levels of knotting of spatial handlebodies if h is a spatial handlebody i e a handlebody embedded in the 3 sphere a spine of h is a graph s3 such that h is a regular neighbourhood of usually h is said to be unknotted if it admits a planar spine this suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties building on this remark we define several levels of knotting of spatial handlebodies and we provide a complete description of the relationships between these levels focusing our attention on the case of genus 2 we also relate the knotting level of a spatial handlebody h to classical topological properties of its complement m s3 h such as its cut number more precisely we show that if h is not highly knotted then m admits special cut systems for m and we discuss the extent to which the converse implication holds along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies these obstructions are based on recent quandle coloring invariants for spatial handlebodies on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links on the analysis of appropriate coverings of handlebody complements and on the study of the classical alexander elementary ideals of their fundamental groups

working correlation structure misspecification estimation and covariate design implications for generalised estimating equations performance the method of generalised estimating equations for regression modelling of clustered outcomes allows for specification of a working matrix that is intended to approximate the true correlation matrix of the observations we investigate the asymptotic relative efficiency of the generalised estimating equation for the mean parameters when the correlation parameters are estimated by various methods the asymptotic relative efficiency depends on three features of the analysis namely i the discrepancy between the working correlation structure and the unobservable true correlation structure ii the method by which the correlation parameters are estimated and iii the design by which we refer to both the structures of the predictor matrices within clusters and distribution of cluster sizes analytical and numerical studies of realistic data analysis scenarios show that choice of working covariance model has a substantial impact on regression estimator efficiency protection against avoidable loss of efficiency associated with covariance misspecification is obtained when a gaussian estimation pseudolikelihood procedure is used with an ar 1 structure

working covariance model selection for generalized estimating equations we investigate methods for data based selection of working covariance models in the analysis of correlated data with generalized estimating equations we study two selection criteria gaussian pseudolikelihood and a geodesic distance based on discrepancy between model sensitive and model robust regression parameter covariance estimators the gaussian pseudolikelihood is found in simulation to be reasonably sensitive for several response distributions and noncanonical mean variance relations for longitudinal data application is also made to a clinical dataset assessment of adequacy of both correlation and variance models for longitudinal data should be routine in applications and we describe open source software supporting this practice

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

symmetry of knots and cyclic surgery if a nontorus knot k admits a symmetry which is not a strong inversion then there exists no nontrivial cyclic surgery on k no surgery on a symmetric knot can produce a fake lens space or a 3 manifold n with 1 m 2 this generalizes the result of culler gordon luecke shalen bleiler scharlemann and supports the conjecture that not nontrivial surgery on a nontrivial knot yields a 3 manifold m with 1 m 5

heuristics of instability and stabilization in model selection in model selection usually a best predictor is chosen from a collection hat mu cdot s of predictors where hat mu cdot s is the minimum least squares predictor in a collection mathsf u s of predictors here s is a complexity parameter that is the smaller s the lower dimensional smoother the models in mathsf u s if mathsf l is the data used to derive the sequence hat mu cdot s the procedure is called unstable if a small change in mathsf l can cause large changes in hat mu cdot s with a crystal ball one could pick the predictor in hat mu cdot s having minimum prediction error without prescience one uses test sets cross validation and so forth the difference in prediction error between the crystal ball selection and the statistician s choice we call predictive loss for an unstable procedure the predictive loss is large this is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods unstable procedures can be stabilized by perturbing the data getting a new predictor sequence hat mu cdot s and then averaging over many such predictor sequences

working correlation structure identification in generalized estimating equations selecting an appropriate working correlation structure is pertinent to clustered data analysis using generalized estimating equations gee because an inappropriate choice will lead to inefficient parameter estimation we investigate the well known criterion of qic for selecting a working correlation structure and have found that performance of the qic is deteriorated by a term that is theoretically independent of the correlation structures but has to be estimated with an error this leads us to propose a correlation information criterion cic that substantially improves the qic performance extensive simulation studies indicate that the cic has remarkable improvement in selecting the correct correlation structures we also illustrate our findings using a data set from the madras longitudinal schizophrenia study

smooth bootstrap methods for analysis of longitudinal data in analysis of longitudinal data the variance matrix of the parameter estimates is usually estimated by the sandwich method in which the variance for each subject is estimated by its residual products we propose smooth bootstrap methods by perturbing the estimating functions to obtain bootstrapped realizations of the parameter estimates for statistical inference our extensive simulation studies indicate that the variance estimators by our proposed methods can not only correct the bias of the sandwich estimator but also improve the confidence interval coverage we applied the proposed method to a data set from a clinical trial of antibiotics for leprosy

a generalized bridge number for links in 3 manifolds in sch schubert introduced a new invariant of knots in the 3 sphere called the bridge number and showed that when reduced by 1 it is an additive invariant under the connected sum operation of knots several generalizations of schubert s theorem can be imagined indeed from schubert s perspective the behaviour of bridge number under addition is only a corollary of a more general theorem about bridge number of satellite knots boileau and lickorish have independently observed b l that the following generalization of schubert s theorem would be useful bridge number as defined by schubert begins with a projection of the knot or link onto a sphere in s 3 project instead onto a higher genus unknotted surface in s 3 one advantage of this approach is that it allows bridge number to be defined for a link in a 3 manifold other than 5 3 by projecting the link onto a heegaard surface for the 3 manifold here we explore how such an invariant behaves under connected sum using a line of argument similar to jaco s proof of haken s theorem that heegaard genus is additive ja a crucial additional ingredient needed is lemma4 6 some of the arguments were motivated by no in which as an aside a different generalization of schubert s theorem is attempted they want to show that the standard bridge number behaves well under knot decomposition even when the knot is decomposed by spheres intersecting the knot more than twice e g under tangle sums l besides recovering a new proof of schubert s theorem more in the spirit of jaco s proof of the haken lemma we are also able to prove an analogous theorem for bridge number defined with respect to genus one heegaard splittings of 5 3 or the lens space and also with respect to irreducible heegaard splittings of closed nonhaken 3 manifolds

analysis of variance of cross validation estimators of the generalization error this paper brings together methods from two different disciplines statistics and machine learning we address the problem of estimating the variance of cross validation cv estimators of the generalization error in particular we approach the problem of variance estimation of the cv estimators of generalization error as a problem in approximating the moments of a statistic the approximation illustrates the role of training and test sets in the performance of the algorithm it provides a unifying approach to evaluation of various methods used in obtaining training and test sets and it takes into account the variability due to different training and test sets for the simple problem of predicting the sample mean and in the case of smooth loss functions we show that the variance of the cv estimator of the generalization error is a function of the moments of the random variables y card sj sj and y card sjc sj c where sj sj are two training sets and sjc sj c are the corresponding test sets we prove that the distribution of y and y is hypergeometric and we compare our estimator with the one proposed by nadeau and bengio 2003 we extend these results in the regression case and the case of absolute error loss and indicate how the methods can be extended to the classification case we illustrate the results through simulation

a caveat concerning independence estimating equations with multivariate binary data clustered binary data occur commonly in both the biomedical and health sciences in this paper we consider logistic regression models for multivariate binary responses where the association between the responses is largely regarded as a nuisance characteristic of the data in particular we consider the estimator based on independence estimating equations iee which assumes that the responses are independent this estimator has been shown to be nearly efficient when compared with maximum likelihood ml and generalized estimating equations gee in a variety of settings the purpose of this paper is to highlight a circumstance where assuming independence can lead to quite substantial losses of efficiency in particular when the covariate design includes within cluster covariates assuming independence can lead to a considerable loss of efficiency in estimating the regression parameters associated with those covariates

akaike s information criterion in generalized estimating equations correlated response data are common in biomedical studies regression analysis based on the generalized estimating equations gee is an increasingly important method for such data however there seem to be few model selection criteria available in gee the well known akaike information criterion aic cannot be directly applied since aic is based on maximum likelihood estimation while gee is nonlikelihood based we propose a modification to aic where the likelihood is replaced by the quasi likelihood and a proper adjustment is made for the penalty term its performance is investigated through simulation studies for illustration the method is applied to a real data set

information ratio test for model misspecification in quasi likelihood inference in this article we focus on the circumstances in quasi likelihood inference that the estimation accuracy of mean structure parameters is guaranteed by correct specification of the first moment but the estimation efficiency could be diminished due to misspecification of the second moment we propose an information ratio ir statistic to test for model misspecification of the variance covariance structure through a comparison between two forms of information matrix the negative sensitivity matrix and the variability matrix we establish asymptotic distributions of the proposed ir test statistics we also suggest an approximation to the asymptotic distribution of the ir statistic via a perturbation resampling method moreover we propose a selection criterion based on the ir test to select the best fitting variance covariance structure from a class of candidates through simulation studies it is shown that the ir statistic provides a powerful statistical tool to detect different scenarios of misspecific

a modified pseudolikelihood approach for analysis of longitudinal data we consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters in general inconsistent estimators of the covariance variance correlation parameters will be produced when the working correlation matrix is misspecified which may result in great loss of efficiency of the mean parameter estimators albeit the consistency is preserved we consider using different working correlation models for the variance and the mean parameters in particular we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified the designated working correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters simulation studies indicate that the proposed algorithm performs very well we also applied different estimation procedures to a data set from a clinical trial for illustration

are large distance heegaard splittings generic in a previous paper we introduced a notion of genericity for countable sets of curves in the curve complex of a surface s based on the lebesgue measure on the space of projective measured laminations in s with this definition we prove that for each fixed g 1 the set of irreducible genus g heegaard splittings of high distance is generic in the set of all irreducible heegaard splittings our definition of genericity is different and more intrinsic then the one given via random walks

the estimation of prediction error having constructed a data based estimation rule perhaps a logistic regression or a classification tree the statistician would like to know its performance as a predictor of future cases there are two main theories concerning prediction error 1 penalty methods such as cp akaike s information criterion and stein s unbiased risk estimate that depend on the covariance between data points and their corresponding predictions and 2 cross validation and related nonparametric bootstrap techniques this article concerns the connection between the two theories a rao blackwell type of relation is derived in which nonparametric methods such as cross validation are seen to be randomized versions of their covariance penalty counterparts the model based penalty methods offer substantially better accuracy assuming that the model is believable

euclidean decompositions of noncompact hyperbolic manifolds on introduit une methode pour diviser une variete hyperbolique non compacte de volume fini en morceaux euclidiens canoniques

on the accuracy of efficiency of estimating equation approach summary liang and zeger 1986 biometrika73 13 22 introduced a generalized estimating equation gee approach based on a working correlation matrix to obtain efficient estimators of regression parameters in the class of generalized linear models for repeated measures data as demonstrated by crowder 1995 biometrika82 407 410 because of uncertainty of the definition of the working correlation matrix the liang zeger approach may in some cases lead to a complete breakdown of the estimation of the regression parameters after taking this comment of crowder into account recently sutradhar and das 1999 biometrika86 459 465 examined the loss of efficiency of the regression estimators due to misspecification of the correlation structures but their study was confined to the regression estimation with cluster level covariates as in the original paper of liang and zeger in this paper we study this efficiency loss problem for the generalized regression models with within cluster covariates by utilizing the approach of sutsadhar and das 1999

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

miscellanea on the efficiency of regression estimators in generalised linear models for longitudinal data summary liang zeger 1986 introduced a generalised estimating equations approach based on a working correlation matrix to obtain consistent and efficient estimators of regression parameters in the class of generalised linear models for repeated measures data as demonstrated by crowder 1995 because of the uncertainty of definition of the working correlation matrix the liang zeger approach may in some cases lead to a complete breakdown of the estimation of the regression parameters in this paper we show that even though the liang zeger approach in many situations yields consistent estimators for the regression parameters these estimators are usually inefficient as compared to the regression estimators obtained by using the independence estimating equations approach

extended generalized estimating equations for clustered data abstract typically analysis of data consisting of multiple observations on a cluster is complicated by within cluster correlation estimating equations for generalized linear modeling of clustered data have recently received much attention this article proposes an extension to the generalized estimating equation method proposed by liang and zeger which treats within cluster correlations as nuisance parameters using ideas from extended quasi likelihood estimating equations for regression and association parameters are provided simultaneously the resulting estimators are proven to be asymptotically normal and consistent under certain conditions the consistency of regression estimators allows incorrect modeling of the correlation among repeated responses the method is illustrated with an analysis of data from a developmental toxicity study

on floer homology and the berge conjecture on knots admitting lens space surgeries we complete the first step in a two part program proposed by baker grigsby and the author to prove that berge s construction of knots in the three sphere which admit lens space surgeries is complete the first step which we prove here is to show that a knot in a lens space with a threesphere surgery has simple in the sense of rank knot floer homology the second conjectured step involves showing that for a fixed lens space the only knots with simple floer homology belong to a simple finite family using results of baker we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots coupled with work of ni these knots provide the first infinite family of non trivial knots which are characterized by their knot floer homology as another application we provide a floer homology proof of a theorem of berge

