bayesian analysis of dichotomous quantal response models abstract qualitative response models qrm s are analyzed from the bayesian point of view using diffuse and informative prior distributions exact finite sample bayesian and large sample bayesian and non bayesian estimation results are compared in addition the paper provides 1 plots and discussion of the properties of likelihood functions for qrm s 2 posterior distributions for logit models derivatives and elasticities 3 bayesian prediction procedures for qrm s 4 new estimates for the median and other fractiles of the logistic distribution 5 posterior odds ratios for model selection problems and 6 comparisons of two alternative monte carlo numerical integration procedures it is concluded that asymptotic approximations are not accurate for small to moderate sized samples even when only a single input variable is used and that operational bayesian methods are available for providing both exact small sample and large sample approximate inferences for drm s

an adaptive inverse iteration for maxwell eigenvalue problem based on edge elements we propose and analyze an adaptive inverse iterative method for solving the maxwell eigenvalue problem with discontinuous physical parameters in three dimensions the adaptive method updates the eigenvalue and eigenfunction based on an a posteriori error estimate of the edge element discretization at each iteration the involved saddle point maxwell system is transformed into an equivalent system consisting of a singular maxwell equation and two poisson equations for both of which preconditioned iterative solvers are available with optimal convergence rate in terms of the total degrees of freedom numerical results are presented which confirms the quasi optimal convergence of the adaptive edge element method in terms of the numerical accuracy and the total degrees of freedom

forecasting multivariate realized stock market volatility we present a new matrix logarithm model of the realized covariance matrix of stock returns the model uses latent factors which are functions of lagged volatility lagged returns and other forecasting variables the model has several advantages it is parsimonious it does not require imposing parameter restrictions and it results in a positive definite estimated covariance matrix we apply the model to the covariance matrix of size sorted stock returns and find that two factors are sufficient to capture most of the dynamics

a posteriori error analysis for the conical diffraction problem abstract maxwell s equations for conical diffraction can be reduced to a system of two helmholtz equations in r 2 coupled via quasi periodic transmission conditions on a set of piecewise smooth interfaces a finite element formulation of the conical diffraction problem is presented in a bounded domain by introducing the nonlocal boundary operators an a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place as our work in wang et al 2015 a duality argument is applied to overcome the difficulty caused by the fact that the truncated pseudo differential mapping does not converge to the original pseudo differential mapping in its operator norm the a posteriori error estimate consists of two parts finite element discretization error and the truncation error of the nonlocal boundary operators in particular the truncation error is exponentially decaying with respect to the truncation parameter

semiparametric estimation of spectral density with irregular observations we propose a semiparametric method for estimating spectral densities of isotropic gaussian processes with scattered data the spectral density function fourier transform of the covariance function is modeled as a linear combination of b splines up to a cutoff frequency and from this point a truncated algebraic tail we calculate an analytic expression for the covariance function and tackle several numerical issues that arise when calculating the likelihood the parameters are estimated by maximizing the likelihood using the simulated annealing method our method directly estimates the tail behavior of the spectral density which has the greatest impact on interpolation properties the use of the likelihood in parameter estimation takes the correlations between observations fully into account we compare our method with a kernel method proposed by hall et al and a parametric method using the matern model simulation results show that our method outperforms the other two by several criteria applicatio

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

adaptive estimation of stationary gaussian fields we study the nonparametric covariance estimation of a stationary gaussian field x observed on a regular lattice in the time series setting some procedures like aic are proved to achieve optimal model selection among autoregressive models however there exists no such equivalent results of adaptivity in a spatial setting by considering collections of gaussian markov random fields gmrf as approximation sets for the distribution of x we introduce a novel model selection procedure for spatial fields for all neighborhoods m in a given collection mathcal m this procedure first amounts to computing a covariance estimator of x within the gmrfs of neighborhood m then it selects a neighborhood m by applying a penalization strategy the so defined method satisfies a nonasymptotic oracle type inequality if x is a gmrf the procedure is also minimax adaptive to the sparsity of its neighborhood more generally the procedure is adaptive to the rate of approximation of the true distribution by gmrfs with growing neighborhoods

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens

serial correlation in unequally spaced longitudinal data summary serial correlation in the within subject error structure in longitudinal data with unequally spaced observations is modelled using continuous time autoregressive moving averages the models considered have both fixed and random effects in addition to serially correlated within subject errors two approaches are presented for calculating the exact likelihood for a model when the errors are gaussian the first calculates the covariance matrices for each subject for assumed values of the unknown parameters and estimates the fixed parameters by weighted least squares the second uses a state space model and the kalman filter to calculate the exact likelihood both methods involve the use of complex arithmetic nonlinear optimization is used to obtain maximum likelihood estimates of the parameters

bayesian inference for a covariance matrix a flexible class of prior distributions is proposed for the covariance matrix of a multivariate normal distribution yielding much more general hierarchical and empirical bayes smoothing and inference when compared with a conjugate analysis involving an inverted wishart distribution a likelihood approximation is obtained for the matrix logarithm of the covariance matrix via bellman s iterative solution to a volterra integral equation exact and approximate bayesian empirical and hierarchical bayesian estimation and finite sample inference techniques are developed some risk and asymptotic frequency properties are investigated a subset of the project talent american high school data is analyzed applications and extensions to multivariate analysis including a generalized linear model for covariance matrices are indicated

large sample properties of the matrix exponential spatial specification with an application to fdi this paper studies large sample properties of the matrix exponential spatial specification mess we find that the quasi maximum likelihood estimator qmle for the mess is consistent under heteroskedasticity a property not shared by the qmle of the sar model for the general model that has mess in both the dependent variable and disturbances labeled mess 1 1 the qmle can be consistent under unknown heteroskedasticity when the spatial weights matrices in the two mess processes are commutative we also consider the generalized method of moments estimator gmme in the homoskedastic case we derive a best gmme that is as efficient as the maximum likelihood estimator under normality and can be asymptotically more efficient than the qmle under non normality in the heteroskedastic case an optimal gmme can be more efficient than the qmle asymptotically the qml approach for the mess model has the computational advantage over that of a sar model the computational simplicity carries over to mess models with any finite order of spatial matrices no parameter range needs to be imposed in order for the model to be stable results of monte carlo experiments for finite sample properties of the estimators are reported finally the mess 1 1 is applied to belgium s outward fdi data and we observe that the dominant motivation of belgium s outward fdi lies in finding cheaper factor inputs

on the nonparametric estimation of covariance functions we describe kernel methods for estimating the covariance function of a stationary stochastic process and show how to ensure that the estimator has the positive semidefiniteness property from a practical viewpoint our method is significant because it does not demand a parametric model for covariance from a technical angle our results exhibit a striking departure from those in more familiar cases of kernel estimation for example in the context of covariance estimation kernel estimators can have the same convergence rates as maximum likelihood estimators and can have exceptionally fast convergence rates when employed to estimate variance

bayesian analysis of covariance matrices and dynamic models for longitudinal data summary parsimonious modelling of the within subject covariance structure while heeding its positive definiteness is of great importance in the analysis of longitudinal data using the cholesky decomposition and the ensuing unconstrained and statistically meaningful reparameterisation we provide a convenient and intuitive framework for developing conditionally conjugate prior distributions for covariance matrices and show their connections with generalised inverse wishart priors our priors offer many advantages with regard to elicitation positive definiteness computations using gibbs sampling shrinking covariances toward a particular structure with considerable flexibility and modelling covariances using covariates bayesian estimation methods are developed and the results are compared using two simulation studies these simulations suggest simpler and more suitable priors for the covariance structure of longitudinal data

nonparametric estimation of the variogram and its spectrum in the study of intrinsically stationary spatial processes a new nonparametric variogram estimator is proposed through its spectral representation the methodology is based on estimation of the variogram s spectrum by solving a regularized inverse problem through quadratic programming the estimated variogram is guaranteed to be conditionally negative definite simulation shows that our estimator is flexible and generally has smaller mean integrated squared error than the parametric estimator under model misspecification our methodology is applied to a spatial dataset of decadal temperature changes copyright 2011 oxford university press

application of copulas to improve covariance estimation for partial least squares dimension reduction techniques such as partial least squares are useful for computing summary measures and examining relationships in complex settings partial least squares requires an estimate of the covariance matrix as a first step in the analysis making this estimate critical to the results in addition the covariance matrix also forms the basis for other techniques in multivariate analysis such as principal component analysis and independent component analysis this paper has been motivated by an example from an imaging study in alzheimer s disease where there is complete separation between alzheimer s and control subjects for one of the imaging modalities this separation occurs in one block of variables and does not occur with the second block of variables resulting in inaccurate estimates of the covariance we propose the use of a copula to obtain estimates of the covariance in this setting where one set of variables comes from a mixture distribution simulation studies show that the proposed estimator is an improvement over the standard estimators of covariance the methods are illustrated from the motivating example from a study in the area of alzheimer s disease

small but slow world how network topology and burstiness slow down spreading while communication networks show the small world property of short paths the spreading dynamics in them turns out slow here the time evolution of information propagation is followed through communication networks by using empirical data on contact sequences and the susceptible infected model introducing null models where event sequences are appropriately shuffled we are able to distinguish between the contributions of different impeding effects the slowing down of spreading is found to be caused mainly by weight topology correlations and the bursty activity patterns of individuals

an approach to the analysis of repeated measurements summary a linear model for repeated measurements is proposed in which the correlation structure within each time sequence of measurements includes parameters for measurement error variation between experimental units and serial correlation within units an approach to data analysis is presented which involves preliminary analysis by ordinary least squares use of the empirical semi variogram of residuals to suggest a suitable correlation structure and formal inference using likelihood based methods applications to two biological data sets are described

arma covariance structures with time heteroscedasticity for repeated measures experiments abstract rochon and helms 1989 presented a model for analyzing repeated measures experiments the general linear model was used to assess the influence of covariate information and arma time series models were put forward to characterize the covariance matrix among the repeated measures practical experience has suggested however that the arma assumption of constant variances and autocovariances over time is too restrictive for many applications for example observations may be relatively stable toward the beginning of the study but become more variable toward the end this article presents a modification to this structure which provides for heteroscedasticity over time maximum likelihood ml estimation procedures are considered and the estimators are found to enjoy optimal large sample properties a scoring algorithm is described for iterating to a solution of the ml equations the model is illustrated with data from a clinical trial investigating human erythropoietin for treating anemia in end

the preconditioned iterative methods with variable parameters for saddle point problem abstract in this paper by transforming the original problem equivalently we propose a new preconditioned iterative method for solving saddle point problem we call the new method as ptu preconditioned transformative uzawa method and we study the convergence of the ptu method under suitable restrictions on the iteration parameters moreover we show the choices of the optimal parameters and the spectrum of the preconditioned matrix deriving from the ptu method based on the ptu iterative method we propose another iterative method nonlinear inexact ptu method for solving saddle point problem we also prove its convergence and study the choices of the optimal parameters in addition we present some numerical results to illustrate the behavior of the considered algorithms

generalized estimating equations for correlated binary data using the odds ratio as a measure of association summary moment methods for analyzing repeated binary responses have been proposed by liang zeger 1986 and extended by prentice 1988 in their generalized estimating equations both liang zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses because the odds ratio has many desirable properties and some investigators may find the odds ratio is easier to interpret we discuss modelling the association between binary responses at pairs of times with the odds ratio we then modify the estimating equations of prentice to estimate the odds ratios in simulations the parameter estimates for the logistic regression model for the marginal probabilities appear slightly more efficient when using the odds ratio parameterization

covariance matrix estimation using repeated measurements when data are incomplete modeling repeated measurements data has been studied extensively lately in the parametric situation however it is significant to study the effect of various treatments over a period of time where the repeated measurements on the same subject are expected to be correlated the correlation among the repeated measurements for all the subjects will be studied via the covariance matrix the positive definite constraint is one of obstacles that encounters modeling the covariance structure however the cholesky decomposition removes this constraint and allows modeling the components of the covariance matrix m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 in this paper we adopt the estimation procedures introduced by diggle and verbyla nonparametric estimation of covariance structure in longitudinal data biometrics 54 1998 401 415 where the variogram cloud as well as the squared residuals are used to estimate the variogram and the variances via the kernel smoothing selecting the appropriate bandwidth value is one of the important steps in the estimation process thus in our data analysis we choose the bandwidth using one of the most simple straight forward methods which is the cross validation method developed by rice and silverman estimating the mean and covariance structure nonparametrically when the data are curves j roy statist soc b 53 1991 233 243 and adapted by others finally we apply these nonparametric techniques as well as a graphical method m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 to a real life data and use the penalized likelihood criterion like aic and bic to compare models of our interest our interest

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

analysis of longitudinal data with unequally spaced observations and time dependent correlated errors a linear model for repeated measurements is proposed in which the correlation structure includes a transformation of the time scale this transformation can produce nonstationary covariance structures within subjects with stationarity as a special case restricted maximum likelihood methods for parameter estimation are discussed the method is applied to simulated data as well as speech recognition data from the iowa cochlear implant project the growth curve for this audiologic performance measure is shown together with estimates of the standard errors of predictions at given months

analysis of longitudinal data unbalanced over time summary the paper considers modelling estimating and diagnostically verifying the response process generating longitudinal data with emphasis on association between repeated measures from unbalanced longitudinal designs our model is based on separate specifications of the moments for the mean standard deviation and correlation with different components possibly sharing common parameters we propose a general class of correlation structures that comprise random effects measurement errors and a serially correlated process these three elements are combined via flexible time varying weights whereas the serial correlation can depend flexibly on the mean time and lag when the measurement schedule is independent of the response process our estimation procedure yields consistent and asymptotically normal estimates for the mean parameters even when the standard deviation and correlation are misspecified and for the standard deviation parameters even when the correlation is misspecified a generic diagnostic method is developed for verifying the models for the mean standard deviation and in particular the correlation which is applicable even when the data are severely unbalanced the methodology is illustrated by an analysis of data from a longitudinal study that was designed to characterize pulmonary growth in girls

correlated bursts and the role of memory range inhomogeneous temporal processes in natural and social phenomena have been described by bursts that are rapidly occurring events within short time periods alternating with long periods of low activity in addition to the analysis of heavy tailed interevent time distributions higher order correlations between interevent times called correlated bursts have been studied only recently as the underlying mechanism behind such correlated bursts is far from being fully understood we devise a simple model for correlated bursts using a self exciting point process with a variable range of memory whether a new event occurs is stochastically determined by a memory function that is the sum of decaying memories of past events in order to incorporate the noise and or limited memory capacity of systems we apply two memory loss mechanisms a fixed number or a variable number of memories by analysis and numerical simulations we find that too much memory effect may lead to a poissonian process implying that there exists an intermediate range of memory effect to generate correlated bursts comparable to empirical findings our conclusions provide a deeper understanding of how long range memory affects correlated bursts

spatial correlation robust inference with errors in location or distance this paper presents results from a monte carlo study concerning inference with spatially dependent data we investigate the impact of location distance measurement errors upon the accuracy of parametric and nonparametric estimators of asymptotic variances nonparametric estimators are quite robust to such errors method of moments estimators perform surprisingly well and mle estimators are very poor we also present and evaluate a specification test based on a parametric bootstrap that has good power properties for the types of measurement error we consider

bayesian nonparametric covariance regression capturing predictor dependent correlations amongst the elements of a multivariate response vector is fundamental to numerous applied domains including neuroscience epidemiology and finance although there is a rich literature on methods for allowing the variance in a univariate regression model to vary with predictors relatively little has been done in the multivariate case as a motivating example we consider the google flu trends data set which provides indirect measurements of influenza incidence at a large set of locations over time our predictor to accurately characterize temporally evolving influenza incidence across regions it is important to develop statistical methods for a time varying covariance matrix importantly the locations provide a redundant set of measurements and do not yield a sparse nor static spatial dependence structure we propose to reduce dimensionality and induce a flexible bayesian nonparametric covariance regression model by relating these location specific trajectories to a lower dimensional subspace through a latent factor model with predictor dependent factor loadings these loadings are in terms of a collection of basis functions that vary nonparametrically over the predictor space such low rank approximations are in contrast to sparse precision assumptions and are appropriate in a wide range of applications our formulation aims to address three challenges scaling to large p domains coping with missing values and allowing an irregular grid of observations the model is shown to be highly exible while leading to a computationally feasible implementation via gibbs sampling the ability to scale to large p domains and cope with missing values is fundamental in analyzing the google flu trends data

maximum likelihood estimation for incomplete repeated measures experiments under an arma covariance structure a stochastic model is presented for the analysis of incomplete repeated measures experiments the general linear model is used to relate the response measures to other variables which are thought to account for inherent variation an autoregressive moving average arma time series representation is used to model disturbance terms maximum likelihood estimation procedures are considered and the properties of these estimators are derived it is concluded that while the assumptions underpinning the arma covariance models may be somewhat restrictive they provide a useful inferential vehicle particularly in the presence of missing values

nonparametric estimation of correlation functions in longitudinal and spatial data with application to colon carcinogenesis experiments in longitudinal and spatial studies observations often demonstrate strong correlations that are stationary in time or distance lags and the times or locations of these data being sampled may not be homogeneous we propose a nonparametric estimator of the correlation function in such data using kernel methods we develop a pointwise asymptotic normal distribution for the proposed estimator when the number of subjects is fixed and the number of vectors or functions within each subject goes to infinity based on the asymptotic theory we propose a weighted block bootstrapping method for making inferences about the correlation function where the weights account for the inhomogeneity of the distribution of the times or locations the method is applied to a data set from a colon carcinogenesis study in which colonic crypts were sampled from a piece of colon segment from each of the 12 rats in the experiment and the expression level of p27 an important cell cycle protein was then measured for each cell within the sampled crypts a simulation study is also provided to illustrate the numerical performance of the proposed method

domain level covariance analysis for multilevel survey data with structured nonresponse health care quality surveys in the united states are administered to individual respondents i e hospital patients health plan members to evaluate the performance of health care organizations i e hospitals health plans which thus constitute estimation domains for better understanding and more parsimonious reporting of dimensions of quality we analyze relationships among quality measures at the domain level rather than specifying a full parametric model for the observed responses and the nonresponse patterns at the lower patient level we first fit generalized variance covariance functions that take into account nonresponse patterns in the survey responses then specify a likelihood function for the domain mean responses using these generalized variance covariance functions this allows us to model directly the relationships among domain means for different items because the response scales are bounded we assume that these means follow a truncated multivariate normal distribution we calcu

evidence of exponential convergence in the computation of maxwell eigenvalues abstract this paper describes a finite element approach for the approximation of maxwell eigenvalues for three dimensional cavity problems hierarchical h 1 and h curl conforming discretisations are employed in conjunction with a technique that reduces the size of the problem to be solved numerical results are given on both polyhedral and curvilinear domains these results indicate that exponential rates of convergence can be obtained with the appropriate combination of h and p refinements

extended generalized estimating equations for clustered data abstract typically analysis of data consisting of multiple observations on a cluster is complicated by within cluster correlation estimating equations for generalized linear modeling of clustered data have recently received much attention this article proposes an extension to the generalized estimating equation method proposed by liang and zeger which treats within cluster correlations as nuisance parameters using ideas from extended quasi likelihood estimating equations for regression and association parameters are provided simultaneously the resulting estimators are proven to be asymptotically normal and consistent under certain conditions the consistency of regression estimators allows incorrect modeling of the correlation among repeated responses the method is illustrated with an analysis of data from a developmental toxicity study

nonparametric estimation of nonstationary spatial covariance structure abstract estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks we introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function z x t observed repeatedly at times ti i 1 t at a finite number of sampling stations xi i 1 2 n in the plane our analyses assume temporal stationarity but do not assume spatial stationarity or isotropy we analyze the spatial dispersions var z xi t z xj t as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs xi xj the model is constructed in two steps first using nonmetric multidimensional scaling mds we compute a two dimensional representation of the sampling stations for which a monotone function of interpoint distances ij approximates the spatial dispersions mds transforms the problem

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

substructuring preconditioners for saddle point problems arising from maxwell s equations in three dimensions this paper is concerned with the saddle point problems arising from edge element discretizations of maxwell s equations in a general three dimensional nonconvex polyhedral domain a new augmented technique is first introduced to transform the problems into equivalent augmented saddle point systems so that they can be solved by some existing preconditioned iterative methods then some substructuring preconditioners are proposed with very simple coarse solvers for the augmented saddle point systems with the preconditioners the condition numbers of the preconditioned systems are nearly optimal namely they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size

nonlinear inexact uzawa algorithms for linear and nonlinear saddle point problems this paper proposes some nonlinear uzawa methods for solving linear and nonlinear saddle point problems a nonlinear inexact uzawa algorithm is first introduced for linear saddle point problems two different pcg techniques are allowed in the inner and outer iterations of the algorithm this algorithm is then extended for a class of nonlinear saddle point problems arising from some convex optimization problems with linear constraints for this extension some pcg method used in the inner iteration needs to be carefully constructed so that it converges in a certain energy norm instead of the usual l 2 norm it is shown that the new algorithm converges under some practical conditions and there is no need for any a priori estimates on the minimal and maximal eigenvalues of the two local preconditioned systems involved the two new methods perform more efficiently than the existing methods in the cases where no good preconditioners are available for the schur complements

a multivariate multilevel gaussian model with a mixed effects structure in the mean and covariance part a traditional gaussian hierarchical model assumes a nested multilevel structure for the mean and a constant variance at each level we propose a bayesian multivariate multilevel factor model that assumes a multilevel structure for both the mean and the covariance matrix that is in addition to a multilevel structure for the mean we also assume that the covariance matrix depends on covariates and random effects this allows to explore whether the covariance structure depends on the values of the higher levels and as such models heterogeneity in the variances and correlation structure of the multivariate outcome across the higher level values the approach is applied to the three dimensional vector of burnout measurements collected on nurses in a large european study to answer the research question whether the covariance matrix of the outcomes depends on recorded system level features in the organization of nursing care but also on not recorded factors that vary with countries hospitals and nursing units simulations illustrate the performance of our modeling approach

least squares estimation of covariance matrices in balanced multivariate variance components models abstract the problem of estimating covariance matrices in balanced multivariate variance components models is discussed as with univariate models it is possible for the traditional estimators based on differences of the mean square matrices to produce estimates that are outside the parameter space in fact in many cases it is extremely likely that traditional estimates of the covariance matrices will not be nonnegative definite nnd in this article we develop an iterative estimation procedure satisfying a least squares criterion that is guaranteed to produce nnd estimates of the covariance matrices discuss the speed of convergence and provide an example to show how the estimates change

local estimation of age dependent variance components from longitudinal twin data summary in the study of longitudinal twin and family data interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function various parametric models for covariance structures have been proposed but e g in studies of children where growth spurts occur at various ages it is difficult to a prioridetermine an appropriate parametric model for the covariance structure in particular there is a general lack of the visualization procedures such as lowess that are invaluable in the initial stages of constructing a parametric model for a mean function here we use kernel smoothing to modify a cross sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data the methods are proposed t o be exploratory as an aid to parametric modeling rather than inferential although approximate asymptotic standard errors are derived in the appendix

a matrix exponential spatial specification we introduce the matrix exponential as a way of modelling spatially dependent data the matrix exponential spatial specification mess simplifies the log likelihood allowing a closed form solution to the problem of maximum likelihood estimation and greatly simplifies the bayesian estimation of the model the mess can produce estimates and inferences similar to those from conventional spatial autoregressive models but has analytical computational and interpretive advantages we present maximum likelihood and bayesian approaches to the estimation of this spatial model specification along with methods of model comparisons over different explanatory variables and spatial specifications

an adaptive perfectly matched layer technique for 3 d time harmonic electromagnetic scattering problems an adaptive perfectly matched layer pml technique for solving the time harmonic electromagnetic scattering problems is developed the pml parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates combined with the adaptive finite element method the adaptive pml technique provides a complete numerical strategy to solve the scattering problem in the framework of fem which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the pml absorbing layer numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

exact predictive densities for linear models with arch disturbances abstract it is shown how exact predictive densities may be formed in the arch linear model by means of monte carlo integration with importance sampling several improvements in computational efficiency over earlier implementations of this procedure are developed including use of the exact likelihood function rather than an asymptotic approximation to construct the importance sampling distribution and antithetic acceleration of convergence a numerical approach to the formulation of posterior odds ratios and the combination of non nested models is also introduced these methods are applied to daily quotations of closing stock prices forecasts are formulated using linear models arch linear models and an integrated model constructed from the posterior probabilities of the respective models the use of the exact predictive density in a decision theoretic context is illustrated by deriving the optimal day to day portfolio adjustments of a trader with constant relative risk aversion

the matrix logarithmic covariance model abstract a flexible method is introduced to model the structure of a covariance matrix c and study the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix a the matrix exponential transformation c exp a is a positive definite matrix because there is no constraint on the possible values of the upper triangular elements on a any possible structure of interest can be imposed on them the method presented here is not intended to replace the existing special models available for a covariance matrix but rather to provide a broad range of further structures that supplements existing methodology maximum likelihood estimation procedures are used to estimate the parameters and the large sample asymptotic properties are obtained a simulation study and two real life examples are given to illustrate the method introduced

efficient semiparametric regression for longitudinal data with nonparametric covariance estimation summary for longitudinal data when the within subject covariance is misspecified the semiparametric regression estimator may be inefficient we propose a method that combines the efficient semiparametric estimator with nonparametric covariance estimation and is robust against misspecification of covariance models we show that kernel covariance estimation provides uniformly consistent estimators for the within subject covariance matrices and the semiparametric profile estimator with substituted nonparametric covariance is still semiparametrically efficient the finite sample performance of the proposed estimator is illustrated by simulation in an application to cd4 count data from an aids clinical trial we extend the proposed method to a functional analysis of the covariance model

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

marginal models for correlated binary responses with multiple classes and multiple levels of nesting a model for correlated binary data is presented marginal probabilities and odds ratios are allowed to have general regression structures that include multiple classes and multiple levels of nesting estimation is done through the generalized estimating equations approach of liang and zeger 1986 biometrika 73 13 22 they are contrasted with conditional models and recommendations for choosing between the two are given examples from genetic epidemiology are presented

on the application of karhunen loeve transform to transient dynamic systems abstract the karhunen loeve transform klt has become a popular method in various fields of engineering science due to its ability to identify the most prominent features in the underlying system dynamics the klt is a favorable method for such tasks as process monitoring model order reduction or optimum control however it is a well known fact that the klt is case sensitive that is that changes in the dynamic system behavior can decisively affect the klt results as much as this property is desired for monitoring problems it limits the performance of klt in model order reduction or optimum control problems if systems are subject to structural changes recent research interest focuses on extending applications of klt to systems with transient dynamic behavior or changing boundary conditions approaches have been published that circumvent the limitations of klt by either assuming reasonable comparability of system dynamics or by measuring the representative performance of klt bases a posteriori however such methods require additional simulations of the full size system and thus jeopardize the idea of model order reduction in this paper we introduce a novel a priori measure to evaluate the performance of the current klt basis this procedure can be of great help in either monitoring or adaptive control of systems that show intermittent transient and quasi stationary dynamic behavior this a priori measure prepares the path for adaptive model order reduction schemes moreover it can be used to measure the stationarity of multidimensional dynamic processes

isotropic spectral additive models of the covariogram summary a class of additive covariance models of an isotropic random process is proposed motivated by the spectral representation of the covariance function model parameters are estimated by using a special case of the minimum norm quadratic estimation estimator whose asymptotic moments have convenient expressions in terms of spectral densities fitting a model in this class is equivalent to fitting an additive model of the spectral density the class of spectral additive models proposed is dense in the set of summable covariance functions having a spectral density allowing approximately unbiased estimation of an arbitrary covariance function and its spectral density theoretical results are supported by numerical comparison with commonly used models a procedure to assist model selection is proposed the techniques are illustrated with an application to contaminant data

measuring burstiness for finite event sequences characterizing inhomogeneous temporal patterns in natural and social phenomena is important to understand underlying mechanisms behind such complex systems and hence even to predict and control them temporal inhomogeneities in event sequences have been described in terms of bursts that are rapidly occurring events in short time periods alternating with long inactive periods the bursts can be quantified by a simple measure called the burstiness parameter which was introduced by goh and barabasi europhys lett 81 48002 2008 euleej0295 507510 1209 0295 5075 81 48002 the burstiness parameter has been widely used due to its simplicity which however turns out to be strongly affected by the finite number of events in the time series as the finite size effects on burstiness parameter have been largely ignored we analytically investigate the finite size effects of the burstiness parameter then we suggest an alternative definition of burstiness that is free from finite size effects and yet simple using our alternative burstiness measure one can distinguish the finite size effects from the intrinsic bursty properties in the time series we also demonstrate the advantages of our burstiness measure by analyzing empirical data sets

covariance regression analysis abstractthis article introduces covariance regression analysis for a p dimensional response vector the proposed method explores the regression relationship between the p dimensional covariance matrix and auxiliary information we study three types of estimators maximum likelihood ordinary least squares and feasible generalized least squares estimators then we demonstrate that these regression estimators are consistent and asymptotically normal furthermore we obtain the high dimensional and large sample properties of the corresponding covariance matrix estimators simulation experiments are presented to demonstrate the performance of both regression and covariance matrix estimates an example is analyzed from the chinese stock market to illustrate the usefulness of the proposed covariance regression model supplementary materials for this article are available online

modeling the random effects covariance matrix for longitudinal data with covariates measurement error longitudinal data occur frequently in practice such as medical studies and life sciences generalized linear mixed models glmms are commonly used to analyze such data it is typically assumed that the random effects covariance matrix is constant among subjects in these models in many situations however the correlation structure may differ among subjects and ignoring this heterogeneity can lead to biases in model parameters estimate recently lee et al developed a heterogeneous random effects covariance matrix for glmms for error free covariates covariates measured with error also happen frequently in the longitudinal data set up eg blood pressure and cholesterol level ignoring this issue in the data may produce bias in model parameters estimate and lead to wrong conclusions in this paper we propose an approach to properly model the random effects covariance matrix based on covariates in the class of glmms where we also have covariates measured with error the resulting parameters from the decomposition of random effects covariance matrix have a sensible interpretation and can be easily modeled without the concern of positive definiteness of the resulting estimator the performance of the proposed approach is evaluated through simulation studies which show that the proposed method performs very well in terms of bias mean squared error and coverage rate an application of the proposed method is also provided using a longitudinal data from manitoba follow up study

two grid methods for maxwell eigenvalue problems two new two grid algorithms are proposed for solving the maxwell eigenvalue problem the new methods are based on the two grid methodology recently proposed by xu and zhou math comp 70 2001 pp 17 25 and further developed by hu and cheng math comp 80 2011 pp 1287 1301 for elliptic eigenvalue problems the new two grid schemes reduce the solution of the maxwell eigenvalue problem on a fine grid to one linear indefinite maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid the new schemes therefore save total computational cost the error estimates reveals that the two grid methods maintain asymptotically optimal accuracy and the numerical experiments presented confirm the theoretical results

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

irregular n2sls and lasso estimation of the matrix exponential spatial specification model abstract in this paper we consider estimation of the matrix exponential spatial specification model with the durbin and endogenous regressors we find that the nonlinear two stage least squares n2sls estimator is in general consistent and asymptotically normal however when the durbin and endogenous regressors are irrelevant the gradient vector of the n2sls criterion function has a singular covariance matrix with probability approaching one w p a 1 some components of the n2sls estimator have slower rates of convergence and their asymptotic distributions are nonstandard the distance difference and gradient test statistics which have irregular asymptotic distributions are derived to test for the irrelevance of the durbin and endogenous regressors as an alternative estimation and model selection approach we propose the adaptive group lasso which penalizes the coefficients of the durbin and endogenous explanatory variables we show that the estimator has the oracle properties so the true model can be selected w p a 1 and the estimator always has the n rate of convergence and asymptotic normal distribution we propose to select the tuning parameter for the adaptive group lasso by minimizing an information criterion

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

nonparametric estimation of covariance structure in longitudinal data summary in longitudinal studies the effect of various treatments over time is usually of prime interest however observations on the same subject are usually correlated and any analysis should account for the underlying covariance structure a nonparametric estimate of the covariance structure is useful either as a guide to the formulation of a parametric model or as the basis for formal inference without imposing parametric assumptions the sample covariance matrix provides such an estimate when the data consist of a short sequence of measurements at a common set of time points on each of many subjects but is impractical when the data are severely unbalanced or when the sequences of measurements on individual subjects are long relative to the number of subjects the variogram of residuals from a saturated model for the mean response has previously been suggested as a nonparametric estimator for covariance structure assuming stationarity in this paper we consider kernel weighted local linear regression smoothing of sample variogram ordinates and of squared residuals to provide a nonparametric estimator for the covariance structure without assuming stationarity the value of the estimator as a diagnostic tool is demonstrated in two applications one to a set of data concerning the blood pressure of newborn babies in an intensive care unit and the other to data on the time evolution of cd4 cell numbers in hiv seroconverters

modelling the random effects covariance matrix in longitudinal data a common class of models for longitudinal data are random effects mixed models in these models the random effects covariance matrix is typically assumed constant across subject however in many situations this matrix may differ by measured covariates in this paper we propose an approach to model the random effects covariance matrix by using a special cholesky decomposition of the matrix in particular we will allow the parameters that result from this decomposition to depend on subject specific covariates and also explore ways to parsimoniously model these parameters an advantage of this parameterization is that there is no concern about the positive definiteness of the resulting estimator of the covariance matrix in addition the parameters resulting from this decomposition have a sensible interpretation we propose fully bayesian modelling for which a simple gibbs sampler can be implemented to sample from the posterior distribution of the parameters we illustrate these models on data from depression studies and examine the impact of heterogeneity in the covariance matrix on estimation of both fixed and random effects

a joint modelling approach for longitudinal studies in longitudinal studies it is of fundamental importance to understand the dynamics in the mean function variance function and correlations of the repeated or clustered measurements for modelling the covariance structure cholesky type decomposition based approaches have been demonstrated to be effective however parsimonious approaches for directly revealing the correlation structure between longitudinal measurements remain less well explored and existing joint modelling approaches may encounter difficulty in interpreting the covariation structure we propose a novel joint mean variance correlation modelling approach for longitudinal studies by applying hyperspherical co ordinates we obtain an unconstrained parameterization for the correlation matrix that automatically guarantees its positive definiteness and we develop a regression approach to model the correlation matrix of the longitudinal measurements by exploiting the parameterization the modelling framework proposed is parsimonious interpretable and flexible for analysing longitudinal data extensive data examples and simulations support the effectiveness of the approach proposed

modelling covariance structure in bivariate marginal models for longitudinal data summary it can be more challenging to efficiently model the covariance matrices for multivariate longitudinal data than for the univariate case due to the correlations arising between multiple responses the positive definiteness constraint and the high dimensionality are further obstacles in covariance modelling in this paper we develop a data based method by which the parametersinthecovariancematricesarereplacedbyunconstrainedandinterpretableparameterswith reduced dimensions the maximum likelihood estimators for the mean and covariance parameters are shown to be consistent and asymptotically normally distributed simulations and real data analysis show that the new approach performs very well even when modelling bivariate nonstationary dependence structures

gmm estimation with cross sectional dependence this paper presents a spatial model of dependence among agents using a metric of economic distance measurements of this economic distance provide cross sectional data with a structure similar to that provided by the time index in time series data generalized method of moments estimators using such dependent data are shown to be consistent and asymptotically normal this paper presents a class of non parametric positive semi definite covariance matrix estimators that allow for general forms of dependence characterized by economic distance these covariance matrix estimators are shown to remain consistent when economic distances are not precisely observed

non markovian infection spread dramatically alters the susceptible infected susceptible epidemic threshold in networks most studies on susceptible infected susceptible epidemics in networks implicitly assume markovian behavior the time to infect a direct neighbor is exponentially distributed much effort so far has been devoted to characterize and precisely compute the epidemic threshold in susceptible infected susceptible markovian epidemics on networks here we report the rather dramatic effect of a nonexponential infection time while still assuming an exponential curing time on the epidemic threshold by considering weibullean infection times with the same mean but different power exponent for three basic classes of graphs the erd s renyi random graph scale free graphs and lattices the average steady state fraction of infected nodes is simulated from which the epidemic threshold is deduced for all graph classes the epidemic threshold significantly increases with the power exponents hence real epidemics that violate the exponential or markovian assumption can behave seriously differently than anticipated based on markov theory

estimation of covariance matrix of multivariate longitudinal data using modified choleksky and hypersphere decompositions linear models are typically used to analyze multivariate longitudinal data with these models estimating the covariance matrix is not easy because the covariance matrix should account for complex correlated structures the correlation between responses at each time point the correlation within separate responses over time and the cross correlation between different responses at different times in addition the estimated covariance matrix should satisfy the positive definiteness condition and it may be heteroscedastic however in practice the structure of the covariance matrix is assumed to be homoscedastic and highly parsimonious such as exchangeable or autoregressive with order one these assumptions are too strong and result in inefficient estimates of the effects of covariates several studies have been conducted to solve these restrictions using modified cholesky decomposition mcd and linear covariance models however modeling the correlation between responses at each time point is not easy because there is no natural ordering of the responses in this paper we use mcd and hypersphere decomposition to model the complex correlation structures for multivariate longitudinal data we observe that the estimated covariance matrix using the decompositions is positive definite and can be heteroscedastic and that it is also interpretable the proposed methods are illustrated using data from a nonalcoholic fatty liver disease study

a comparison between different discrete ambiguity domain definitions in stochastic time frequency analysis the ambiguity domain plays a central role in estimating the time varying spectrum and in estimating the covariance function of nonstationary random processes in continuous time for processes in discrete time there exist different definitions of the ambiguity domain but it is well known that neither of these definitions perfectly resembles the usefulness of the continuous ambiguity domain in this paper we present some of the most frequently used definitions of the ambiguity domain in discrete time the claasen mecklenbrauker the jeong williams and the nuttall definitions for the first time we prove their equivalence within some necessary conditions and we present theorems that justify their usage

generalized least squares taylor series linearization and fisher s scoring in multivariate nonlinear regression in this article we consider a general multivariate nonlinear regression setting in which the marginal mean and variance covariance structure share a common set of regression parameters estimation is carried out via iteratively reweighted generalized least squares irgls that entails repeated application of taylor series linearization and estimated generalized least squares egls under normality this irgls procedure is equivalent to fisher s method of scoring and hence maximum likelihood estimation mle however estimates from this procedure are also shown to minimize a bias corrected generalized least squares objective function that does not require the assumption of normality under fairly mild regularity conditions the resulting estimates are consistent asymptotically normal and under normality assumptions asymptotically efficient the estimates are compared against those obtained as solutions to the usual generalized estimating equations gee using both simulation and numerical examples

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

random effects selection in linear mixed models nous presentons le probleme pratique important de la selection des effets aleatoires dans un modele lineaire mixte un modele bayesien hierarchique est utilise pour identifier tout effet aleatoire devariance nulle l approche proposee reparametrise le modele mixte de telle sorte que les fonctions des parametres de covariance de la distribution des effets aleatoires soient incorporees comme des coefficients de regression pour des variables latentes normales nous permettons aux effets aleatoires de sortir du modele en choisissant des melanges d a priori avec des masses ponctuelles en zero pour les variances des effets aleatoires du fait de la reparametrisation le modele prend une structure lineaire conditionnelle qui facilite l utilisation de lois a priori conjuguees et normales nous demontrons que les calculs a posteriori peuvent etre effectues via un algorithme simple et efficace de monte carlo par chaine de markov les methodes sont illustrees par des donnees simulees et par des donnees reelles d exposition prenatale au polychlorobiphenile et des donnees sur le developpement psychomoteur des enfants

a nonparametric prior for simultaneous covariance estimation in the modeling of longitudinal data from several groups appropriate handling of the dependence structure is of central importance standard methods include specifying a single covariance matrix for all groups or independently estimating the covariance matrix for each group without regard to the others but when these model assumptions are incorrect these techniques can lead to biased mean effects or loss of efficiency respectively thus it is desirable to develop methods to simultaneously estimate the covariance matrix for each group that will borrow strength across groups in a way that is ultimately informed by the data in addition for several groups with covariance matrices of even medium dimension it is difficult to manually select a single best parametric model among the huge number of possibilities given by incorporating structural zeros and or commonality of individual parameters across groups in this paper we develop a family of nonparametric priors using the matrix stick breaking process of dunson et al 2008 that seeks to accomplish this task by parameterizing the covariance matrices in terms of the parameters of their modified cholesky decomposition pourahmadi 1999 we establish some theoretic properties of these priors examine their effectiveness via a simulation study and illustrate the priors using data from a longitudinal clinical trial

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

intrinsic regression models for positive definite matrices with applications to diffusion tensor imaging the aim of this paper is to develop an intrinsic regression model for the analysis of positive definite matrices as responses in a riemannian manifold and their association with a set of covariates such as age and gender in a euclidean space the primary motivation and application of the proposed methodology is in medical imaging because the set of positive definite matrices do not form a vector space directly applying classical multivariate regression may be inadequate in establishing the relationship between positive definite matrices and covariates of interest such as age and gender in real applications our intrinsic regression model which is a semiparametric model uses a link function to map from the euclidean space of covariates to the riemannian manifold of positive definite matrices we develop an estimation procedure to calculate parameter estimates and establish their limiting distributions we develop score statistics to test linear hypotheses on unknown parameters and develop a test pr

shifted inverse iteration based multigrid methods for the quad curl eigenvalue problem abstract this paper is devoted to multgrid methods for solving the quad curl eigenvalue problem the multgrid methods based on the rayleigh quotient iteration and the inverse iteration with fixed shift are proposed we prove the error estimates for both methods numerical experiments confirm our theoretical analysis and validate the efficiency of both methods

group comparison of eigenvalues and eigenvectors of diffusion tensors diffusion tensor imaging dti data differ from most medical images in that values at each voxel are not scalars but 3 3 symmetric positive definite matrices called diffusion tensors dts the anatomic characteristics of the tissue at each voxel are reflected by the dt eigenvalues and eigenvectors in this article we consider the problem of testing whether the means of two groups of dt images are equal at each voxel in terms of the dt s eigenvalues eigenvectors or both because eigendecompositions are highly nonlinear existing likelihood ratio statistics lrts for testing differences in eigenvalues or eigenvectors of means of gaussian symmetric matrices assume an orthogonally invariant covariance structure between the matrix entries while retaining the form of the lrts we derive new approximations to their true distributions when the covariance between the dt entries is arbitrary and possibly different between the two groups the approximate distributions are those of similar lrt statistics comput