counter example in three dimension and homogenization of geometric motions in two dimension in this paper we give a counter example to the homogenization of the forced mean curvature motion in a periodic setting in dimension n geqq 3 when the forcing is positive we also prove a general homogenization result for geometric motions in dimension n 2 under the assumption that there exists a constant 0 such that every straight line moving with a normal velocity equal to is a subsolution for the motion we also present a generalization in dimension 2 where we allow sign changing normal velocity and still construct bounded correctors when there exists a subsolution with compact support expanding in all directions

optimal convex error estimators for classification a cross validation error estimator is obtained by repeatedly leaving out some data points deriving classifiers on the remaining points computing errors for these classifiers on the left out points and then averaging these errors the 0 632 bootstrap estimator is obtained by averaging the errors of classifiers designed from points drawn with replacement and then taking a convex combination of this zero bootstrap error with the resubstitution error for the designed classifier this gives a convex combination of the low biased resubstitution and the high biased zero bootstrap another convex error estimator suggested in the literature is the unweighted average of resubstitution and cross validation this paper treats the following question given a feature label distribution and classification rule what is the optimal convex combination of two error estimators i e what are the optimal weights for the convex combination this problem is considered by finding the weights to minimize the mse of a convex estimator it also considers optimality under the constraint that the resulting estimator be unbiased owing to the large amount of results coming from the various feature label models and error estimators a portion of the results are presented herein and the main body of results appears on a companion website in the tabulated results each table treats the classification rules considered for the model various bayes errors and various sample sizes each table includes the optimal weights mean errors and standard deviations for the relevant error measures and the mse and mae for the optimal convex estimator many observations can be made by considering the full set of experiments some general trends are outlined in the paper the general conclusion is that optimizing the weights of a convex estimator can provide substantial improvement depending on the classification rule data model sample size and component estimators optimal convex bootstrap estimators are applied to feature set ranking to illustrate their potential advantage over non optimized convex estimators

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

on the sampling distribution of resubstitution and leave one out error estimators for linear classifiers error estimation is a problem of high current interest in many areas of application this paper concerns the classical problem of determining the performance of error estimators in small sample settings under a gaussianity parametric assumption we provide here for the first time the exact sampling distribution of the resubstitution and leave one out error estimators for linear discriminant analysis lda in the univariate case which is valid for any sample size and combination of parameters including unequal variances and sample sizes for each class in the multivariate case we provide a quasi binomial approximation to the distribution of both the resubstitution and leave one out error estimators for lda under a common but otherwise arbitrary class covariance matrix which is assumed to be known in the design of the lda discriminant we provide numerical examples using both synthetic and real data that indicate that these approximations are accurate provided that lda classification error is not too large

analytic representation of bayes labeling and bayes clustering operators for random labeled point processes clustering algorithms typically group points based on some similarity criterion but without reference to an underlying random process to make clustering algorithms rigorously predictive in fact there exists a probabilistic theory of clustering in the context of random labeled point sets in which clustering error is defined in terms of the process in the present paper given an underlying point process we develop a general analytic procedure for finding an optimal clustering operator the bayes clusterer that corresponds to the bayes classifier in classification theory we provide detailed solutions under gaussian models owing to computational complexity we also develop approximations of the bayes clusterer

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part i representation in recent years biomedicine has been faced with difficult high throughput small sample classification problems in such settings classifier error estimation becomes a critical issue because training and testing must be done on the same data a recently proposed error estimator places the problem in a signal estimation framework in the presence of uncertainty permitting a rigorous solution optimal in a minimum mean square error sense the uncertainty in this model is relative to the parameters of the feature label distributions resulting in a bayesian approach to error estimation closed form solutions are available for two important problems discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors in this work part i of a two part study we introduce the theoretical mean square error mse conditioned on the observed sample of any estimate of the classifier error including the bayesian error estimator for both bayesian models thus bayesian error estimation has a unique advantage in that its mathematical framework naturally gives rise to a practical expected measure of performance given an observed sample in part ii of the study we examine consistency of the error estimator demonstrate various mse properties and apply the conditional mse to censored sampling

bayesian minimum mean square error estimation for classification error part i definition and the bayesian mmse error estimator for discrete classification with the advent of high throughput genomic and proteomic technologies in conjunction with the difficulty in obtaining even moderately sized samples small sample classifier design has become a major issue in the biological and medical communities with small samples training data error estimation becomes mandatory yet none of the popular error estimation techniques have been rigorously designed based on statistical inference and optimization in this investigation we place classifier error estimation into the framework of optimal mean square error mse signal estimation in the presence of uncertainty which results in a bayesian approach to error estimation based on a parameterized family of feature label distributions with the prior distribution of the parameters governing the choice of feature label distribution these bayesian error estimators are optimal when averaged over a given family of distributions unbiased when averaged over a given family and all samples and analytically address a trade off between robustness modeling assumptions and accuracy minimum mean square error in this paper part i of a two part study we define the minimum mean square error mmse error estimator discuss its basic properties provide closed form analytic estimator representation for discrete classifiers with both non informative and informative prior distributions and examine the performance and robustness of the mmse error estimator via simulations in part ii of this paper in this same issue of ieee transactions on signal processing we address all of these issues in particular closed form representation for linear classification in the gaussian model with known and unknown covariance matrices for both the discrete and gaussian cases the mmse error estimator has especially good performance for distributions having moderate true errors

homogenization and enhancement of the g equation in random environments we study the homogenization of a g equation which is advected by a divergence free stationary vector field in a general ergodic random environment we prove that the averaged equation is an anisotropic deterministic g equation and we give necessary and sufficient conditions in order to have enhancement since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions in addition as we show the associated minimal first passage time function does not satisfy in general the uniform integrability condition which is necessary to apply the sub additive ergodic theorem we overcome these obstacles by i establishing a new reachability controllability estimate for the minimal function and ii constructing for each direction and almost surely a random sequence which has both a long time averaged limit due to the sub additive ergodic theorem and stays in the same sense asymptotically close to the minimal time

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

event based state estimation of linear dynamic systems with unknown exogenous inputs in this work an event based optimal state estimation problem for linear time varying systems with unknown inputs is investigated by treating the unknown input as a process with a non informative prior the event based minimum mean square error mmse estimator is obtained in a recursive form it is shown that for the general time varying case the closed loop matrix of the optimal event based estimator is exponentially stable and the estimation error covariance matrix is asymptotically bounded for each sample path of the event triggering process the results are also extended to the multiple sensor scenario where each sensor is allowed to have its own event triggering condition the efficiency of the proposed results is illustrated by a numerical example and comparative simulation with the mmse estimators obtained based on time triggered measurements the results are potentially applicable to event based secure state estimation of cyber physical systems

discrete optimal bayesian classification with error conditioned sequential sampling when in possession of prior knowledge concerning the feature label distribution in particular when it is known that the feature label distribution belongs to an uncertainty class of distributions governed by a prior distribution this prior knowledge can be used in conjunction with the training data to construct the optimal bayesian classifier obc whose performance is on average optimal among all classifiers relative to the posterior distribution derived from the prior distribution and the data typically in classification theory it is assumed that sampling is performed randomly in accordance with the prior probabilities on the classes and this has heretofore been true in the case of obc in the present paper we propose to forego random sampling and utilize the prior knowledge and previously collected data to determine which class to sample from at each step of the sampling specifically we choose to sample from the class that leads to the smallest expected classification error with the addition of the new sample point we demonstrate the superiority of the resulting nonrandom sampling procedure to random sampling on both synthetic data and data generated from known biological pathways highlightsa sampling algorithm for training the optimal bayesian classifier is introduced the algorithm works based on minimization of the expected error on the uncertainty class of prior knowledge using a zipf model we show that our sampling algorithm leads to a less true error on average than random sampling our algorithm shows robustness even in case when prior knowledge drifts away from true distributions an example on data from p53 network shows that our method works well on from real pathway data as well

homogenization of first order equations with u varepsilon periodic hamiltonians part i local equations in this paper we present a result of homogenization of first order hamilton jacobi equations with u varepsilon periodic hamiltonians on the one hand under a coercivity assumption on the hamiltonian and some natural regularity assumptions we prove an ergodicity property of this equation and the existence of non periodic approximate correctors on the other hand the proof of the convergence of the solution usually based on the introduction of a perturbed test function in the spirit of evans work uses here a twisted perturbed test function for a higher dimensional problem

