approximation of linear programs by bregman s df projections abstract the motivation of this paper is twofold to contribute to the theory of bregman s df projections and to point out its link to finding optimal solutions to linear programming problems a symmetric primal dual pair of the df projection problem is presented we call it young programming problem because the usual inequality that relates the primal and dual objective functions reduces to the well known young inequality some basic properties of young programs are derived and an associated linear program is defined in a natural way it will be shown that an optimal solution of the associated linear program is obtained by solving an parametric young program and the optimal solutions of parametric young programs converge to an optimal solution of the associated linear program when approaches 0 it may also be interesting that the explicit formulation of the dual program enables us to give a dual interpretation of row action methods a powerful tool for solving df projection problems in general and allows us to find new functions satisfying sufficient conditions for the convergence of row action methods

using relative entropy to find optimal approximations an application to simple fluids abstract we develop a maximum relative entropy formalism to generate optimal approximations to probability distributions the central results consist of a justifying the use of relative entropy as the uniquely natural criterion to select a preferred approximation from within a family of trial parameterized distributions and b to obtain the optimal approximation by marginalizing over parameters using the method of maximum entropy and information geometry as an illustration we apply our method to simple fluids the exact canonical distribution is approximated by that of a fluid of hard spheres the proposed method first determines the preferred value of the hard sphere diameter and then obtains an optimal hard sphere approximation by a suitably weighed average over different hard sphere diameters this leads to a considerable improvement in accounting for the soft core nature of the interatomic potential as a numerical demonstration the radial distribution function and the equation of state for a lennard jones fluid argon are compared with results from molecular dynamics simulations

high breakdown linear discriminant analysis abstract the classification rules of linear discriminant analysis are defined by the true mean vectors and the common covariance matrix of the populations from which the data come because these true parameters are generally unknown they are commonly estimated by the sample mean vector and covariance matrix of the data in a training sample randomly drawn from each population however these sample statistics are notoriously susceptible to contamination by outliers a problem compounded by the fact that the outliers may be invisible to conventional diagnostics high breakdown estimation is a procedure designed to remove this cause for concern by producing estimates that are immune to serious distortion by a minority of outliers regardless of their severity in this article we motivate and develop a high breakdown criterion for linear discriminant analysis and give an algorithm for its implementation the procedure is intended to supplement rather than replace the usual sample moment methodology of discri

the entropic penalty approach to stochastic programming a penalty type decision theoretic approach to nonlinear programming problems with stochastic constraints is introduced the stochastic program sp is replaced by a deterministic program dp by adding a term to the objective function to penalize solutions which are not feasible in the mean the special feature of our approach is the choice of the penalty function pe which is given in terms of the relative entropy functional and is accordingly called entropic penalty it is shown that pe has properties which make it suitable to treat stochastic programs some of these properties are derived via a dual representation of the entropic penalty which also enable one to compute pe more easily in particular if the constraints in sp are stochastically independent the dual representation is also used to express the deterministic problem dp as a saddle function problem for problems in which the randomness occurs in the rhs of the constraints it is shown that the dual problem of dp is equivalent to expected utility maximization of the classical lagrangian dual function of sp with the utility being of the constant risk aversion type finally mean variance approximations of pe and the induced approximating deterministic program are considered

using angles to identify concentrated multivariate outliers this article describes a procedure for the detection of multivariate outliers based on the analysis of certain angular properties of the observations the method is simple exploratory in nature and particularly well suited for the detection of concentrated contamination patterns in which the outliers appear to form a cluster separated from the sample it is shown that it presents good properties for the identification of contaminations on high dimensional sample spaces and for high contamination levels including some cases in which methods based on robust estimators the minimum covariance determinant and minimum volume ellipsoid estimators the stahel donoho estimator or other recent proposals may fail the use of the procedure is illustrated through several examples

alpha divergence is unique belonging to both f divergence and bregman divergence classes a divergence measure between two probability distributions or positive arrays positive measures is a useful tool for solving optimization problems in optimization signal processing machine learning and statistical inference the csiszar f divergence is a unique class of divergences having information monotonicity from which the dual alpha geometrical structure with the fisher metric is derived the bregman divergence is another class of divergences that gives a dually flat geometrical structure different from the alpha structure in general csiszar gave an axiomatic characterization of divergences related to inference problems the kullback leibler divergence is proved to belong to both classes and this is the only such one in the space of probability distributions this paper proves that the alpha divergences constitute a unique class belonging to both classes when the space of positive measures or positive arrays is considered they are the canonical divergences derived from the dually flat geometrical structure of the space of positive measures

