a stable truncated series approximation of the reduction to the pole operator we combine a stabilized reduction to the pole and an upward continuation filter to produce meaningful reduced to the pole fields at low magnetic latitudes the stabilizing procedure is based on the development in taylor s series of the theoretical expression for the reduction to the pole filter in the wavenumber domain the filter instability is caused by the huge filter amplitudes along the magnetization azimuth which are expressed by the infinite sum of terms close to unity the stabilizing procedure reduces to simply truncating the infinite series the upward continuation filter attenuates the high wavenumber component of the noise and allows us to design a stabilized filter closer to the theoretical one besides quantitative interpretations of source depth based on the filtered field are more reliable when using upward continuation as compared with arbitrary low pass filters the proposed filter was applied to synthetic data of a single prism uniformly magnetized along a supposedly known direction

geometric exponents and kleinian groups suppose lambda is the limit set of an analytically finite kleinian group and that omega j is an enumeration of the components of omega s 2 setminus lambda then sum j diam omega j 2 infty this had been conjectured by maskit we also define a number of different geometric critical exponents associated to a compact set in the plane which generalize the index of besicovitch and taylor on the line although these exponents may differ for general sets we show that they are all equal when lambda is the limit set of a non elementary analytically finite kleinian group and they agree with the classical poincare exponent

reduction to the pole at low latitudes by wiener filtering using simple estimates of the signal and noise power from gridded magnetic data we design regulated frequency domain operators for reduction to the pole at low magnetic latitudes these operators suppress the artifacts along the direction of the magnetic declination associated with the conventional reduction to the pole procedure with negligible increase in computational load the new procedure is applied to produce high quality reductions to the pole for noisy low latitude synthetic data and for magnetic data from the dixon seamount

on general resampling algorithms and their performance in distribution estimation recent work of several authors has focussed on first order properties e g consistency of general bootstrap algorithms where the numbers of times that data values are resampled form an exchangeable sequence in the present paper we develop second order properties of such algorithms in a very general setting performance is discussed in the context of distribution estimation and formulae for higher order moments and cumulants are developed arguing thus necessary and sufficient conditions are given for general resampling algorithms to correctly capture second order properties

a scattered equivalent source method for interpolation and gridding of potential field data in three dimensions potential field geophysical data observed at scattered discrete points in three dimensions can be interpolated gridded for example onto a level surface by relating the point data to a continuous function of equivalent discrete point sources the function used here is the inverse distance newtonian potential the sources located beneath some of the data points at a depth proportional to distance to the nearest neighboring data point are determined iteratively areas of no data are filled by minimum curvature for two dimensional 2 d data all data points at the same elevation grids calculated by minimum curvature and by equivalent sources are similar but the equivalent source method can be tuned to reduce aliasing gravity data in an area of high topographic relief in southwest u s a were gridded by minimum curvature a 2 d algorithm and also by equivalent sources 3 d the minimum curvature grid shows strong correlation with topography as expected because variation in gravity effect due to

rapid construction of equivalent sources using wavelets we have developed a fast algorithm for generating an equivalent source by using fast wavelet transforms based on orthonormal compactly supported wavelets we apply a 2d wavelet transform to each row and column of the coefficient matrix and subsequently threshold the transformed matrix to generate a sparse representation in the wavelet domain the algorithm then uses this sparse matrix to construct the the equivalent source directly in the wavelet domain performing an inverse wavelet transform then yields the equivalent source in the space domain using upward continuation of total field magnetic data between uneven surfaces as examples we have compared this approach with the direct solution using the dense matrix in the space domain we have shown that the wavelet approach can reduce the cpu time by as many as two orders of magnitude

a weighted bootstrap approach to bootstrap iteration the operation of resampling from a bootstrap resample encountered in applications of the double bootstrap may be viewed as resampling directly from the sample but using probability weights that are proportional to the numbers of times that sample values appear in the resample this suggests an approximate approach to double bootstrap monte carlo simulation where weighted bootstrap methods are used to circumvent much of the labour involved in compounded monte carlo approximation in the case of distribution estimation or equivalently confidence interval calibration the new method may be used to reduce the computational labour moreover the method produces the same order of magnitude of coverage error for confidence intervals or level error for hypothesis tests as a full application of the double bootstrap

relations between potential fields and some equivalent sources the concept of equivalent sources is described from first principles two types of sources that produce the same magnetic field are studied in detail thin sheets and uniaxially magnetized half spaces simple analytic solutions show that their magnetization variations are essentially given by the pseudogravity field and the magnetic field respectively while the former varies slowly to account for the slow variation in the vertically integrated magnetization the latter varies rapidly to account for the spatial variation in rock magnetization close to the observation points an extension of these simple distributions is the sandwich distribution using statistical arguments we construct sandwich distributions the depths and mean magnetizations of each layer are found from the decay of the azimuthally averaged power spectrum the equivalent layer magnetizations closely resemble the magnetic field anomaly when upward continued to a height equal to the mean depth of the layer

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

principal weighted support vector machines for sufficient dimension reduction in binary classification sufficient dimension reduction is popular for reducing data dimensionality without stringent model assumptions however most existing methods may work poorly for binary classification for example sliced inverse regression li 1991 can estimate at most one direction if the response is binary in this paper we propose principal weighted support vector machines a unified framework for linear and nonlinear sufficient dimension reduction in binary classification its asymptotic properties are studied and an efficient computing algorithm is proposed numerical examples demonstrate its performance in binary classification

bounded geodesics of riemann surfaces and hyperbolic manifolds we study the set of bounded geodesics of hyperbolic manifolds for general riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its hausdorff dimension some applications to diophantine approximation are included

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

hausdorff dimension and dendritic limit sets let be a singly degenerate closed surface group acting properly discontinuously on hyperbolic 3 space h3 such that h3 has positive injectivity radius it is known that the limit set is a dendrite of hausdorff dimension 2 we show that the cut point set of the limit set has hausdorff dimension strictly less than 2

using an equivalent source with positivity for low latitude reduction to the pole without striation we present a reformulation of reduction to the pole rtp of magnetic data at low latitudes and the equator using equivalent sources the proposed method addresses both the theoretical difficulty of low latitude instability and the practical issue of computational cost we prove that a positive equivalent source exists when the magnetic data are produced by normal induced magnetization and we show that the positivity is sufficient to overcome the low latitude instability in the space domain we further apply a regularization term directly to the recovered rtp field to improve the solution the use of equivalent source also naturally enables the processing of data acquired on uneven surface the result is a practical algorithm that is effective at the equatorial region and can process large scale data sets with uneven observation heights

overfitting in making comparisons between variable selection methods this paper addresses a common methodological flaw in the comparison of variable selection methods a practical approach to guide the search or the selection process is to compute cross validation performance estimates of the different variable subsets used with computationally intensive search algorithms these estimates may overfit and yield biased predictions therefore they cannot be used reliably to compare two selection methods as is shown by the empirical results of this paper instead like in other instances of the model selection problem independent test sets should be used for determining the final performance the claims made in the literature about the superiority of more exhaustive search algorithms over simpler ones are also revisited and some of them infirmed

improvements on cross validation the 632 bootstrap method abstract a training set of data has been used to construct a rule for predicting future responses what is the error rate of this rule this is an important question both for comparing models and for assessing a final selected model the traditional answer to this question is given by cross validation the cross validation estimate of prediction error is nearly unbiased but can be highly variable here we discuss bootstrap estimates of prediction error which can be thought of as smoothed versions of cross validation we show that a particular bootstrap method the 632 rule substantially outperforms cross validation in a catalog of 24 simulation experiments besides providing point estimates we also consider estimating the variability of an error rate estimate all of the results here are nonparametric and apply to any possible prediction rule however we study only classification problems with 0 1 loss in detail our simulations include smooth prediction rules like fisher s linear discriminant fun

strange actions of groups on spheres in fs we investigated certain topological analogs of schottky groups called admissible actions and their compatibility with various structures on spheres we constructed an action f 2 s 3 s 3 which was not topologically conjugate to a uniformly quasiconformal action also there was an example f r 2 s 3 s 3 r sufficiently large which was smooth and uniformly quasiconformal but not topologically conjugate to a conformal action and we gave examples of admissible actions on higher dimensional spheres but did not analyse the structures preserved

cusps are dense we show cusps are dense in bers boundary for teichmuller space the proof rests on an estimate for the algebraic effect of a unit quasiconformal deformation supported in the thin part of a hyperbolic riemann surface

a transcendental julia set of dimension 1 we construct a non polynomial entire function whose julia set has finite 1 dimensional spherical measure and hence hausdorff dimension 1 in 1975 baker proved the dimension of such a julia set must be at least 1 but whether this minimum could be attained has remained open until now our example also has packing dimension 1 and is the first transcendental julia set known to have packing dimension strictly less than 2 it is also the first example with a multiply connected wandering domain where the dynamics can be completely described

limit sets of discrete groups of isometries of exotic hyperbolic spaces let be a geometrically finite discrete group of isometries of hyperbolic space hf where f r c h or o in which case n 2 we prove that the critical exponent of equals the hausdorff dimension of the limit sets and that the smallest eigenvalue of the laplacian acting on square integrable functions is a quadratic function of either of them when they are sufficiently large a generalization of hopf ergodicity theorem for the geodesic flow with respect to the bowen margulis measure is also proven

quasiconformal groups patterson sullivan theory and local analysis of limit sets we extend the part of patterson sullivan theory to discrete quasiconformal groups that relates the exponent of convergence of the poincare series to the hausdorff dimension of the limit set in doing so we define new bi lipschitz invariants that localize both the exponent of convergence and the hausdorff dimension we find these invariants help to expose and explain the discrepancy between the conformal and quasiconformal setting of patterson sullivan theory

estimating subject specific survival functions under the accelerated failure time model summary we use the semiparametric accelerated failure time model to predict the survival function and its related quantities for future subjects with a given set of covariates we derive the large sample distribution for the subject specific cumulative hazard function estimate we then propose a simple resampling technique for constructing pointwise confidence intervals and simultaneous bands for the corresponding survival function and its quantile function over a properly selected time interval the new proposals are illustrated with the mayo primary biliary cirrhosis data

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

geometrical finiteness for hyperbolic groups abstract in this paper we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension we prove the equivalence of various definitions of geometrical finiteness and describe the geometry of fundamental domains we give a complete account of when dirichiet domains are finite sided

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

the patterson measure for geometrically finite groups with parabolic elements new and old we study the patterson measure on the limit set of a geometrically finite kleinian group with parabolic elements using the conformality of the patterson measure in combination with basic hyperbolic geometry a global shadow lemma is obtained this macroscopic measure estimate is exploited to derive a khintchine type theorem which is then used for a finer analysis of the density of the limit set in terms of hausdorff measure

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

bootstrap model selection abstract in a regression problem typically there are p explanatory variables possibly related to a response variable and we wish to select a subset of the p explanatory variables to fit a model between these variables and the response a bootstrap variable model selection procedure is to select the subset of variables by minimizing bootstrap estimates of the prediction error where the bootstrap estimates are constructed based on a data set of size n although the bootstrap estimates have good properties this bootstrap selection procedure is inconsistent in the sense that the probability of selecting the optimal subset of variables does not converge to 1 as n this inconsistency can be rectified by modifying the sampling method used in drawing bootstrap observations for bootstrapping pairs response explanatory variable it is found that instead of drawing n bootstrap observations a customary bootstrap sampling plan much less bootstrap observations should be sampled the bootstrap selection p

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

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

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

related aspects of positivity in riemannian geometry pour une variete de riemann on etudie le lien entre le point 0 du l 2 spectre le plus proche de zero pour l extension de friedrich d un operateur semi defini symetrique et le taux exponentiel de transition d un operateur de markoff conservant la positivite

representation theoretic rigidity in psl 2 r let g be a connected simple lie group with trivial center let f be an abstract group and let 51 and 2 be inclusions of f in g assume throughout that each of the images 5j f is a lattice subgroup meaning that j f is discrete and that the g invariant measure on g s j f has total finite mass we say that tl and are equivalent if there is some automorphism 0 of g so that 2 qo51 if g is not isomorphic to psl 2 r then the mostow rigidity theorem see 18 19 16 and 24 says that 1 and t2 axe necessarily equivalent alternatively this says that any isomorphism between lattice subgroups of g extends to an automorphism of the whole group this remarkable result fails for psl 2 r see section 2 nonetheless taking g psl 2 r we have

quasi actions on trees i bounded valence given a bounded valence bushy tree t we prove that any cobounded quasi action of a group g on t is quasiconjugate to an action of g on another bounded valence bushy tree t this theorem has many applications quasi isometric rigidity for fundamental groups of finite bushy graphs of coarse pd n groups for each fixed n a generalization to actions on cantor sets of sullivan s theorem about uniformly quasiconformal actions on the 2 sphere and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice finally we give the first examples of two finitely generated groups which are quasi isometric and yet which cannot act on the same proper geodesic metric space properly discontinuously and cocompactly by isometries

fuchsian groups quasiconformal groups and conical limit sets we construct examples showing that the normalized lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one this is in contrast to the cases of fuchsian groups acting on the disc conformal groups acting discontinuously on the ball in dimension three or higher uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions in these cases the normalized lebesgue measure is either zero or one

estimating the confidence interval for prediction errors of support vector machine classifiers support vector machine svm is one of the most popular and promising classification algorithms after a classification rule is constructed via the svm it is essential to evaluate its prediction accuracy in this paper we develop procedures for obtaining both point and interval estimators for the prediction error under mild regularity conditions we derive the consistency and asymptotic normality of the prediction error estimators for svm with finite dimensional kernels a perturbation resampling procedure is proposed to obtain interval estimates for the prediction error in practice with numerical studies on simulated data and a benchmark repository we recommend the use of interval estimates centered at the cross validated point estimates for the prediction error further applications of the proposed procedure in model evaluation and feature selection are illustrated with two examples

the hausdorff dimension of the limit set of a geometrically finite kleinian group we consider in this paper groups of mobius transformations of l r tj zc such a group is called kleinian if it acts discontinuously somewhere in 1 the action of g extends to the n d dimensional hyperbolic space h i r zc and g is geometrically finite if there is a hyperbolic fundamental polyhedron with a finite number of faces for the action of g in h t for a more precise definition see 15 1 b we prove in this paper that the hausdor f dimension dim l g of the limit set l g of a geometrically finite kleinian group g of 1 is less than n theorem d our proof of this theorem is based on the following obser ation assume that g is of compact t3 pe i e if izi i h iui then i l g g is compact then there is an integer q such that if we divide any n cube q of r into q equal subcubes then at least one of these subcubes does not touch l g let l4 q be the family of these subcubes which touch l g then the n measures of q e le q do not add up to the n measure of q and we get

continuation of potential fields between arbitrary surfaces an equivalent source algorithm is described for continuing either one or two dimensional potential fields between arbitrary surfaces in the two dimensional case the dipole surface is approximated as a set of plane faces with constant moments over each face in the one dimensional case the plane faces of the dipole surface reduce to straight line segments application of the algorithm to model and field examples of aeromagnetic data shows the method to be effective and accurate even when the terrain has strong topographic relief and is composed of highly magnetic volcanic rocks

semiparametric box cox power transformation models for censored survival observations the accelerated failure time model specifies that the logarithm of the failure time is linearly related to the covariate vector without assuming a parametric error distribution in this paper we consider the semiparametric box cox transformation model which includes the above regression model as a special case to analyse possibly censored failure time observations inference procedures for the transformation and regression parameters are proposed via a resampling technique prediction of the survival function of future subjects with a specific covariate vector is also provided via pointwise and simultaneous interval estimates all the proposals are illustrated with datasets from two clinical studies

bootstrap procedures under some non i i d models it is shown in this article that the classical i i d bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location as long as the random observations are independently drawn from distributions with essentially a common location this may be viewed as a robust property of the classical i i d bootstrap also included is a study of the second order properties of a different bootstrap procedure proposed by wu in the context of heteroscedasticity in regression