blow up of critical semilinear wave equations on schwarzschild spacetime abstract in this paper we established the blow up theorem for critical semilinear wave equations with focusing nonlinear term on schwarzschild spacetime concavity method is used to prove the main result which was introduced by levine payne in the papers levine and payne 1974 11 12 and levine 1973 7 in 1970s also a new auxiliary function with parameter is constructed following the idea from payne 1975 13 in order to guarantee that the result holds without any assumption on the initial data and initial energy

sufficient dimension reduction via squared loss mutual information estimation the goal of sufficient dimension reduction in supervised learning is to find the low dimensional subspace of input features that contains all of the information about the output values that the input features possess in this letter we propose a novel sufficient dimension reduction method using a squared loss variant of mutual information as a dependency measure we apply a density ratio estimator for approximating squared loss mutual information that is formulated as a minimum contrast estimator on parametric or nonparametric models since cross validation is available for choosing an appropriate model our method does not require any prespecified structure on the underlying distributions we elucidate the asymptotic bias of our estimator on parametric models and the asymptotic convergence rate on nonparametric models the convergence analysis utilizes the uniform tail bound of a u process and the convergence rate is characterized by the bracketing entropy of the model we then develop a natural gradient algorithm on the grassmann manifold for sufficient subspace search the analytic formula of our estimator allows us to compute the gradient efficiently numerical experiments show that the proposed method compares favorably with existing dimension reduction approaches on artificial and benchmark data sets

identification of linear directions in multivariate adaptive spline models identifying linear directions in multivariate regression has been a statistical challenge and has attracted particular attention since projection pursuit was developed to this end we propose and investigate the use of linear discriminant analysis and projection hessian directions in conjunction with multivariate adaptive regression splines simulation studies in a variety of settings demonstrate the usefulness of our approach in revealing both the functional forms and the linear substructures based on the observed data mathematical results are also provided to support our approach comparisons are made between our approach and existing approaches and the improvements are evident depending on the circumstance the extent of improvement can be substantial

splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

a two stage filter for smoothing multivariate noisy data on unstructured grids abstract experimental data as well as numerical simulations are very often affected by noise random fluctuations that distort the final output or the intermediate products of a numerical process in this paper a new method for smoothing data given on nonstructured grids is proposed it takes advantage of the smoothing properties of kernel weighted averaging and least squares techniques the weighted averaging is performed following a shepard like procedure where gaussian kernels are employed while least squares fitting is reduced to the use of very few basis functions so as to improve smoothness though at the price of interpolation accuracy once we have defined an n point grid and want to make a smoothed fit at a given grid point this method reduces to consideration of all m point stencils for a given value of m where m n that include that point and to make a least squares fitting for that particular point in each of those stencils finally the various results thus obtained are weight averaged the weights being inversely proportional to the distance of the point to the middle of the stencil then this process is repeated for all the grid points so as to obtain the smoothing of the input function though this method is generalized for the multivariate case one two and five dimensional test cases are shown as examples of the performance of this method

blow up of solutions to semilinear wave equations with variable coefficients and boundary this paper is devoted to studying initial boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions we will establish blow up results for the initial boundary value problems it is proved that there can be no global solutions no matter how small the initial data are and also we give the life span estimate of solutions for the problems

flexible discriminant analysis by optimal scoring abstract fisher s linear discriminant analysis is a valuable tool for multigroup classification with a large number of predictors one can find a reduced number of discriminant coordinate functions that are optimal for separating the groups with two such functions one can produce a classification map that partitions the reduced space into regions that are identified with group membership and the decision boundaries are linear this article is about richer nonlinear classification schemes linear discriminant analysis is equivalent to multiresponse linear regression using optimal scorings to represent the groups in this paper we obtain nonparametric versions of discriminant analysis by replacing linear regression by any nonparametric regression method in this way any multiresponse regression technique such as mars or neural networks can be postprocessed to improve its classification performance

nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions abstract the aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions n 5 we prove that solutions cannot exist globally in time the novelty is that we first use a cutoff function to transform the exterior problem to a cauchy problem then we divide the time into two intervals in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used furthermore we obtain the upper bound of the lifespan by following the idea of zhou and han 2014