optimal robust classifiers qualitatively a filter is said to be robust if its performance degradation is acceptable for distributions close to the one for which it is optimal that is the one for which it has been designed this paper adapts the signal processing theory of optimal robust filters to classifiers the distribution class conditional distributions to which the classifier is to be applied is parameterized by a state vector and the principle issue is to choose a design state that is optimal in comparison to all other states relative to some measure of robustness a minimax robust classifier is one whose worst performance over all states is better than the worst performances of the other classifiers defined at the other states a bayesian robust classifier is one whose expected performance is better than the expected performances of the other classifiers the state corresponding to the bayesian robust classifier is called the maximally robust state minimax robust classifiers tend to give too much weight to states for which classification is very difficult and therefore our effort is focused on bayesian robust classifiers whereas the signal processing theory of robust filtering concentrates on design with full distributional knowledge and a fixed number of observation variables features design via training from sample data and feature selection are so important for classification that robustness optimality must be considered from these perspectives in particular for small samples in this context for a given sample size we will be concerned with the maximally robust state feature pair all definitions are independent of the classification rule however applications are only considered for linear and quadratic discriminant analysis for which there are parametric forms for the optimal discriminants

optimal mean square error calibration of classifier error estimators under bayesian models a recently proposed bayesian modeling framework for classification facilitates both the analysis and optimization of error estimation performance the bayesian error estimator is then defined to have optimal mean square error performance but in many situations closed form representations are unavailable and approximations may not be feasible to address this we present a method to optimally calibrate arbitrary error estimators for minimum mean square error performance within a supposed bayesian framework assuming a fixed sample size classification rule and error estimation rule as well as a fixed bayesian model the calibration is done by first computing a calibration function that maps error estimates to their optimally calibrated values off line once found this calibration function may be easily applied to error estimates on the fly whenever the assumptions apply we demonstrate that calibrated error estimators offer significant improvement in performance relative to classical error estimators under bayesian models with both linear and non linear classification rules

exact performance of error estimators for discrete classifiers discrete classification problems abound in pattern recognition and data mining applications one of the most common discrete rules is the discrete histogram rule this paper presents exact formulas for the computation of bias variance and rms of the resubstitution and leave one out error estimators for the discrete histogram rule we also describe an algorithm to compute the exact probability distribution of resubstitution and leave one out as well as their deviations from the true error rate using a parametric zipf model we compute the exact performance of resubstitution and leave one out for varying expected true error number of samples and classifier complexity number of bins we compare this to approximate performance measures computed by monte carlo sampling of 10 repeated 4 fold cross validation and the 0 632 bootstrap error estimator our results show that resubstitution is low biased but much less variable than leave one out and is effectively the superior error estimator between the two provided classifier complexity is low in addition our results indicate that the overall performance of resubstitution as measured by the rms can be substantially better than the 10 repeated 4 fold cross validation estimator and even comparable to the 0 632 bootstrap estimator provided that classifier complexity is low and the expected error rates are moderate in addition to the results discussed in the paper we provide an extensive set of plots that can be accessed on a companion website at the url http ee tamu edu edward exact discrete

distributed gaussian polynomials and associated gaussian quadratures an orthogonal function set called distributed gaussian orthogonal polynomials is constructed from equally spaced gaussians and the corresponding quadrature is studied the infinite chain limit of both polynomials and the quadrature is studied and analytical expressions are found for polynomials quadrature points weights and lagrange functions the connections to wannier functions are explored and a way of constructing wannier functions from gaussians is proposed

analytic study of performance of error estimators for linear discriminant analysis we derive double asymptotic analytical expressions for the first moments second moments and cross moments with the actual error for the resubstitution and leave one out error estimators in the case of linear discriminant analysis in the multivariate gaussian model under the assumption of a common known covariance matrix and a fixed mahalanobis distance as dimensionality approaches infinity sample sizes for the two classes need not be the same they are only assumed to reach a fixed but arbitrary asymptotic ratio with the dimensionality from the asymptotic moment representations we directly obtain double asymptotic expressions for the bias variance and rms of the error estimators the asymptotic expressions presented here generally provide good small sample approximations as demonstrated via numerical experiments the applicability of the theoretical results is illustrated by finding the minimum sample size to bound the rms in gene expression classification

analytical study of performance of linear discriminant analysis in stochastic settings this paper provides exact analytical expressions for the first and second moments of the true error for linear discriminant analysis lda when the data are univariate and taken from two stochastic gaussian processes the key point is that we assume a general setting in which the sample data from each class do not need to be identically distributed or independent within or between classes we compare the true errors of designed classifiers under the typical i i d model and when the data are correlated providing exact expressions and demonstrating that depending on the covariance structure correlated data can result in classifiers with either greater error or less error than when training with uncorrelated data the general theory is applied to autoregressive and moving average models of the first order and it is demonstrated using real genomic data

planelike minimizers in periodic media we show that given an elliptic integrand in d that is periodic under integer translations and given any plane in d there is at least one minimizer of that remains at a bounded distance from this plane this distance can be bounded uniformly on the planes we also show that when folded back to d d the minimizers we construct give rise to a lamination one particular case of these results is minimal surfaces for metrics invariant under integer translations the same results hold for other functionals that involve volume terms small and average zero in such a case the minimizers satisfy the prescribed mean curvature equation a further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that their fundamental group and universal cover satisfy some hypotheses 2001 john wiley sons inc

homogenization of the hele shaw problem in periodic spatiotemporal media we consider the homogenization of the hele shaw problem in periodic media that are inhomogeneous both in space and time after extending the theory of viscosity solutions into this context we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous hele shaw type problem with a general possibly nonlinear dependence of the free boundary velocity on the gradient moreover the free boundaries converge locally uniformly in hausdorff distance

bloch and wannier functions composed of gaussian orbitals analyticity localization and expectation values we discuss bloch and wannier functions composed of gaussian orbitals the fact that the bloch functions may be expressed in terms of functions is used to discuss them in detail we study not only their analyticity properties but also features of expectation values of some specifically chosen hamiltonians we also investigate the localization properties of various choices of wannier functions and compare them with the original gauss functions likewise we discuss in a more general context the relationship between the analyticity properties of bloch functions and the localization properties of corresponding wannier functions

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the perturbed test function method for viscosity solutions of nonlinear pde the method of viscosity solutions for nonlinear partial differential equations pdes justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions we describe in this paper a perturbed test function device which entails various modifications of the test functions by lower order correctors applications include homogenisation for quasilinear elliptic pdes and approximation of quasilinear parabolic pdes by systems of hamilton jacobi equations

bayesian minimum mean square error estimation for classification error part ii linear classification of gaussian models in this paper part ii of a two part study we derive a closed form analytic representation of the bayesian minimum mean square error mmse error estimator for linear classification assuming gaussian models this is presented in a general framework permitting a structure on the covariance matrices and a very flexible class of prior parameter distributions with four free parameters closed form solutions are provided for known scaled identity and arbitrary covariance matrices we examine performance in small sample settings via simulations on both synthetic and real genomic data and demonstrate the robustness of these error estimators to false gaussian modeling assumptions by applying them to johnson distributions

classifier performance as a function of distributional complexity abstract when choosing a classification rule it is important to take into account the amount of sample data available this paper examines the performances of classifiers of differing complexities in relation to the complexity of feature label distributions in the case of small samples we define the distributional complexity of a feature label distribution to be the minimal number of hyperplanes necessary to achieve the bayes classifier if the bayes classifier is achievable by a finite number of hyperplanes and infinity otherwise our approach is to choose a model and compare classifier efficiencies for various sample sizes and distributional complexities simulation results are obtained by generating data based on the model and the distributional complexities a linear support vector machine svm is considered along with several nonlinear classifiers for the most part we see that there is little improvement when one uses a complex classifier instead of a linear svm for higher levels of distributional complexity the linear classifier degrades but so do the more complex classifiers owing to insufficient training data hence if one were to obtain a good result with a more complex classifier it is most likely that the distributional complexity is low and there is no gain over using a linear classifier hence under the model it is generally impossible to claim that use of the nonlinear classifier is beneficial in essence the sample sizes are too small to take advantage of the added complexity an exception to this observation is the behavior of the three nearest neighbor 3nn classifier in the case of two variables but not three when there is very little overlap between the label distributions and the sample size is not too small with a sample size of 60 the 3nn classifier performs close to the bayes classifier even for high levels of distributional complexity consequently if one uses the 3nn classifier with two variables and obtains a low error then the distributional complexity might be large and if such is the case there is a significant gain over using a linear classifier

on cell problems for hamilton jacobi equations with non coercive hamiltonians and their application to homogenization problems abstract we study a cell problem arising in homogenization for a hamilton jacobi equation whose hamiltonian is not coercive we introduce a generalized notion of effective hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective hamiltonian under some sufficient conditions the result is applied to the associated homogenization problem we also show that homogenization for non coercive equations fails in general

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part ii consistency and performance analysis in part i of a two part study on the mse performance of bayesian error estimation we have derived analytical expressions for mse conditioned on the sample for bayesian error estimators and arbitrary error estimators in two bayesian models discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors here in part ii we examine the consistency of bayesian error estimation and provide several simulation studies that illustrate the concept of conditional mse and how it may be used in practice a salient application is censored sampling where sample points are collected one at a time until the conditional mse reaches a stopping criterion

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

homogenization and enhancement for the g equation we consider the so called g equation a level set hamilton jacobi equation used as a sharp interface model for flame propagation perturbed by an oscillatory advection in a spatio temporal periodic environment assuming that the advection has suitably small spatial divergence we prove that as the size of the oscillations diminishes the solutions homogenize average out and converge to the solution of an effective anisotropic first order spatio temporal homogeneous level set equation moreover we obtain a rate of convergence and show that under certain conditions the averaging enhances the velocity of the underlying front we also prove that at scale one the level sets of the solutions of the oscillatory problem converge at long times to the wulff shape associated with the effective hamiltonian finally we also consider advection depending on position at the integral scale

complete sets of bloch and wannier functions composed of oscillator eigenfunctions we discuss bloch and wannier functions related to oscillator eigenfunctions in particular we construct complete sets of mutually orthogonal bloch and wannier functions we show that they can be expressed in several ways in terms of theta functions and their derivatives we also analyze their localization properties and discuss expectation values for specifically chosen hamiltonians

moments and root mean square error of the bayesian mmse estimator of classification error in the gaussian model the most important aspect of any classifier is its error rate because this quantifies its predictive capacity thus the accuracy of error estimation is critical error estimation is problematic in small sample classifier design because the error must be estimated using the same data from which the classifier has been designed use of prior knowledge in the form of a prior distribution on an uncertainty class of feature label distributions to which the true but unknown feature distribution belongs can facilitate accurate error estimation in the mean square sense in circumstances where accurate completely model free error estimation is impossible this paper provides analytic asymptotically exact finite sample approximations for various performance metrics of the resulting bayesian minimum mean square error mmse error estimator in the case of linear discriminant analysis lda in the multivariate gaussian model these performance metrics include the first second and cross moments of the bayesian mmse error estimator with the true error of lda and therefore the root mean square rms error of the estimator we lay down the theoretical groundwork for kolmogorov double asymptotics in a bayesian setting which enables us to derive asymptotic expressions of the desired performance metrics from these we produce analytic finite sample approximations and demonstrate their accuracy via numerical examples various examples illustrate the behavior of these approximations and their use in determining the necessary sample size to achieve a desired rms the supplementary material contains derivations for some equations and added figures highlightsthe bayesian minimum mean square error bmmse error estimator of lda is studied the first second and cross moments of the bmmse estimator with true error are considered both conditional and unconditional performance metrics are analyzed for the first time the kolmogorov double asymptotic is used in a bayesian setting asymptotically exact finite sample approximations of performance metrics are derived

relationship between the accuracy of classifier error estimation and complexity of decision boundary error estimation is a crucial part of classification methodology and it becomes problematic with small samples we demonstrate here that the complexity of the decision boundary plays a key role on the performance of error estimation methods first a model is developed which quantifies the complexity of a classification problem purely in terms of the geometry of the decision boundary without relying on the bayes error then this model is used in a simulation study to analyze the bias and root mean square rms error of a few widely used error estimation methods relative to the complexity of the decision boundary resubstitution leave one out 10 fold cross validation with repetition 0 632 bootstrap and bolstered resubstitution in two and three dimensional spaces each estimator is implemented with three classification rules quadratic discriminant analysis qda 3 nearest neighbor 3nn and two layer neural network nnet the results show that all the estimation methods lose accuracy as complexity increases

the reliability of estimated confidence intervals for classification error rates when only a single sample is available error estimation accuracy is the salient issue regarding the validity of a classifier model when samples are small training data based error estimates tend to suffer from inaccuracy and quantification of error estimation accuracy is difficult numerous methods have been proposed for estimating confidence intervals for the true error based on the estimated error this paper surveys proposed methods and quantifies their performance monte carlo methods are used to obtain accurate estimates of the true confidence intervals and compare these to the intervals estimated from samples we consider different error estimators and several proposed confidence bound estimators both synthetic and real genomic data are employed our simulations show the majority of the confidence intervals methods have poor performance because of the difference of shape between true and estimated intervals according to our results the best estimation strategy is to use the 10 time 10 fold cross validation with a confidence interval based on the standard deviation

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

on the relevance of linear discriminative features linear discriminant analysis lda has been widely used to extract linear features for classification in real applications the usefulness of the extracted features usually needs to be confirmed using an error rate of classification embedded in a classifier little attention has been paid to whether and how discriminative features themselves can be interpreted as indicators of usefulness we refer to this as relevance i e the capability of discriminative features to characterize the contribution of the original variables to classification we approach the relevance by considering how it could be lost while extracting optimal discriminative features then the discrepancy between the relevance and optimality of discriminative features is shown to originate from the angle between the space spanned by eigenvectors of the within class scattering matrix and the primary space in which the original variables reside in particular for a given dataset the larger the angle the less evident is the relevance discovered from optimal discriminative features furthermore the relevance and optimality are regarded as two constraint conditions or a tradeoff in order to extract relevant discriminative features at last a simulated experiment is used to show how the relevance is lost when the angle is changed experimental results based on both usps handwritten digitals and pie face databases show that a maximum margin criterion is a reasonable compromise between the relevance and optimality since it approximates the averaged class margin using euclidean distance measured in the primary space

optimal classifiers with minimum expected error within a bayesian framework part ii properties and performance analysis in part i of this two part study we introduced a new optimal bayesian classification methodology that utilizes the same modeling framework proposed in bayesian minimum mean square error mmse error estimation optimal bayesian classification thus completes a bayesian theory of classification where both the classifier error and our estimate of the error may be simultaneously optimized and studied probabilistically within the assumed model having developed optimal bayesian classifiers in discrete and gaussian models in part i here we explore properties of optimal bayesian classifiers in particular invariance to invertible transformations convergence to the bayes classifier and a connection to bayesian robust classifiers we also explicitly derive optimal bayesian classifiers with non informative priors and explore relationships to linear and quadratic discriminant analysis lda and qda which may be viewed as plug in rules under gaussian modeling assumptions finally we present several simulations addressing the robustness of optimal bayesian classifiers to false modeling assumptions companion website http gsp tamu edu publications supplementary dalton12a

generalized consistent error estimator of linear discriminant analysis a classifier is epistemologically vacuous without an accurate estimate of its true error rate in situations where the number of sample points is of the same order of magnitude as the dimension of observations serious issues arise with respect to the performance of error estimators in this paper we place the problem of synthesizing an error rate estimator of a common linear classifier in an asymptotic setting in which the number of sample points is kept comparable in magnitude to the dimension of observations double asymptotic we construct a generalized consistent estimator of the true error rate for linear discriminant analysis in the multivariate gaussian model under the assumption of a common covariance matrix in other words the estimator converges to true error rate in the double asymptotic sense we employ simulations using both synthetic and real data to compare the performance of the new estimator to the classical consistent estimator of the true error plug in estimator as well as other well known estimators we observe that the constructed estimator can outperform other estimators of the true error in many situations in terms of bias and root mean square rms error

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

error estimation based on variance analysis of k fold cross validation when the numbers of samples and folds are both large enough we proved that cv variance and its accuracy have the quadratic relationship the relationships between cv variance and its factors have been derived allowing to predict which variance is less before applying k fold cv theoretical explanations have been given for some empirical evidences of rodriguez and kohavi from the respect of variance analysis the proposed normalized variance has significant correlation with the error and is unrelated to k so that it can serve as a stable error measurement cross validation cv is often used to estimate the generalization capability of a learning model the variance of cv error has a considerable impact on the accuracy of cv estimator and the adequacy of the learning model so it is very important to analyze cv variance the aim of this paper is to investigate how to improve the accuracy of the error estimation based on variance analysis we first describe the quantitative relationship between cv variance and its accuracy which can provide guidance for improving the accuracy by reducing the variance we then study the relationships between variance and relevant variables including the sample size the number of folds and the number of repetitions these form the basis of theoretical strategies of regulating cv variance our classification results can theoretically explain the empirical results of rodrguez and kohavi finally we propose a uniform normalized variance which not only measures model accuracy but also is irrelative to fold number therefore it simplifies the selection of fold number in k fold cv and normalized variance can serve as a stable error measurement for model comparison and selection we report the results of experiments using 5 supervised learning models and 20 datasets the results indicate that it is reliable to determine which variance is less before k fold cv by the proposed theorems and thus the accuracy of error estimation can be promoted by reducing variance in so doing we are more likely to select the best parameter or model

