corrected confidence bands for functional data using principal components functional principal components fpc analysis is widely used to decompose and express functional observations curve estimates implicitly condition on basis functions and other quantities derived from fpc decompositions however these objects are unknown in practice in this article we propose a method for obtaining correct curve estimates by accounting for uncertainty in fpc decompositions additionally pointwise and simultaneous confidence intervals that account for both model and decomposition based variability are constructed standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition iterated expectation and variance formulas combine model based conditional estimates across the distribution of decompositions a bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions we apply our method to sparse observations of cd4 cell counts and to dense white matter tract profiles code for the analyses and simulations is publicly available and our method is implemented in the r package refund on cran

joint modelling of paired sparse functional data using principal components we propose a modelling framework to study the relationship between two paired longitudinally observed variables the data for each variable are viewed as smooth curves measured at discrete time points plus random errors while the curves for each variable are summarized using a few important principal components the association of the two longitudinal variables is modelled through the association of the principal component scores we use penalized splines to model the mean curves and the principal component curves and cast the proposed model into a mixed effects model framework for model fitting prediction and inference the proposed method can be applied in the difficult case in which the measurement times are irregular and sparse and may differ widely across individuals use of functional principal components enhances model interpretation and improves statistical and numerical stability of the parameter estimates

shadowed c means integrating fuzzy and rough clustering a new method of partitive clustering is developed in the framework of shadowed sets the core and exclusion regions of the generated shadowed partitions result in a reduction in computations as compared to conventional fuzzy clustering unlike rough clustering here the choice of threshold parameter is fully automated the number of clusters is optimized in terms of various validity indices it is observed that shadowed clustering can efficiently handle overlapping among clusters as well as model uncertainty in class boundaries the algorithm is robust in the presence of outliers a comparative study is made with related partitive approaches experimental results on synthetic as well as real data sets demonstrate the superiority of the proposed approach

modified profile likelihoods in models with stratum nuisance parameters it is well known at least through many examples that when there are many nuisance parameters modified profile likelihoods often perform much better than the profile likelihood ordinary asymptotics almost totally fail to deal with this issue for this reason we study asymptotic properties of the profile and modified profile likelihoods in models for stratified data in a two index asymptotics setting this means that both the sample size of the strata m and the dimension of the nuisance parameter q may increase to infinity it is shown that in this asymptotic setting modified profile likelihoods give improvements with respect to the profile likelihood in terms of consistency of estimators and of asymptotic distributional properties in particular the modified profile likelihood based statistics have the usual asymptotic distribution provided that l m o q 1 3 while the analogous condition for the profile likelihood is l m o q 1

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

flexible regression models with cubic splines we describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates this simple method can help prevent the problems that result from inappropriate linearity assumptions we compare restricted cubic spline regression to non parametric procedures for characterizing the relationship between age and survival in the stanford heart transplant data we also provide an illustrative example in cancer therapeutics

multiscale fractional step schemes for the numerical simulation of the rotating shallow water flows with complex periodic topography in this paper we study several multiscale fractional step schemes for the numerical solution of the rotating shallow water equations with complex topography we consider the case of periodic boundary conditions f plane model spatial discretization is obtained using a fourier spectral galerkin method for the schemes presented in this paper we consider two approaches the first approach multiscale schemes is based on topography scale separation and the numerical time integration is function of the scales the second approach is based on a splitting of the operators and the time integration method is function of the operator considered fractional step schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme with these multiscale fractional step schemes the objective is to propose new schemes giving numerical results similar to those obtained using only one uniform fine grid n nn n and a time step t but with a cpu time near the cpu time needed when using only one coarse grid n1 n1n1 n1 n1 t t t

the analysis of transformed data abstract recently it was suggested bickel and doksum 1981 that when data are used to select a transformation the post transformation analysis of those data may need to be modified considerably from standard form so as to allow for the selection we argue that common sense and the work of box and cox 1964 point to a contrary conclusion our argument is based on considerations of parameter interpretation and subsequent bayesian analysis within the context of fitting normal error linear models numerical examples are used to illustrate the main points