median based classifiers for high dimensional data conventional distance based classifiers use standard euclidean distance and so can suffer from excessive volatility if vector components have heavy tailed distributions this difficulty can be alleviated by replacing the l2 distance by its l1 counterpart for example the l1 version of the popular centroid classifier would allocate a new data value to the population to whose centroid it was closest in l1 terms however this approach can lead to inconsistency because the centroid is defined using l2 rather than l1 distance in particular by mixing l1 and l2 approaches we produce a classifier that can seriously misidentify data in cases where the means and medians of marginal distributions take different values these difficulties motivate replacing centroids by medians however in the very high dimensional settings commonly encountered today this can be problematic if we attempt to work with a conventional spatial median therefore we suggest using componentwise medians to construct a robust class

parametric variance function estimation for nonnormal repeated measurement data zeger and liang 1986 biometrics 42 121 130 proposed a procedure for analyzing nonnormal longitudinal data in the context of the generalized linear model this procedure is extended to model variance heterogeneity allowing the observations to come from distributions with different scale parameters loss of efficiency is evaluated when heterogeneity of scale factor is ignored

essential surfaces of non negative euler characteristic in genus two handlebody exteriors we provide a classification of the essential surfaces of non negative euler characteristic in the exteriors of genus two handlebodies embedded in the 3 sphere

limiting subhessians limiting subjets and their calculus we study calculus rules for limiting subjets of order two these subjets are obtained as limits of sequences of subjets a subjet of a function f at some point x being the taylor expansion of a twice differentiable function which minorizes f and coincides with f at x these calculus rules are deduced from approximate or fuzzy calculus rules for subjets of order two in turn these rules are consequences of delicate results of crandall ishii lions we point out the similarities and the differences with the case of first order limiting subdifferentials

block regularized m 2 cross validated estimator of the generalization error a cross validation method based on replications of two fold cross validation is called an cross validation an cross validation is used in estimating the generalization error and comparing of algorithms performance in machine learning however the variance of the estimator of the generalization error in cross validation is easily affected by random partitions poor data partitioning may cause a large fluctuation in the number of overlapping samples between any two training test sets in cross validation this fluctuation results in a large variance in the cross validated estimator the influence of the random partitions on variance becomes serious as increases thus in this study the partitions with a restricted number of overlapping samples between any two training test sets are defined as a block regularized partition set the corresponding cross validation is called block regularized cross validation bcv it can effectively reduce the influence of random partitions we prove that the variance of the bcv estimator of the generalization error is smaller than the variance of cross validated estimator and reaches the minimum in a special situation an analytical expression of the variance can also be derived in this special situation this conclusion is validated through simulation experiments furthermore a practical construction method of bcv by a two level orthogonal array is provided finally a conservative estimator is proposed for the variance of estimator of the generalization error

a two dimensional accuracy based measure for classification performance abstract accuracy has been used traditionally to evaluate the performance of classifiers however it is well known that accuracy is not able to capture all the different factors that characterize the performance of a multiclass classifier in this manuscript accuracy is studied and analyzed as a weighted average of the classification rate of each class this perspective allows us to propose the dispersion of the classification rate of each class as its complementary measure in this sense a graphical performance metric which is defined in a two dimensional space composed by accuracy and dispersion is proposed to evaluate the performance of classifiers we show that the combined values of accuracy and dispersion must fall within a clearly bounded two dimensional region different for each problem the nature of this region depends only on the a priori probability of each class and not on the classifier used thus the performance of multiclassifiers is represented in a two dimensional space where the models can be compared in a more fair manner providing greater awareness of the strategies that are more accurate when trying to improve the performance of a classifier furthermore we experimentally analyze the behavior of seven different performance metrics based on the computation of the confusion matrix values in several scenarios identifying clusters and relationships between measures as shown in the experimentation the graphical metric proposed is specially suitable in challenging highly imbalanced and with a high number of classes datasets the approach proposed is a novel point of view to address the evaluation of multiclassifiers and it is an alternative to other evaluation measures used in machine learning

exceptional and cosmetic surgeries on knots we show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and heegaard surfaces in the manifolds that result from non trivial dehn surgeries on the knot in particular knots with high bridge distance do not admit non trivial non hyperbolic surgeries or non trivial cosmetic surgeries we further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of seifert genus or indeed by the genus of almost any essential surface or heegaard surface in the surgered manifold

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 repeated measures analysis with misspecified covariance structure in recent years various sophisticated methods have been developed for the analysis of repeated measures or longitudinal data the more traditional approach based on a normal likelihood function has been shown to be unsatisfactory in the sense of yielding asymptotically biased estimates when the covariance structure is misspecified more recent methodology based on generalized linear models and quasi likelihood estimation has gained widespread acceptance as generalized estimating equations however this also has theoretical problems in this paper a suggestion is made for improving the asymptotic behaviour of estimators by using the older approach implemented via gaussian estimation the resulting estimating equations include the quasi score function as one component so the methodology proposed can be viewed as a combination of gaussian estimation and generalized estimating equations which has a firmer asymptotic basis than either alone has

effects of variance function misspecification in analysis of longitudinal data the approach of generalized estimating equations gee is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within subject correlations the variance is often assumed to be a known function of the mean this article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses our numerical studies indicate that 1 correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified 2 misspecification of the variance function impacts much more on estimators for within cluster covariates than for cluster level covariates and 3 if the variance function is misspecified correct choice of the correlation structure may not necessarily improve estimation efficiency we illustrate impacts of different variance functions using a real data set from cow growth

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties

robust estimating functions and bias correction for longitudinal data analysis robust methods are useful in making reliable statistical inferences when there are small deviations from the model assumptions the widely used method of the generalized estimating equations can be robustified by replacing the standardized residuals with the m residuals if the pearson residuals are assumed to be unbiased from zero parameter estimators from the robust approach are asymptotically biased when error distributions are not symmetric we propose a distribution free method for correcting this bias our extensive numerical studies show that the proposed method can reduce the bias substantially examples are given for illustration

efficiency of regression estimates for clustered data statistical methods for clustered data such as generalized estimating equations gee and generalized least squares gls require selecting a correlation or convariance structure to specify the dependence between observations within a cluster valid regression estimates can be obtained that do not depend on correct specification of the true correlation but inappropriate specifications can result in a loss of efficiency we derive general expressions for the asymptotic relative efficiency of gee and gls estimators under nested correlation structures efficiency is shown to depend on the covariate distribution the cluster sizes the response variable correlation and the regression parameters the results demonstrate that efficiency is quite sensitive to the between and within cluster variation of the covariates and provide useful characterizations of models for which upper and lower efficiency bounds are attained efficiency losses for simple working correlation matrices such as independence can be large even for small to moderate correlations and cluster sizes

no unbiased estimator of the variance of k fold cross validation most machine learning researchers perform quantitative experiments to estimate generalization error and compare the performance of different algorithms in particular their proposed algorithm in order to be able to draw statistically convincing conclusions it is important to estimate the uncertainty of such estimates this paper studies the very commonly used k fold cross validation estimator of generalization performance the main theorem shows that there exists no universal valid under all distributions unbiased estimator of the variance of k fold cross validation the analysis that accompanies this result is based on the eigen decomposition of the covariance matrix of errors which has only three different eigenvalues corresponding to three degrees of freedom of the matrix and three components of the total variance this analysis helps to better understand the nature of the problem and how it can make naive estimators that don t take into account the error correlations due to the overlap between training and test sets grossly underestimate variance this is confirmed by numerical experiments in which the three components of the variance are compared when the difficulty of the learning problem and the number of folds are varied

cusp structures of alternating links summary an alternating link lfr is canonically associated with every finite connected planar graph f the natural ideal polyhedral decomposition of the complement of 5 is investigated natural singular geometric structures exist on s3 r with respect to which the geometry of the cusp has a shape reflecting the combinatorics of the underlying link projection for the class of balanced graphs this induces a flat structure on peripheral tori modelled on the tessellation of the plane by equilateral triangles examples of links containing immersed closed zl injective surfaces in their complements are given these surfaces persist after most surgeries on the link the resulting closed 3 manifolds consequently being determined by their fundamental groups

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

algorithmic stability and sanity check bounds for leave one out cross validation in this article we prove sanity check bounds for the error of the leave one out cross validation estimate of the generalization error that is bounds showing that the worst case error of this estimate is not much worse than that of the training error estimate the name sanity check refers to the fact that although we often expect the leave one out estimate to perform considerably better than the training error estimate we are here only seeking assurance that its performance will not be considerably worse perhaps surprisingly such assurance has been given only for limited cases in the prior literature on cross validation any nontrivial bound on the error of leave one out must rely on some notion of algorithmic stability previous bounds relied on the rather strong notion of hypothesis stability whose application was primarily limited to nearest neighbor and other local algorithms here we introduce the new and weaker notion of error stability and apply it to obtain sanity check bounds for leave one out for other classes of learning algorithms including training error minimization procedures and bayesian algorithms we also provide lower bounds demonstrating the necessity of some form of error stability for proving bounds on the error of the leave one out estimate and the fact that for training error minimization algorithms in the worst case such bounds must still depend on the vapnik chervonenkis dimension of the hypothesis class

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

essential laminations in 3 manifolds it is a well established principle in manifold topology that by studying codimension 1 objects and their complementary pieces one obtains a great deal of topological and geometric information about the manifold itself this is particularly true for 3 dimensional manifolds using incompressible surfaces a great many advances have been made during the last 30 years the most spectacular results were obtained by haken ha waldhausen w and thurston t1 more recently work t2 g14 using taut foliations in 3 manifolds has proven fruitful unfortunately in some sense most closed 3 manifolds do not contain incompressible surfaces and it is currently not known exactly which 3 manifolds have taut foliations the purpose of this paper is to study another codimension 1 object the essential lamination which is a generalization of the incompressible surface as well as the taut foliation we will show that the existence of an essential lamination in a 3 manifold m implies that m has useful properties similar to those of manifolds having either an incompressible surface or a taut foliation we will also show that the universal cover of a manifold containing an essential lamination is r3 precise definitions of terms used in the following theorem will be given in section 1 here we give only a rough idea we begin with two alternative approximate definitions of essential lamination i an essential lamination in a 3 manifold m is a lamination satisfying four conditions the inclusion of leaves of the lamination into m should induce an injection on stj the complement of the lamination should be irreducible no leaf should be a sphere and the lamination should be end incompressible to say that a lamination is end incompressible means roughly that a folded leaf can be straightened using an isotopy there are no infinite folds ii alternatively

generalized least squares with misspecified serial correlation structures summary the regression literature contains hundreds of studies on serially correlated disturbances most of these studies assume that the structure of the error covariance matrix q is known or can be estimated consistently from data surprisingly few studies investigate the properties of estimated generalized least squares gls procedures when the structure of q is incorrectly identified and the parameters are inefficiently estimated we compare the finite sample efficiencies of ordinary least squares ols gls and incorrect gls igls estimators we also prove new theorems establishing theoretical efficiency bounds for igls relative to gls and ols results from an exhaustive simulation study are used to evaluate the finite sample performance and to demonstrate the robustness of igls estimates vis a vis ols and gls estimates constructed for models with known and estimated but correctly identified q some of our conclusions for finite samples differ from established asymptotic results

on the use of a working correlation matrix in using generalised linear models for repeated measures summary in a seminal paper liang zeger 1986 extended the use of generalised linear models to repeated measures data they based the analysis on specifications for the means and variances of the observations as usual for generalised linear models but showed how specifications for the correlations between measurements made on the same unit could be avoided by using a working correlation matrix in some cases the parameters involved in this matrix are subject to an uncertainty of definition which can lead to a breakdown of the asymptotic properties of the estimators

rank based regression for analysis of repeated measures we consider rank based regression models for repeated measures to account for possible within subject correlations we decompose the total ranks into between and within subject ranks and obtain two different estimators based on between and within subject ranks a simple perturbation method is then introduced to generate bootstrap replicates of the estimating functions and the parameter estimates this provides a convenient way for combining the corresponding two types of estimating function for more efficient estimation

stable functions and common stabilizations of heegaard splittings we present a new proof of reidemeister and singer s theorem that any two heegaard splittings of the same 3 manifold have a common stabilization the proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type two cusps in a rubinstein scharlemann graphic for the two splittings

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

unbiased estimating equations from working correlation models for irregularly timed repeated measures the method of generalized estimating equations gees has been criticized recently for a failure to protect against misspecification of working correlation models which in some cases leads to loss of efficiency or infeasibility of solutions however the feasibility and efficiency of gee methods can be enhanced considerably by using flexible families of working correlation models we propose two ways of constructing unbiased estimating equations from general correlation models for irregularly timed repeated measures to supplement and enhance gee the supplementary estimating equations are obtained by differentiation of the cholesky decomposition of the working correlation or as score equations for decoupled gaussian pseudolikelihood the estimating equations are solved with computational effort equivalent to that required for a first order gee full details and analytic expressions are developed for a generalized markovian model that was evaluated through simulation large sample sandwich standard errors for working correlation parameter estimates are derived and shown to have good performance the proposed estimating functions are further illustrated in an analysis of repeated measures of pulmonary function in children

block regularized repeated learning testing for estimating generalization error abstract repeated learning testing rlt is a popular cross validation method that is usually adopted in estimation and comparison of algorithm performance in machine learning however the variance of the estimator of the generalization error in the rlt method is easily affected by random partitions poor data partitioning may cause a large overlap between any two training sets of rlt and enlarge the variance in the rlt estimator of the generalization error thus in this study we constrain numbers of overlapping samples between any two training sets and construct a novel data partitioning schema in which rlt is called the block regularized rlt brlt we theoretically prove that the brlt estimator has a smaller variance than does the rlt estimator specifically the variance of the rlt estimator reaches its minimum when samples in a data set equally occur in all training sets and all the numbers of overlapping samples are equal furthermore we provide easy to use construction algorithms of brlt s partition set for several training set size and partition count settings by adopting two level orthogonal arrays we also illustrate brlt s optimal properties with several simulated and real world examples

power and sample size calculation incorporating misspecifications of the variance function in comparative clinical trials with over dispersed count data over dispersed count data are frequently observed in clinical trials where the primary endpoint is occurrence of clinical events sample sizes of comparative clinical trials with these data are typically calculated under negative binomial models or quasi poisson models with specified variance functions or under the assumption that the specified working variance functions are correctly specified in this article we propose a sample size formula anticipating misspecifications of the working variance function we derived a method based on the asymptotic distribution of a wald test statistic with a sandwich type robust variance estimator under quasi poisson models under misspecifications of the working variance function the asymptotic variance of the estimator of the treatment effect is expressed as a form involving both true and working variance functions our sample size formula includes several existing formulas as special cases when the working variance function is correctly specified as the true variance function we also consider a sensitivity analysis for possible misspecifications of the true variance function when estimating sample sizes using our formula a simulation study demonstrated the adequacy of our formulas in finite sample size settings an application to a clinical trial to evaluate the treatment effect on prevention of copd exacerbation is provided

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

credible intervals for precision and recall based on a k fold cross validated beta distribution in typical machine learning applications such as information retrieval precision and recall are two commonly used measures for assessing an algorithm s performance symmetrical confidence intervals based on k fold cross validated t distributions are widely used for the inference of precision and recall measures as we confirmed through simulated experiments however these confidence intervals often exhibit lower degrees of confidence which may easily lead to liberal inference results thus it is crucial to construct faithful confidence credible intervals for precision and recall with a high degree of confidence and a short interval length in this study we propose two posterior credible intervals for precision and recall based on k fold cross validated beta distributions the first credible interval for precision or recall is constructed based on the beta posterior distribution inferred by all k data sets corresponding to k confusion matrices from a k fold cross validation second considering that each data set corresponding to a confusion matrix from a k fold cross validation can be used to infer a beta posterior distribution of precision or recall the second proposed credible interval for precision or recall is constructed based on the average of k beta posterior distributions experimental results on simulated and real data sets demonstrate that the first credible interval proposed in this study almost always resulted in degrees of confidence greater than 95 with an acceptable degree of confidence both of our two proposed credible intervals have shorter interval lengths than those based on a corrected k fold cross validated t distribution meanwhile the average ranks of these two credible intervals are superior to that of the confidence interval based on a k fold cross validated t distribution for the degree of confidence and are superior to that of the confidence interval based on a corrected k fold cross validated t distribution for the interval length in all 27 cases of simulated and real data experiments however the confidence intervals based on the k fold and corrected k fold cross validated t distributions are in the two extremes thus when focusing on the reliability of the inference for precision and recall the proposed methods are preferable especially for the first credible interval

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

gaussian estimation for correlated binomial data summary the mean variance structure of correlated binomial data is discussed and whittle s 1961 gaussian estimation is suggested as a useful general method in this context comparisons are made with other monrent methods in current use