traveling waves of a curvature flow in almost periodic media abstract in a plane media with almost periodic vertical striations we study a curvature flow and construct two kinds of traveling waves one having a straight line like profile and the other having a v shaped profile for each of the first kind traveling waves its profile is the graph of a function whose derivative is almost periodic for each of the second kind traveling waves its profile is like a pulsating cone whose two tails approach asymptotically the profiles of the first kind traveling waves also we consider a homogenization problem and provide an explicit formula for the homogenized traveling speed

joint sampling distribution between actual and estimated classification errors for linear discriminant analysis error estimation must be used to find the accuracy of a designed classifier an issue that is critical in biomarker discovery for disease diagnosis and prognosis in genomics and proteomics this paper presents for what is believed to be the first time the analytical formulation for the joint sampling distribution of the actual and estimated errors of a classification rule the analysis presented here concerns the linear discriminant analysis lda classification rule and the resubstitution and leave one out error estimators under a general parametric gaussian assumption exact results are provided in the univariate case and a simple method is suggested to obtain an accurate approximation in the multivariate case it is also shown how these results can be applied in the computation of condition bounds and the regression of the actual error given the observed error estimate in contrast to asymptotic results the analysis presented here is applicable to finite training data in particular it applies in the small sample settings commonly found in genomics and proteomics applications numerical examples which include parameters estimated from actual microarray data illustrate the analysis throughout

exact representation of the second order moments for resubstitution and leave one out error estimation for linear discriminant analysis in the univariate heteroskedastic gaussian model this paper provides exact analytical expressions for the bias variance and rms for the resubstitution and leave one out error estimators in the case of linear discriminant analysis lda in the univariate heteroskedastic gaussian model neither the variances nor the sample sizes for the two classes need be the same the generality of heteroskedasticity unequal variances is a fundamental feature of the work presented in this paper which distinguishes it from past work the expected resubstitution and leave one out errors are represented by probabilities involving bivariate gaussian distributions their second moments and cross moments with the actual error are represented by 3 and 4 variate gaussian distributions from these the bias deviation variance and rms for resubstitution and leave one out as estimators of the actual error can be computed the rms expressions are applied to the determination of sample size and illustrated in biomarker classification

optimal bayesian classification with vector autoregressive data dependency in classification theory it is generally assumed that the data are independent and identically distributed however in many practical applications we face a set of observations that are collected sequentially with a dependence structure among samples the primary focus of this investigation is to construct the optimal bayesian classifier obc when the training observations are serially dependent to model the effect of dependency we assume the training observations are generated from text var p which is a multidimensional vector autoregressive process of order p at the same time we assume there exists uncertainty about parameters governing the text var p model to model this uncertainty we assume that model parameters coefficient matrices are random variables with a prior distribution and find the resulting obc under the assumption of known covariance matrices of white noise processes we employ simulations using both synthetic and real data to demonstrate the efficacy of the constructed obc

fuzzy rough regression with application to wind speed prediction accurate wind speed prediction is a prerequisite of large scale wind power generation there are several uncertain factors which degrade the performance of the current wind speed prediction systems fuzzy rough sets are considered as a powerful tool to deal with uncertainty and have been widely discussed and applied in classification learning in this work we describe a regression algorithm based on fuzzy rough sets consisting of fuzzy partition fuzzy approximation and estimation of regression values in this algorithm the training set is divided into k fuzzy classes with fuzzy partition and then the predicted values of test samples are determined in the finite intervals with fuzzy rough approximation finally they are estimated with lower and upper limits of the intervals numerical experiments on uci data sets and wind speed prediction show the effectiveness of the proposed algorithm