leverage local influence and curvature in nonlinear regression summary the connections between measures of leverage statistical curvature and the local influence of a mean shift perturbation are explored in nonlinear regression models the circumstances under which the jacobian leverage reduces to a commonly used linear approximation are found to be connected to the effective residual curvature of the nonlinear model the local influence of a mean shift perturbation in the model function is seen to be closely related to the jacobian leverage

anisotropic goal oriented mesh adaptivity for elliptic problems we propose in this paper an anisotropic adaptive finite element algorithm for steady linear advection diffusion reaction problems with strong anisotropic features the error analysis is based on the dual weighted residual methodology allowing us to perform goal oriented adaptation of a certain functional j u of the solution and derive an optimal metric tensor for local mesh adaptation with linear and quadratic finite elements as a novelty and to evaluate the weights of the error estimator on unstructured meshes composed of anisotropic triangles we make use of a patchwise higher order interpolation recovery readily extendable to finite elements of arbitrary order we carry out a number of numerical experiments in two dimensions so as to prove the capabilities of the goal oriented adaptive method we compute the convergence rate and the effectivity index for a series of output functionals of the solution the results show the good performance of the algorithm with linear as well as quadratic

leverage and superleverage in nonlinear regression abstract several measures of the leverage of an observation in a nonlinear regression model are defined and developed in contrast to the upper bound on the leverage in a linear model it is found that in a nonlinear model the leverage of an observation may exceed 1 such a case is said to exhibit superleverage relationships between the leverage measures are explored and several examples are developed to illustrate the proposed methodology

adaptivity with dynamic meshes for space time finite element discretizations of parabolic equations in this paper we develop an error estimator and an adaptive algorithm for efficient solution of parabolic partial differential equations the error estimator assesses the discretization error with respect to a given quantity of physical interest and separates the influence of the time and space discretizations this allows us to set up an efficient adaptive strategy producing economical locally refined meshes for each time step and an adapted time discretization the space and time discretization errors are equilibrated leading to an efficient method

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

estimation of higher sobolev norm from lower order approximation given an accurate piecewise constant flux approximation q of a smooth but unknown exact flux p how can one compute a sobolev norm of p over a small domain omega this question arises in a duality argument of goal oriented finite element a posteriori error analysis two equivalent practical estimators eta m and eta mathcal e are presented for an approximation q of p h 1 omega and model situations are addressed where q is asymptotically an upper and lower bound of p h 1 omega

remark on the rate of decay of higher order derivatives for solutions to the navier stokes equations in rn we present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced navier stokes equations in rn the method based on so called gevrey estimates also yields explicit bounds on the growth of the radius of analyticity of the solution in time moreover under the assumption that the navier stokes solution stays sufficiently close to a solution of the heat equation in the l2 norm a result known to be true for a large class of initial data lower bounds on the decay of higher order derivatives can be obtained

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

goal oriented error estimation in the analysis of fluid flows with structural interactions abstract we present some developments for the extension of goal oriented error estimation procedures to the analysis of navier stokes incompressible fluid flows with structural interactions particular focus is given on error assessment of specific quantities of interest defined on the structural part the goal is to establish relatively coarse meshes to model the fluid flow but achieve acceptable accuracy in the quantities of interest a nonlinear goal oriented error estimation procedure is presented which is applicable to general nonlinear analyses some illustrative solutions using adina are given

bias in nonlinear regression abstract this document investigates the biases of the residuals and the maximum likelihood parameters estimates from standard normal theory nonlinear regression models emphasis is placed on determining the influence of individual cases on the biases and on understanding how the residual biases can affect the usefulness of standard diagnostic methods it is shown that the various bias expressions in the literature are equivalent that the biases in nonlinear regression can be studied usefully in the context of linear regression and that diagnostic plots using residuals can be misleading because of substantial residual biases for a class of partially nonlinear models it is shown that the maximum intrinsic curvature is closely related to the residual expectations finally the model associated with power transformations of single explanatory variables in linear regression is investigated in further detail and several numerical illustrations are presented author

a note on jorgensen s iteratively defined statistics summary the newton raphson and fisher scoring iteratively reweighted least squares methods are two fundamental tools for computing parameter estimators and deletion diagnostics jorgensen 1993 proposed a modified method to compute deletion diagnostics in this paper we study the relationship between jorgensen s newton raphson and fisher scoring one step deletion estimators by using the observed and expected fisher information matrices in addition we extend jorgensen s true leverage and approximate leverage from binary regression models to general model settings

a second order investigation of asymptotic ancillarity the paper deals with approximate ancillarity as discussed by efron and hinkley 1978 in the multivariate i i d case we derive the second order edgeworth expansion of the mle given a normalized version of the second derivative of the log likelihood at its maximum the expansion agrees with the one derived by amari 1982a for curved exponential families but holds for any family satisfying the regularity conditions given in the paper it is shown that the fisher information lost by reducing the data to the mle is recovered by the conditioning and it is sketched how the loss of information relates to the deficiency as defined by lecam finally we investigate some properties of three test statistics proving a conjecture by efron and hinkley 1978 concerning the conditional null distribution of the likelihood ratio test statistic and establishing a kind of superiority of the observed fisher information over the expected one as estimate of the inverse variance of the mle

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

variational localizations of the dual weighted residual estimator the dual weighted residual method dwr and its localization for mesh adaptivity applied to elliptic partial differential equations are investigated the contribution of this paper is twofold first we introduce a novel localization technique based on the introduction of a partition of unity this new technique is very easy to apply as neither strong residuals nor jumps over element edges are required second we compare and analyze theoretically and numerically different localization techniques used for mesh adaptivity with respect to their effectivity here we focus on localizations in variational formulations that do not require the evaluation of the corresponding differential operator in the classical strong formulation in our mathematical analysis we show for different localization techniques established methods and our new approach that the local error indicators used for mesh adaptivity converge with proper order in the error functional several numerical tests substantiate our theoretical investigations

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

space time decay of navier stokes flows invariant under rotations we show that the solutions to the non stationary navier stokes equations in open image in new window d 2 3 which are left invariant under the action of discrete subgroups of the orthogonal group o d decay much faster as open image in new window than in generic case and we compute for each subgroup the precise decay rates in space time of the velocity field

influence analysis for linear mixed effects models in this paper we extend several regression diagnostic techniques commonly used in linear regression such as leverage infinitesimal influence case deletion diagnostics cook s distance and local influence to the linear mixed effects model in each case the proposed new measure has a direct interpretation in terms of the effects on a parameter of interest and collapses to the familiar linear regression measure when there are no random effects the new measures are explicitly defined functions and do not necessitate re estimation of the model especially for cluster deletion diagnostics the basis for both the cluster deletion diagnostics and cook s distance is a generalization of miller s simple update formula for case deletion for linear models pregibon s infinitesimal case deletion diagnostics is adapted to the linear mixed effects model a simple compact matrix formula is derived to assess the local influence of the fixed effects regression coefficients finally a link between the local influence approach and cook s distance is established these influence measures are applied to an analysis of 5 year medicare reimbursements to colon cancer patients to identify the most influential observations and their effects on the fixed effects coefficients

on two new inequalities for leray solutions of the navier stokes equations in rn abstract we derive two asymptotic inequalities not previously observed that are valid for spatial derivatives of arbitrary order of leray solutions to the incompressible navier stokes equations in r n n 4 they are important to the decay investigation of solution derivatives

lower bounds of rates of decay for solutions to the navier stokes equations where a p 3 2 p 1 and the constant c depends on the l2 and lnorms of the initial data u0 this paper deals with the more subtle problem of deriving lower bounds on the energy decay rates we show that for a certain class of initial data the solutions u x t to the 2d and 3d navier stokes equations admit an algebraic lower bound on the energy decay specifically there are two cases to consider in the first case the average of the initial data f u0 dx is nonzero this case was treated in the earlier paper 5 where it was established that

leverage in least squares additive plus multiplicative fits for two way tables abstract an additive plus multiplicative model can describe both main effects and row x column interactions in two way tables of data when each cell contains exactly one observation a least squares fit for this nonlinear model calculates the main effects using means of rows and columns and then fits a multiplicative term to the additive residuals using the singular value decomposition a natural extension of the hat matrix for a linear model yields a definition of leverage that provides insights about the impact of erroneous data values on the fit theoretical and numerical investigations reveal the complex nature of leverage for this nonlinear model