the spectrum of an infinite graph abstract the spectrum of a locally finite countable graph is defined some theorems known from the theory of spectra of finite graphs are extended to infinite graphs

on the dual of linear inverse problems abstract in linear inverse problems considered in this paper a vector with positive components is to be selected from a feasible set defined by linear constraints the selection rule involves minimization of a certain function which is a measure of distance from a priori guess csiszar made an axiomatic approach towards defining a family of functions we call it divergence that can serve as logically consistent selection rules in this paper we present an explicit and perfect dual of the resulting convex programming problem prove the corresponding duality theorem and optimality criteria and make some suggestions on an algorithmic solution

entropy like proximal methods in convex programming we study an extension of the proximal method for convex programming where the quadratic regularization kernel is substituted by a class of convex statistical distances called divergences which are typically entropy like in form after establishing several basic properties of these quasi distances we present a convergence analysis of the resulting entropy like proximal algorithm applying this algorithm to the dual of a convex program we recover a wide class of nonquadratic multiplier methods and prove their convergence

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

deriving robust counterparts of nonlinear uncertain inequalities in this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters we use convex analysis support functions conjugate functions fenchel duality and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints it turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function conveniently these two computations are completely independent this approach has several advantages first it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities second it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived we also study some cases where the inequality is nonconcave in the uncertain parameters

an inequality for logarithmic mapping and applications for the shannon entropy using the concavity property of the log mapping and the weighted arithmetic mean geometric mean inequality we point out an analytic inequal ity for the logarithmic map and apply it for the kullback leibler distance in information theory some applications for shannon s entropy are given as well

maximum entropy and conditional probability it is well known that maximum entropy distributions subject to appropriate moment constraints arise in physics and mathematics in an attempt to find a physical reason for the appearance of maximum entropy distributions the following theorem is offered the conditional distribution of x l given the empirical observation 1 n sum n i l h x i alpha where x 1 x 2 cdots are independent identically distributed random variables with common density g converges to f lambda x e lambda t h x g x suitably normalized where lambda is chosen to satisfy int f lambda x h x dx alpha thus the conditional distribution of a given random variable x is the normalized product of the maximum entropy distribution and the initial distribution this distribution is the maximum entropy distribution when g is uniform the proof of this and related results relies heavily on the work of zabell and lanford

certainty equivalents and information measures duality and extremal principles given a convex function ir ir the csiszar divergence csiszar 1978 is a function i ir n ir n ir

general entropy criteria for inverse problems with applications to data compression pattern classification and cluster analysis minimum distance approaches are considered for the reconstruction of a real function from finitely many linear functional values an optimal class of distances satisfying an orthogonality condition analogous to that enjoyed by linear projections in hilbert space is derived these optimal distances are related to measures of distances between probability distributions recently introduced by c r rao and t k nayak 1985 and possess the geometric properties of cross entropy useful in speech and image compression pattern classification and cluster analysis several examples from spectrum estimation and image processing are discussed

comparison of model misspecification diagnostics using residuals from least mean of squares and least median of squares fits abstract this article explores model misspecification diagnostics based on least squares and least median of squares fits it shows that in some circumstances least median of squares methods or any other estimator with the exact fit property fail to reveal an incorrectly specified mean function but least squares methods succeed

fast very robust methods for the detection of multiple outliers abstract a few repeats of a simple forward search from a random starting point are shown to provide sufficiently robust parameter estimates to reveal masked multiple outliers the stability of the patterns obtained is exhibited by the stalactite plot the robust estimators used are least median of squares for regression and the minimum volume ellipsoid for multivariate outliers the forward search also has potential as an algorithm for calculation of these parameter estimates for large problems parallel computing provides appreciable reduction in computational time

quadratically constrained minimum cross entropy analysis quadratically constrained minimum cross entropy problem has recently been studied by zhang and brockett through an elaborately constructed dual in this paper we take a geometric programming approach to analyze this problem unlike zhang and brockett we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem furthermore by using the dual perturbation method we directly prove the strong duality theorem and derive a dual to primal conversion formula as a by product the perturbation proof gives us insights to develop a computation procedure that avoids dual non differentiability and allows us to use a general purpose optimizer to find ane optimal solution for the quadratically constrained minimum cross entropy analysis

the polya information divergence extensions of sanov s theorem and the conditional limit theorem colt are established for a multicolor polya eggenberger pe urn sampling scheme giving the polya information divergence and a polya extension to the maximum relative entropy maxent method polya maxent includes the standard maxent as well as its variants used in bose einstein fermi dirac and intermediate acharya swamy statistics as special cases in the pe setting standard maxent is in general asymptotically inconsistent

controlled dual perturbations for posynomial programs abstract with geometric programs getting larger and larger the dual based solution methods which optimize a less complicated problem than the original problem become more important however there are two major difficulties related to the development of a dual method one is the non differentiability of the dual objective function over its feasible region and the other one is the conversion of a dual solution to a primal solution in this paper we address these issues for the posynomial programs a well controlled dual perturbation method which guarantees optimal primal dual solution pair is proposed to overcome the nondifferentiability difficulty and a simple linear programming method is introduced to generate a quick dual to primal conversion

unmasking outliers and leverage points a confirmation abstract identification of multiple outliers and leverage points is difficult because of the masking effect recently rousseeuw and van zomeren suggested using high breakdown robust estimation methods the least median of squares and minimum volume ellipsoid for unmasking these observations these methods tend to declare too many observations as extreme however a stepwise analysis is proposed here for confirmation of outliers and leverage points detected using the robust methods diagnostic measures are constructed for observations added back to the reduced sample they are shown graphically the complementary use of robust and diagnostic methods gives satisfactory results in analyzing two data sets one data set consists often bad and four good leverage points four or 10 using a different cutoff extreme observations of the other data set of size 28 are identified using the robust methods but the stepwise analysis confirms only one the limitations of atkinson s confirmatory approach are discusse

investigating the geometry of a p dimensional data set abstract this article concerns itself with exploratory data analysis in hyperspace and discusses a method of data reduction intended to allow insight into the geometry of a p dimensional data set where p 2 the essential idea is to examine various views through the data set using a suitably chosen line the baseline for defining a viewing direction for a given data point its distance from a suitable point on the baseline called the vertex and the angular separation between the baseline and the line connecting the vertex to the data point are taken as coordinates of an ordinary polar coordinate plot in two dimensions the bulk of the article discusses empirical evidence for the utility of such a plot which can be referred to as a coneplot because points are essentially identified with the intersection in p dimensions of a hypersphere and a ray of a hypercone

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

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

pseudoconvex optimization for a special problem of paint industry abstract a new mathematical programming model is presented for the computer color formulation problem the model is essentially based on the two constant kubelka munk theory that describes most of the necessary physical properties of this problem the model is a nonconvex programming problem it has a nonconvex objective function with some nice pseudoconvexity properties the set of feasible solutions is a relatively simple polyhedron an algorithm is also proposed to solve the problem

convex solutions of a functional equation arising in information theory abstract given a convex function f defined for positive real variables the so called csiszar f divergence is a function i f defined for two n dimensional probability vectors p p 1 p n and q q 1 q n as i f p q i 1 n q i f p i q i for this generalized measure of entropy to have distance like properties especially symmetry it is necessary for f to satisfy the following functional equation f x x f 1 x for all x 0 in the present paper we determine all the convex solutions of this functional equation by proposing a way of generating all of them in doing so existing usual f divergences are recovered and new ones are proposed

robpca a new approach to robust principal component analysis we introduce a new method for robust principal component analysis pca classical pca is based on the empirical covariance matrix of the data and hence is highly sensitive to outlying observations two robust approaches have been developed to date the first approach is based on the eigenvectors of a robust scatter matrix such as the minimum covariance determinant or an s estimator and is limited to relatively low dimensional data the second approach is based on projection pursuit and can handle high dimensional data here we propose the robpca approach which combines projection pursuit ideas with robust scatter matrix estimation robpca yields more accurate estimates at noncontaminated datasets and more robust estimates at contaminated data robpca can be computed rapidly and is able to detect exact fit situations as a by product robpca produces a diagnostic plot that displays and classifies the outliers we apply the algorithm to several datasets from chemometrics and engineering

estimating bayes factors via posterior simulation with the laplace metropolis estimator abstract the key quantity needed for bayesian hypothesis testing and model selection is the integrated or marginal likelihood of a model we describe a way to use posterior simulation output to estimate integrated likelihoods we describe the basic laplace metropolis estimator for models without random effects for models with random effects we introduce the compound laplace metropolis estimator we apply this estimator to data from the world fertility survey and show it to give accurate results batching of simulation output is used to assess the uncertainty involved in using the compound laplace metropolis estimator the method allows us to test for the effects of independent variables in a random effects model and also to test for the presence of the random effects

regularization of noisy isar images containing extended features we present a regularization method that employs a cross entropy functional and which deals with the three issues of complex valued data prior information and noise mitigation the basic model of this approach is similar to that used in usual maximum a posteriori analysis and allows for a more general relationship between the image and its configuration entropy than that usually employed

bayesian exponentially tilted empirical likelihood summary while empirical likelihood has been shown to exhibit many of the properties of conven tional parametric likelihoods a formal probabilistic interpretation has so far been lacking we show that a likelihood function very closely related to empirical likelihood naturally arises from a nonparametric bayesian procedure which places a type of noninformative prior on the space of distributions this prior gives preference to distributions having a small support and among those sharing the same support it favours entropy maximising distributions the resulting nonparametric bayesian procedure admits a computationally convenient representation as an empirical likelihood type likelihood where the probability weights are obtained via exponential tilting the proposed methodology provides an attractive alternative to the bayesian bootstrap as a nonparametric limit of a bayesian procedure for moment condition models

identifying multiple outliers in multivariate data summary we propose a procedure for the detection of multiple outliers in multivariate data let xbe an n x p data matrix representing n observations onp variates we first order the n observations using an appropriately chosen robust measure of outlyingness then divide the data set into two initial subsets a basic subset which containsp 1 good observations and a nonbasic subset which contains the remaining n p 1 observations second we compute the relative distance from each point in the data set to the centre of the basic subset relative to the possibly singular covariance matrix of the basic subset third we rearrange the n observations in ascending order accordingly then divide the data set into two subsets a basic subset which contains the first p 2 observations and a non basic subset which contains the remaining n p 2 observations this process is repeated until an appropriately chosen stopping criterion is met the final non basic subset of observations is declared an outlying subset the procedure proposed is illustrated and compared with existing methods by using several data sets the procedure is simple computationally inexpensive suitable for automation computable with widely available software packages effective in dealing with masking and swamping problems and most importantly successful in identifying multivariate outliers

ambiguity aversion robustness and the variational representation of preferences we characterize in the anscombe aumann framework the preferences for which there are a utility functionu on outcomes and an ambiguity indexc on the set of probabilities on the states of the world such that for all acts f and g the function u represents the decision maker s risk attitudes while the index c captures his ambiguity attitudes these preferences include the multiple priors preferences of gilboa and schmeidler and the multiplier preferences of hansen and sargent this provides a rigorous decision theoretic foundation for the latter model which has been widely used in macroeconomics and finance

updating subjective probability abstract jeffrey s rule for revising a probability p to a new probability p based on new probabilities p ei on a partition ei i 1 n is p a p a ei p ei jeffrey s rule is applicable if it is judged that p a ei p a ei for all a and i this article discusses some of the mathematical properties of this rule connecting it with sufficient partitions and maximum entropy updating of contingency tables the main results concern simultaneous revision on two partitions

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

procedures for the identification of multiple outliers in linear models abstract we consider the problem of identifying and testing multiple outliers in linear models the available outlier identification methods often do not succeed in detecting multiple outliers because they are affected by the observations they are supposed to identify we introduce two test procedures for the detection of multiple outliers that appear to be less sensitive to this problem both procedures attempt to separate the data into a set of clean data points and a set of points that contain the potential outliers the potential outliers are then tested to see how extreme they are relative to the clean subset using an appropriately scaled version of the prediction error the procedures are illustrated and compared to various existing methods using several data sets known to contain multiple outliers also the performances of both procedures are investigated by a monte carlo study the data sets and the monte carlo indicate that both procedures are effective in the detection of multiple outliers

the identification of influential subsets in regression using a fuzzy clustering strategy a fuzzy clustering strategy is used to identify subsets of influential observations in regression as part of the fuzzy clustering strategy the analyst can explore the uniqueness of selected subsets and the degree of belonging of observations to selected subsets this is accomplished by either varying a fuzzy parameter or the number of clusters once the observations or subsets have been identified the analyst can then compute regression diagnostics to confirm their degree of influence in regression bootstrapping and high breakdown procedures were used to confirm the influence of the previously identified subsets this fuzzy clustering strategy is applied to the modified data on wood specific gravity and an augmented production dataset both datasets have been previously presented in the literature

an approximate algorithm for computing multidimensional convex hulls to find a convex hull for n points in d dimensional space the optimal algorithm has time complexity o n d 2 when n and d are large the execution time is very long in this paper we propose an approximate algorithm for computing multidimensional convex hulls this algorithm finds quasi two side approximation to the hull to reduce the time for computing the exact hull boundary to yield an e approximate convex hull it has time complexity o e 1 d 1 n and storage complexity o e 1 d 1 the approximate algorithm has several advantages 1 it can easily be implemented 2 it is suitable for parallel implementation 3 it is much faster than the exact algorithm 4 the user can choose to get more accurate results using longer computation time and 5 it can be applied to solve many problems related to convex hull computation

orthogonal polynomials with a constant recursion formula and an application to harmonic analysis on calcule la mesure pour laquelle une suite de polynomes avec une formule de recurrence constante est orthogonale

robust solutions of optimization problems affected by uncertain probabilities in this paper we focus on robust linear optimization problems with uncertainty regions defined by divergences for example chi squared hellinger kullback leibler we show how uncertainty regions based on divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector such problems frequently occur in for example optimization problems in inventory control or finance that involve terms containing moments of random variables expected utility etc we show that the robust counterpart of a linear optimization problem with divergence uncertainty is tractable for most of the choices of typically considered in the literature we extend the results to problems that are nonlinear in the optimization variables several applications including an asset pricing example and a numerical multi item newsvendor example illustrate the relevance of the proposed approach this paper was accepted by gerard p cachon optimization

divisors and the spectrum of infinite graphs abstract let g be a connected locally finite graph a matrix d d ij is a front divisor of g if there is a partition of v g into classes v 1 v 2 v 3 such that 1 for each i j and each v i there are exactly d ij edges emanating from and having the terminal vertex in v j and 2 for each i the set v i is finite let d be the matrix determined by d ij d ij d ji 1 2 if x is a graph or a matrix let x denote its spectrum it is shown that d g if g is a distance regular graph then the distance partition of g determines a front divisor p it is shown that in this case p g as an application the spectra of all infinite locally finite distance regular graphs are determined

linearly constrained estimation by mathematical programming abstract some mathematical programming models of the mixing problem are discussed in this paper five models based on different discrepancies are considered and their fundamental properties are examined using variational and smirnov distances linear programming models are obtained pearson divergence leads to quadratic programming hellinger divergence leads to l p programming and the kullback leibler information divergence gives a special geometric programming model finally some computational experiences are presented

approximation theoretic derivation of logarithmic entropy principles for inverse problems and unique extension of the maximum entropy method to incorporate prior knowledge the minimum distance approach for reconstructing a positive function based on knowledge of finitely many linear functional values is examined two important classes of directed distances for signal processing and statistical inference are discussed by imposing conditions analogous to those satisfied by linear projections in hilbert space two logarithmic entropy principles are derived one of these involves the itakura saito distortion measure of communication theory and uniquely extends burg s maximum entropy method to incorporate prior knowledge the other uses the kullback leibler distance of statistics

line integral alternating minimization algorithm for dual energy x ray ct image reconstruction we propose a new algorithm called line integral alternating minimization liam for dual energy x ray ct image reconstruction instead of obtaining component images by minimizing the discrepancy between the data and the mean estimates liam allows for a tunable discrepancy between the basis material projections and the basis sinograms a parameter is introduced that controls the size of this discrepancy and with this parameter the new algorithm can continuously go from a two step approach to the joint estimation approach liam alternates between iteratively updating the line integrals of the component images and reconstruction of the component images using an image iterative deblurring algorithm an edge preserving penalty function can be incorporated in the iterative deblurring step to decrease the roughness in component images images from both simulated and experimentally acquired sinograms from a clinical scanner were reconstructed by liam while varying the regularization parameters to identify good choices the results from the dual energy alternating minimization algorithm applied to the same data were used for comparison using a small fraction of the computation time of dual energy alternating minimization liam achieves better accuracy of the component images in the presence of poisson noise for simulated data reconstruction and achieves the same level of accuracy for real data reconstruction

walk generating functions and spectral measures of infinite graphs let g be a locally finite graph with bounded vertex degrees its adjacency matrix a gives rise to a bounded self adjoint operator on l2 v g let e be the resolution of the identity for a and ev the vector in l2 v g having the v coordinate 1 and all other coordinates 0 the function uv e eu ev is called the spectral measure of g corresponding to vertices u v of g it is related to the generating function wuv z for the number of walks of given length between u and v the spectral measures are used to show the following the pairing theorem characterizing finite bipartite graphs in terms of their spectrum is generalized from finite to infinite graphs the average closed walk generating function can be defined in some cases generalizing the plancherel measure to not necessarily vertex transitive graphs some theorems about the convergence of graph measures are given with applications to the calculation of the expected eigenvalue distribution of a large finite random regular or bipartite semiregular graph and the calculation of spectral measures of several examples

on one step gm estimates and stability of inferences in linear regression abstract the folklore on one step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster than n 1 4 where n is the sample size we investigate the extent to which this folklore is valid for one step gm estimators and their associated standard errors in linear regression we find that one step gm estimates based on newton raphson or scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points moreover these estimators have bounded influence functions and their standard errors can have high breakdown points the folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic

quasi newton projection methods and the discrepancy principle in image restoration abstract in this work the problem of the restoration of images corrupted by space invariant blur and noise is considered this problem is ill posed and regularization is required the image restoration problem is formulated as a nonnegatively constrained minimization problem whose objective function depends on the statistical properties of the noise corrupting the observed image the cases of gaussian and poisson noise are both considered a newton like projection method with early stopping of the iterates is proposed as an iterative regularization method in order to determine a nonnegative approximation to the original image a suitable approximation of the hessian of the objective function is proposed for a fast solution of the newton system the results of the numerical experiments show the effectiveness of the method in computing a good solution in few iterations when compared with some methods recently proposed as best performing

radial functions on free products on considere les groupes dont les fonctions radiales forment une algebre on decrit les mesures de plancherel sur une grande classe de ces groupes radiaux

an optimal technique for constraint based image restoration and reconstruction a new technique for finding an optimal feasible solution to the general image reconstruction and restoration problem is described this method allows the use of prior knowledge of the properties of both the solution and any noise present on the data the problem is formulated as the optimization of a cost function over the intersection of a number of convex constraint sets each set being defined as containing those solutions consistent with a particular constraint a duality theorem is then applied to yield a dual problem in which the unknown image is replaced by a model defined in terms of a finite dimensional parameter vector and the kernels of the integral equations relating the data and solution the dual problem may then be solved for the model parameters using a gradient descent algorithm this method serves as an alternative to the primal constrained optimization and projection onto convex sets pocs algorithms problems in which this new approach is appropriate are discussed an example is given for image reconstruction from noisy projection data applying the dual method results in a fast nonlinear algorithm simulation results demonstrate the superiority of the optimal feasible solution over one obtained using a suboptimal approach

robust clustering with applications in computer vision a clustering algorithm based on the minimum volume ellipsoid mve robust estimator is proposed the mve estimator identifies the least volume region containing h percent of the data points the clustering algorithm iteratively partitions the space into clusters without prior information about their number at each iteration the mve estimator is applied several times with values of h decreasing from 0 5 a cluster is hypothesized for each ellipsoid the shapes of these clusters are compared with shapes corresponding to a known unimodal distribution by the kolmogorov smirnov test the best fitting cluster is then removed from the space and a new iteration starts constrained random sampling keeps the computation low the clustering algorithm was successfully applied to several computer vision problems formulated in the feature space paradigm multithresholding of gray level images analysis of the hough space and range image segmentation

breakdown in nonlinear regression abstract the breakdown point is considered an important measure of the robustness of a linear regression estimator this article addresses the concept of breakdown in nonlinear regression because it is not invariant to nonlinear reparameterization the usual definition of the breakdown point in linear regression is inadequate for nonlinear regression the original definition of breakdown due to hampel is more suitable for nonlinear problems but may indicate breakdown when the fitted values change very little we introduce breakdown functions which measure breakdown of the fitted values using the breakdown functions we introduce a new definition of the breakdown point for the linear regression model our definition of the breakdown point coincides with the usual definition for linear regression as well as with hampel s definition for most nonlinear regression functions we show that the breakdown point of the least squares estimator is 1 n we prove that for a large class of unbounded regression func

the asymptotics of rousseeuw s minimum volume ellipsoid estimator rousseeuw s minimum volume estimator for multivariate location and dispersion parameters has the highest possible breakdown point for an affine equivariant estimator in this paper we establish that it satisfies a local holder condition of order 1 2 and converges weakly at the rate of n 1 3 to a non gaussian distribution

penalty functions and duality in stochastic programming via φ divergence functionals the paper considers stochastically constrained nonlinear programming problems a penalty function is constructed in terms of a distance between random variables defined in terms of the divergence functional a generalization of the relative entropy a duality theory is developed in which a general relation between divergence and utility functions is revealed via the conjugate transform and a new type of certainty equivalent concept emerges

principal information theoretic approaches 2000 principal information theoretic approaches journal of the american statistical association vol 95 no 452 pp 1349 1353

practical bayesian density estimation using mixtures of normals abstract mixtures of normals provide a flexible model for estimating densities in a bayesian framework there are some difficulties with this model however first standard reference priors yield improper posteriors second the posterior for the number of components in the mixture is not well defined if the reference prior is used third posterior simulation does not provide a direct estimate of the posterior for the number of components we present some practical methods for coping with these problems finally we give some results on the consistency of the method when the maximum number of components is allowed to grow with the sample size

finding an unknown number of multivariate outliers we use the forward search to provide robust mahalanobis distances to detect the presence of outliers in a sample of multivariate normal data theoretical results on order statistics and on estimation in truncated samples provide the distribution of our test statistic we also introduce several new robust distances with associated distributional results comparisons of our procedure with tests using other robust mahalanobis distances show the good size and high power of our procedure we also provide a unification of results on correction factors for estimation from truncated samples

on lp programming abstract our paper treats the primal and dual program of lp programming lp programming is a generalization of lp approximation problems there is a strict connection between lp programming and geometrical programming because in both of them geometrical inequality plays a fundamental role the structure of our paper follows that of klafszkys 1 in the first sections duality theorems are proved which play an important role in mathematical programming most of these results can be found in petersons and eckers 3 4 5 but our proofs are much more simple and we show these fundamental properties more detailed afterwards the relation between the lagrange function and the optimal solution pair is investigated regularity is investigated as well and we show the marginal value of lp programming in the end linear programming lp constrained lp approximation problems the quadratically constrained quadratic programming and compromise programming are shown as special cases of lp programming

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

minimum variance quadratic unbiased estimators as a tool to identify compound normal distributions abstract we derive the minimum variance quadratic unbiased estimator mivque of the variance of the components of a random vector having a compound normal distribution cnd we show that the mivque converges in probability to a random variable whose distribution is essentially the mixing distribution characterising the cnd this fact is very important because the mivque allows us to make out the signature of a particular cnd and notably allows us to check if an hypothesis of normality for multivariate observations y1 ym is plausible

on the convexity of some divergence measures based on entropy functions three measures of divergence between vectors in a convex set of a n dimensional real vector space are defined in terms of certain types of entropy functions and their convexity property is studied among other results a classification of the entropies of degree alpha is obtained by the convexity of these measures these results have applications in information theory and biological studies

expected utility penalty functions and duality in stochastic nonlinear programming we consider nonlinear programming problem p with stochastic constraints the lagrangean corresponding to such problems has a stochastic part which in this work is replaced by its certainty equivalent in the sense of expected utility theory it is shown that the deterministic surrogate problem ce p thus obtained contains a penalty function which penalizes violation of the constraints in the mean the approach is related to several known methods in stochastic programming such as chance constraints stochastic goal programming reliability programming and mean variance analysis the dual problem of ce p is studied for problems with stochastic righthand sides in the constraints and a comprehensive duality theory is developed by introducing a new certainty equivalent nce concept motivation for the nce and its potential role in decision theory are discussed as well as mean variance approximations

computation of high breakdown nonlinear regression parameters most algorithms for estimating high breakdown regression estimators in linear regression rely on finding the least squares fit to many p point elemental sets where p is the dimension of the x matrix such an approach is computationally infeasible in nonlinear regression this article presents a new algorithm for computing high breakdown estimates in nonlinear regression that requires only a small number of least squares fits to p points the algorithm is used to compute rousseeuw s least median of squares lms estimate and yohai s mm estimate in both simulations and examples it is also used to compute bootstrapped and monte carlo standard error estimates for mm estimates which are compared with asymptotic standard errors ase s using the progress algorithm for a two parameter nonlinear model with sample size 30 would require finding the least squares fit to 435 two point subsets of the data in the settings considered in this article the proposed algorithm performs just as well with 25 as with 435

on bregman distances and divergences of probability measures this paper introduces scaled bregman distances of probability distributions which admit nonuniform contributions of observed events they are introduced in a general form covering not only the distances of discrete and continuous stochastic observations but also the distances of random processes and signals it is shown that the scaled bregman distances extend not only the classical ones studied in the previous literature but also the information divergence and the related wider class of convex divergences of probability measures an information processing theorem is established too but only in the sense of invariance w r t statistically sufficient transformations and not in the sense of universal monotonicity pathological situations where coding can increase the classical bregman distance are illustrated by a concrete example in addition to the classical areas of application of the bregman distances and convex divergences such as recognition classification learning and evaluation of proximity of various features and signals the paper mentions a new application in 3 d exploratory data analysis explicit expressions for the scaled bregman distances are obtained in general exponential families with concrete applications in the binomial poisson and rayleigh families and in the families of exponential processes such as the poisson and diffusion processes including the classical examples of the wiener process and geometric brownian motion

the accuracy of elemental set approximations for regression abstract the elemental set algorithm involves performing many fits to a data set each fit made to a subsample of size just large enough to estimate the parameters in the model elemental sets have been proposed as a computational device to approximate estimators in the areas of high breakdown regression and multivariate location scale estimation where exact optimization of the criterion function is computationally intractable although elemental set algorithms are used widely and for a variety of problems the quality of the approximation they give has not been studied this article shows that they provide excellent approximations for the least median of squares least trimmed squares and ordinary least squares criteria it is suggested that the approach likely will be equally effective in the other problem areas in which exact optimization of a criterion is difficult or impossible

unmasking multivariate outliers and leverage points comment detecting outliers in a multivariate point cloud is not trivial especially when there are several outliers the classical identification method does not always find them because it is based on the sample mean and covariance matrix which are themselves affected by the outliers to avoid this masking effect we propose to compute distances based on very robust estimates of location and covariance in the case of regression data the outliers also may be unmasked by using a highly robust regression method a new display is proposed in which the robust regression residuals are plotted versus the robust distances

new formulas for the legendre fenchel transform we present new formulas for the legendre fenchel transform of functions they concern the following three operations inverting a strictly monotone convex function post composing an arbitrary function with a strictly monotone concave function multiplying two positively valued strictly monotone convex functions

the impact of outliers and leverage points for technical efficiency measurement using high breakdown procedures given that most data used for production studies have not been accumulated for such purposes it is important that the quantitative tools for messy data which can affect the accuracy of computed technical efficiency measures be found in this study high breakdown robust methods are used in conjunction with a robust distance measure defined relative to the minimum volume ellipsoid estimator the standardized robust residuals from the high breakdown estimators and the robust distance measures are used to statistically and graphically depict both multivariate outliers and leverage points once these points are found their relationship to those observations that exhibit strong technically efficient or inefficient behavior scale inefficiency and or unusual production characteristics is analyzed for three linerboard manufacturing facilities additionally the impact of the outliers and leverage points on the estimated least squares coefficients which are used by the corrected ordinary least squares methodology to compute the full frontier technical efficiency measures is explored finally a sensitivity analysis of the impact of outliers and leverage points on the computed linear programming based technical efficiency measures is presented