covariate information matrix for sufficient dimension reduction building upon recent research on the applications of the density information matrix we develop a tool for sufficient dimension reduction sdr in regression problems called covariate information m

dimension reduction for nonelliptically distributed predictors sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor or when such conditions are not satisfied rely on marginal transformation or reweighting to fulfill them approximately for example a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distribution in this paper we reformulate the commonly used dimension reduction methods via the notion of central solution space so as to circumvent the requirements of such strong assumptions while at the same time preserve the desirable properties of the classical methods such as n consistency and asymptotic normality imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the curse of dimensionality but the development of this paper shows that this need not be the case the new methods will be compared with existing methods by simulation and applied to a data set

contour regression a general approach to dimension reduction we propose a novel approach to sufficient dimension reduction in regression based on estimating contour directions of small variation in the response these directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response leading to simple and general contour regression scr and gcr methodology in comparison with existing sufficient dimension reduction techniques this contour based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions while maintaining sqrt n consistency and computational ease moreover it proves robust to departures from ellipticity we establish population properties for both scr and gcr and asymptotic properties for scr simulations to compare performance with that of standard techniques such as ordinary least squares sliced inverse regression principal hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses we demonstrate the use of contour based methods on a data set concerning soil evaporation

neumann problems with time singularities in this paper we study the existence and uniqueness of solutions to a nonlinear neumann problem for a scalar second order ordinary differential equation u atu f t u u where a 0 and f t x y satisfies the local caratheodory conditions on 0 t xrxr

using the bootstrap to select one of a new class of dimension reduction methods dimension reduction in a regression analysis of response y given a p dimensional vector of predictors x reduces the dimension of x by replacing it with a lower dimensional linear combination x of the x s without specifying a parametric model and without loss of information about the conditional distribution of y given x we unify three existing methods sliced inverse regression sir sliced average variance estimate save and principal hessian directions phd into a larger class of methods each method estimates a particular candidate matrix essentially a matrix of parameters we introduce broad classes of dimension reduction candidate matrices and we distinguish estimators of the matrices from the matrices themselves given these classes of methods and several ways to estimate any matrix we now have the problem of selecting a particular matrix and estimation method we propose bootstrap methodology to select among candidate matrices estimators and dimension and in particular we investigate l

nonparametric variable transformation in sufficient dimension reduction sufficient dimension reduction sdr techniques have proven to be very useful data analysis tools in various applications underlying many sdr techniques is a critical assumption that the predictors are elliptically contoured when this assumption appears to be wrong practitioners usually try variable transformation such that the transformed predictors become nearly normal the transformation function is often chosen from the log and power transformation family as suggested in the celebrated box cox model however any parametric transformation can be too restrictive causing the danger of model misspecification we suggest a nonparametric variable transformation method after which the predictors become normal to demonstrate the main idea we combine this flexible transformation method with two well established sdr techniques sliced inverse regression sir and inverse regression estimator ire the resulting sdr techniques are referred to as tsir and tire respectively both simulation and real da

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

an effective semiparametric estimation approach for the sufficient dimension reduction model abstractin the exploratory data analysis the sufficient dimension reduction model has been widely used to characterize the conditional distribution of interest different from the existing approaches our main achievement is to simultaneously estimate two essential elements basis and structural dimension of the central subspace and the bandwidth of a kernel distribution estimator through a single estimation criterion with an appropriate order of kernel function the proposed estimation procedure can be effectively carried out by starting with a dimension of zero until the first local minimum is reached meanwhile the optimal bandwidth selector is ensured to be a valid tuning parameter for the central subspace estimator an important advantage of this estimation technique is its flexibility to allow a response to be discrete and some of covariates to be discrete or categorical providing that a certain continuity condition holds under very mild assumptions we further derive the uniform consistency of

wellposedness of a nonlocal nonlinear diffusion equation of image processing abstract existence and uniqueness of solutions to non smooth initial data are established for a slight modification of the degenerate regularization of the well known perona malik equation first proposed in guidotti and lambers 2009 the results heavily rely on the choice of an appropriate functional setting inspired by a recent approach to degenerate parabolic equations via so called singular riemannian manifolds amann 2013 2016

asymptotics for kernel estimate of sliced inverse regression to explore nonlinear structures hidden in high dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression li and duan proposed the sliced inverse regression sir method which is simple to use in this paper the asymptotic properties of the kernel estimate of sliced inverse regression are investigated it turns out that regardless of the kernel function the asymptotic distribution remains the same for a wide range of smoothing parameters

estimating central subspaces via inverse third moments summary modern graphical tools have enhanced our ability to learn many things from data directly in recent years dimension reduction has proven to be an effective tool for generating low dimensional summary plots without appreciable loss of information some well known inverse regression methods for dimension reduction such as sliced inverse regression li 1991 and sliced average variance estimation cook weisberg 1991 have been developed to estimate summary plots for regression and discriminant analysis in this paper we suggest a new method that makes use of inverse third moments this method can find structure beyond that found by sliced inverse regression and sliced average variance estimation particularly regression mixtures illustrative examples are presented

on the conditional distributions of low dimensional projections from high dimensional data we study the conditional distribution of low dimensional projections from high dimensional data where the conditioning is on other low dimensional projections to fix ideas consider a random d vector z that has a lebesgue density and that is standardized so that ez 0 and ezz id moreover consider two projections defined by unit vectors and namely a response y z and an explanatory variable x z it has long been known that the conditional mean of y given x is approximately linear in x under some regularity conditions cf hall and li 1993 however a corresponding result for the conditional variance has not been available so far we here show that the conditional variance of y given x is approximately constant in x again under some regularity conditions these results hold uniformly in and for most s provided only that the dimension of z is large in that sense we see that most linear submodels of a high dimensional overall model are approximately correct our findings provide new insights in a variety of modeling scenarios we discuss several examples including sparse linear modeling generalized linear models under potential link violation sliced inverse regression sliced average variance estimation and kernel learning machines

trace pursuit a general framework for model free variable selection abstractwe propose trace pursuit for model free variable selection under the sufficient dimension reduction paradigm two distinct algorithms are proposed stepwise trace pursuit and forward trace pursuit stepwise trace pursuit achieves selection consistency with fixed p forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality the screening consistency property of forward trace pursuit based on sliced inverse regression is established finite sample performances of trace pursuit and other model free variable selection methods are compared through numerical studies supplementary materials for this article are available online

dimension reduction for the conditional kth moment in regression the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction here we introduce the central kth moment subspace to capture information from the mean variance and so on up to the kth conditional moment of the regression new methods are studied for estimating these subspaces connections with sliced inverse regression are established and examples illustrating the theory are presented

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 perona malik paradox the perona malik equation is a formally ill posed parabolic equation for which simple discretizations are nevertheless numerically found to be stable after discussing the background of this parad

group analysis of a conformal perfect fluid spacetime we find new exact solutions of the einstein field equations for a perfect fluid metric conformal to a spacetime of type d in the petrov classification scheme we analyse the complete system of equations using lie group analysis while previous work was confined to conformal factors of the form u u t x we investigate the complete situation u u t x y z as well as an auxiliary integrability condition new classes of solutions are generated via the symmetry generators the resulting solutions are examined for physical plausibility expressions for the energy density and pressure are obtained explicitly and empirical results suggest that these dynamical quantities are positive as expected

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

efficiency loss caused by linearity condition in dimension reduction linearity sometimes jointly with constant variance is routinely assumed in the context of sufficient dimension reduction it is well understood that when these conditions do not hold blindly using them may lead to inconsistency in estimating the central subspace and the central mean subspace surprisingly we discover that even if these conditions do hold using them will bring efficiency loss this paradoxical phenomenon is illustrated through sliced inverse regression and principal hessian directions the efficiency loss also applies to other dimension reduction procedures we explain this empirical discovery by theoretical investigation

on estimation efficiency of the central mean subspace summary we investigate the estimation efficiency of the central mean subspace in the framework of sufficient dimension reduction we derive the semiparametric efficient score and study its practical applicability despite the difficulty caused by the potential high dimension issue in the variance component we show that locally efficient estimators can be constructed in practice we conduct simulation studies and a real data analysis to demonstrate the finite sample performance and gain in efficiency of the proposed estimators in comparison with several existing methods

estimation of mean response via the effective balancing score we introduce effective balancing scores for estimation of the mean response under a missing at random mechanism unlike conventional balancing scores the effective balancing scores are constructed via dimension reduction free of model specification three types of effective balancing scores are introduced those that carry the covariate information about the missingness the response or both they lead to consistent estimation with little or no loss in efficiency compared to existing estimators the effective balancing score based estimator relieves the burden of model specification and is the most robust it is a near automatic procedure which is most appealing when high dimensional covariates are involved we investigate both the asymptotic and the numerical properties and demonstrate the proposed method in a study on human immunodeficiency virus disease

functional calculus for semigroup generators via transference abstract in this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators in particular it is proved that if a generates a c 0 semigroup on a hilbert space then for each 0 the operator a has a bounded calculus for the closed ideal of bounded holomorphic functions on a sufficiently large right half plane that satisfy f z o e re z as z the bound of this calculus grows at most logarithmically as 0 as a consequence f a is a bounded operator for each holomorphic function f on a right half plane with polynomial decay at then we show that each semigroup generator has a so called strong m bounded calculus for all m n and that this property characterizes semigroup generators similar results are obtained if the underlying banach space is a umd space upon restriction to so called bounded semigroups the hilbert space results actually hold in general banach spaces

on the interpretation of regression plots abstract a framework is developed for the interpretation of regression plots including plots of the response against selected covariates residual plots added variable plots and detrended added variable plots it is shown that many of the common interpretations associated with these plots can be misleading the framework also allows for the generalization of standard plots and the development of new plotting paradigms a paradigm for graphical exploration of regression problems is sketched

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

dimension reduction for conditional mean in regression in many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors and less concerned with the other aspects of the conditional distribution in this paper we develop dimension reduction methods that incorporate this consideration we introduce the notion of the central mean subspace cms a natural inferential object for dimension reduction when the mean function is of interest we study properties of the cms and develop methods to estimate it these methods include a new class of estimators which requires fewer conditions than phd and which displays a clear advantage when one of the conditions for phd is violated cms also reveals a transparent distinction among the existing methods for dimension reduction ols phd sir and save we apply the new methods to a data set involving recumbent cows

efficient estimation in sufficient dimension reduction we develop an efficient estimation procedure for identifying and estimating the central subspace using a new way of parameterization we convert the problem of identifying the central subspace to the problem of estimating a finite dimensional parameter in a semiparametric model this conversion allows us to derive an efficient estimator which reaches the optimal semiparametric efficiency bound the resulting efficient estimator can exhaustively estimate the central subspace without imposing any distributional assumptions our proposed efficient estimation also provides a possibility for making inference of parameters that uniquely identify the central subspace we conduct simulation studies and a real data analysis to demonstrate the finite sample performance in comparison with several existing methods

determining the dimension of iterative hessian transformation the central mean subspace cms and iterative hessian transformation iht have been introduced recently for dimension reduction when the conditional mean is of interest suppose that x is a vector valued predictor and y is a scalar response the basic problem is to find a lower dimensional predictor t x such that e y x e y t x the cms defines the inferential object for this problem and iht provides an estimating procedure compared with other methods iht requires fewer assumptions and has been shown to perform well when the additional assumptions required by those methods fail in this paper we give an asymptotic analysis of iht and provide stepwise asymptotic hypothesis tests to determine the dimension of the cms as estimated by iht here the original iht method has been modified to be invariant under location and scale transformations to provide empirical support for our asymptotic results we will present a series of simulation studies these agree well with the theory the method is applied to analyze an ozone data set

one dimensional quaternion homogeneous polynomial differential equations in this paper we study the dynamics of real four dimensional polynomial differential equations of the forms q aqkq m or q qkaqm where q a are quaternions q is the conjugate of q and k and m are non negative integers in many parts of the work the quaternion structure is used along the proofs

symmetry properties and solutions of nonlinear dispersive thin film equations in three dimensions the lie point symmetries are obtained for each one of two nonlinear dispersive thin film equations in 2 1 variables by means of the invariants of the one and two dimensional subgroups all reductions to ordinary differential equations and to partial differential equations in two variables are obtained solutions are provided for a number of the reduced equations it is also ascertained that neither one of the two considered thin film equations is integrable in some cases solitary and periodic wave solutions having curved wavecrests are obtained

nonparametric estimation of conditional expectation with auxiliary information and dimension reduction nonparametric estimation of the conditional expectation e y u of an outcome y given a covariate vector u is of primary importance in many statistical applications such as prediction and personaliz

counting process based dimension reduction methods for censored outcomes we propose counting process based dimension reduction methods for right censored survival data semiparametric estimating equations are constructed to estimate the dimension reduction subspace for the failure time model our methods address two limitations of existing approaches first using the counting process formulation they do not require estimation of the censoring distribution to compensate for the bias in estimating the dimension reduction subspace second the nonparametric estimation involved adapts to the structural dimension so our methods circumvent the curse of dimensionality asymptotic normality is established for the estimators we propose a computationally efficient approach that requires only a singular value decomposition to estimate the dimension reduction subspace numerical studies suggest that our new approaches exhibit significantly improved performance the methods are implemented in the formula see text package formula see text

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

dimension reduction for non elliptically distributed predictors second order methods summary many classical dimension reduction methods especially those based on inverse conditional moments require the predictors to have elliptical distributions or at least to satisfy a linearity condition such conditions however are too strong for some applications li and dong 2009 introduced the notion of the central solution space and used it to modify first order methods such as sliced inverse regression so that they no longer rely on these conditions in this paper we generalize this idea to second order methods such as sliced average variance estimation and directional regression in doing so we demonstrate that the central solution space is a versatile framework we can use it to modify essentially all inverse conditional moment based methods to relax the distributional assumption on the predictors simulation studies and an application show a substantial improvement of the modified methods over their classical counterparts

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

conformal vector fields in general relativity a general discussion of conformal vector fields in space times is given amongst the topics considered are the maximum dimension of the conformal algebra for space times that are not conformally flat the nature of conformal isotropies and a new approach to the theorem of bilyalov and defrise carter concerning the reduction of the conformal algebra to a killing or homothetic algebra some deficiencies in the original statements of this theorem are discussed with reference to a general class of counterexamples and corrected the proof offered is geometrical in nature and has the advantage of displaying some of the more general features and properties of conformal vector fields and the ways in which they can differ from killing vector fields

singular parabolic equations of second order on manifolds with singularities abstract the main aim of this article is to establish an l p theory for elliptic operators on manifolds with singularities the particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the singular ends of the manifolds such a theory is of importance for the study of elliptic and parabolic equations on non compact or even incomplete manifolds with or without boundary

estimating the structural dimension of regressions via parametric inverse regression a new estimation method for the dimension of a regression at the outset of an analysis is proposed a linear subspace spanned by projections of the regressor vector x which contains part or all of the modelling information for the regression of a vector y on x and its dimension are estimated via the means of parametric inverse regression smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model no restrictions are placed on the distribution of the regressors the estimate of the dimension of the regression is based on optimal estimation procedures a simulation study shows the method to be more powerful than sliced inverse regression in some situations

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

a boundary element numerical scheme for the two dimensional convection diffusion equation in this study the theoretical and numerical fundamentals of biem techniques for the two dimensional convection diffusion problem are presented after an extended presentation of the basic integral formulation the discretizing and iterative processes for its resolution are introduced interesting remarks on general expressions versus previously published particularized results are worth mentioning a numerical solution scheme is provided which has been completely developed and designed to the physical problem posed a novel scheme based in the simultaneous solving of the potential and the gradient of the potential boundary integral equations is included a diversity of problems is tested to prove the possibilities of the method copyright 2008 john wiley sons ltd

a new nonlocal nonlinear diffusion of image processing abstract a novel nonlocal nonlinear diffusion is analyzed which has proven useful as a denoising tool in image processing the equation can be viewed as a new paradigm for the regularization of the well known perona malik equation the regularization is implemented via nonlinearity intensity reduction through fractional derivatives well posedness in the weak setting is established global existence and convergence to the average holds in the purely diffusive limit whereas an interesting dynamic behavior is engendered by the presence of nontrivial equilibria as the intensity of the nonlinearity is increased and comes close to the one of perona malik

on an instability exhibited by the ballistic diffusive heat conduction model of xu and hu we show that the constitutive relation for the thermal flux proposed by xu hu 2011 admits an unconditional instability we also highlight the difference between mathematical models containing delay and those that include relaxation effects

fused estimators of the central subspace in sufficient dimension reduction when studying the regression of a univariate variable y on a vector x of predictors most existing sufficient dimension reduction sdr methods require the construction of slices of y to estimate moments of the conditional distribution of x given y but there is no widely accepted method for choosing the number of slices while a poorly chosen slicing scheme may produce miserable results we propose a novel and easily implemented fusing method that can mitigate the problem of choosing a slicing scheme and improve estimation efficiency at the same time we develop two fused estimators called fire and dire based on an optimal inverse regression estimator the asymptotic variance of fire is no larger than that of the original methods regardless of the choice of slicing scheme while dire is less computational intense and more robust simulation studies show that the fused estimators perform effectively the same as or substantially better than the parent methods fused estimators based on other methods can be

a convex formulation for high dimensional sparse sliced inverse regression summarysliced inverse regression is a popular tool for sufficient dimension reduction which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the covariates the estimated linear combinations include all covariates making results difficult to interpret and perhaps unnecessarily variable particularly when the number of covariates is large in this paper we propose a convex formulation for fitting sparse sliced inverse regression in high dimensions our proposal estimates the subspace of the linear combinations of the covariates directly and performs variable selection simultaneously we solve the resulting convex optimization problem via the linearized alternating direction methods of multiplier algorithm and establish an upper bound on the subspace distance between the estimated and the true subspaces through numerical studies we show that our proposal is able to identify the correct covariates in the high dimensional setting

a semiparametric approach to dimension reduction we provide a novel and completely different approach to dimension reduction problems from the existing literature we cast the dimension reduction problem in a semiparametric estimation framework and derive estimating equations viewing this problem from the new angle allows us to derive a rich class of estimators and obtain the classical dimension reduction techniques as special cases in this class the semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact the common assumption of linearity and or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression the semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example this article has online supplementary material

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

estimating sufficient reductions of the predictors in abundant high dimensional regressions we study the asymptotic behavior of a class of methods for sufficient dimension reduction in high dimension regressions as the sample size and number of predictors grow in various alignments it is demonstrated that these methods are consistent in a variety of settings particularly in abundant regressions where most predictors contribute some information on the response and oracle rates are possible simulation results are presented to support the theoretical conclusion