares the model selection procedure has two stages its first stage is so designed that we expect a model of a somewhat larger order than the truth to be selected in the second stage the final model is obtained using backwards deletion this results in not only an estimate of the model order but also an estimate of peak locations and heights with much smaller bias and variation than that found in a direct trigonometric moment estimate a more efficient maximum likelihood estimate can be calculated from these estimates using a gauss newton algorithm we also present some relevant results concerning the spectral structure of toeplitz matrices which play a key role in the estimation finally we illustrate the behavior of these estimates using simulated and real dna sequencing data

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

which bridge estimator is the best for variable selection we study the problem of variable selection for linear models under the high dimensional asymptotic setting where the number of observations n grows at the same rate as the number of predictors p we consider two stage variable selection techniques tvs in which the first stage uses bridge estimators to obtain an estimate of the regression coefficients and the second stage simply thresholds this estimate to select the important predictors the asymptotic false discovery proportion operatorname afdp and true positive proportion atpp of these tvs are evaluated we prove that for a fixed atpp in order to obtain a smaller operatorname afdp one should pick a bridge estimator with smaller asymptotic mean square error in the first stage of tvs based on such principled discovery we present a sharp comparison of different tvs via an in depth investigation of the estimation properties of bridge estimators rather than orderwise error bounds with loose constants our analysis focuses on precise error characterization various interesting signal to noise ratio and sparsity settings are studied our results offer new and thorough insights into high dimensional variable selection for instance we prove that a tvs with ridge in its first stage outperforms tvs with other bridge estimators in large noise settings two stage lasso becomes inferior when the signal is rare and weak as a by product we show that two stage methods outperform some standard variable selection techniques such as operatorname lasso and sure independence screening under certain conditions

on optimality of bayesian testimation in the normal means problem we consider a problem of recovering a high dimensional vector observed in white noise where the unknown vector is assumed to be sparse the objective of the paper is to develop a bayesian formalism which gives rise to a family of l 0 type penalties the penalties are associated with various choices of the prior distributions n on the number of nonzero entries of and hence are easy to interpret the resulting bayesian estimators lead to a general thresholding rule which accommodates many of the known thresholding and model selection procedures as particular cases corresponding to specific choices of n furthermore they achieve optimality in a rather general setting under very mild conditions on the prior we also specify the class of priors n for which the resulting estimator is adaptively optimal in the minimax sense for a wide range of sparse sequences and consider several examples of such priors

a radon nikodym theorem for fuzzy valued measures abstract we study measures mapping into the set of fuzzy real numbers and in particular prove a radon nikodym theorem where the radon nikodym derivatives are fuzzy valued functions

on bayesian testimation and its application to wavelet thresholding summary we consider the problem of estimating the unknown response function in the gaussian white noise model we first utilize the recently developed bayesian maximum a posteriori testimation procedure of abramovich et al 2007 for recovering an unknown high dimensional gaussian mean vector the existing results for its upper error bounds over various sparse lp balls are extended to more general cases we show that for a properly chosen prior on the number of nonzero entries of the mean vector the corresponding adaptive estimator is asymptotically minimax in a wide range of sparse and dense lp balls the proposed procedure is then applied in a wavelet context to derive adaptive global and level wise wavelet estimators of the unknown response function in the gaussian white noise model these estimators are then proven to be respectively asymptotically near minimax and minimax in a wide range of besov balls these results are also extended to the estimation of derivatives of the response function simulated examples are conducted to illustrate the performance of the proposed level wise wavelet estimator in finite sample situations and to compare it with several existing counterparts

self affine tiles in ℝn abstract a self affine tile in r n is a set t of positive measure with a t d d t d where a is an expanding n n real matrix with det a m an integer and d d d 2 d m r n is a set of m digits it is known that self affine tiles always give tilings of r n by translation this paper extends known characterizations of digit sets d yielding self affine tiles it proves several results about the structure of tilings of r n possible using such tiles and gives examples showing the possible relations between self replicating tilings and general tilings which clarify results of kenyon on self replicating tilings