the ii p method for estimating multivariate functions from noisy data the method for estimating an underlying smooth function of m variables x l xm using noisy data is based on approximating it by a sum of products of the form m m x m the problem is then reduced to estimating the univariate functions in the products a convergent algorithm is described the method keeps tight control on the degrees of freedom used in the fit many examples are given the quality of fit given by the method is excellent usually only a few products are enough to fit even fairly complicated functions the coding into products of univariate functions allows a relatively understandable interpretation of the multivariate fit

applying experimental design and regression splines to high dimensional continuous state stochastic dynamic programming in stochastic dynamic programming sdp with continuous state and decision variables the future value function is computed at discrete points in the state spac e interpolation can be used to approximate the values of the future value function between these discrete points however for large dimensional problems the number of discrete points required to obtain a good approximation of the future value function can be prohibitively large statistical methods of experimental design and function estimation may be employed to overcome this curse of dimensionality in this paper we describe a method for estimating the future value function by multivariate adaptive regression splines mars fit over a discretization scheme based on orthogonal array oa experimental designs because orthogonal arrays only grow polynomially in the state space dimension our oa mars method is accurately able to solve higher dimensional sdp problems than previously possible to our knowledge the most efficient method published prior to this work employs tensor product cubic splines to approximate the future value function johnson et al 1993 the computational advantages of oa ma rs are demonstrated in comparisons with the method using tensor product cubic splines for applications of an inventory forecasting sdp with up to nine state variables computed on a small workstation in particular the storage of an adequate tensor product cubic spline for six dimensions exceeds the memory of our workstation and the run time for an accurate oa mars sdp solution would be at least an order of magnitude faster than using tensor product cubic splines for higher than six dimensions

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

the glassey conjecture on asymptotically flat manifolds we verify the 3 dimensional glassey conjecture on asymptotically flat manifolds r1 3 g where the metric g is a small space time perturbation or nontrapping time independent perturbation of the minkowski metric and approaches the minkowski metric like 1 x with 0 moreover for radial asymptotically flat manifolds r1 n g with n 3 we verify the glassey conjecture in the radial case high dimensional wave equations with higher regularity are also discussed the main idea is to exploit local energy and kss estimates with variable coefficients together with the weighted sobolev estimates including trace estimates

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

a thurstonian pairwise choice model with univariate and multivariate spline transformations a probabilistic choice model is developed for paired comparisons data about psychophysical stimuli the model is based on thurstone s law of comparative judgment case v and assumes that each stimulus is measured on a small number of physical variables the utility of a stimulus is related to its values on the physical variables either by means of an additive univariate spline model or by means of multivariate spline model in the additive univariate spline model a separate univariate spline transformation is estimated for each physical dimension and the utility of a stimulus is assumed to be an additive combination of these transformed values in the multivariate spline model the utility of a stimulus is assumed to be a general multivariate spline function in the physical variables the use of b splines for estimating the transformation functions is discussed and it is shown how b splines can be generalized to the multivariate case by using as basis functions tensor products of the univariate basis functions a maximum likelihood estimation procedure for the thurstone case v model with spline transformation is described and applied for illustrative purposes to various artificial and real data sets finally the model is extended using a latent class approach to the case where there are unreplicated paired comparisons data from a relatively large number of subjects drawn from a heterogeneous population an em algorithm for estimating the parameters in this extended model is outlined and illustrated on some real data

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

tree structured classification via generalized discriminant analysis abstract the problem of constructing classification rules that can be represented as decision trees is considered each object to be classified has an associated x vector containing possibly incomplete covariate information tree construction is based on the information provided in a learning sample of objects with known class identities the x vectors in the learning sample may have missing values as well procedures are proposed for each of the components of classifier construction such as split selection tree size determination treatment of missing values and ranking of variables the main idea is recursive application of linear discriminant analysis with the variables at each stage being appropriately chosen according to the data and the type of splits desired standard statistical techniques used as basic building blocks include analysis of variance linear and canonical discriminant analysis and principal component analysis a new method of tree structured classification is obtained by assem

efficient algorithms for function approximation with piecewise linear sigmoidal networks this paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes a one hidden layer network is constructed one node at a time using the well known method of fitting the residual the task of fitting an individual node is accomplished using a new algorithm that searches for the best fit by solving a sequence of quadratic programming problems this approach offers significant advantages over derivative based search algorithms e g backpropagation and its extensions unique characteristics of this algorithm include finite step convergence a simple stopping criterion solutions that are independent of initial conditions good scaling properties and a robust numerical implementation empirical results are included to illustrate these characteristics

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

interactive tree structured regression via principal hessian directions abstract an interactive approach to tree structured regression is introduced unlike other procedures driven by cost optimization this approach focuses on the exploration of geometric information in the data the procedure begins with finding a direction along which the regression surface bends the most this direction is used for splitting the data into two regions within each region another direction is found and the partition is made in the same manner the process continues recursively until the entire regressor domain is decomposed into regions wherein the surface no longer bends significantly and linear regression fit becomes appropriate for implementing the direction search the method of principal hessian directions is applied several simulation and empirical results are reported comparison with three methods cart support and mars is made the benefit of using geometric information is highlighted

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

the l 1 method for robust nonparametric regression abstract consider the problem of estimating the regression function underlying a set of data that is contaminated by a heavy tailed error distribution the two standard approaches to such a problem are each flawed robust parametric least squares is appropriate only if there is good reason to believe that the underlying function has some particular form whereas most nonparametric regression methods are asymptotically equivalent to kernel regression methods which are not resistant against outliers existing algorithms for robustifying nonparametric regression procedures use either nonlinear optimization of an influence function or iterative solution of local polynomial fitting using reweighted least squares neither of these two approaches combines computational ease with asymptotic theoretical results furthermore application of the robust procedure has been limited almost exclusively to the case of a single explanatory variable with the response variable in this article a new hybrid method is propose

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

on abstract strichartz estimates and the strauss conjecture for nontrapping obstacles we establish the strauss conjecture concerning small data global existence for nonlinear wave equations in the setting of exterior domains to compact obstacles for space dimensions n 3 and 4 the obstacle is assumed to be nontrapping and the solution is assumed to satisfy either dirichlet or neumann conditions along the boundary of the obstacle the key step in the proof is establishing certain abstract strichartz estimates for the linear wave equation on exterior domains

a piecewise single index model for dimension reduction in this article we propose to use different single index models for observations in different regions of the sample space this approach inherits the estimation efficiency of the single index model in each region and at the same time allows the global model to have multidimensionality in the sense of conventional dimension reduction on the other hand the model can be seen as an extension of cart and a piecewise linear model proposed modeling procedures including identifying the region for every single index model and estimation of the single index models are developed simulation studies and real data analysis are employed to demonstrate the usefulness of the approach computer code and technical details of the method are provided as supplementary material online

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

concerning the strauss conjecture on asymptotically euclidean manifolds abstract in this paper we verify the strauss conjecture for semilinear wave equations on asymptotically euclidean manifolds when n 3 4 we also give an almost sharp lifespan for the subcritical case 2 p p c when n 3 the main ingredients include a keel smith sogge type estimate with 0 1 2 and weighted strichartz estimates of order two

local discriminative learning for pattern recognition abstract local discriminative learning methods approximate a target function a posteriori class probability function directly by partitioning the feature space into a set of local regions and appropriately modeling a simple input output relationship function in each one this paper presents a new method for judiciously partitioning the input feature space in order to accurately represent the target function the method accomplishes this by approximating not only the target function itself but also its derivatives as such the method partitions the input feature space along those dimensions for which the class probability function changes most rapidly thus minimizing bias the efficacy of the method is validated using a variety of simulated and real world data

some diagnostics for binary logistic regression via smoothing summary the paper discusses some diagnostic tools for binary logistic regression which use smoothing techniques smoothed binary data are employed to devise a battery of diagnostic plots from simple procedures which indicate only the adequacy or inadequacy of a maximum likelihood logistic fit to partial residual plots and delta local deviance plots which reveal the type of model inadequacy suggesting perhaps the addition of nonlinear terms or interactions properties of the methods are studied by simulation the diagnostic tools are illustrated on a set of data in which the probability of having telephone service is related to the demographic variables income and education

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

an adaptable k nearest neighbors algorithm for mmse image interpolation we propose an image interpolation algorithm that is nonparametric and learning based primarily using an adaptive k nearest neighbor algorithm with global considerations through markov random fields the empirical nature of the proposed algorithm ensures image results that are data driven and hence reflect ldquoreal worldrdquo images well given enough training data the proposed algorithm operates on a local window using a dynamic k nearest neighbor algorithm where k differs from pixel to pixel small for test points with highly relevant neighbors and large otherwise based on the neighbors that the adaptable k provides and their corresponding relevance measures a weighted minimum mean squared error solution determines implicitly defined filters specific to low resolution image content without yielding to the limitations of insufficient training additionally global optimization via single pass markov approximations similar to cited nearest neighbor algorithms provides additional weighting for filter generation the approach is justified in using a sufficient quantity of training per test point and takes advantage of image properties for in depth analysis we compare to existing methods and draw parallels between intuitive concepts including classification and ideas introduced by other nearest neighbor algorithms by explaining manifolds in low and high dimensions

nonnegative ranks decompositions and factorizations of nonnegative matrices the nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank one matrices into which the matrix can be decomposed additively such decompositions are useful in diverse scientific disciplines we obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm

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

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

locally weighted polynomial regression parameter choice and application to forecasts of the great salt lake 1 relationships between hydrologic variables are often nonlinear usually the functional form of such a relationship is not known a priori a multivariate nonparametric regression methodology is provided here for approximating the underlying regression function using locally weighted polynomials locally weighted polynomials consider the approximation of the target function through a taylor series expansion of the function in the neighborhood of the point of estimate cross validatory procedures for the selection of the size of the neighborhood over which this approximation should take place and for the order of the local polynomial to use are provided and shown for some simple situations the utility of this nonparametric regression approach is demonstrated through an application to nonparametric short term forecasts of the biweekly great salt lake volume blind forecasts up to 1 year in the future using the 1847 2004 time series of the great salt lake are presented

acoustic articulatory mapping in vowels by locally weighted regression a method for mapping between simultaneously measured articulatory and acoustic data is proposed the method uses principal components analysis on the articulatory and acoustic variables and mapping between the domains by locally weighted linear regression or loess cleveland w s 1979 j am stat assoc 74 829 836 the latter method permits local variation in the slopes of the linear regression assuming that the function being approximated is smooth the methodology is applied to vowels of four speakers in the wisconsin x ray microbeam speech production database with formant analysis results are examined in terms of 1 examples of forward articulation to acoustics mappings and inverse mappings 2 distributions of local slopes and constants 3 examples of correlations among slopes and constants 4 root mean square error and 5 sensitivity of formant frequencies to articulatory change it is shown that the results are qualitatively correct and that loess performs better than global regression the forward mappings show different root mean square error properties than the inverse mappings indicating that this method is better suited for the forward mappings than the inverse mappings at least for the data chosen for the current study some preliminary results on sensitivity of the first two formant frequencies to the two most important articulatory principal components are presented

regression by local fitting methods properties and computational algorithms abstract local regression is a procedure for estimating regression surfaces by the local fitting of linear or quadratic functions of the independent variables in a moving fashion that is analogous to how a moving average is computed for a time series the advantage of the methodology over the global fitting of parametric functions of the independent variables by least squares the current paradigm in regression studies is that a much wider class of regression functions can be estimated without distortion in this paper we discuss the methods their statistical properties and computational algorithms

the strauss conjecture on kerr black hole backgrounds we examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the strauss conjecture on the schwarzschild and kerr with small angular momentum black hole backgrounds the key estimates are a class of weighted strichartz estimates which are used near infinity where the metrics can be viewed as small perturbations of the minkowski metric and a localized energy estimate on the black hole background which handles the behavior in the remaining compact set

sparsity oriented importance learning for high dimensional linear regression abstractwith now well recognized nonnegligible model selection uncertainty data analysts should no longer be satisfied with the output of a single final model from a model selection process regar

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

a systematic approach to the analysis of complex interaction patterns in two level factorial designs analysis of data from two level full factorial designs often ends up with a final prediction equation that gives only the significant main effect and interaction terms when the number of interactions is small simple and useful interpretation of the equation can then be drawn immediately this article addresses a different situation in which the number of significant interactions may be large so that additional efforts are needed to sort out the pattern and the relationship between them in particular we bring out a class of models in which most interactions can be attributed to just one or two or very few factors and conditional on these factors the models become essentially linear we offer a strategy for uncovering this structure by linear domain splitting whereby a complicated global model is replaced by a series of local domain specific linear models we present a recommended methodology phd principal hessian direction for systematically proceeding from the global equation to local split dom

a tree structured adaptive network for function approximation in high dimensional spaces nonlinear function approximation is often solved by finding a set of coefficients for a finite number of fixed nonlinear basis functions however if the input data are drawn from a high dimensional space the number of required basis functions grows exponentially with dimension leading many to suggest the use of adaptive nonlinear basis functions whose parameters can be determined by iterative methods the author proposes a technique based on the idea that for most of the data only a few dimensions of the input may be necessary to compute the desired output function additional input dimensions are incorporated only where needed the learning procedure grows a tree whose structure depends upon the input data and the function to be approximated this technique has a fast learning algorithm with no local minima once the network shape is fixed and it can be used to reduce the number of required measurements in situations where there is a cost associated with sensing three examples are given controlling the dynamics of a simulated planar two joint robot arm predicting the dynamics of the chaotic mackey glass equation and predicting pixel values in real images from pixel values above and to the left

blow up of solutions to critical semilinear wave equations with variable coefficients abstract we verify the critical case p p 0 n of strauss conjecture 30 concerning the blow up of solutions to semilinear wave equations with variable coefficients in r n where n 2 the perturbations of laplace operator are assumed to be smooth and decay exponentially fast at infinity we also obtain a sharp lifespan upper bound for solutions with compactly supported data when p p 0 n the unified approach to blow up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of zhou 43 and zhou han 45 exponential eigenfunctions of the laplacian 37 are used to construct the test function q for linear wave equation with variable coefficients and john s method of iterations 13 is augmented with the slicing method of agemi kurokawa and takamura 1 for lower bounds in the critical case

filtering out high frequencies in time series using f transform abstract in this paper we focus on application of fuzzy transform f transform to analysis of time series under the assumption that the latter can be additively decomposed into trend cycle seasonal component and noise we prove that when setting properly width of the basic functions the inverse f transform of the time series closely approximates its trend cycle this means that the f transform almost completely removes the seasonal component and noise the obtained theoretical results are demonstrated on two artificial time series whose trend cycle is precisely known and on three real time series at the same time comparison with three classical methods namely stl method ssa method and low pass butterworth filter is also provided

ats methods nonparametric regression for non gaussian data abstract ats methods provide an approach to fitting curves and surfaces to data using nonparametric regression when distributions are not necessarily gaussian first a small amount of local averaging the a in ats is carried out then a variance stabilizing transformation is applied t and finally the result is smoothed s using a nonparametric regression procedure ats methods are quite broad in terms of applications in this article we show how they can be used for fitting a surface when the response is binary for estimating density and for estimating the spectrum of a time series we also present some theoretical investigations that give guidance on how to choose the amount of averaging and how efficient the methods are

conditional parametric models for storm sewer runoff 1 the method of conditional parametric modeling is introduced for flow prediction in a sewage system it is a well known fact that in hydrological modeling the response runoff to input precipitation varies depending on soil moisture and several other factors consequently nonlinear input output models are needed the model formulation described in this paper is similar to the traditional linear models like final impulse response fir and autoregressive exogenous arx except that the parameters vary as a function of some external variables the parameter variation is modeled by local lines using kernels for local linear regression as such the method might be referred to as a nearest neighbor method the results achieved in this study were compared to results from the conventional linear methods fir and arx the increase in the coefficient of determination is substantial furthermore the new approach conserves the mass balance better hence this new approach looks promising for various hydrological models and analysis

maximal correlation and adaptive splines friedman and silverman proposed the turbo algorithm for univariate additive modeling and also introduced the concept of minimum span as an important ingredient of the algorithm recently friedman adopted the idea of minimum span to develop his mars algorithm for multivariate additive modeling based on the observation that the correlation between basis functions on a single variable increases as their knots get closer this article presents a maximal correlation constraint to replace the minimum span condition in these algorithms limiting the maximal correlation between a new basis function and the set already in the model is a more effective way of avoiding degeneracies involved in the multivariate regression for a univariate model we show by example that mars version 3 5 and the modified algorithm proposed herein are competitive and that the latter is sometimes superior we also show that the running time of the new algorithm is linear in the sample size even though we locate the optimal knot size w

slicing regression a link free regression method slicing regression a link free regression m e t h o d author s naihua duan and k e r c h a u l i s o u r c e the annals of statistics v o l 19 n o 2 t u n 1991 p p 5 0 5 5 3 0 p u b l i s h e d b y institute of mathematical statistics s t a b l e u r l http www jstor org stable 2242072 accessed 16 05 2011 your use of the jstor archive indicates your acceptance of jstor s terms and conditions of use available at http www jstor org page info about policies terms jsp jstor s terms and conditions of use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the jstor archive only for your personal non commercial use please contact the publisher regarding any further use of this work publisher contact information may be obtained at http www jstor org action showpublisher publishercode ims each copy of any part of a jstor transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jstor is a not for profit service that helps scholars researchers and students discover use and build upon a wide range of content in a trusted digital archive we use information technology and tools to increase productivity and facilitate new forms of scholarship for more information about jstor please contact support jstor org http www jstor org

semiparametric analysis of variance with tensor product thin plate splines we consider fitting multivariate models to response data based on analysis of variance anova like decompositions of functions of several variables f t 1 t d c f t f t t a theory for fitting some components of these models with polynomial smoothing splines exists when each t is in a subset of the real line in this case the various estimated components turn out to be certain tensor sums and products of polynomial splines this approach may not be natural when one or more of the variables are geographic in particular where nature does not know north from east

exact smoothing properties of schrodinger semigroups we study schrodinger semigroups in the scale of sobolev spaces and show that for kato class potentials the range of such semigroups in lp has exactly two more derivatives than the potential this proves a conjecture of b simon we show that eigenfunctions of schrodinger operators are generically smoother by exactly two derivatives in given sobolev spaces than their potentials we give applications to the relation between the potential s smoothness and particle kinetic energy in the context of quantum mechanics and characterize kinetic energies in coulomb systems the techniques of proof involve leibniz and chain rules for fractional derivatives which are of independent interest as well as a new characterization of the kato class 1 introduction in this paper we attempt a precise study of the action of schrodinger semigroups in the scale of sobolev spaces work in this area was begun by b simon si4 in 1985 he proved a number of positive and negative results on such smoothing and also presented an open question in the form of a conjecture regarding necessary and sufficient conditions on the potential v for smoothing of order s by the semigroup of functions in l2 rd study of schrodinger semigroups is of interest for a number of reasons see

preaveraged localized orthogonal polynomial estimators for surface smoothing and partial differentiation abstract we propose a multivariate smoothing method based on products of localized orthogonal polynomial series estimators for a smooth regression surface in the fixed design regression model the estimation of partial derivatives is included the proposed method provides for automatic and efficient boundary modifications near the edges of the surface assuming that the boundary of the support of the regression function satisfies some regularity conditions by allowing for a preaveraging step the corresponding algorithms are speeded up considerably and are easy to implement computation of special boundary kernels as required by the kernel method to avoid edge effects is not necessary it is shown that under sufficient smoothness assumptions the global average mean squared error has the same optimal rate of convergence as the mean squared error at an interior point that is the boundary correction is asymptotically effective the method depends on two smoothing parameters one determining the amount

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat