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

numerical solution to a sideways parabolic equation the inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane this is a model of a problem where one wants to determine the temperature on both sides of a thick wall but one side is inaccessible to measurements a numerical procedure for this severely ill posed problem is suggested which consists of two steps namely a mollification of the data and a marching difference scheme the numerical method is proved to be stable several computational results are presented and discussed copyright 2001 john wiley sons ltd

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

reflection subgroups of coxeter systems let w r be a coxetcr system and t ij w wrw denote the corresponding set of reflections for u e w define n w tv tlf tw l w h w ere i denotes the length function of w r the principal result 3 3 of this paper is that if w is a reflection subgroup of w i e w w n t then e ti n z n w 1 is a set of coxeter generators for w in particular w is a coxeter group in 6 v deodhar gives a geometric proof of this last fact using properties of the root system of w it is easily seen that the coxeter gcncrators for w defined by deodhar coincide with those described above the idea of the proof here is that n may be regarded as a cocycle with n r r rer and coxeter systems are characterized among groups generated by involutions r by the existence of such a cocycle one then checks that w h n w n w w e w gives a suitable cocycle for w the same argument applies to a wider class of reflection systems g x characterized by the cxistcncc of a certain cocycle and so we give the proof for these the coxeter systems are precisely the reflection systems g x in which x consists of involutions the arrangement of this paper is as follows in section 2 we describe the main facts about reflection systems closely paralleling 1 chap iv sect 11 in section 3 we prove that reflection subgroups of a reflection system are themselves reflection systems in a canonical way 3 3 and obtain some additional facts about their canonical generators 3 4 3 11 in section 4 we specialize to the case of a coxeter system and give a geometric criterion 4 4 for a set of reflections to be the set of canonical generators of a reflection subgroup in terms of the inner products of the

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

automorphisms of coxeter groups of rank 3 with infinite bonds abstract if w is a rank 3 coxeter group whose coxeter diagram contains at least one infinite bond then the automorphism group of w is larger than the group generated by the inner automorphisms and the automorphisms induced from automorphisms of the coxeter diagram in each case the automorphism group of w is described

infinitely divisible pulses continuous deconvolution and the characterization of linear time invariant systems this paper addresses the problem of determining the impulse response of a linear time invariant system by probing the system with a causal c infty approximation to the dirac delta function we analyze the ill posed deconvolution problem which results from a wide choice of possibly multimodal infinitely divisible probe pulses the notion of infinite divisibility is shown to play a key role when the system s response is suspected of having nondifferentiable singularities we reformulate the volterra integral equation as a cauchy problem for a linear partial differential equation in two independent variables and introduce the concepts of partial and continuous deconvolution we then show that partial deconvolution of the output waveform results in l infty error bounds for the regularized solution and its derivatives under l 2 a priori bounds on the data noise and the unknown system response using the poisson summation formula and fft algorithms we construct an efficient computational alg

impulse response acquisition as an inverse heat conduction problem the impulse response of a linear time invariant system can be obtained by deconvolution of the response to a physically realizable smooth approximation to the dirac delta function as shown in siam j app math 47 1987 pp 892 927 it is advantageous to synthesize infinitely divisible probe waveforms and recover the system s impulse response by solving the related cauchy problem for a generalized diffusion equation the present paper is concerned with perturbations of the originally designed pulse shape caused by amplifiers transducers and other interfacing devices that may destroy infinite divisibility it is shown that in a large number of cases a simple linear transformation of the noisy output data can be found that converts the deconvolution problem with the perturbed pulse back into one for which the diffusion theory applies the search for this transformation is accomplished in the fourier domain by comparing the perturbed pulse with the designed pulse as a by product it is shown

automorphisms of coxeter groups we compute aut w for any even coxeter group whose coxeter diagram is connected contains no edges labeled 2 and cannot be separated into more than 2 connected components by removing a single vertex the description is given explicitly in terms of the given presentation for the coxeter group and admits an easy characterization of those groups w for which out w is finite

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

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

two variations on a3 a1 a1 1 type discrete painlev equations by considering the normalizers of reflection subgroups of types a 1 1 and a 1 3 in w d5 1 two subgroups w a3 a1 1 w a1 1 and w a1 a1 1 w a3 1 can be constructed from a a3a1a1 1 ty

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

reflection rigidity of 2 spherical coxeter groups we prove that each finitely generated irreducible and 2 spherical coxeter system w s is strongly reflection rigid whenever the group w is of infinite order this means in particular that all reflection preserving automorphisms of such a group are inner by graph our result can be seen as a first major step towards a proof of the conjecture that all infinite irreducible coxeter systems are strongly reflection rigid if they do not admit diagram twists

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

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

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

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

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

normalizers of parabolic subgroups in coxeter groups we shall describe a simple procedure for obtaining presentations of normalizers of parabolic subgroups of arbitrary coxeter groups in each case the normalizer is the semidirect product of the parabolic subgroup by a complement which acts by permuting the simple re ections this complement may be viewed in a natural way as a maximal subgroup of a certain groupoid for which we obtain a set of generators and de ning relations moreover the de ning relations are all derived from subgroupoids of ranks one and two so that the groupoid can be viewed as a generalization of a coxeter group

a quiver presentation for solomon s descent algebra the descent algebra w is a subalgebra of the group algebra qw of a finite coxeter group w which supports a homomorphism with nilpotent kernel and commutative image in the character ring of w thus w is a basic algebra and as such it has a presentation as a quiver with relations here we construct w as a quotient of a subalgebra of the path algebra of the hasse diagram of the boolean lattice of all subsets of s the set of simple reflections in w from this construction we obtain some general information about the quiver of w and an algorithm for the construction of a quiver presentation for the descent algebra w of any given finite coxeter group w

probe waveforms and deconvolution in the experimental determination of elastic green s functions we propose a new time domain method for the experimental determination of the impulse response of linear systems the technique centers around the use of specifically designed probe waveforms these waveforms are particular c infty approximations to the dirac delta function and the heaviside unit step function and lead to a subsequent time domain deconvolution problem which can be implemented as a cauchy initial value problem this approach allows for continuous deconvolution a powerful option in the presence of noise we orient the discussion to the context of acoustic emission and elastic green s functions and present several numerical reconstructions of sharp signals from smooth synthetic data

locally parabolic subgroups in coxeter groups of arbitrary ranks abstract despite the significance of the notion of parabolic closures in coxeter groups of finite ranks the parabolic closure is not guaranteed to exist as a parabolic subgroup in a general case in this paper first we give a concrete example to clarify that the parabolic closure of even an irreducible reflection subgroup of countable rank does not necessarily exist as a parabolic subgroup then we propose a generalized notion of locally parabolic closure by introducing a notion of locally parabolic subgroups which involves parabolic ones as a special case and prove that the locally parabolic closure always exists as a locally parabolic subgroup it is a subgroup of parabolic closure and we give another example to show that the inclusion may be strict in general our result suggests that locally parabolic closure has more natural properties and provides more information than parabolic closure we also give a result on maximal locally finite locally parabolic subgroups in coxeter groups which generalizes a similar well known fact on maximal finite parabolic subgroups

normalisers of parabolic subgroups in finite unitary reflection groups abstract it is well known that the normaliser of a parabolic subgroup of a finite coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots we show that a similar result holds for all finite unitary reflection groups namely the normaliser of a parabolic subgroup is a semidirect product and in many but not all cases the complement can be obtained as the stabiliser of a set of roots in addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup

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

parabolic deligne lusztig varieties abstract motivated by the broue conjecture on blocks with abelian defect groups for finite reductive groups we study parabolic deligne lusztig varieties and construct on those which occur in the broue conjecture an action of a braid monoid whose action on their l adic cohomology will conjecturally factor through a cyclotomic hecke algebra in order to construct this action we need to enlarge the set of varieties we consider to varieties attached to a ribbon category this category has a garside family which plays an important role in our constructions so we devote the first part of our paper to the necessary background on categories with garside families