minimizers of cost functions involving nonsmooth data fidelity terms application to the processing of outliers we present a theoretical study of the recovery of an unknown vector x in mathbb r p such as a signal or an image from noisy data y in mathbb r q by minimizing with respect to x a regularized cost function cal f x y psi x y alpha phi x where psi is a data fidelity term phi is a smooth regularization term and alpha 0 is a parameter typically psi x y ax y 2 where a is a linear operator the data fidelity terms psi involved in regularized cost functions are generally smooth functions only a few papers make an exception to this and they consider restricted situations nonsmooth data fidelity terms are avoided in image processing in spite of this we consider both smooth and nonsmooth data fidelity terms our goal is to capture essential features exhibited by the local minimizers of regularized cost functions in relation to the smoothness of the data fidelity term in order to fix the context of our study we consider psi x y sum i psi a i tx y i where a i t are the rows of a and psi is cal c m on mathbb r setminus 0 we show that if psi 1 0 psi 1 0 then typical data y give rise to local minimizers hat x of cal f y which fit exactly a certain number of the data entries there is a possibly large set hat h of indexes such that a i t hat x y i for every i in hat h in contrast if psi is smooth on mathbb r for almost every y the local minimizers of cal f y do not fit any entry of y thus the possibility that a local minimizer fits some data entries is due to the nonsmoothness of the data fidelity term this is a strong mathematical property which is useful in practice by way of application we construct a cost function allowing aberrant data outliers to be detected and to be selectively smoothed our numerical experiments advocate the use of nonsmooth data fidelity terms in regularized cost functions for special purposes in image and signal processing

line detection in images through regularized hough transform the problem of determining the location and orientation of straight lines in images is of great importance in the fields of computer vision and image processing traditionally the hough transform a special case of the radon transform has been widely used to solve this problem for binary images in this paper we pose the problem of detecting straight lines in gray scale images as an inverse problem our formulation is based on use of the inverse radon operator which relates the parameters determining the location and orientation of the lines in the image to the noisy input image the advantage of this formulation is that we can then approach the problem of line detection within a regularization framework and enhance the performance of the hough based line detector through the incorporation of prior information in the form of regularization we discuss the type of regularizers that are useful for this problem and derive efficient computational schemes to solve the resulting optimization problems enabling their use in large applications finally we show how our new approach can be alternatively viewed as one of finding an optimal representation of the noisy image in terms of elements chosen from a dictionary of lines this interpretation relates the problem of hough based line finding to the body of work on adaptive signal representation

on use of the em algorithm for penalized likelihood estimation summary the em algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation this paper discusses properties of the em algorithm in such contexts concentrating on rates of conver gence and presents an alternative that is usually more practical and converges at least as quickly the em algorithm is a general approach to maximum likelihood estimation rather than a specific algorithm dempster et al 1977 discussed the method and derived basic properties demonstrating that a variety of procedures previously developed rather informally could be unified the common strand to problems where the approach is applicable is a notion of incomplete data this includes the conventional sense of missing data but is much broader than that the em algorithm demon strates its strength in situations where some hypothetical experiment yields data from which estimation is particularly convenient and economical the incomplete data actually at hand are regarded as observable functions of these complete data the resulting algorithms while usually slow to converge are often extremely simple and remain practical in large problems where no other approaches may be feasible dempster et al 1977 briefly refer to the use of the same approach to the problem of finding the posterior mode maximum a posteriori estimate in a bayesian estima

a new look at entropy for solving linear inverse problems entropy based methods are widely used for solving inverse problems particularly when the solution is known to be positive here we address linear ill posed and noisy inverse problems of the form z ax n with a general convex constraint x spl isin x where x is a convex set although projective methods are well adapted to this context we study alternative methods which rely highly on some information based criteria our goal is to clarify the role played by entropy in this field and to present a new point of view on entropy using general tools and results coming from convex analysis we present then a new and broad scheme for entropic based inversion of linear noisy inverse problems this scheme was introduced by navaza in 1985 in connection with a physical modeling for crystallographic applications and further studied by dacunha castelle and gamboa 1990 important features of this paper are i a unified presentation of many well known reconstruction criteria ii proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics iii a description of practical inversion of data using the aforementioned criteria and iv a presentation of some reconstruction results

information and entropy econometrics editor s view amos golan department of economics american university roper 200 4400 massachusetts ave nw washington dc 20016 usa

a maximum entropy approach to recovering information from multinomial response data abstract the classical maximum entropy me approach to estimating the unknown parameters of a multinomial discrete choice problem which is equivalent to the maximum likelihood multinomial logit ml estimator is generalized the generalized maximum entropy gme model includes noise terms in the multinomial information constraints each noise term is modeled as the mean of a finite set of a priori known points in the interval 1 1 with unknown probabilities where no parametric assumptions about the error distribution are made a gme model for the multinomial probabilities and for the distributions associated with the noise terms is derived by maximizing the joint entropy of multinomial and noise distributions under the assumption of independence the gme formulation reduces to the me in the limit as the sample grows large or when no noise is included in the entropy maximization further even though the gme and the logit estimators are conceptually different the dual gme model is related to a gener

superresolution in the markov moment problem abstract suppose that bar x is a characteristic function and a 1 a 2 a n are weight functions on a finite measure space recent work of gamboa and gassiat observed conditions guaranteeing that x bar x 1 is small whenever 0 x 1 a e and the moment errors int x bar x a i are small i 1 2 n using concise and elementary techniques we obtain similar results under very mild assumptions we also provide precise error bounds

characterizations of some fuzzy topological notions in probabilistic metric spaces abstract since hohle s work in 1982 on the probabilistic metrization of fuzzy uniformities menger s probabilistic metric spaces considered here under the triangular norm min only have been endowed with a number of fuzzy topological structures we introduce topology free characterizations of the three so far known types of continuity there directly in terms of probabilistic metrics we show that the associated level topologies are pseudo metrizable and in particular that a fuzzy neighbourhood space is probabilistic pseudo metrizable iff its level topologies for all 0

conditional measures based on archimedean semigroups abstract the starting point of our measure theory are decomposable measures m for crisp events i e those characterized by the property with respect to an archimidean t conorm in the present paper we pay attention to the nsa case i e where the measures can be transformed into probabilities a generalized bayesian model will be presented based on inconditional and conditional decomposable measures section 1 conditional measures as functions of the condition can be interpreted as fuzzy sets crisp and fuzzy valued measures for such fuzzy sets will be discussed section 2 the model and the measures will be applied to sugeno s problem of buying residences section 3 the model will be generalized section 4

fuzzy cardinals based on the generalized equality of fuzzy subsets the purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality the proposed idea will be compared with several approaches which were earlier offered in fuzzy literature using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy semicardinals i e we construct fuzzy natural numbers defining degrees to which a finite fuzzy subset has respectively at most least and less than more than k elements various properties of different fuzzy cardinals and semicardinals as well as some interdependences between them are also given

a generalized information theoretical approach to tomographic reconstruction a generalized maximum entropy based approach to noisy inverse problems such as the abel problem tomography or deconvolution is presented in this generalized method instead of employing a regularization parameter each unknown parameter is redefined as a proper probability distribution within a certain pre specified support then the joint entropies of both the noise and signal probabilities are maximized subject to the observed data after developing the method information measures basic statistics and the covariance structure are developed as well this method is contrasted with other approaches and includes the classical maximum entropy formulation as a special case the method is then applied to the tomographic reconstruction of the soft x ray emissivity of the hot fusion plasma

model selection and sharp asymptotic minimaxity we obtain sharp minimax results for estimation of an n dimensional normal mean under quadratic loss the estimators are chosen by penalized least squares with a penalty that grows like ck log n k for k equal to the number of nonzero elements in the estimating vector for a wide range of sparse parameter spaces we show that the penalized estimator achieves the exact minimax rate with the correct multiplication constant if and only if c equals 2 our results unify the theory obtained by many other authors for penalized estimation of normal means in particular we establish that a conjecture by abramovich et al ann stat 34 584 653 2006 is true

does the maximum entropy method improve sensitivity maximum entropy reconstruction has been used in several fields to produce visually striking reconstructions of positive objects images densities spectra from noisy indirect measurements in magnetic resonance spectroscopy this technique is notable for its apparent noise suppression and its avoidance of the artifacts that affect discrete fourier transform spectra of short zero extended data records in the general case where the length of the reconstructed spectrum exceeds that of the data record or where a convolution kernel is incorporated in the reconstruction no known analytical solution to the reconstruction problem exists consequently knowledge of the properties of maximum entropy reconstruction has been mainly anecdotal based on a small selection of published reconstructions however in the limiting case where the lengths of the reconstructed spectrum and the data record are the same and a convolution kernel is not applied the problem can be solved analytically the solution has a simple structure that helps explain several commonly observed features of maximum entropy reconstructions for example the biases in the recovered intensities and the fact that noise near the baseline is more successfully suppressed than is noise superimposed on broad features in the spectrum the solution also shows that the noise suppression offered by maximum entropy reconstruction could in this special case be equally well obtained by a cosmetic device simply displaying the conventional fourier transform reconstruction using a certain nonlinear plotting scale for the vertical y coordinate

statistical regularization of inverse problems in experimental sciences we often need to solve inverse problems that is we want to obtain information about the internal structure of a physical system from indirect noisy observations often the problem is not whether a solution exists on the contrary there are too many solutions that fit the data to a chosen tolerance level the goal is to use prior information to determine a physically meaningful solution here we present some of the basic questions that arise we describe methods that can be used to find inversion estimates as well as ways to assess their performance

self affine manifolds abstract this paper studies closed 3 manifolds which are the attractors of a system of finitely many affine contractions that tile r 3 such attractors are called self affine tiles effective characterization and recognition theorems for these 3 manifolds as well as theoretical generalizations of these results to higher dimensions are established the methods developed build a bridge linking geometric topology with iterated function systems and their attractors a method to model self affine tiles by simple iterative systems is developed in order to study their topology the model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self affine tile it has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold the induced map between the attractor of the model and the self affine tile is a quotient map and can be checked in certain cases to be monotone or cell like deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self affine tile is a topological or generalized manifold in all dimensions these new tools are used to check that several self affine tiles in the literature are 3 balls an example of a wild 3 dimensional self affine tile is given whose boundary is a topological 2 sphere but which is not itself a 3 ball the paper describes how any 3 dimensional handlebody can be given the structure of a self affine 3 manifold it is conjectured that every self affine tile which is a manifold is a handlebody

estimating anomalies from indirect observations indirect observations often lead to ill posed inverse problems where the use of prior information plays a crucial role when correctly used informative priors should be based on our belief of what the typical situation looks like while the aim of the measurement procedure is often to find deviations from the typical this work proposes a method of implementing priors that on one hand reflects the belief concerning the typical situation and on the other retains sensitivity to anomalies of the prescribed type numerical results are based on bayesian sampling methods

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

analysis of variance why it is more important than ever analysis of variance anova is an extremely important method in exploratory and confirmatory data analysis unfortunately in complex problems e g split plot designs it is not always easy to set up an appropriate anova we propose a hierarchical analysis that automatically gives the correct anova comparisons even in complex scenarios the inferences for all means and variances are performed under a model with a separate batch of effects for each row of the anova table we connect to classical anova by working with finite sample variance components fixed and random effects models are characterized by inferences about existing levels of a factor and new levels respectively we also introduce a new graphical display showing inferences about the standard deviations of each batch of effects we illustrate with two examples from our applied data analysis first illustrating the usefulness of our hierarchical computations and displays and second showing how the ideas of anova are helpful in understanding a previously fit hierarchical model

connectedness of a class of planar self affine tiles we consider a class of planar self affine tiles t that are generated by the lower triangular expanding matrices and the product form digit sets we give necessary and sufficient conditions for t to be connected and disk like also for the disconnect case we give a condition that enumerates the number of connected components of t

fuzzy measures assuming their values in the set of fuzzy numbers abstract fuzzy valued fuzzy measures are defined in an axiomatic way extending a result of klement j math anal appl 75 1980 330 339 it is shown that they can be characterized by a suitable family of ordinary measures and markov kernels

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

the maximum entropy method on the mean applications to linear programming and superresolution in this paper we give two different results we propose new methods to solve classical optimization problems in linear programming we also obtain precise quantitative results for the superresolution phenomenon as observed earlier by practical searchers on specific algorithms the common background of our work is the generalized moment problem which is known to be connected with linear programming and superresolution we describe the maximum entropy method on the mean that provides solution to the problem and leads to computational criteria to decide the existence of solutions or not

a simultaneous estimation and variable selection rule a new data based method of estimation and variable selection in linear statistical models is proposed this method is based on a generalized maximum entropy formalism and makes use of both sample and non sample information in determining a basis for coefficient shrinkage and extraneous variable identification in contrast to tradition shrinkage and variable selection are achieved on a coordinate by coordinate basis and the procedure works well for both ill and well posed statistical models analytical asymptotic results are presented and sampling experiments are used as a basis for determining finite sample behavior and comparing the sampling performance of the new estimation rule with traditional competitors solution algorithms for the non linear inversion problem that results are simple to implement

projection pursuit in high dimensions projection pursuit is a classical exploratory data analysis method to detect interesting low dimensional structures in multivariate data originally projection pursuit was applied mostly to data of moderately low dimension motivated by contemporary applications we here study its properties in high dimensional settings specifically we analyze the asymptotic properties of projection pursuit on structureless multivariate gaussian data with an identity covariance as both dimension p and sample size n tend to infinity with p n 0 our main results are that i if then there exist projections whose corresponding empirical cumulative distribution function can approximate any arbitrary distribution and ii if 0 not all limiting distributions are possible however depending on the value of various non gaussian distributions may still be approximated in contrast if we restrict to sparse projections involving only a few of the p variables then asymptotically all empirical cumulative distribution functions are gaussian and iii if 0 then asymptotically all projections are gaussian some of these results extend to mean centered sub gaussian data and to projections into k dimensions hence in the small n large p setting unless sparsity is enforced and regardless of the chosen projection index projection pursuit may detect an apparent structure that has no statistical significance furthermore our work reveals fundamental limitations on the ability to detect non gaussian signals in high dimensional data in particular through independent component analysis and related non gaussian component analysis

exact minimax estimation of the predictive density in sparse gaussian models we consider estimating the predictive density under kullback leibler loss in an l0 sparse gaussian sequence model explicit expressions of the first order minimax risk along with its exact constant asymptotically least favorable priors and optimal predictive density estimates are derived compared to the sparse recovery results involving point estimation of the normal mean new decision theoretic phenomena are seen suboptimal performance of the class of plug in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk we find that minimax optimal strategies lie outside the gaussian family but can be constructed with threshold predictive density estimates novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates

higher criticism thresholding optimal feature selection when useful features are rare and weak in important application fields today genomics and proteomics are examples selecting a small subset of useful features is crucial for success of linear classification analysis we study feature selection by thresholding of feature z scores and introduce a principle of threshold selection based on the notion of higher criticism hc for i 1 2 p let i denote the two sided p value associated with the ith feature z score and i denote the ith order statistic of the collection of p values the hc threshold is the absolute z score corresponding to the p value maximizing the hc objective i p i i p 1 i p we consider a rare weak rw feature model where the fraction of useful features is small and the useful features are each too weak to be of much use on their own hc thresholding hct has interesting behavior in this setting with an intimate link between maximizing the hc objective and minimizing the error rate of the designed classifier and very different behavior from popular threshold selection procedures such as false discovery rate thresholding fdrt in the most challenging rw settings hct uses an unconventionally low threshold this keeps the missed feature detection rate under better control than fdrt and yields a classifier with improved misclassification performance replacing cross validated threshold selection in the popular shrunken centroid classifier with the computationally less expensive and simpler hct reduces the variance of the selected threshold and the error rate of the constructed classifier results on standard real datasets and in asymptotic theory confirm the advantages of hct

minimax risk of matrix denoising by singular value thresholding an unknown m by n matrix x 0 is to be estimated from noisy measurements y x 0 z where the noise matrix z has i i d gaussian entries a popular matrix denoising scheme solves the nuclear norm penalization problem operatorname min x y x f 2 2 lambda x where x denotes the nuclear norm sum of singular values this is the analog for matrices of ell 1 penalization in the vector case it has been empirically observed that if x 0 has low rank it may be recovered quite accurately from the noisy measurement y in a proportional growth framework where the rank r n number of rows m n and number of columns n all tend to infty proportionally to each other r n m n rightarrow rho m n n rightarrow beta we evaluate the asymptotic minimax mse mathcal m rho beta lim m n n rightarrow infty inf lambda sup operatorname rank x leq r n operatorname mse x 0 hat x lambda our formulas involve incomplete moments of the quarter and semi circle laws beta 1 square case and the marcenko pastur law beta 1 nonsquare case for finite m and n we show that mse increases as the nonzero singular values of x 0 grow larger as a result the finite n worst case mse a quantity which can be evaluated numerically is achieved when the signal x 0 is infinitely strong the nuclear norm penalization problem is solved by applying soft thresholding to the singular values of y we also derive the minimax threshold namely the value lambda rho which is the optimal place to threshold the singular values all these results are obtained for general nonsquare nonsymmetric real matrices comparable results are obtained for square symmetric nonnegative definite matrices

capturing the intangible concept of information abstract the purpose of this article is to discuss the intricacies of quantifying information in some statistical problems the aim is to develop a general appreciation for the meanings of information functions rather than their mathematical use this theme integrates fundamental aspects of the contributions of kullback lindley and jaynes and bridges chaos to probability modeling a synopsis of information theoretic statistics is presented in the form of a pyramid with shannon at the vertex and a triangular base that signifies three distinct variants of quantifying information discrimination information kullback mutual information lindley and maximum entropy information jaynes examples of capturing information by the maximum entropy me method are discussed it is shown that the me approach produces a general class of logit models capable of capturing various forms of sample and nonsample information diagnostics for quantifying information captured by the me logit models are given and decom

convergence rates for maximum entropy regularization the authors prove results about stability convergence and convergence rates for the maximum entropy method by interpreting it as a nonlinear regularization method the results are applied to a widely studied problem from physical chemistry namely the reconstruction of fluorescence lifetime distributions from the distribution of the delays between absorption and emission of photons

application of the pade method to solving the noisy trigonometric moment problem some initial results the noisy trigonometric moment problem for a finite linear combination of box functions is considered and a research program possibly leading to a superresolving method is outlined and some initial steps are performed the method is based on the remark that the poles of the pade approximant to the z transform of the noiseless moments show asymptotically a regular pattern in the complex plane the pattern can be described by a set of arcs connecting points on the unit circle and a pole density function defined on the arcs when a moderate noise affects the moments more arcs are needed to describe the pole pattern but the noiseless pattern is slightly deformed still allowing its identification when this identification is possible a very effective noise filter and moment extrapolator should be easily constructed in this paper only some preliminary steps of the above research program are performed specifically the case of one box function is considered a method for computing the pole patterns

maximum entropy image reconstruction from sparsely sampled coherent field data there are many practical problems in which it is required to detect and characterize hidden structures or remote objects by virtue of the scattered acoustic or electromagnetic fields they generate it remains an open question however as to which reconstruction algorithms offer the most informative images for a given set of field measurements commonly used time domain beamforming techniques and their equivalent frequency domain implementations are conceptually simple and stable in the presence of noise however large proportions of missing measurements can quickly degrade the image quality we apply a new algorithm based on the maximum entropy method mem to the reconstruction of images from sparsely sampled coherent field data the general principles and limitations of the new method are discussed in the framework of regularization theory and the results of monostatic imaging experiments confirm that superior resolution and artifact suppression are obtained relative to a commonly used linear inverse filtering approach

on minimax estimation of a sparse normal mean vector mallows has conjectured that among distributions which are gaussian but for occasional contamination by additive noise the one having least fisher information has two sided geometric contamination a very similar problem arises in estimation of a non negative vector parameter in gaussian white noise when it is known also that most i e 1i components are zero we provide a partial asymptotic expansion of the minimax risk as 0 while the conjecture seems unlikely to be exactly true for flnite we verify it asymptotically up to the accuracy of the expansion numerical work suggests the expansion is accurate for as large as 05 the best 1 estimation rule is flrst but not second order minimax the results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases

fuzzy measures as extensions of stochastic measures abstract links between fuzzy measures cf hohle z wahrsch verw gebiete 36 1976 179 188 and stochastic measures cf morando lecture notes in mathematics no 88 pp 190 216 springer verlag berlin new york 1969 are specified in particular a class of stochastic measures from which fuzzy measures can be derived quite naturally is exhibited

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed