on degrees of freedom of projection estimators with applications to multivariate nonparametric regression abstract in this article we consider the nonparametric regression problem with multivariate predictors we provide a characterization of the degrees of freedom and divergence for estimators of the

effective degrees of freedom a flawed metaphor to most applied statisticians a fitting procedure s degrees of freedom is synonymous with its model complexity or its capacity for overfitting to data in particular it is often used to parameterize the bias variance tradeoff in model selection we argue that on the contrary model complexity and degrees of freedom may correspond very poorly we exhibit and theoretically explore various fitting procedures for which degrees of freedom is not monotonic in the model complexity parameter and can exceed the total dimension of the ambient space even in very simple settings we show that the degrees of freedom for any non convex projection method can be unbounded

the structure of symmetric attractors we consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems we show for example that the symmetry of an attractor cannot in general be an arbitrary subgroup of the group of symmetries in addition there are group theoretic restrictions on the symmetry of connected components of a symmetric attractor the symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time average of a chaotic dynamical system

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

on the existence and weak stability of solutions to the compressible semigeostrophic equations abstract in this article existence and weak stability of solutions to the compressible semigeostrophic equations are studied both in physical and in dual spaces we present a proof of the existence of a weak solution to these equations in dual space assuming initial data in l 1 r 3 extending previous results cullen and maroofi 2003 8 an extension to the concept of lagrangian solutions due to cullen and feldman 2006 6 is presented to the compressible semigeostrophic system and it is shown the existence of lagrangian solutions in physical space with initial potential vorticities in l 1 as well a weak stability result with respect to perturbations of the potential vorticity in l 1

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation

random calibration with many measurements an application of stein estimation in the problem considered a vector of many imprecise measurements e g spectroscopic is used to linearly predict a quantity whose precise measurement is difficult or expensive the regression vector is estimated from a calibration experiment having both types of measurements for a random sample most previous approaches to this problem adjust for approximate multicollinearity which often results from the correlations among the imprecise measurements by inverting an approximation e g factor analytic to their covariance matrix in the approach here it is argued that the regression vector should lie in a lower dimensional suhspace determined by the principal components of the covariance matrix it is then estimated by applying a stein contraction of the least squares estimator to the principal components regression estimator examples using real data are presented in which the proposed estimator substantially improves on the ordinary least squares estimation

a variational approach for the 2 dimensional semi geostrophic shallow water equations existence of weak solutions to the 3 d semi geostrophic equations with rigid boundaries was proved by benamou and brenier 3 using monge transport theory this paper extends the results to a free surface boundary condition which is more physically appropriate this extension is at present for the 2 d shallow water case only in addition we establish stronger time regularity than was possible in 3

holder regularity of the 2d dual semigeostrophic equations via analysis of linearized monge ampere equations we obtain the holder regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity our main tool is an interior holder estimate in two dimensions for an inhomogeneous linearized monge ampere equation with right hand side being the divergence of a bounded vector field as a further application of our holder estimate we prove the holder regularity of the polar factorization for time dependent maps in two dimensions with densities bounded away from zero and infinity our applications improve previous work by g loeper who considered the cases of densities sufficiently close to a positive constant

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

how many variables should be entered in a regression equation abstract the optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points as the number of regressors grows large the sp criterion provides an asymptotically optimal rule for the number of variables to enter

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

the semigeostrophic equations discretized in reference and dual variables we study the evolution of a system of n particles x i v i i 1 n in mathbb r 2d that system is a conservative system with a hamiltonian of the form h mu w 2 2 mu nu n where w2 is the wasserstein distance and is a discrete measure concentrated on the set x i v i i 1 n typically 0 is a discrete measure approximating an initial l density and can be chosen randomly when d 1 our results prove convergence of the discrete system to a variant of the semigeostrophic equations we obtain that the limiting densities are absolutely continuous with respect to the lebesgue measure when nu n n 1 infty converges to a measure concentrated on a special d dimensional set we obtain the vlasov monge ampere vma system when d 1 the vma system coincides with the standard vlasov poisson system

hamiltonian odes in the wasserstein space of probability measures in this paper we consider a hamiltonian h on p 2 r 2d the set of probability measures with finite quadratic moments on the phase space r 2d r d r d which is a metric space when endowed with the wasserstein distance w 2 we study the initial value problem d t dt j d v t t 0 where j d is the canonical symplectic matrix 0 is prescribed and v t is a tangent vector to p 2 r 2d at t belonging to h t the subdifferential of h at t two methods for constructing solutions of the evolutive system are provided the first one concerns only the case where 0 is absolutely continuous it ensures that fit remains absolutely continuous and v t h t is the element of minimal norm in h t the second method handles any initial measure 0 if we further assume that h is convex proper and lower semicontinuous on p 2 r 2d we prove that the hamiltonian is preserved along any solution of our evolutive system h t h 0

a formula for a symmetry detective detectives are used to determine the symmetry of attractors in finite dimensional systems we derive an explicit formula for a symmetry detective for each representation of a finite group

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

when does more regularization imply fewer degrees of freedom sufficient conditions and counterexamples regularization aims to improve prediction performance by trading an increase in training error for better agreement between training and prediction errors which is often captured through decreased degrees of freedom in this paper we give examples which show that regularization can increase the degrees of freedom in common models including the lasso and ridge regression in such situations both training error and degrees of freedom increase making the regularization inherently without merit two important scenarios are described where the expected reduction in degrees of freedom is guaranteed all symmetric linear smoothers and convex constrained linear regression models like ridge regression and the lasso when compared to unconstrained linear regression

phase space reconstruction for symmetric dynamical systems abstract since the pioneering work of packard et al and takens it has become customary to reconstruct the topology of attractors in phase space from a time series of one dimensional experimental observations by using delay coordinates many practical refinements of the original methods have been developed many experimental systems possess symmetry and bifurcations can cause changes in the symmetry of observed states these changes are quite subtle when the dynamics is chaotic it is therefore important to reconstruct not just the topology of the attractor but its symmetry we indicate how this can be done by extending the packard takens approach to a single equivariant observation taking values not in the real numbers r but in a linear representation v of the symmetry group g in effect a single set of symmetrically related observations is required our central point is that not all plausible choices for such a set can generate embeddings in order for the method to produce an embedding it is necessary that v should be sufficiently complicated more precisely the phase space m must be subordinate to v in a sense introduced by wassermann this concept is technical but unavoidable in this context and it greatly clarifies the issue of embeddability using it we state a symmetric version of the takens embedding theorem and sketch the proof we also discuss the issue of setwise versus pointwise symmetry of an attractor and relate this to the transition from spatial order to spatial disorder in temporally chaotic systems

better subset regression using the nonnegative garrote a new method called the nonnegative nn garrote is proposed for doing subset regression it both shrinks and zeroes coefficients in tests on real and simulated data it produces lower prediction error than ordinary subset selection it is also compared to ridge regression if the regression equations generated by a procedure do not change drastically with small changes in the data the procedure is called stable subset selection is unstable ridge is very stable and the nn garrote is intermediate simulation results illustrate the effects of instability on prediction error

inverse groundwater modeling for hydraulic conductivity estimation using bayesian model averaging and variance window 1 this study proposes a bayesian model averaging bma method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods bma is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates two major issues in applying bma to hydraulic conductivity estimation are discussed the first problem is using occam s window in usual bma applications to measure approximated posterior model probabilities occam s window only accepts models in a very narrow range tending to single out the best method and discard other good methods we propose a variance window to replace occam s window to cope with this problem the second problem is the kashyap information criterion kic in the approximated posterior model probabilities which tends to prefer highly uncertain parameterization methods by considering the fisher information matrix with sufficient amounts of observation data the bayesian information criterion bic is a good approximation and is able to avoid controversial results from using kic this study adopts multiple generalized parameterization gp methods such as the bma models to estimate spatially correlated hydraulic conductivity numerical examples illustrate the issues of using kic and occam s window and show the advantages of using bic and the variance window in bma application finally we apply bma to the hydraulic conductivity estimation of the 1500 foot sand in east baton rouge parish louisiana

range image segmentation using surface selection criterion in this paper we address the problem of recovering the true underlying model of a surface while performing the segmentation first and in order to solve the model selection problem we introduce a novel criterion which is based on minimising strain energy of fitted surfaces we then evaluate its performance and compare it with many other existing model selection techniques using this criterion we then present a robust range data segmentation algorithm capable of segmenting complex objects with planar and curved surfaces the presented algorithm simultaneously identifies the type order and geometric shape of each surface and separates all the points that are part of that surface this paper includes the segmentation results of a large collection of range images obtained from objects with planar and curved surfaces the resulting segmentation algorithm successfully segments various possible types of curved objects more importantly the new technique is capable of detecting the association between separated parts of a surface which has the same cartesian equation while segmenting a scene this aspect is very useful in some industrial applications of range data analysis

detection of symmetry of attractors from observations i theory abstract barany et al 1993 proposed a method for determining the symmetries of attractors of equivariant systems by averaging certain classes of equivariant maps we use an idea in barany et al to re cast definitions of symmetry detectives assuming that we only have access to equivariant observations from the system detecting from observations allows one to perform averaging in spaces that may have much lower dimension than the phase space this paper generalises and develops their suggestion among the generalisations we consider is the use of nonpolynomial detectives and we show using the notion of prevalence of hunt et al 1992 that our detectives and the detectives of barany et al 1993 dellnitz et al 1994 and golubitsky and nicol 1995 give the correct symmetry of attractors almost certainly in a measure theoretic sense we show that detectives can persistently give incorrect symmetries at isolated points in parametrised systems and discuss how to overcome this we show how one can find the symmetry of an attractor from examination of a poincare section in part ii of this article ashwin and tomes apply these results to find symmetries of attractors in a physical system of four coupled electronic oscillators with s4 symmetry

regularized liml for many instruments the use of many moment conditions improves the asymptotic efficiency of the instrumental variables estimators however in finite samples the inclusion of an excessive number of moments increases the bias to solve this problem we propose regularized versions of the limited information maximum likelihood liml based on three different regularizations tikhonov landweber fridman and principal components our estimators are consistent and asymptotically normal under heteroskedastic error moreover they reach the semiparametric efficiency bound assuming homoskedastic error we show that the regularized liml estimators possess finite moments when the sample size is large enough the higher order expansion of the mean square error mse shows the dominance of regularized liml over regularized two staged least squares estimators we devise a data driven selection of the regularization parameter based on the approximate mse a monte carlo study and two empirical applications illustrate the relevance of our estimators

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

analysis of the semi geostrophic shallow water equations abstract the semi geostrophic shallow water equations are an accurate approximation to the rotating shallow water equations on scales larger than the rossby radius of deformation the global existence of weak solutions of the lagrangian form of the equations is known however uniqueness is not known except for short times for smooth initial data similarly a rigorous proof of existence of solutions has not been achieved for variable rotation rate which is important for atmospheric applications the obstacle is the lack of regularity of solutions to the monge ampere equation which underlies the solution procedure in this note it is shown formally that the modified form of the monge ampere equation that appears in the solution procedure for the semi geostrophic shallow water equations is better controlled on large scales than that of the standard equation that is used in the incompressible case this suggests that more progress should be possible with the rigorous theory and in particular that it should be possible to prove uniqueness

detecting the symmetry of attractors abstract this paper addresses the issue of how to determine numerically the symmetry of an attractor for dynamical systems the symmetries of attractors in phase space are related to patterns in the time average of the solution our approach to this question proceeds in two parts first we prove a general theorem based on group theoretic and differential topological ideas which states that generically the symmetry of a thickened attractor can be computed from the symmetries of a point in an auxilliary space this theorem proceeds by integrating an equivariant mapping over the thickened attractor once this is done the numerical computation of symmetries reduces to showing that a certain nonnegative number is zero numerically demonstrating that this number is zero can be difficult thus the second part of the algorithm is to consider how this number varies with parameters and noting that sudden jumps towards zero can be associated with increases in symmetry the paper is divided into two parts in the first we prove the general theorem and in the second we illustrate how the numerical techniques work on several examples including discrete dynamical systems with tetrahedral symmetry in r 3 and systems of three coupled cells in high dimensions the integral mentioned previously is difficult to compute for such examples we assume that an ergodic theorem is valid and that symmetries can be computed using a time average we compare both of these methods on the low dimensional examples as well as detect points of symmetry creation for a reaction diffusion equation on an interval this technique can also be used in principle to compute the symmetries of an attractor in an experiment from a time series

symmetry groups of attractors for maps equivariant under the action of a finite group on n the possible symmetries of fixed points are known and correspond to the isotropy subgroups this paper investigates the possible symmetries of arbitrary possibly chaotic attractors and finds that the necessary conditions of melbourne dellnitz golubitsky 15 are sufficient at least for continuous maps

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

on the degrees of freedom of the lasso we study the effective degrees of freedom of the lasso in the framework of stein s unbiased risk estimation sure we show that the number of nonzero coefficients is an unbiased estimate forthe degrees of freedom of the lasso a conclusion that requires no special assumption on the predictors in addition the unbiased estimator is shown to be asymptotically consistent with these results on hand various model selection criteria cp aic and bic are available which along with the lars algorithm provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least squares fit

degrees of freedom in lasso problems we derive the degrees of freedom of the lasso fit placing no assumptions on the predictor matrix x like the well known result of zou et al 2007 which gives the degrees of freedom of the lasso fit when x has full column rank we express our result in terms of the active set of a lasso solution we extend this result to cover the degrees of freedom of the generalized lasso fit for an arbitrary predictor matrix x and an arbitrary penalty matrix d though our focus is degrees of freedom we establish some intermediate results on the lasso and generalized lasso that may be interesting on their own

moreau yosida approximation and convergence of hamiltonian systems on wasserstein space abstract in this paper we study the stability property of hamiltonian systems on the wasserstein space let h be a given hamiltonian satisfying certain properties we regularize h using the moreau yosida approximation and denote it by h we show that solutions of the hamiltonian system for h converge to a solution of the hamiltonian system for h as converges to zero we provide sufficient conditions on h to carry out this process

two nonparametric methods for identifying the impulse response of linear systems this paper addresses the problem of impulse response identification using nonparametric methods although the techniques developed herein apply to the truncated untruncated and circulant models we focus on the truncated model which is useful in certain applications two methods of impulse response identification will be presented the first is based on the minimization of the c l statistic which is an estimate of the mean square prediction error the second is a bayesian approach for both of these methods we consider the effects of using both the identity matrix and the laplacian matrix as weights on the energy in the impulse response in addition we present a method for estimating the effective length of the impulse response estimating the length is particularly important in the truncated case finally we develop a method for estimating the noise variance at the output which is needed in the approach involving the c l statistic often prior information on the noise variance is not available and a good estimate is crucial to the success of estimating the impulse response with a nonparametric technique

toward optimal model averaging in regression models with time series errors abstract consider a regression model with infinitely many parameters and time series errors we are interested in choosing weights for averaging across generalized least squares gls estimators obtained from a set of approximating models however gls estimators depending on the unknown inverse covariance matrix of the errors are usually infeasible we therefore construct feasible generalized least squares fgls estimators using a consistent estimator of the unknown inverse matrix based on this inverse covariance matrix estimator and fgls estimators we develop a feasible autocovariance corrected mallows model averaging criterion to select weights thereby providing an fgls model averaging estimator of the true regression function we show that the generalized squared error loss of our averaging estimator is asymptotically equivalent to the minimum one among those of gls model averaging estimators with the weight vectors belonging to a continuous set which includes the discrete weight set used in hansen 2007 as its proper subset

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

the stability principle and global weak solutions of the free surface semi geostrophic equations in geostrophic coordinates the semi geostrophic equations are used widely in the modelling of large scale atmospheric flows in this note we prove the global existence of weak solutions of the incompressible semi geostrophic equations in geostrophic coordinates in a three dimensional domain with a free upper boundary the proof based on an energy minimization argument originally inspired by the stability principle as studied by cullen purser and others uses optimal transport techniques as well as the analysis of hamiltonian odes in spaces of probability measures as studied by ambrosio and gangbo we also give a general formulation of the stability principle in a rigorous mathematical framework

detecting symmetry breaking bifurcations in the system describing the dynamics of coupled arrays of josephson junctions symmetry breaking and symmetry increasing bifurcations in a system of globally coupled josephson junctions which possesses sn symmetry have been investigated by constructing a detective for the group sn the permutation group of n elements in addition with help of this detective several bifurcations are followed numerically in two parameters an algorithm to obtain a detective for any finite subgroup of o n is given

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

excess optimism how biased is the apparent error of an estimator tuned by sure abstractnearly all estimators in statistical prediction come with an associated tuning parameter in one way or another common practice given data is to choose the tuning parameter value that minimizes a constructed estimate of the prediction error of the estimator we focus on stein s unbiased risk estimator or sure which forms an unbiased estimate of the prediction error by augmenting the observed training error with an estimate of the degrees of freedom of the estimator parameter tuning via sure minimization has been advocated by many authors in a wide variety of problem settings and in general it is natural to ask what is the prediction error of the sure tuned estimator an obvious strategy would be simply use the apparent error estimate as reported by sure that is the value of the sure criterion at its minimum to estimate the prediction error of the sure tuned estimator but this is no longer unbiased in fact we would expect that the minimum of the sure criterion is systematically bias

oracle inequalities for inverse problems we consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations let a finite set a of linear estimators be given our aim is to mimic the estimator in a that has the smallest risk on the true f under general conditions we show that this can be achieved by simple minimization of an unbiased risk estimator provided the singular values of the operator of the inverse problem decrease as a power law the main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact this inequality can also be used to obtain sharp minimax adaptive results in particular we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures

pointwise influence matrices for functional response regression we extend the notion of an influence or hat matrix to regression with functional responses and scalar predictors for responses depending linearly on a set of predictors our definition is shown to reduce to the conventional influence matrix for linear models the pointwise degrees of freedom the trace of the pointwise influence matrix are shown to have an adaptivity property that motivates a two step bivariate smoother for modeling nonlinear dependence on a single predictor this procedure adapts to varying complexity of the nonlinear model at different locations along the function and thereby achieves better performance than competing tensor product smoothers in an analysis of the development of white matter microstructure in the brain

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

piecewise polynomial estimation of a regression function we deal with the problem of choosing a piecewise polynomial estimator of a regression function s mapping 0 1 p into r in a first part of this paper we consider some collection of piecewise polynomial models each model is defined by a partition m of 0 1 p and a series of degrees d d j j m nm we propose a penalized least squares criterion which selects a model whose associated piecewise polynomial estimator performs approximately as well as the best one in the sense that its quadratic risk is close to the infimum of the risks the risk bound we provide is nonasymptotic in a second part we apply this result to tree structured collections of partitions which look like the one constructed in the first step of the cart algorithm and we propose an extension of the cart algorithm to build a piecewise polynomial estimator of a regression function

on the semi geostrophic system in physical space with general initial data in order to accommodate general initial data an appropriately relaxed notion of renormalized lagrangian solutions for the semi geostrophic system in physical space is introduced this is shown to be consistent with previous notions generalizing them a weak stability result is obtained first followed by a general existence result whose proof employs said stability and approximating solutions with regular initial data the renormalization property ensures the return from physical to dual space as consequences we get conservation of hamiltonian energy and some weak time regularity of solutions

why bandwidth selectors tend to choose smaller bandwidths and a remedy summary this note is concerned with the problem of automatic data driven bandwidth selectors for nonparametric regression though some selectors were shown to be consistent and asymptotically unbiased by rice 1984 and hardle hall marron 1988 it is often observed in simulation studies that most selectors are biased toward undersmoothing an explanation for this is given the source of the large sample variation in the bandwidth estimates is also pointed out this leads to the consideration of a new procedure which is a simple modification of a classical selector a simulation study suggests that the proposed selector is much more consistent than the classical one

relative rates of convergence for efficient model selection criteria in linear regression summary two approaches to estimating a smooth regression function not specified by a finite number of parameters are i to fit a parametric model to the data with the number of parameters selected by the akaike information criterion aic and ii to use nonparametric smoothing techniques with the smoothing parameter selected by cross validation or some other automatic method we consider the relative rate of convergence of the mean integrated squared error of the selected estimator compared to the best possible estimator in the class of candidates under consideration we extend results of shibata 1981 to show that the relative rate of convergence using aic in the parametric case can be as fast as op n for arbitrary positive d we also show that this rate can be attained in the special cases of polynomial and trigonometric regression

detection of symmetry of attractors from observations ii an experiment with s 4 symmetry abstract in part i of this paper we adapt the detectives of barany et al 1993 for symmetries of attractors of symmetric dynamical systems to apply to experiments as an illustration of this theory we consider the dynamics of a system of four forced coupled electronic oscillators with full permutation symmetry s4 on varying parameters in the circuit we obtain attractors with many different symmetry types and detect these using the methods described in part i

optimal bounds for aggregation of affine estimators we study the problem of aggregation of estimators when the estimators are not independent of the data used for aggregation and no sample splitting is allowed if the estimators are deterministic vectors it is well known that the minimax rate of aggregation is of order log m where m is the number of estimators to aggregate it is proved that for affine estimators the minimax rate of aggregation is unchanged it is possible to handle the linear dependence between the affine estimators and the data used for aggregation at no extra cost the minimax rate is not impacted either by the variance of the affine estimators or any other measure of their statistical complexity the minimax rate is attained with a penalized procedure over the convex hull of the estimators for a penalty that is inspired from the q aggregation procedure the results follow from the interplay between the penalty strong convexity and concentration

parametric or nonparametric a parametricness index for model selection in model selection literature two classes of criteria perform well asymptotically in different situations bayesian information criterion bic as a representative is consistent in selection when the true model is finite dimensional parametric scenario akaike s information criterion aic performs well in an asymptotic efficiency when the true model is infinite dimensional nonparametric scenario but there is little work that addresses if it is possible and how to detect the situation that a specific model selection problem is in in this work we differentiate the two scenarios theoretically under some conditions we develop a measure parametricness index pi to assess whether a model selected by a potentially consistent procedure can be practically treated as the true model which also hints on aic or bic is better suited for the data for the goal of estimating the regression function a consequence is that by switching between aic and bic based on the pi the resulting regression estimator is simultaneously asymptotically efficient for both parametric and nonparametric scenarios in addition we systematically investigate the behaviors of pi in simulation and real data and show its usefulness

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

nonparametric variable selection the earth algorithm we consider regression experiments involving a response variable y and a large number of predictor variables x1 xd many of which may be irrelevant for the prediction of y and thus must be removed before y can be predicted from the x s we consider two procedures that select variables by using importance scores that measure the strength of the relationship between predictor variables and a response and keep those variables whose importance scores exceed a threshold in the first of these procedures scores are obtained by randomly drawing subregions tubes of the predictor space that constrain all but one predictor and in each subregion computing a signal to noise ratio efficacy based on a nonparametric univariate regression of y on the unconstrained variable the subregions are adapted to boost weak variables iteratively by searching hunting for the subregions in which the efficacy is maximized the efficacy can be viewed as an approximation to a one to one function of the probability of identif

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

observing the symmetry of attractors abstract we show how the symmetry of attractors of equivariant dynamical systems can be observed by equivariant projections of the phase space equivariant projections have long been used but they can give misleading results if used improperly and have been considered untrustworthy we find conditions under which an equivariant projection generically shows the correct symmetry of the attractor

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

linear model selection by cross validation abstract we consider the problem of selecting a model having the best predictive ability among a class of linear models the popular leave one out cross validation method which is asymptotically equivalent to many other model selection methods such as the akaike information criterion aic the c p and the bootstrap is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observations n we show that the inconsistency of the leave one out cross validation can be rectified by using a leave n v out cross validation with n v the number of observations reserved for validation satisfying n v n 1 as n this is a somewhat shocking discovery because nv n 1 is totally opposite to the popular leave one out recipe in cross validation motivations justifications and discussions of some practical aspects of the use of the leave n v out cross validation method are provided and results

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159