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

nested hierarchical functional data modeling and inference for the analysis of functional plant phenotypes abstractin a plant science root image study the process of seedling roots bending in response to gravity is recorded using digital cameras and the bending rates are modeled as functional plant phenotype data the functional phenotypes are collected from seeds representing a large variety of genotypes and have a three level nested hierarchical structure with seeds nested in groups nested in genotypes the seeds are imaged on different days of the lunar cycle and an important scientific question is whether there are lunar effects on root bending we allow the mean function of the bending rate to depend on the lunar day and model the phenotypic variation between genotypes groups of seeds imaged together and individual seeds by hierarchical functional random effects we estimate the covariance functions of the functional random effects by a fast penalized tensor product spline approach perform multi level functional principal component analysis fpca using the best linear unbiased predictor of the pri

asymptotically unbiased inference for a dynamic panel model with fixed effects when both n and t are large we consider a dynamic panel ar 1 model with fixed effects when both n and t are large under the t fixed n large asymptotic approximation the ordinary least squares ols or gaussian maximum likelihood estimator mle is known to be inconsistent due to the well known incidental parameter problem we consider an alternative asymptotic approximation where n and t grow at the same rate it is shown that although ols or the mle is asymptotically biased a relatively simple fix to ols or the mle results in an asymptotically unbiased estimator under the assumption of gaussian innovations the bias corrected mle is shown to be asymptotically efficient by a hajek type convolution theorem

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

functional mixed effects models in this article a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced the linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects which are modeled as realizations of zero mean stochastic processes the fixed functional effects and the random functional effects are modeled in the same functional space which guarantee the population average and subject specific curves have the same smoothness property these models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures can include continuous covariates as well as dummy factors in both the fixed or random design matrices and include the nested curves models as special cases two estimation procedures are proposed the first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software the second procedure is a sequential estimation procedure using kalman filtering this algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets a generalized maximum likelihood gml ratio test is proposed for inference and model selection an application to comparison of cortisol profiles is used as an illustration

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

silhouettes a graphical aid to the interpretation and validation of cluster analysis a new graphical display is proposed for partitioning techniques each cluster is represented by a so called silhouette which is based on the comparison of its tightness and separation this silhouette shows which objects lie well within their cluster and which ones are merely somewhere in between clusters the entire clustering is displayed by combining the silhouettes into a single plot allowing an appreciation of the relative quality of the clusters and an overview of the data configuration the average silhouette width provides an evaluation of clustering validity and might be used to select an appropriate number of clusters

a semiparametric approach to simultaneous covariance estimation for bivariate sparse longitudinal data estimation of the covariance structure for irregular sparse longitudinal data has been studied by many authors in recent years but typically using fully parametric specifications in addition when data are collected from several groups over time it is known that assuming the same or completely different covariance matrices over groups can lead to loss of efficiency and or bias nonparametric approaches have been proposed for estimating the covariance matrix for regular univariate longitudinal data by sharing information across the groups under study for the irregular case with longitudinal measurements that are bivariate or multivariate modeling becomes more difficult in this article to model bivariate sparse longitudinal data from several groups we propose a flexible covariance structure via a novel matrix stick breaking process for the residual covariance structure and a dirichlet process mixture of normals for the random effects simulation studies are performed to investigate the effectiveness of the proposed approach over more traditional approaches we also analyze a subset of framingham heart study data to examine how the blood pressure trajectories and covariance structures differ for the patients from different bmi groups high medium and low at baseline

three part joint modeling methods for complex functional data mixed with zero and one inflated proportions and zero inflated continuous outcomes with skewness we take a functional data approach to longitudinal studies with complex bivariate outcomes this work is motivated by data from a physical activity study that measured 2 responses over time in 5 minute intervals one response is the proportion of time active in each interval a continuous proportions with excess zeros and ones the other response energy expenditure rate in the interval is a continuous variable with excess zeros and skewness this outcome is complex because there are 3 possible activity patterns in each interval inactive partially active and completely active and those patterns which are observed induce both nonrandom and random associations between the responses more specifically the inactive pattern requires a zero value in both the proportion for active behavior and the energy expenditure rate a partially active pattern means that the proportion of activity is strictly between zero and one and that the energy expenditure rate is greater than zero and likely to be moderate and the completely active pattern means that the proportion of activity is exactly one and the energy expenditure rate is greater than zero and likely to be higher to address these challenges we propose a 3 part functional data joint modeling approach the first part is a continuation ratio model to reorder the ordinal valued 3 activity patterns the second part models the proportions when they are in interval 0 1 the last component specifies the skewed continuous energy expenditure rate with box cox transformations when they are greater than zero in this 3 part model the regression structures are specified as smooth curves measured at various time points with random effects that have a correlation structure the smoothed random curves for each variable are summarized using a few important principal components and the association of the 3 longitudinal components is modeled through the association of the principal component scores the difficulties in handling the ordinal and proportional variables are addressed using a quasi likelihood type approximation we develop an efficient algorithm to fit the model that also involves the selection of the number of principal components the method is applied to physical activity data and is evaluated empirically by a simulation study

preconditioned newton methods using incremental unknowns methods for the resolution of a steady state navier stokes like problem in a previous work one of the authors has studied a numerical treatment by fully implicit discretizations of a two dimensional navier stokes like problem and has proved existence and convergence results for the resulting discretized systems with homogeneous dirichlet boundary conditions in this work we propose some new preconditioned multilevel versions of inexact newton algorithms to solve these equations we also develop another multilevel preconditioner for a nonlinear gmres algorithm all of the preconditioners are based on incremental unknowns formulations

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

on profile likelihood abstract we show that semiparametric profile likelihoods where the nuisance parameter has been profiled out behave like ordinary likelihoods in that they have a quadratic expansion in this expansion the score function and the fisher information are replaced by the efficient score function and efficient fisher information the expansion may be used among others to prove the asymptotic normality of the maximum likelihood estimator to derive the asymptotic chi squared distribution of the log likelihood ratio statistic and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance

numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds we consider the numerical solution of nonlinear dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds coarse and fine grids are employed with a nonlinear mapping to relate the solutions computed on these grids the approach is illustrated by consideration of the kuramoto sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions numerical results are presented on the ability of the method to mimic the time evolution of the infinite dimensional dynamical system the results demonstrate that the coarse fine grid method is more accurate than the standard pseudospectral method on the same coarse grid however time integration studies suggest that the increased accuracy is bought at an increase in computational cost to achieve a specified accuracy the computational cost of the two grid method exceeds that of a standard pseudospectral method for this spatially periodic problem

a joint modeling and estimation method for multivariate longitudinal data with mixed types of responses to analyze physical activity data generated by accelerometers a mixed effect model is proposed to jointly analyze multivariate longitudinal data with continuous proportion count and binary responses the association of the variables is modeled through the correlation of random effects we use a quasi likelihood type approximation for nonlinear variables and transform the proposed model into a multivariate linear mixed model framework for estimation and inference via an extension to the em approach an efficient algorithm is developed to fit the model the method is applied to physical activity data which uses a wearable accelerometer device to measure daily movement and energy expenditure information our approach is also evaluated by a simulation study

wavelet based functional mixed models increasingly scientific studies yield functional data in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid we present new methodology that generalizes the linear mixed model to the functional mixed model framework with model fitting done by using a bayesian wavelet based approach this method is flexible allowing functions of arbitrary form and the full range of fixed effects structures and between curve covariance structures that are available in the mixed model framework it yields nonparametric estimates of the fixed and random effects functions as well as the various between curve and within curve covariance matrices the functional fixed effects are adaptively regularized as a result of the non linear shrinkage prior that is imposed on the fixed effects wavelet coefficients and the random effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient because we have posterior samples for all model quantities we can perform pointwise or joint bayesian inference or prediction on the quantities of the model the adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks

nonnegative decomposition of functional count data we present a novel decomposition of nonnegative functional count data that draws on concepts from nonnegative matrix factorization our decomposition which we refer to as narfd nonnegative and regularized function decomposition enables the study of patterns in variation across subjects in a highly interpretable manner prototypic modes of variation are estimated directly on the observed scale of the data are local and are transparently added together to reconstruct observed functions this contrasts with generalized functional principal component analysis an alternative approach that estimates functional principal components on a transformed scale produces components that typically vary across the entire functional domain and reconstructs observations using complex patterns of cancellation and multiplication of functional principal components narfd is implemented using an alternating minimization algorithm and we evaluate our approach in simulations we apply narfd to an accelerometer dataset comprising observations of physical activity for healthy older americans

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

granular box regression granular computing grc has gained increasing attention in the past decade although not uniquely defined its basic idea is to approximate detailed machine like information by a coarser presentation on a human like level within granular computing the mapping of continuous variables into intervals plays an important role these intervals are often prerequisites for the formulation of linguistic variables in this paper we suggest a piecewise interval approximation and propose granular box regression its objective is to establish relationships between independent and dependent variables by multidimensional boxes we interpret granular box regression as interval regression and show its potential for the extraction of fuzzy rules from data in two experiments we apply granular box regression to an artificial as well as to a real dataset in the field of finance and evaluate its properties

joint modeling and clustering paired generalized longitudinal trajectories with application to cocaine abuse treatment data in a cocaine dependence treatment study we have paired binary longitudinal trajectories that record the cocaine use patterns of each patient before and after a treatment to better understand the drug using behaviors among the patients we propose a general framework based on functional data analysis to jointly model and cluster these paired non gaussian longitudinal trajectories our approach assumes that the response variables follow distributions from the exponential family with the canonical parameters determined by some latent gaussian processes to reduce the dimensionality of the latent processes we express them by a truncated karhunen loeve kl expansion allowing the mean and covariance functions to be different across clusters we further represent the mean and eigenfunctions functions by flexible spline bases and determine the orders of the truncated kl expansions using data driven methods by treating the cluster membership as a missing value we cluster the cocaine use trajectories by a l

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

registration for exponential family functional data we consider the problem of aligning curves from exponential family distributions the approach is based on the combination of alignment and functional principal components analysis and is facilitated by recent extensions of fpca to non gaussian settings our work is motivated by the study of physical activity using accelerometers wearable devices that provide around the clock monitoring of activity and produce non gaussian measurements we apply the proposed methods to activity counts using a poisson distribution and to a binary active vs inactive indicator using a binomial distribution after alignment the trajectories show clear peaks of activity in the morning and afternoon with a dip in the middle of the day

an approximation to the modified profile likelihood function an approximation to the modified profile likelihood function is proposed this approximation is invariant under interest respecting reparameterisations it agrees with the modified profile likelihood function to order o n 1 in the moderate deviation sense and to order o n 1 2 in the large deviation sense and it is stable in the sense of conditional inference for the case of a curved exponential family model this approximation agrees with the one proposed by barndorff nielsen 1995 a general expression for the approximate modified profile likelihood function is given for nonlinear regression models and for generalised linear models with known dispersion parameter

multilevel schemes for the shallow water equations in this paper we study a number of multilevel schemes for the numerical solution of the shallow water equations new schemes and new perspectives of known schemes are examined we consider the case of periodic boundary conditions spatial discretization is obtained using a fourier spectral galerkin method for the time integration two strategies are studied the first one is based on scale separation and we choose the time scheme explicit or semi implicit as a function of the spatial scales multilevel schemes the second approach is based on a splitting of the operators and we choose the time integration method as a function of the operator considered multistep or fractional schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme and with the semi implicit scheme leap frog scheme with crank nicholson scheme for the gravity terms both computed with a similar mesh the drawback of the explicit reference scheme being the numerical stability constraint on the time step and the drawback of the semi implicit scheme being the dispersive error the aim with the new schemes is to obtain schemes with less dispersive error than the semi implicit scheme and with better stability properties than the explicit reference scheme the numerical results obtained show that the schemes proposed allow one to reduce the dispersive error and to increase the numerical stability at reduced cost

conditional second order generalized estimating equations for generalized linear and nonlinear mixed effects models generalized linear and nonlinear mixed effects models are used extensively in health care research including applications in pharmacokinetics clinical trials and epidemiology because the underlying model may be nonlinear in the random effects there will generally be no closed form expression for the marginal likelihood or indeed for the marginal moments consequently estimation is often carried out either using numerical integration techniques or by approximating the marginal likelihood and or marginal moments using first order expansion methods an advantage of the first order methods is that they do not necessarily require specification of a conditional like lihood to estimate the regression parameters of interest however in many cases they may not take full advantage of the fact that the conditional variance depends on both fixed and random effects in this article we propose using conditional second order generalized estimating equations cgee2 to estimate both fixed and random effects param

equation based interpolation and incremental unknowns for solving the three dimensional helmholtz equation abstract in an earlier paper poullet and boag 2007 1 we developed an efficient incremental unknowns iu preconditioner for solving the two dimensional 2d helmholtz problem in both high and low frequency wavenumber regimes the multilevel preconditioning scheme involves separation of each grid into a coarser grid of the following level and a complementary grid on which the ius are defined by interpolation this approach is efficient as long as the mesh size of the coarsest grid is sufficiently small compared to the wavelength in order to overcome this restriction the authors introduced recently in poullet and boag 2010 2 a modified iu method combining the conventional interpolation with the helmholtz equation based interpolation ebi the ebi coefficients are derived numerically using a sufficiently large set of analytic solutions of the helmholtz equation on a special hierarchy of stencils the modified ius using helmholtz ebi are shown to provide improved preconditioning on the coarse scales where the conventional interpolation can not be employed this study deals with the extension of this idea for solving the three dimensional 3d helmholtz equation

calibrated model based evidential clustering using bootstrapping abstract evidential clustering is an approach to clustering in which cluster membership uncertainty is represented by a collection of dempster shafer mass functions forming an evidential partition in this paper we propose to construct these mass functions by bootstrapping finite mixture models in the first step we compute bootstrap percentile confidence intervals for all pairwise probabilities the probabilities for any two objects to belong to the same class we then construct an evidential partition such that the pairwise belief and plausibility degrees approximate the bounds of the confidence intervals this evidential partition is calibrated in the sense that the pairwise belief plausibility intervals contain the true probabilities most of the time i e with a probability close to the defined confidence level this frequentist property is verified by simulation and the practical applicability of the method is demonstrated using several real datasets

effects of not knowing the order of an autoregressive process on the mean squared error of prediction i abstract given a realization of t consecutive observations of an autoregressive process x of unknown order m a process of an arbitrary order p is assumed to be fitted for predicting the future values of another process xtilde t where xtilde t is independent of xt but otherwise has identical probabilistic structure the mean squared error of prediction when the autoregressive coefficients are estimated by least squares is obtained to terms of order t 1 the bias in estimating the autoregressive coefficients is examined the results are illustrated for three processes with m 2 by considering fitting of a process with p 1

robust priors in nonlinear panel data models many approaches to estimation of panel models are based on an average or integrated likelihood that assigns weights to different values of the individual effects fixed effects random effects and bayesian approaches all fall in this category we provide a characterization of the class of weights or priors that produce estimators that are first order unbiased we show that such bias reducing weights must depend on the data unless an orthogonal reparameterization or an essentially equivalent condition is available two intuitively appealing weighting schemes are discussed we argue that asymptotically valid confidence intervals can be read from the posterior distribution of the common parameters when n and t grow at the same rate finally we show that random effects estimators are not bias reducing in general and discuss important exceptions three examples and some monte carlo experiments illustrate the results

nonlinear galerkin method in the finite difference case and wavelet like incremental unknowns the img algorithm inertial manifold multigrid algorithm which uses the first order incremental unknowns was introduced in 20 the img algorithm is aimed at numerically implementing inertial manifolds see e g 19 when finite difference discretizations are used for that purpose it is necessary to decompose the unknown function into its long wavelength and its short wavelength components first order incremental unknowns iu were proposed in 20 as a means to realize this decomposition our aim in the present article is to propose and study other forms of incremental unknowns in particular the wavelet like incremental unknowns wiu so called because of their oscillatory nature in this report we first extend the general convergence results in 20 by proving them under slightly weaker conditions we then present three sets of incremental unknowns i e the first order as in 20 the second order and wavelet like incremental unknowns we show that these incremental unknown can be used to construct convergent img algorithms special stress is put on the wavelet like incremental unknowns since this set of unknowns has thel 2 orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

the ecme algorithm a simple extension of em and ecm with faster monotone convergence a generalisation of the ecm algorithm meng rubin 1993 which is itselfan extension of the em algorithm dempster laird rubin 1977 can be obtained by replacing some cm steps of ecm which maximise the constrained expected complete data loglikelihood function with steps that maximise the correspondingly constrained actual likelihood function this algorithm which we call ecme algorithm for expectation conditional maximisation either shares with both em and ecm their stable monotone convergence and basic simplicity of implementation relative to competing faster converging methods moreover ecme can have a substantially faster convergence rate than either em or ecm measured using either the number of iterations or actual computer time

a correlated probit model for joint modeling of clustered binary and continuous responses a difficulty in joint modeling of continuous and discrete response variables is the lack of a natural multivariate distribution for joint modeling of clustered observations on binary and continuous responses we study a correlated probit model that has an underlying normal latent variable for the binary responses catalano and ryan have factored the model into a marginal and a conditional component and used generalized estimating equations methodology to estimate the effects we propose a monte carlo expectation conditional maximization algorithm for finding maximum likelihood estimates of the mixed model itself extending and accelerating an algorithm for models with binary responses we demonstrate the methodology with a developmental toxicity study measuring fetal weight and a binary malformation status for several litters of mice a simulation study suggests that efficiency gains of joint fittings over separate fittings of the response variables occur mainly for small datasets with strong correlation

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

jackknife and analytical bias reduction for nonlinear panel models fixed effects estimators of panel models can be severely biased because of the well known incidental parameters problem we show that this bias can be reduced by using a panel jackknife or an analytical bias correction motivated by large t we give bias corrections for averages over the fixed effects as well as model parameters we find large bias reductions from using these approaches in examples we consider asymptotics where t grows with n as an approximation to the properties of the estimators in econometric applications we show that if t grows at the same rate as n the fixed effects estimator is asymptotically biased so that asymptotic confidence intervals are incorrect but that they are correct for the panel jackknife we show t growing faster than n 1 3 suffices for correctness of the analytic correction a property we also conjecture for the jackknife

modelling sparse generalized longitudinal observations with latent gaussian processes summary in longitudinal data analysis one frequently encounters non gaussian data that are repeatedly collected for a sample of individuals over time the repeated observations could be binomial poisson or of another discrete type or could be continuous the timings of the repeated measurements are often sparse and irregular we introduce a latent gaussian process model for such data establishing a connection to functional data analysis the functional methods proposed are non parametric and computationally straightforward as they do not involve a likelihood we develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations this method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model these scores can then be used for further statistical analysis such as inference regression discriminant analysis or clustering we illustrate these non parametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with generalized estimating equations and generalized linear mixed models

model selection in the presence of incidental parameters this paper considers model selection in panels where incidental parameters are present primary interest centers on selecting a model that best approximates the underlying structure involving parameters that are common within the panel it is well known that conventional model selection procedures are often inconsistent in panel models and this can be so even without nuisance parameters modifications are then needed to achieve consistency new model selection information criteria are developed here that use either the kullback leibler information criterion based on the profile likelihood or the bayes factor based on the integrated likelihood with a bias reducing prior these model selection criteria impose heavier penalties than those associated with standard information criteria such as aic and bic the additional penalty which is data dependent properly reflects the model complexity arising from the presence of incidental parameters a particular example is studied in detail involving lag order selection in dynamic panel models with fixed effects the new criteria are shown to control for over under selection probabilities in these models and lead to consistent order selection criteria

the asymptotic variance of semiparametric estimators this paper derives a general formula for the asymptotic variance of semiparametric estimators that accounts for the presence of nonparametric estimators of functions the general formula is specialized to show invariance of the asymptotic variance to the type of nonparametric estimator and to obtain correction terms for estimation of densities and mean square projections including conditional expectations regularity conditions for the validity of the formula are also given including primitive conditions for asymptotic normality when series estimators are present new examples considered include a semiparametric panel probit estimator and a series estimator of the average derivative copyright 1994 by the econometric society

generalised information criteria in model selection summary the problem of evaluating the goodness of statistical models is investigated from an information theoretic point of view information criteria are proposed for evaluating models constructed by various estimation procedures when the specified family of probability distributions does not contain the distribution generating the data the proposed criteria are applied to the evaluation of models estimated by maximum likelihood robust penalised likelihood bayes procedures etc we also discuss the use of the bootstrap in model evaluation problems and present a variance reduction technique in the bootstrap simulation

incremental unknowns multilevel methods and the numerical simulation of turbulence abstract the purpose of this monograph is to describe the dynamic multilevel dml methodology applied to the numerical simulation of turbulence the general setting of the navier stokes equations is recalled and also a number of basic notions on the statistical theory of turbulence the practical limitations of direct numerical simulation dns and the needs for modeling are emphasized the objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds aim this mathematical concept stemming from the dynamical systems theory is briefly presented aim are based on a decomposition of the velocity field into small and large scale components they give a slaving law of the small scales as a function of the large scales the novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods indeed the dml methodology is based on the properties of this decomposition of the velocity field and on numerical arguments numerical analysis of multilevel methods for a simple system is proposed dml methods with some numerical and physical justifications are described for homogeneous turbulence the numerical results obtained are discussed

can the strengths of aic and bic be shared a conflict between model indentification and regression estimation a traditional approach to statistical inference is to identify the true or best model first with little or no consideration of the specific goal of inference in the model identification stage can the pursuit of the true model also lead to optimal regression estimation in model selection it is well known that bic is consistent in selecting the true model and aic is minimax rate optimal for estimating the regression function a recent promising direction is adaptive model selection in which in contrast to aic and bic the penalty term is data dependent some theoretical and empirical results have been obtained in support of adaptive model selection but it is still not clear if it can really share the strengths of aic and bic model combining or averaging has attracted increasing attention as a means to overcome the model selection uncertainty can bayesian model averaging be optimal for estimating the regression function in a minimax sense we show that the answers to these questions are basically in the negative for any model selection criterion to be consistent it must behave suboptimally for estimating the regression function in terms of minimax rate of covergence and bayesian model averaging cannot be minimax rate optimal for regression estimation

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

incremental unknowns for solving partial differential equations incremental unknowns have been proposed in t as a method to approximate fractal attractors by using finite difference approximations of evolution equations in the case of linear elliptic problems the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization the method is similar but different from the classical multigrid method in this article we describe the application of incremental unknowns for solving laplace equations in dimensions one and two we provide theoretical results concerning two level approximations and we report on numerical tests done with multi level approximations

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

multiparameter iterative schemes for the solution of systems of linear and nonlinear equations in this paper we introduce multiparameter generalizations of the linear and nonlinear iterative richardson methods for solving systems of linear and nonlinear equations the new algorithms are based on using a optimal matricial relaxation instead of the optimal scalar relaxation of the steepest descent method the optimal matrix which is defined at each iteration by minimizing the current residual is computed as the least squares solution of an associated problem whose dimension is generally much lower than that of the original problem in particular thanks to this approach we construct multiparameter versions of the delta k method introduced for solving nonlinear fixed point problems various numerical results illustrate the implementation of the new schemes they concern the solution of a linear problem and of a nonlinear one which comes out from a reaction diffusion problem which exhibits bifurcations in both cases the optimal multiparameter relaxation improves the convergence as compared to the optimal scalar one

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

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

a novel kernel fuzzy clustering algorithm for geo demographic analysis we proposed a novel kernel based fuzzy clustering for geo demographic analysis it relied on gaussian kernel function updated elements and standardized weights the algorithm was both theoretical and experimental validated its clustering quality was better than those of other relevant algorithms some properties of solutions were examined geo demographic analysis gda is a major concentration of various interdisciplinary researches and has been used in many decision making processes regarding the provision and distribution of products and services in society machine learning methods namely principal component analysis self organizing map k means fuzzy clustering and fuzzy geographically weighted clustering were proposed to enhance the quality of gda among them the state of the art method modified intuitionistic possibilistic fuzzy geographically weighted clustering mipfgwc has some drawbacks such as i using the euclidean similarity measure often results in high error rate and sensitivity to noises and outliers ii updating the membership matrix by the spatial interaction modification model sim2 model leads to new centers not being geographically aware in this paper we present a novel fuzzy clustering algorithm named as kernel fuzzy geographically clustering kfgc that utilizes both the kernel similarity function and the new update mechanism of the sim2 model to remedy the disadvantages of mipfgwc some supported properties and theorems of kfgc are also examined in the paper specifically the differences between solutions of kfgc and those of mipfgwc and of some variants of kfgc are theoretically validated lastly experimental analysis is performed to compare the performance of kfgc with those of the relevant algorithms in terms of clustering quality

counting degrees of freedom in hierarchical and other richly parameterised models summary drawing on linear model theory we rigorously extend the notion of degrees of freedom to richly parameterised models including linear hierarchical and random effect models some smoothers and spatial models and combinations of these the number of degrees of freedom is often much smaller than the number of parameters our notion of degrees of freedom is compatible with similar ideas long associated with smoothers but is applicable to new classes of models and can be interpreted using the projection theory of linear models we use an example to illustrate the two applications of setting prior distributions for variances and fixing model complexity by fixing degrees of freedom

profile likelihood and conditionally parametric models in this paper we outline a general approach to estimating the parametric component of a semiparametric model for the case of a scalar parametric component the method is based on the idea of first estimating a one dimensional subproblem of the original problem that is least favorable in the sense of stein the likelihood function for the scalar parameter along this estimated subproblem may be viewed as a generalization of the profile likelihood for the problem the scalar parameter is then estimated by maximizing this generalized profile likelihood this method of estimation is applied to a particular class of semiparametric models where it is shown that the resulting estimator is asymptotically efficient

reduced rank mixed effects models for spatially correlated hierarchical functional data hierarchical functional data are widely seen in complex studies where subunits are nested within units which in turn are nested within treatment groups we propose a general framework of functional mixed effects model for such data within unit and within subunit variations are modeled through two separate sets of principal components the subunit level functions are allowed to be correlated penalized splines are used to model both the mean functions and the principal components functions where roughness penalties are used to regularize the spline fit an expectation maximization em algorithm is developed to fit the model while the specific covariance structure of the model is utilized for computational efficiency to avoid storage and inversion of large matrices our dimension reduction with principal components provides an effective solution to the difficult tasks of modeling the covariance kernel of a random function and modeling the correlation between functions the proposed methodology is illus

evaluating and comparing soft partitions an approach based on dempster shafer theory in evidential clustering cluster membership uncertainty is represented by dempster shafer mass functions the notion of evidential partition generalizes other soft clustering structures such as fuzzy possibilistic or rough partitions in this paper we propose two extensions of the rand index for evaluating and comparing evidential partitions called similarity and consistency indices the similarity index is suitable for measuring the closeness of two soft partitions whereas the consistency index allows one to assess the agreement or lack of conflict between a soft partition and the true hard partition simulation experiments illustrate some applications of these indices

generalized multilevel function on scalar regression and principal component analysis this manuscript considers regression models for generalized multilevel functional responses functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects this data structure is increasingly common across scientific domains and is exemplified by our motivating example in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days we use a generalized linear model to incorporate scalar covariates into the mean structure and decompose subject specific and subject day specific deviations using multilevel functional principal components analysis thus functional fixed effects are estimated while accounting for within function and within subject correlations and major directions of variability within and between subjects are identified fixed effect coefficient functions and principal component basis functions are estimated using penalized splines model parameters are estimated in a bayesian framework using stan a programming language that implements a hamiltonian monte carlo sampler simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets in both cross sectional and multilevel scenarios code is publicly available in the application we identify effects of age and bmi on the time specific change in probability of being active over a 24 hour period in addition the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

implementation of a modified marder weitzner method for solving nonlinear eigenvalue problems our goal is to propose four versions of modified marder weitzner methods and to present the implementation of the new type methods with incremental unknowns for solving nonlinear eigenvalue problems by combining with compact schemes and modified marder weitzner methods six schemes well suited for the calculation of unstable solutions are obtained we illustrate the efficiency of the new algorithms by using numerical computations and by comparing them with existing methods for some two dimensional problems

asymptotics for semiparametric econometric models via stochastic equicontinuity this paper provides a general framework for proving the square root of t consistency and asymptotic normality of a wide variety of semiparametric estimators the class of estimators considered consists of estimators that can be defined as the solution to a minimization problem based on a criterion function that may depend on a preliminary infinite dimensional nuisance parameter estimator the method of proof exploits results concerning the stochastic equicontinuity of stochastic processes the results are applied to the problem of semiparametric weighted least squares estimation of partially parametric regression models primitive conditions are given for square root of t consistency and asymptotic normality of this estimator copyright 1994 by the econometric society

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

a modified crank nicolson scheme with incremental unknowns for convection dominated diffusion equations a modified crank nicolson scheme based on one sided difference approximations is proposed for solving time dependent convection dominated diffusion equations in two dimensional space the modified scheme is consistent and unconditionally stable a priori l 2 error estimate for the fully discrete modified scheme is derived with the use of the incremental unknowns preconditioner at each time step a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified crank nicolson scheme

high reynolds number solutions of navier stokes equations using incremental unknowns abstract the unsteady 2d navier stokes equations on an unregularized driven cavity are solved in vorticity streamfunction variables using incremental unknowns periodic asymptotic solutions have been found for re 10000 and re 12500

hierarchical functional data with mixed continuous and binary measurements summary motivated by objective measurements of physical activity we take a functional data approach to longitudinal data with simultaneous measurement of a continuous and a binary outcomes the regression structures are specified as smooth curves measured at various time points with random effects that have a hierarchical correlation structure the random effect curves for each variable are summarized using a few important principal components and the association of the two longitudinal variables is modeled through the association of the principal component scores we use penalized splines to model the mean curves and the principal component curves and cast the proposed model into a mixed effects model framework for model fitting prediction and inference via a quasilikelihood type approximation for the binary component we develop an algorithm to fit the model data based transformation of the continuous variable and selection of the number of principal components are incorporated into the algorithm the method is applied to the motivating physical activity data and is evaluated empirically by a simulation study extensions for different types of outcomes are also discussed

the focused information criterion a variety of model selection criteria have been developed of general and specific types most of these aim at selecting a single model with good overall properties for example formulated via average prediction quality or shortest estimated overall distance to the true model the akaike the bayesian and the deviance information criteria along with many suitable variations are examples of such methods these methods are not concerned however with the actual use of the selected model which varies with context and application the present article takes the view that the model selector should instead focus on the parameter singled out for interest in particular a model that gives good precision for one estimand may be worse when used for inference for another estimand we develop a method that for a given focus parameter estimates the precision of any submodel based estimator the framework is that of large sample likelihood inference using an unbiased estimate of limiting risk we propose a foc

rough clustering utilizing the principle of indifference abstract clustering is one of the most widely used method in data mining with applications in virtually any domain its main objective is to group similar objects into the same cluster while dissimilar objects should belong to different clusters in particular k means clustering as member of the partitioning clustering family has obtained great popularity the classic hard k means assigns an object unambiguously to one and only one cluster to address uncertainty soft clustering has been introduced using concepts like fuzziness possibility or roughness a decade ago lingras and west introduced a k means approach based on the interval interpretation of rough sets theory in the past years their rough k means has gained increasing attention in our paper we propose a refined rough k means algorithm that utilizes laplace s principle of indifference to calculate the means as we will discuss this provides a sounder justification for the impacts of the objects in the approximations in comparison to established rough k means algorithms furthermore the weighting in the mean function is based on individual objects rather than on aggregated sub means in experiments we compare the refined algorithm to related approaches

asymptotic optimality for c_p c_l cross validation and generalized cross validation discrete index set c p c l cross validation and generalized cross validation are useful data driven techniques for selecting a good estimate from a proposed class of linear estimates the asymptotic behaviors of these procedures are studied some easily interpretable conditions are derived to demonstrate the asymptotic optimality it is argued that cross validation and generalized cross validation can be viewed as some special ways of applying c l applications in nearest neighbor nonparametric regression and in model selection are discussed in detail

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded