probability estimation in the rare events regime we address the problem of estimating the probability of an observed string that is drawn i i d from an unknown distribution motivated by models of natural language we consider the regime in which the length of the observed string and the size of the underlying alphabet are comparably large in this regime the maximum likelihood distribution tends to overestimate the probability of the observed letters so the good turing probability estimator is typically used instead we show that when used to estimate the sequence probability the good turing estimator is not consistent in this regime we then introduce a novel sequence probability estimator that is consistent this estimator also yields consistent estimators for other quantities of interest and a consistent universal classifier

nonparametric estimation of conditional quantile functions in the presence of irrelevant covariates abstract allowing for the existence of irrelevant covariates we study the problem of estimating a conditional quantile function nonparametrically with mixed discrete and continuous data we estimate the conditional quantile regression function using the check function based kernel method and suggest a data driven cross validation cv approach to simultaneously determine the optimal smoothing parameters and remove the irrelevant covariates when the number of covariates is large we first use a screening method to remove the irrelevant covariates and then apply the cv criterion to those that survive the screening procedure simulations and an empirical application demonstrate the usefulness of the proposed methods

numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

asymptotic normality of a nonparametric estimator of sample coverage this paper establishes a necessary and sufficient condition for the asymptotic normality of the nonparametric estimator of sample coverage proposed by good biometrica 40 1953 237 264 this new necessary and sufficient condition extends the validity of the asymptotic normality beyond the previously proven cases

a global jacobian method for mortar discretizations of nonlinear porous media flows we describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method there are two main ideas 1 linearize the global system in both subdomain and interface variables simultaneously to yield a single newton iteration and 2 algebraically eliminate subdomain velocities and optionally subdomain pressures to solve linear systems for the 1st or the 2nd schur complements solving the 1st schur complement system gives the multiscale solution without the need to solve an interface iteration solving the 2nd schur complement system gives a linear interface problem for a nonlinear model the methods are less complex than a previously developed nonlinear mortar algorithm which requires two nested newton iterations and a forward difference approximation furthermore efficient linear preconditioners can be applied to speed up the iteration the methods are implemented in parallel and a numerical study is performed

multiscale seamless domain method for solving nonlinear problems using statistical estimation methodology this article presents a nonlinear solver combining regression analysis and a multiscale simulation scheme first the proposed method repeats microscopic analysis of a local simulation domain which is extracted from the entire global domain to statistically estimate the relation s between the value of a dependent variable at a point and values at surrounding points the relation is called regression function subsequent global analysis reveals the behavior of the global domain with only coarse grained points using the regression function quickly at low computational cost which can be accomplished using a multiscale numerical solver called the seamless domain method sdm the objective of the study is to solve a nonlinear problem accurately and at low cost by combining the two techniques we present an example problem of a nonlinear steady state heat conduction analysis of a heterogeneous material the proposed model using fewer than 1000 points generates a solution with precision similar to that of a standard finite element solution using hundreds of thousands of nodes to investigate the relationship between the accuracy and computational time we apply the sdm under varying conditions such as the number of iterations of the prior analysis for statistical data learning

estimating functions of probability distributions from a finite set of samples this paper addresses the problem of estimating a function of a probability distribution from a finite set of samples of that distribution a bayesian analysis of this problem is presented the optimal properties of the bayes estimators are discussed and as an example of the formalism closed form expressions for the bayes estimators for the moments of the shannon entropy function are derived then numerical results are presented that compare the bayes estimator to the frequency counts estimator for the shannon entropy we also present the closed form estimators all derived elsewhere for the mutual information mathrm ensuremath chi 2 covariance and some other statistics c 1995 the american physical society

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

a local error estimate of the method of multi scale asymptotic expansions for elliptic problems with rapidly oscillatory coefficients in this paper on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 we present a local error estimate of the method of multi scale asymptotic expansions for second order elliptic problems with rapidly oscillatory coefficients

investigation on the high order approximation of the entropy bias abstract the estimation of entropy from experimental data has a considerable bias when the discretization of the variable domain is comparable to the sample size in this case the source of the bias is the difference between the a priori distribution and the observed distribution from sampled data in this paper we estimate the entropy bias considering an infinite sum of central moments of the binomial distribution using two probability mass functions we analyze the bias in the light of the ratio between the number of the partition of the domain and the sample size the main motivation of this study is improving statistical hypothesis testing in which probabilities are conceived beforehand we examine the adequacy of high order approximation according to the ratio between the sample size and the number of domain partitions finally we expand the analysis to the entropy derived mutual information and present an application for network reconstruction

dynamic analyses of information encoding in neural ensembles neural spike train decoding algorithms and techniques to compute shannon mutual information are important methods for analyzing how neural systems represent biological signals decoding algorithms are also one of several strategies being used to design controls for brain machine interfaces developing optimal strategies to desig n decoding algorithms and compute mutual information are therefore important problems in computational neuroscience we present a general recursive filter decoding algorithm based on a point process model of individual neuron spiking activity and a linear stochastic state space model of the biological signal we derive from the algorithm new instantaneous estimates of the entropy entropy rate and the mutual information between the signal and the ensemble spiking activity we assess the accuracy of the algorithm by computing along with the decoding error the true coverage probability of the approximate 0 95 confidence regions for the individual signal estimates we illustrate the new algorithm by reanalyzing the position and ensemble neural spiking activity of ca1 hippocampal neurons from two rats foraging in an open circular environment we compare the performance of this algorithm with a linear filter constructed by the widely used reverse correlation method the median decoding error for animal 1 2 during 10 minutes of open foraging was 5 9 5 5 cm the median entropy was 6 9 7 0 bits the median information was 9 4 9 4 bits and the true coverage probability for 0 95 confidence regions was 0 67 0 75 using 34 32 neurons these findings improve significantly on our previous results and suggest an integrated approach to dynamically reading neural codes measuring their properties and quantifying the accuracy with which encoded information is extracted

testing the assumptions of linear prediction analysis in normal vowels in this paper we develop an improved surrogate data test to show experimental evidence for all the simple vowels of u s english for both male and female speakers that gaussian linear prediction analysis a ubiquitous technique in current speech technologies cannot be used to extract all the dynamical structure of real speech time series the test provides robust evidence undermining the validity of these linear techniques supporting the assumptions of either dynamical nonlinearity and or non gaussianity common to more recent complex efforts at dynamical modeling speech time series however an additional finding is that the classical assumptions cannot be ruled out entirely and plausible evidence is given to explain the success of the linear gaussian theory as a weak approximation to the true nonlinear non gaussian dynamics this supports the use of appropriate hybrid linear nonlinear non gaussian modeling with a calibrated calculation of statistic and particular choice of experimental protocol some of the known systematic problems of the method of surrogate data testing are circumvented to obtain results to support the conclusions to a high level of significance

a new approximate method for second order elliptic problems with rapidly oscillating coefficients based on the method of multiscale asymptotic expansions abstract in this paper we consider second order elliptic problems with rapidly oscillating coefficients on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 wen ming he jun zhi cui a pointwise estimate on the 1 order approximation of g x 0 e ima j appl math 70 2005 241 269 we propose a new approximate method to solve these problems of course we present its error estimate

application of the finite difference heterogeneous multiscale method to the richards equation 1 this paper extends the finite difference heterogeneous multiscale method fdhmm to simulate transient unsaturated water flow problems in random porous media the numerical method is based on the use of two different schemes for the original equation at different grid levels which allows numerical results at a lower cost than solving the original equations the main feature of fdhmm is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain the generated code is verified by applying the linearization model of the richards equation considering two different constitutive relationships this method is applied to several test examples with different soil textures and boundary conditions both the dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by fdhmm the numerical experiments demonstrate that fdhmm can effectively simulate the transient unsaturated water flow in the specific soils the numerical experiments also demonstrate that fdhmm can achieve accurate global mass balance and is a globally convergent algorithm and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method

false discovery rate control for high dimensional networks of quantile associations conditioning on covariates motivated by gene coexpression pattern analysis we propose a novel sample quantile contingency squac statistic to infer quantile associations conditioning on covariates it features enhanced flexibility in handling variables with both arbitrary distributions and complex association patterns conditioning on covariates we first derive its asymptotic null distribution and then develop a multiple testing procedure based on the squac statistic to test simultaneously the independence between one pair of variables conditioning on covariates for all p p 1 2 pairs here p is the length of the outcomes and could exceed the sample size the testing procedure does not require resampling or perturbation and thus is computationally efficient we prove by theory and numerical experiments that this testing method asymptotically controls the false discovery rate it outperforms all alternative methods when the complex association patterns exist applied to a gastric cancer data set this testing method successfully inferred the gene coexpression networks of early and late stage patients it identified more changes in the networks which are associated with cancer survivals we extend our method to the case that both the length of the outcomes and the length of covariates exceed the sample size and show that the asymptotic theory still holds

measuring synchronization in coupled model systems a comparison of different approaches the investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation for experimental data the underlying dynamics are usually not completely known therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis in this study we use three different coupled model systems to create a controlled setting for a comparison of six different measures of synchronization all measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications although certain dynamical features of a system under investigation e g power spectra dimension may render certain measures more suitable than others c 2006 elsevier b v all rights reserved

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

first order corrections to the homogenised eigenvalues of a periodic composite medium a convergence proof let e be a dirichlet eigenvalue of the periodically rapidly oscillating elliptic operator a x e and let be a corresponding simple eigenvalue of the homogenised operator a we characterise the possible limit points of the ratio e e as e 0 our characterisation is quite explicit when the underlying domain is a planar convex classical polygon with sides of rational or infinite slopes in particular in this case it implies that there is often a continuum of such limit points

estimating information rates with confidence intervals in neural spike trains information theory provides a natural set of statistics to quantify the amount of knowledge a neuron conveys about a stimulus a related work kennel shlens abarbanel chichilnisky 2005 demonstrated how to reliably estimate with a bayesian confidence interval the entropy rate from a discrete observed time series we extend this method to measure the rate of novel information that a neural spike train encodes about a stimulus the average and specific mutual information rates our estimator makes few assumptions about the underlying neural dynamics shows excellent performance in experimentally relevant regimes and uniquely provides confidence intervals bounding the range of information rates compatible with the observed spike train we validate this estimator with simulations of spike trains and highlight how stimulus parameters affect its convergence in bias and variance finally we apply these ideas to a recording from a guinea pig retinal ganglion cell and compare results to a simple linear decoder

multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains the multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains are discussed in this paper a multiscale asymptotic analysis formulation for this problem is presented by constructing properly the boundary layer a multiscale numerical algorithm and a postprocessing technique are given finally numerical results show that the method presented in this paper is effective and reliable

asymptotic bias in information estimates and the exponential bell polynomials we present a new derivation of the asymptotic correction for bias in the estimate of information from a finite sample the new derivation reveals a relationship between information estimates and a sequence of polynomials with combinatorial significance the exponential bell polynomials and helps to provide an understanding of the form and behavior of the asymptotic correction for bias

adaptive iterative multiscale finite volume method the multiscale finite volume msfv method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients it has been shown for a wide range of test cases that the msfv results are in close agreement with those obtained with a classical computationally expensive technique the method however fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions recently a convergent iterative msfv i msfv method was developed to enhance the quality of the multiscale results by improving the localization conditions although the i msfv method proved to be efficient for most practical problems it is still favorable to improve the localization condition adaptively i e only for a sub domain where the original msfv localization conditions are not acceptable e g near shale layers and long coherent structures with high permeability contrasts in this paper a space time adaptive i msfv ai msfv method is introduced it is shown how to improve the msfv results adaptively in space and simulation time the fine scale smoother which is necessary for convergence of the i msfv method is also applied locally finally for multiphase flow problems two criteria are investigated for adaptively updating the msfv interpolation functions 1 a criterion based on the total mobility change for the transient coefficients and 2 a criterion based on the pressure equation residual for the accuracy of the results for various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub domains and not everywhere the findings show that the error introduced in the msfv framework can be controlled and improved very efficiently with very little additional computational cost compared to the original non iterative msfv method

entropy estimates of small data sets estimating entropies from limited data series is known to be a non trivial task naive estimations are plagued with both systematic bias and statistical errors here we present a new balanced estimator for entropy functionals shannon renyi and tsallis specially devised to provide a compromise between low bias and small statistical errors for short data series this new estimator outperforms other currently available ones when the data sets are small and the probabilities of the possible outputs of the random variable are not close to zero otherwise other well known estimators remain a better choice the potential range of applicability of this estimator is quite broad specially for biological and digital data series

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

particle method approximation of oscillatory solutions to hyperbolic differential equations particle methods approximating hyperbolic partial diferential equations with oscillatory solutions are studied convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid highly oscillatory solutions to the broadwell and variable coefficient carleman models are considered homogenization results are given and the approximations of more general systems are discussed numerical examples are presented

coverage adjusted entropy estimation summary data on neural coding have frequently been analyzed using information theoretic measures these formulations involve the fundamental and generally difficult statistical problem of estimating entropy we review briefly several methods that have been advanced to estimate entropy and highlight a method the coverage adjusted entropy estimator cae due to chao and shen that appeared recently in the environmental statistics literature this method begins with the elementary horvitz thompson estimator developed for sampling from a finite population and adjusts for the potential new species that have not yet been observed in the sample these become the new patterns or words in a spike train that have not yet been observed the adjustment is due to i j good and is called the good turing coverage estimate we provide a new empirical regularization derivation of the coverage adjusted probability estimator which shrinks the maximum likelihood estimate we prove that the cae is consistent and first order optimal with rate op 1 logn in the class of distributions with finite entropy variance and that within the class of distributions with finite qth moment of the log likelihood the good turing coverage estimate and the total probability of unobserved words converge at rate op 1 logn q we then provide a simulation study of the estimator with standard distributions and examples from neuronal data where observations are dependent the results show that with a minor modification the cae performs much better than the mle and is better than the best upper bound estimator due to paninski when the number of possible words m is unknown or infinite copyright q 2007 john wiley sons ltd

an error analysis for the finite element method applied to convection diffusion problems abstract this paper analyzes the finite element method applied to a convection diffusion model problem linear elements are used for the trial space the error is measured in a norm closely related to the l sup p norm when the test space is composed of linear elements with parabolic upwinding the method is shown to be optimal when the input data is piecewise smooth a condition which is usually observed in practice without these smoothness assumptions the method is shown to be non optimal even if the class of test spaces is extended to include any elements which have a shape independent of the mesh size author

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

bayesian entropy estimation for countable discrete distributions we consider the problem of estimating shannon s entropy h from discrete data in cases where the number of possible symbols is unknown or even countably infinite the pitman yor process a generalization of dirichlet process provides a tractable prior distribution over the space of countably infinite discrete distributions and has found major applications in bayesian non parametric statistics and machine learning here we show that it provides a natural family of priors for bayesian entropy estimation due to the fact that moments of the induced posterior distribution over h can be computed analytically we derive formulas for the posterior mean bayes least squares estimate and variance under dirichlet and pitman yor process priors moreover we show that a fixed dirichlet or pitman yor process prior implies a narrow prior distribution over h meaning the prior strongly determines the entropy estimate in the under sampled regime we derive a family of continuous measures for mixing pitman yor processes to produce an approximately flat prior over h we show that the resulting pitman yor mixture pym entropy estimator is consistent for a large class of distributions finally we explore the theoretical properties of the resulting estimator and show that it performs well both in simulation and in application to real data

robust multilevel methods for general symmetric positive definite operators a general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed the term robust refers to the convergence rate of the method being independent of discretization parameters i e the problem size and problem parameters important instances of such problem parameters are in particular highly varying coefficients the method belongs to the class of nonlinear algebraic multilevel iterations the crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems the method is analyzed in a rather general setting applying to the scalar elliptic equation the equations of linear elasticity and equations arising in the solution of maxwell s equations numerical results for the scalar elliptic equation are presented showing its robust convergence behavior

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

size biased sampling of poisson point processes and excursions some general formulae are obtained for size biased sampling from a poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function these formulae explain why in certain cases gamma and stable the size biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking residual allocation scheme defined by independent beta random variables an application is made to length biased sampling of excursions of a markov process away from a recurrent point of its statespace with emphasis on the brownian and bessel cases when the associated inverse local time is a stable subordinator results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by brownian motion

estimation of the entropy based on its polynomial representation estimating entropy from empirical samples of finite size is of central importance for information theory as well as the analysis of complex statistical systems yet this delicate task is marred by intrinsic statistical bias here we decompose the entropy function into a polynomial approximation function and a remainder function the approximation function is based on a taylor expansion of the logarithm given n observations we give an unbiased linear estimate of the first n power series terms based on counting sets of k coincidences for the remainder function we use nonlinear bayesian estimation with a nearly flat prior distribution on the entropy that was developed by nemenman shafee and bialek our simulations show that the combined entropy estimator has reduced bias in comparison to other available estimators

a multiscale a posteriori error estimate we introduce a hierarchic a posteriori error estimate for singularly perturbed reaction diffusion problems the estimator is based on a petrov galerkin method in which the trial space is enriched with nonpolynomial functions or multiscale functions we study the equivalence between the a posteriori estimate and the exact error in the energy norm moreover we prove a relationship between the hierarchic estimator and an explicit residual estimator the approach provides accurate estimates for the boundary layer regions which is confirmed by numerical experiments 2004 elsevier b v all rights reserved

a note on entropy estimation we compare an entropy estimator recently discussed by zhang 2012 with two estimators and introduced by grassberger 2003 and schurmann 2004 we prove the identity which has not been taken into account by zhang 2012 then we prove that the systematic error bias of is less than or equal to the bias of the ordinary likelihood or plug in estimator of entropy finally by numerical simulation we verify that for the most interesting regime of small sample estimation and large event spaces the estimator has a significantly smaller statistical error than

nonparametric estimation of kullback leibler divergence in this letter we introduce an estimator of kullback leibler divergence based on two independent samples we show that on any finite alphabet this estimator has an exponentially decaying bias and that it is consistent and asymptotically normal to explain the importance of this estimator we provide a thorough analysis of the more standard plug in estimator we show that it is consistent and asymptotically normal but with an infinite bias moreover if we modify the plug in estimator to remove the rare events that cause the bias to become infinite the bias still decays at a rate no faster than further we extend our results to estimating the symmetrized kullback leibler divergence we conclude by providing simulation results which show that the asymptotic properties of these estimators hold even for relatively small sample sizes

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

global 2d digital image correlation for motion estimation in a finite element framework a variational formulation and a regularized pyramidal multi grid implementation in this study the inverse problem of reconstructing the in plane 2d displacements of a monitored surface through a sequence of two dimensional digital images severely ill posed in hadamard s sense is deeply investigated a novel variational formulation is presented for the continuum 2d digital image correlation problem and critical issues such as semi coercivity and solution multiplicity are discussed by functional analysis tools in the framework of a galerkin finite element discretization of the displacement field a robust implementation for 2d digital image correlation is outlined aiming to attenuate the spurious oscillations which corrupt the deformation scenario especially when very fine meshes are adopted recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid multi grid cycles are performed ascending and descending along the pyramid with only one newton iteration per level irrespective of the tolerance satisfaction as if the problem were linear at each level the conventional least square matching functional is herein enriched by a tychonoff regularization provision preserving the solution against an unstable response the algorithm is assessed on the basis of both synthetic and truly experimental image pairs copyright 2013 john wiley sons ltd

lack of fit tests for quantile regression models the paper novelly transforms lack of fit tests for parametric quantile regression models into checking the equality of two conditional distributions of covariates accordingly by applying some successful two sample test statistics in the literature two tests are constructed to check the lack of fit for low and high dimensional quantile regression models the low dimensional test works well when the number of covariates is moderate whereas the high dimensional test can maintain the power when the number of covariates exceeds the sample size the null distribution of the high dimensional test has an explicit form and the p values or critical values can then be calculated directly the finite sample performance of the tests proposed is examined by simulation studies and their usefulness is further illustrated by two real examples

estimating the shannon entropy recurrence plots versus symbolic dynamics recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics a few years later the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis among the different measures introduced a shannon entropy was found to be correlated with the inverse of the largest lyapunov exponent the discrepancy between this and the usual interpretation of a shannon entropy is solved here by using a new definition still based on the recurrence plots and it is verified that this new definition is correlated with the largest lyapunov exponent as expected from the pesin conjecture a comparison with a shannon entropy computed from symbolic dynamics is also provided

a normal limit law for a nonparametric estimator of the coverage of a random sample the coverage of a multinomial random sample is the sum of the probabilities of the observed classes a normal limit law is rigorously proved for good s 1953 coverage estimator the result is valid under very general conditions and all terms except the coverage itself are observable nevertheless the implied confidence intervals are not much wider than those developed under restrictive assumptions such as in the classical occupancy problem the asymptotic variance is somewhat unexpected the proof utilizes a method of holst 1979

zonal multiscale finite volume framework in this work the zonal multiscale finite volume zmsfv framework is presented the framework generalizes previous extensions of the multiscale finite volume msfv method and allows to integrate additional models in a unified way the new approach is based on splitting the domain of interest into a classical msfv domain and additional zones within each zone the solution space is spanned by so called zonal functions coupling of the zonal solutions to the surrounding domain is achieved via extended zonal functions the construction of basis zonal and extended zonal functions can be performed such that the functions form a partition of unity the zmsfv framework allows to include the influence of the zonal functions directly in the coarse scale problem similar to previous msfv method formulations the zmsfv framework can ensure conservative coarse scale solutions and allows for the construction of conservative fine scale solutions the main features of the zmsfv framework are used to derive robust preconditioners for iterative solutions of high contrast problems as encountered in subsurface flow and transport simulations it is shown that the obtained convergence rates are independent of the contrast

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

first order corrections to the homogenized eigenvalues of a periodic composite medium this paper studies the eigenvalue problem associated with the vibration of a com posite medium assumed to have a periodic microstructure the investigation concentrates on the first order correction to the homogenized eigenvalues a major difficulty arises due to the interaction of the periodic microstructure with the boundary of the medium the way in which the boundary intersects the underlying periodic structure strongly influences the value of the correction a rigorous derivation is given of the eigenvalue correction formula the properties of the eigenvalue correction are studied in one and two dimensional numerical calculations

bayesian non parametric inference for species variety with a two parameter poisson dirichlet process prior summary a bayesian non parametric methodology has been recently proposed to deal with the issue of prediction within species sampling problems such problems concern the evaluation conditional on a sample of size n of the species variety featured by an additional sample of size m genomic applications pose the additional challenge of having to deal with large values of both n and m in such a case the computation of the bayesian non parametric estimators is cumbersome and prevents their implementation we focus on the two parameter poisson dirichlet model and provide completely explicit expressions for the corresponding estimators which can be easily evaluated for any sizes of n and m we also study the asymptotic behaviour of the number of new species conditionally on the observed sample such an asymptotic result combined with a suitable simulation scheme allows us to derive asymptotic highest posterior density intervals for the estimates of interest finally we illustrate the implementation of the proposed methodology by the analysis of five expressed sequence tags data sets

asymptotic normality of an entropy estimator with exponentially decaying bias this paper establishes the asymptotic normality of an entropy estimator with an exponentially decaying bias on any finite alphabet furthermore it is shown that the nonparametric estimator is asymptotically efficient

globally adaptive quantile regression with ultra high dimensional data quantile regression has become a valuable tool to analyze heterogeneous covaraite response associations that are often encountered in practice the development of quantile regression methodology for high dimensional covariates primarily focuses on examination of model sparsity at a single or multiple quantile levels which are typically prespecified ad hoc by the users the resulting models may be sensitive to the specific choices of the quantile levels leading to difficulties in interpretation and erosion of confidence in the results in this article we propose a new penalization framework for quantile regression in the high dimensional setting we employ adaptive l1 penalties and more importantly propose a uniform selector of the tuning parameter for a set of quantile levels to avoid some of the potential problems with model selection at individual quantile levels our proposed approach achieves consistent shrinkage of regression quantile estimates across a continuous range of quantiles levels enhancing the flexibility and robustness of the existing penalized quantile regression methods our theoretical results include the oracle rate of uniform convergence and weak convergence of the parameter estimators we also use numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposal

quantile regression for analyzing heterogeneity in ultra high dimension ultra high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of non location scale covariate effects to accommodate heterogeneity we advocate a more general interpretation of sparsity which assumes that only a small number of covariates influence the conditional distribution of the response variable given all candidate covariates however the sets of relevant covariates may differ when we consider different segments of the conditional distribution in this framework we investigate the methodology and theory of nonconvex penalized quantile regression in ultra high dimension the proposed approach has two distinctive features 1 it enables us to explore the entire conditional distribution of the response variable given the ultra high dimensional covariates and provides a more realistic picture of the sparsity pattern 2 it requires substantially weaker conditions compared with alternative methods in the literature thus it greatly alleviates the

homogenization and convergence of the vortex method for 2 d euler equations with oscillatory vorticity fields the 2 d incompressible euler equations with oscillatory vorticity fields are studied a homogenization result for 2 d euler equations in velocity vorticity formulation is obtained and weak continuity of the equations is proved convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles numerical results are given comparisons are made with the corresponding finite difference approximation

modeling multiscale diffusion processes in random heterogeneous media a simple stochastic up scaling strategy is presented for incorporating the effects of micro scale variations of thermal conductivity in the thermal analysis of macro scale systems incomplete characterization of the fine scale variation necessitates posing the thermal conductivity as a stochastic multiscale random field a data driven approach coupled with a non linear dimension reduction algorithm is used to construct a low dimensional stochastic descriptor of the fine scale variation of the stochastic conductivity a representative volume element rve argument is utilized to construct a simple model for up scaling the stochastic thermal conductivity in this model the effective stochastic thermal conductivity is estimated by virtually interrogating the stochastic rv a realistic example of the proposed stochastic multiscale diffusion strategy is illustrated utilizing limited experimental data

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

bayesian empirical likelihood for quantile regression bayesian inference provides a flexible way of combining data with prior information however quantile regression is not equipped with a parametric likelihood and therefore bayesian inference for quantile regression demands careful investigation this paper considers the bayesian empirical likelihood approach to quantile regression taking the empirical likelihood into a bayesian framework we show that the resultant posterior from any fixed prior is asymptotically normal its mean shrinks toward the true parameter values and its variance approaches that of the maximum empirical likelihood estimator a more interesting case can be made for the bayesian empirical likelihood when informative priors are used to explore commonality across quantiles regression quantiles that are computed separately at each percentile level tend to be highly variable in the data sparse areas e g high or low percentile levels through empirical likelihood the proposed method enables us to explore various forms of commonality across quantiles for efficiency gains by using an mcmc algorithm in the computation we avoid the daunting task of directly maximizing empirical likelihood the finite sample performance of the proposed method is investigated empirically where substantial efficiency gains are demonstrated with informative priors on common features across several percentile levels a theoretical framework of shrinking priors is used in the paper to better understand the power of the proposed method

multiscale finite volume method for compressible multiphase flow in porous media the multiscale finite volume msfv method has been recently developed and tested for multiphase flow problems with simplified physics i e incompressible flow without gravity and capillary effects and proved robust accurate and efficient however applications to practical problems necessitate extensions that enable the method to deal with more complex processes in this paper we present a modified version of the msfv algorithm that provides a suitable and natural framework to include additional physics the algorithm consists of four main steps computation of the local basis functions which are used to extract the coarse scale effective transmissibilities solution of the coarse scale pressure equation reconstruction of conservative fine scale fluxes and solution of the transport equations within this framework we develop a msfv method for compressible multiphase flow the basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure the effects of compressibility are taken into account in the solution of the coarse scale pressure equation and if necessary in the reconstruction of the fine scale fluxes we consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows tracer transport in gas flow imbibition and drainage of partially saturated reservoirs depletion of gas water reservoirs and flooding of oil gas reservoirs we demonstrate that the msfv method provides accurate solutions for compressible multiphase flow problems whereas slightly compressible flows can be treated with a very simple model a more sophisticate flux reconstruction is needed to obtain accurate fine scale saturation fields in highly compressible flows

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

entropy estimation in turing s perspective a new nonparametric estimator of shannon s entropy on a countable alphabet is proposed and analyzed against the well known plug in estimator the proposed estimator is developed based on turing s formula which recovers distributional characteristics on the subset of the alphabet not covered by a size n sample the fundamental switch in perspective brings about substantial gain in estimation accuracy for every distribution with finite entropy in general a uniform variance upper bound is established for the entire class of distributions with finite entropy that decays at a rate of o ln n n compared to o ln n 2 n for the plug in in a wide range of subclasses the variance of the proposed estimator converges at a rate of o 1 n and this rate of convergence carries over to the convergence rates in mean squared errors in many subclasses specifically for any finite alphabet the proposed estimator has a bias decaying exponentially in n several new bias adjusted estimators are also discussed

analysis of the multiscale finite element method for nonlinear and random homogenization problems in this paper we study the convergence of the multiscale finite element method for nonlinear and random homogenization problems error estimates similar to those for linear homogenization problems are obtained here

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

composite quantile regression and the oracle model selection theory coefficient estimation and variable selection in multiple linear regression is routinely done in the penalized least squares ls framework the concept of model selection oracle introduced by fan and li j amer statist assoc 96 2001 1348 1360 characterizes the optimal behavior of a model selection procedure however the least squares oracle theory breaks down if the error variance is infinite in the current paper we propose a new regression method called composite quantile regression cqr we show that the oracle model selection theory using the cqr oracle works beautifully even when the error variance is infinite we develop a new oracular procedure to achieve the optimal properties of the cqr oracle when the error variance is finite cqr still enjoys great advantages in terms of estimation efficiency we show that the relative efficiency of cqr compared to the least squares is greater than 70 regardless the error distribution moreover cqr could be much more efficient and sometimes arbitrarily more efficient than the least squares the same conclusions hold when comparing a cqr oracular estimator with a ls oracular estimator

the upward bias in measures of information derived from limited data samples extracting information measures from limited experimental samples such as those normally available when using data recorded in vivo from mammalian cortical neurons is known to be plagued by a systematic error which tends to bias the estimate upward we calculate here the average of the bias under certain conditions as an asymptotic expansion in the inverse of the size of the data sample the result agrees with numerical simulations and is applicable as an additive correction term to measurements obtained under such conditions moreover we discuss the implications for measurements obtained through other usual procedures

a multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media 1 numerical solution of unsaturated water flow in heterogeneous porous media is difficult not only because of the spatial variability in the parameters used to characterize the relevant physical properties of the natural porous media but also because of the mathematical problems that arise in dealing with the nonlinearities in this study a multiscale finite element linearization scheme is presented for effectively simulating unsaturated flow in heterogeneous porous media spanning over many scales the central goal of this method is to obtain the large scale solution of richards equation with heterogeneous coefficients accurately and efficiently on a coarse grid without resolving all the small scale details the underlying idea is to employ the slodicka linear relaxation approximation scheme to address the nonlinear characteristics of the equation and to use the multiscale finite element base functions to account for the spatial variability in the equation coefficients we describe the principle for constructing such a method and give an algorithm for implementing it numerical experiments are carried out for the unsaturated flow equation with both periodic and randomly generated lognormal hydraulic conductivity to demonstrate the efficiency and accuracy of the proposed method

multi scale methods in this paper four multiple scale methods are proposed the meshless hierarchical partition of unity is used as a multiple scale basis the multiple scale analysis with the introduction of a dilation parameter to perform multiresolution analysis is discussed the multiple field based on a 1 d gradient plasticity theory with material length scale is also proposed to remove the mesh dependency difficulty in softening localization problems a non local smoothing particle integration procedure with its multiple scale analysis are then developed these techniques are described in the context of the reproducing kernel particle method results are presented for elastic plastic one dimensional problems and 2 d large deformation strain localization problems to illustrate the effectiveness of these methods copyright 2000 john wiley sons ltd

a mixed multiscale finite element method for convex optimal control problems with oscillating coefficients abstract we study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients since the objective functional contains flux we approximate the problems using the mixed finite element methods we first analyze the standard mixed finite element approximation schemes then motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media we use a multiscale mixed finite element method to solve the state equations the multiscale finite element bases are constructed by locally solving neumann boundary value problems the analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic which allows us to use homogenization theory to obtain the asymptotic structure of the solutions although the numerical schemes are designed for the general case

an analysis of the nlmc upscaling method for high contrast problems abstract in this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects in our method each coarse block is decomposed into various regions according to the contrast ratio and we require that the contrast ratio should be relatively small within each region the basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise numerical analysis shows that the resulting basis functions can be localizable and have a decay property the convergence of the multiscale solution is also proved finally some numerical experiments are carried out to illustrate the performances of the proposed method they show that the proposed method can solve problem with high contrast medium efficiently in particular if the oversampling size is large enough then we can achieve the desired error

application of a conservative generalized multiscale finite element method to flow models in this paper we propose a method for the construction of locally conservative flux fields from generalized multiscale finite element method gmsfem pressure solutions the flux values are obtained from an element based postprocessing procedure in which an independent set of 4x4 linear systems need to be solved to test the performance of the method we consider two heterogeneous permeability coefficients and couple the resulting fluxes to a two phase flow model the increase in accuracy associated with the computation of the gmsfem pressure solutions is inherited by the postprocessed flux fields and saturation solutions and is closely correlated to the size of the reduced order systems in particular the addition of more basis functions to the enriched coarse space yields solutions that more accurately capture the behavior of the fine scale model a number of numerical examples are offered to validate the performance of the method

testing for marginal linear effects in quantile regression this paper develops a new marginal testing procedure to detect the presence of significant predictors associated with the conditional quantiles of a scalar response the idea is to fit the marginal quantile regression on each predictor one at a time and then base the test on the t statistics associated with the most predictive predictors a resampling method is devised to calibrate this test statistic which has non regular limiting behavior due to the selection of the most predictive variables asymptotic validity of the procedure is established in a general quantile regression setting in which the marginal quantile regression models can be misspecified even though a fixed dimension is assumed to derive the asymptotic results the proposed test is applicable and computationally feasible for large dimensional predictors the method is more flexible than existing marginal screening test methods based on mean regression and has the added advantage of being robust against outliers in the response the approach is illustrated using an application to an hiv drug resistance dataset

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

simulation of the multi scale convergence in computational homogenization approaches although the asymptotic homogenization is known to explicitly predict the thermo mechanical behaviors of an overall structure as well as the microstructures the current developments in engineering fields introduce some kinds of approximation about the microstructural geometry in order for the homogenization method for periodic media to apply for general heterogeneous ones the problems arising from mathematical modeling are examined in the framework of representative volume element rve analyses here the notion of homogenization convergence allows us to eliminate the geometrical periodicity requirement when the size of rve is sufficiently large then the numerical studies in this paper realize the multi scale nature of the convergence of overall material properties as the unit cell size is increased in addition to such dependency of the macroscopic field variables on the selected size of unit cells the convergence nature of microscopic stress values is also studied quantitatively via the computational homogenization method similar discussions are made for the elastoplastic mechanical responses in both macro and microscopic levels in these multi scale numerical analyses the specific effects of the microstructural morphology are reflected by using the digital image based dib finite element fe modeling technique which enables the construction of accurate microstructural models

bias analysis in entropy estimation we consider the problem of finite sample corrections for entropy estimation new estimates of the shannon entropy are proposed and their systematic error the bias is computed analytically we find that our results cover correction formulae of current entropy estimates recently discussed in the literature the trade off between bias reduction and the increase of the corresponding statistical error is analysed

regularization parameter selections via generalized information criterion we apply the nonconcave penalized likelihood approach to obtain variable selections as well as shrinkage estimators this approach relies heavily on the choice of regularization parameter which controls the model complexity in this paper we propose employing the generalized information criterion encompassing the commonly used akaike information criterion aic and bayesian information criterion bic for selecting the regularization parameter our proposal makes a connection between the classical variable selection criteria and the regularization parameter selections for the nonconcave penalized likelihood approaches we show that the bic type selector enables identification of the true model consistently and the resulting estimator possesses the oracle property in the terminology of fan and li 2001 in contrast however the aic type selector tends to overfit with positive probability we further show that the aic type selector is asymptotically loss efficient while the bic type selector is not

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

minimax estimation of functionals of discrete distributions we propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters and elaborate on the case of discrete distributions where the support size s is unknown and may be comparable with or even much larger than the number of observations n we treat the respective regions where the functional is nonsmooth and smooth separately in the nonsmooth regime we apply an unbiased estimator for the best polynomial approximation of the functional whereas in the smooth regime we apply a bias corrected version of the maximum likelihood estimator mle we illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures 1 the entropy h p sum i 1 s p i ln p i and 2 f alpha p sum i 1 s p i alpha alpha 0 we obtain the minimax l 2 rates for estimating these functionals in particular we demonstrate that our estimator achieves the optimal sample complexity n asymp s ln s for entropy estimation we also demonstrate that the sample complexity for estimating f alpha p 0 is n asymp s 1 alpha ln s which can be achieved by our estimator but not the mle for 1 we show the minimax l 2 rate for estimating f alpha p is n ln n 2 alpha 1 for infinite support size while the maximum l 2 rate for the mle is n 2 alpha 1 for all the above cases the behavior of the minimax rate optimal estimators with n samples is essentially that of the mle plug in rule with n ln n samples which we term effective sample size enlargement we highlight the practical advantages of our schemes for the estimation of entropy and mutual information we compare our performance with various existing approaches and demonstrate that our approach reduces running time and boosts the accuracy moreover we show that the minimax rate optimal mutual information estimator yielded by our framework leads to significant performance boosts over the chow liu algorithm in learning graphical models the wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable