rates of convergence in stochastic programs with complete integer recourse the stability of stochastic programs with mixed integer recourse and random right hand sides under perturbations of the integrating probability measure is considered from a quantitative viewpoint objective function values of perturbed stochastic programs are related to each other via a variational distance of probability measures based on a suitable vapnik cervonenkis class of borel sets in a euclidean space this leads to holder continuity of local optimal values in the context of estimation via empirical measures the general results imply qualitative and quantitative statements on the asymptotic convergence of local optimal values and optimal solutions

on the glivenko cantelli problem in stochastic programming linear recourse and extensions integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure under fairly broad conditions it is proved that uniform convergence of empirical approximations of the right hand sides of the constraints implies uniform convergence of the optimal values in the linear and convex case

nonlinear manifold representations for functional data for functional data lying on an unknown nonlinear low dimensional space we study manifold learning and introduce the notions of manifold mean manifold modes of functional variation and of functional manifold components these constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods which we modify to address functional data settings in simulations and applications we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross sectional mean and functional principal components we also include consistency proofs for our estimators under certain assumptions

synchronizing sample curves nonparametrically more and more often the outcome of a study is not a random variable but a noisy function for each experimental unit resulting in a sample of curves typically the individual curves vary not only in amplitude or intensity but also with respect to the time axis different subjects experience certain events sooner or later analyzing such data involves finding out the time changes or curve registration among curves following our previous work where modified dynamic time warping is applied to align two curves we formulate a global minimization problem to align all curves in a sample and to compute the aligned average curve algorithms for solving the minimization problem are presented and tested with simulated and real data the test results are promising the method which involves kernel smoothing of regression functions estimates the time changes and the average of the aligned curves from noisy data large sample asymptotics is derived

functional additive models in commonly used functional regression models the regression of a scalar or functional response on the functional predictor is assumed to be linear this means that the response is a linear function of the functional principal component scores of the predictor process we relax the linearity assumption and propose to replace it by an additive structure leading to a more widely applicable and much more flexible framework for functional regression models the proposed functional additive regression models are suitable for both scalar and functional responses the regularization needed for effective estimation of the regression parameter function is implemented through a projection on the eigenbasis of the covariance operator of the functional components in the model the use of functional principal components in an additive rather than linear way leads to substantial broadening of the scope of functional regression models and emerges as a natural approach because the uncorrelatedness of the functional pr

a stochastic approach to stability in stochastic programming abstract a random optimization problem p 0 min x 0 0 x is approximated by a sequence of random surrogate problems p n n n with p n min x n n x p a given probability space we investigate the convergence almost surely and in probability of the optimal values and the solution sets the results can be regarded as random versions of well known stability statements of parametric programming semicontinuous convergence almost surely in probability of sequences of random functions is a crucial assumption in this framework and will be investigated in more detail

lipschitz stability for stochastic programs with complete recourse this paper investigates the stability of optimal solution sets to stochastic programs with complete recourse where the underlying probability measure is understood as a parameter varying in some space of probability measures in math programming 67 1994 pp 99 108 sha piro proved lipschitz upper semicontinuity of the solution set mapping inspired by this result we introduce a subgradient distance for probability distributions and establish the persistence of optimal solutions for a subclass of recourse models we show that the solution set mapping is hausdorff lipschitz continuous with respect to the subgradient distance moreover the subgradient distance is estimated above by the kolmogorov smirnov distance of certain distribution functions related to the recourse model the lipschitz continuity result is illustrated by verifiable sufficient conditions for stochastic programs to belong to the mentioned subclass and by examples showing its validity and limitations finally the lipschitz continuity result is used to derive some new results on the asymptotic behavior ofoptimal solutions when the probability measure underlying the recourse model is estimated via empirical measures law of iterated logarithm large deviation estimate estimate for asymptotic distribution

on structure and stability in stochastic programs with random technology matrix and complete integer recourse for two stage stochastic programs with integrality constraints in the second stage we study continuity properties of the expected recourse as a function both of the first stage policy and the integrating probability measure

methodology and convergence rates for functional linear regression in functional linear regression the slope parameter is a function therefore in a nonparametric context it is determined by an infinite number of unknowns its estimation involves solving an ill posed problem and has points of contact with a range of methodologies including statistical smoothing and deconvolution the standard approach to estimating the slope function is based explicitly on functional principal components analysis and consequently on spectral decomposition in terms of eigenvalues and eigen functions we discuss this approach in detail and show that in certain circumstances optimal convergence rates are achieved by the pca technique an alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view

nonparametric maximum likelihood estimation of the structural mean of a sample of curves a random sample of curves can be usually thought of as noisy realisations of a compound stochastic process x t z w t where z t produces random amplitude variation and w t produces random dynamic or phase variation in most applications it is more important to estimate the so called structural mean t e z t than the crosssectional mean e x t but this estimation problem is difficult because the process z t is not directly observable in this paper we propose a nonparametric maximum likelihood estimator of t this estimator is shown to be surd n consistent and asymptotically normal under the assumed model and robust to model misspecification simulations and a realdata example show that the proposed estimator is competitive with landmark registration often considered the benchmark and has the advantage of avoiding time consuming and often infeasible individual landmark identification

on stability in multiobjective programming a stochastic approach in this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints making use of a large deviations approach we investigate the behaviour of the constraint sets the sets of efficient points and the solution sets if the size of the underlying sample tends to infinity the results are illustrated by applying them to stochastic programming with chance constraints where i the distribution function of the random variable is estimated by the empirical distribution function ii certain parameters have to be estimated

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

bayesian inference in models with euclidean structures abstract when a parameter space is endowed with a euclidean structure there is a euclidean measure called the structural or geometric prior which is an obvious candidate to represent ignorance concerning an unknown parameter the rationale for using this prior is strengthened when the unknown parameter is the canonical parameter of an exponential family then the structural prior is the uniform prior for the canonical parameter it is shown also that euclidean structures can be combined in different ways with group structures the structural or geometric prior is then the product of the invariant prior for the group structure and the structural or geometric prior for the euclidean structure

on rates of convergence and asymptotic normality in the multiknapsack problem in meanti et al 1990 an almost sure asymptotic characterization has been derived for the optimal solution value as function of the knapsack capacities when the profit and requirement coefficients of items to be selected from are random variables in this paper we establish a rate of convergence for this process using results from the theory of empirical processes

the topology of probability distributions on manifolds let mathcal p be a set of n random points in mathbb r d generated from a probability measure on a m dimensional manifold mathcal m subset mathbb r d in this paper we study the homology of mathcal u mathcal p r the union of d dimensional balls of radius r around mathcal p as n rightarrow infty and r rightarrow 0 in addition we study the critical points of d mathcal p the distance function from the set mathcal p these two objects are known to be related via morse theory we present limit theorems for the betti numbers of mathcal u mathcal p r as well as for number of critical points of index k for d mathcal p depending on how fast r decays to zero as n grows these two objects exhibit different types of limiting behavior in one particular case n r m ge c log n we show that the betti numbers of mathcal u mathcal p r perfectly recover the betti numbers of the original manifold mathcal m a result which is of significant interest in topological manifold learning

asymptotic behavior of optimal solutions in stochastic programming asymptotic behavior of optimal solutions xi n of a sequence of stochastic programming problems is studied variational and generalized equations approaches are discussed an expansion of xi n in terms of a parametrized mathematical programming problem depending on a single random vector is given when optimal solutions of the parametrized program are directionally differentiable this expansion leads to a close form expression for the asymptotic distribution of xi n applicability of the involved regularity conditions to nondifferentiable cases and in particular to stochastic programming with recourse is discussed

stability of varepsilon approximate solutions to convex stochastic programs an analysis of convex stochastic programs is provided when the underlying probability distribution is subjected to small perturbations it is shown in particular that varepsilon approximate solution sets of convex stochastic programs behave lipschitz continuously with respect to certain distances of probability distributions that are generated by the relevant integrands it is shown that these results apply to linear two stage stochastic programs with random recourse we discuss the consequences on associating fortet mourier metrics to two stage models and on the asymptotic behavior of empirical estimates of such models respectively

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

statistical tools to analyze data representing a sample of curves the paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological chemical ldots process such samples of curves often possess the following features there is a typical structural pattern common to all curves of the sample on the other hand individual realizations of the typical shape show different dynamics and intensity in particular typical peaks are shifted from individual to individual differences in dynamics complicate the analysis of samples of curves for example the cross sectional average usually does not reflect an average pattern due to shifts structure is smeared or might even disappear our approach consists in synchronizing the individual curves before determining the average or any further statistics pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity the method requires the introduction of new statistical objects they are defined mathematically their properties are discussed and possible estimators are proposed the asymptotic bias and variance of the estimators are derived an application to visually evoked brain potentials illustrates the approach

stability results for stochastic programs and sensors allowing for discontinuous objective functions the paper examines the stability of the optimal value and the solutions of stochastic programming problems stability is checked with respect to variations in both the problem formulation and the probability distribution that describes the uncertainty of particular interest is the case where the payoff functions may be discontinuous these results are applied to analyze the stability of sensors that model the possibility of making inquiries to improve the probabilistic information available about the uncertain quantities

asymptotic theory for solutions in statistical estimation and stochastic programming new techniques of local sensitivity analysis for nonsmooth generalized equations are applied to the study of sequences of statistical estimates and empirical approximations to solutions of stochastic programs consistency is shown to follow from a certain local invertibility property and asymptotic distributions are derived from a generalized implicit function theorem that characterizes asymptotic behavior in situations where estimates are subjected to constraints and estimation functionals are nonsmooth

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

asymptotic behavior of solutions an application to stochastic nlp in this article we study the consistency of optimal and stationary kkt points of a stochastic non linear optimization problem involving expectation functionals when the underlying probability distribution associated with the random variable is weakly approximated by a sequence of random probability measures the optimization model includes constraints with expectation functionals those are not captured in direct application of the previous results on optimality conditions exist in the literature we first study the consistency of stationary points of a general nlp problem with convex and locally lipschitz data and then apply those results to the stochastic nlp problem and stochastic minimax problem moreover we derive an exponential bound for such approximations using a large deviation principle

the glivenko cantelli problem we give a new type of characterization of the glivenko cantelli classes in the case of a class mathscr l of sets the characterization is closely related to the configuration that the sets of mathscr l can have it allows one to decide simply whether a given class is a glivenko cantelli class the characterization is based on a new measure theoretic analysis of sets of measurable functions this analysis also gives an approximation theorem for glivenko cantelli classes sharpenings of the vapnik cervonenkis criteria and the value of the asymptotic discrepancy for classes that are not glivenko cantelli an application is given to the law of large numbers in a banach space for functions that need not be random variables

local epi continuity and local optimization one of the fundamental questions in nonlinear optimization is how optimization problems behave when the functions defining them change e g by continuous deformation recently the study of epi continuity has somewhat unified the results in this area here we show how to localize the concept of epi continuity and how to apply these localized ideas to ensure persistence and stability of local optimizing sets we also show how these conditions follow from known properties of nonlinear programming problems

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

inference for density families using functional principal component analysis we consider t 1 t samples of iid observations x1t xntt from unknown population densities ft to characterize differences and similarities of ft we assume their expansions into the first l principal components from the given observations xit we study inference on the components and on their required number l a detailed asymptotic theory is presented our method is applied in the analysis of yearly cross sectional samples of british households interpretation of the estimated principal components and their scores provides new insights into the evolution and interplay of household income and age distributions from 1968 1988 from estimating their required numbersl we draw conclusions on the dimensionality of mixture models for describing the densities

principal modes of variation for processes with continuous sample curves analysis of a process with continuous sample curves can be carried out in a manner similar to principal components analysis of vector processes by appropriate definition of a best linear model in the continuous case we show that principal modes of variation consist of eigenfunctions of the process covariance function c s t procedures for estimation of these eigenfunctions from a finite sample of observed curves are given and results are compared with principal components analysis of the same data

self modelling warping functions the paper introduces a semiparametric model for functional data the warping functions are assumed to be linear combinations ofqcommon components which are estimated from the data hence the name self modelling even small values ofqprovide remarkable model flexibility comparable with nonparametric methods at the same time this approach avoids overfitting because the common components are estimated combining data across individuals as a convenient by product component scores are often interpretable and can be used for statistical inference an example of classification based on scores is given abstract from author

stability in two stage stochastic programming we analyze the effect of changes in problem functions and or distributions in certain two stage stochastic programming problems with recourse under reasonable assumptions the locally optimal value of the perturbed problem will be continuous and the corresponding set of local optimizers will be upper semicontinuous with respect to the parameters including the probability distribution in the second stage