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

quantiles expectiles and splines abstract a time varying quantile can be fitted by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm it is shown that such quantiles satisfy the defining property of fixed quantiles in having the appropriate number of observations above and below like quantiles time varying expectiles can be estimated by a state space signal extraction algorithm and they satisfy properties that generalize the moment conditions associated with fixed expectiles because the state space form can handle irregularly spaced observations the proposed algorithms can be adapted to provide a viable means of computing spline based non parametric quantile and expectile regressions

on the estimation of a probability density function by the maximum penalized likelihood method abstract a class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density the limiting case of the estimates as the amount of smoothing increasing has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty the estimates are shown to be the solution of an unconstrained convex optimization problem and mild natural conditions are given for them to exist rates of consistency in various norms and conditions for asymptotic normality and approximation by a gaussian process are given thus breaking new ground in the theory of maximum penalized likelihood density estimation author

semiparametric nonlinear mixed effects models and their applications nonlinear mixed effects models nlmms and self modeling nonlinear regression semor models are often used to fit repeated measures data they use a common function shared by all subjects to model variation within each subject and some fixed and or random parameters to model variation between subjects the parametric nlmm may be too restrictive and the semiparametric semor model ignores correlations within each subject in this article we propose a class of semiparametric nonlinear mixed effects models snmms that extend nlmms semor models and many other existing models in a natural way a snmm assumes that the mean function depends on some parameters and nonparametric functions the parameters provide an interpretable data summary the nonparametric functions provide flexibility to allow the data to decide some unknown or uncertain components such as the shape of the mean response over time a second stage model with fixed and random effects is used to model the parameters smoothing splines are us

additive nonparametric regression with autocorrelated errors a bayesian approach is presented for nonparametric estimation of an additive regression model with autocorrelated errors each of the potentially non linear components is modelled as a regression spline using many knots while the errors are modelled by a high order stationary autoregressive process parameterized in terms of its autocorrelations the distribution of significant knots and partial autocorrelations is accounted for using subset selection our approach also allows the selection of a suitable transformation of the dependent variable all aspects of the model are estimated simultaneously by using the markov chain monte carlo method it is shown empirically that the approach proposed works well on several simulated and real examples

a fast and efficient cross validation method for smoothing parameter choice in spline regression abstract the spline smoothing approach to nonparametric regression is considered with particular reference to the problem of choosing how much to smooth both computational and statistical aspects of the method of cross validation are discussed an approximate cross validation method is proposed that requires a small linear amount of computer time a simulation study indicates that the method has very good statistical properties some mathematical justification and motivation are also provided

the analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems we prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems we provide results for the so called symmetric multigrid schemes we show that for the variable v script cycle and the w script cycle schemes multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns provided that the initial grid is sufficiently fine we show that the v script cycle algorithm also converges under appropriate assumptions on the coarsest grid but at a rate which may deteriorate as the number of levels increases this deterioration for the v script cycle may occur even in the case of full elliptic regularity finally the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory

smoothing splines for longitudinal data in a longitudinal data model with fixed and random effects polynomials are used to model the fixed effects and smoothing polynomial splines are used to model the within subject random effect curves the splines are generated by modelling the data for each subject as observations of an integrated random walk with observational error the initial conditions for each subject s deviation from the fixed effect curve are assumed to have zero mean and arbitrary covariance matrix which is estimated by maximum likelihood producing an empirical bayes estimate this is in contrast to modelling a single curve using a diffuse prior an example is presented using unbalanced longitudinal data from a pilot study in breast cancer patients

on the use of rational iterations and domain decomposition methods for the helmholtz problem an iterative algorithm for the numerical solution of the helmholtz problem is considered it is difficult to solve the problem numerically in particular when the imaginary part of the wave number is zero or small we develop a parallel iterative algorithm based on a rational iteration and a nonoverlapping domain decomposition method for such a non hermitian non coercive problem algorithm parameters artificial damping and relaxation are introduced to accelerate the convergence speed of the iteration convergence analysis and effective strategies for finding efficient algorithm parameters are presented numerical results carried out on an ncube2 are given to show the efficiency of the algorithm to reduce the boundary reflection we employ a hybrid absorbing boundary condition abc which combines the first order abc and the physical q abc computational results comparing the hybrid abc with non hybrid ones are presented

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

the scan sampler for time series models summary an algorithm called the scan sampler is developed and discussed the scan sampler has a variety of uses for time series analysis based on the state space model with nongaussian observations the algorithm is based on the kalman filter smoothing algorithm it can be used for bayesian inference using markov chain monte carlo and to find posterior modes

sample function properties of a class of smoothed estimates the structural properties of sample functions of the solution to the fixed interval continuous discrete smoothing problem are examined it is shown that every sample function has a functional form determined by the poles of the signal model and a degree of smoothness determined by the relative order of the model

the estimation of residual variance in nonparametric regression summary a wide class of estimators of the residual variance in nonparametric regression is considered namely those that are quadratic in the data unbiased for linear regression and always nonnegative the minimax mean squared error estimator over a natural class of regression functions is derived this optimal estimator has an interesting structure and is closely related to a minimax estimator of the regression curve itself

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

the nonparametric estimation of branching curves abstract the spline smoothing approach to nonparametric regression is extended to cover the estimation of curves that branch at certain specified points in time this extension is motivated and illustrated by data from an experiment on sunflowers apart from providing insight into this particular data analytic problem the work described demonstrates the power of the roughness penalty approach in coping with curve estimation of a nonstandard kind particularly in the context of certain experiments in horticulture and arboriculture the experimental material may be treated identically to the control up to a certain known time part of the way through the experiment to display and investigate the effect of treatments applied during growth it is of interest to estimate a branching curve in the simplest case we want to study two growth curves that are common up to a certain known time and then diverge or branch after the application of a treatment to part of the experimental material the spline smoothing

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

a bayesian approach to additive semiparametric regression abstract we present a bayesian approach to estimating an additive semiparametric regression model which is robust to outliers the unknown curves are estimated by posterior means and are shown to be smoothing splines by using markov chain monte carlo an o mn algorithm is produced where n is the sample size and m is the total number of markov chain iterations previous exact approaches required o n3 operations making the estimation of large data sets impractical efficient methods for estimating the posterior means using mixture and backfitting estimates are developed the properties of the curve estimates are studied empirically using both simulated and real examples

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

a flexible model for human circadian rhythms many hormones and other physiological processes vary in a circadian pattern although a sine cosine function can be used to model these patterns this functional form is not appropriate when there is asymmetry between the peak and nadir phases in this paper we describe a semiparametric periodic spline function that can be fit to circadian rhythms the model includes both phase and amplitude so that the time and the magnitude of the peak or nadir can be estimated we also describe tests of fit for components in the model data from an experiment to study immunological responses in humans are used to demonstrate the methods

proprietes de convergence presque complete du predicteur a noyau let z n be a mixing process which is valued in mathbb e subset mathbb r q g be a real measurable function defined on mathbb e s and k be two positive integers we suppose the existence of a function r satisfying r e g z n s z n k 1 z n n n k and estimate the function r from a sequence z i i 1 n by r n with r n u frac sum limits i k n s g z i s k u z i k 1 cdots z i h n 1 sum limits i k n s k u z i k 1 cdots z i h n 1 forall u in mathbb e k where k is a kernel of kq and h n h n 0 we give various conditions on both the sequences h n and the sequence n which is associated with z n for the two following properties uniform complete convergence to r for the functional estimator r n and complete convergence to 0 for the real random variable r n z n k 1 z n r zn k 1 z n this last result concerns the predictor r n z n k 1 z n of g z n s from z i i 1 n when the process z n is stationary and markovian of order k

greater robustness of second order statistics than higher order statistics algorithms to distortions of the mixing matrix in blind source separation of human eeg implications for single subject and group analyses a mandatory assumption in blind source separation bss of the human electroencephalogram eeg is that the mixing matrix remains invariant i e that the sources electrodes and geometry of the head do not change during the experiment actually this is not often the case for instance it is common that some electrodes slightly move during eeg recording this issue is even more critical for group independent component analysis gica a method of growing interest in which only one mixing matrix is estimated for several subjects indeed because of interindividual anatomo functional variability this method violates the mandatory principle of invariance here using simulated experiments 1 and 2 and real experiment 3 eeg data we test how eleven current bss algorithms undergo distortions of the mixing matrix we show that this usual kind of perturbation creates non gaussian features that are virtually added to all sources impairing the estimation of real higher order statistics hos features of the actual sources by hos algorithms e g ext infomax fastica hos based methods are likely to identify more components with similar properties than actual neurological sources a problem frequently encountered by bss users in practice the quality of the recovered signal and the efficiency of subsequent source localization are substantially impaired performing dimensionality reduction before applying hos based bss does not seem to be a safe strategy to circumvent the problem second order statistics sos based bss methods belonging to the less popular sobi family class are much less sensitive to this bias

nonparametric spline regression with prior information summary by using prior information about the regression curve we propose new nonparametric regression estimates we incorporate two types of information first we suppose that the regression curve is similar in shape to a family of parametric curves characterized as the solution to a linear differential equation the regression curve is estimated by penalized least squares with the differential operator defining the smoothness penalty we discuss in particular growth and decay curves and take a time transformation to obtain a tractable solution the second type of prior information is linear equality constraints we estimate unknown parameters by generalized cross validation or maximum likelihood and obtain efficient 0 n algorithms to compute the estimate of the regression curve and the cross validation and maximum likelihood criterion functions

a frequency domain selection criterion for regression with autocorrelated errors abstract we consider the regression model yt t e t t 0 1 n 1 where y t are scalar observations t is the unknown regression function and e t are unobservable errors generated by a zero mean weakly stationary process independent of t and with completely unknown autocorrelation structure we propose a data driven method for selecting a parametric or nonparametric estimator of t the method is based on cross validation in the frequency domain and requires no assumptions about the form of the estimator or the error correlations it does however require the discrete fourier transform dft of the signal t to be a smooth complex function of frequency as is the case for example with transient signals or growth and decay curves after giving some general motivations for the method we focus on the special case of linear estimators of a nonparametric regression function including both kernel and spline estimators for these estimators we develop efficient methods of evaluating t

a fast algorithm for signal extraction influence and cross validation in state space models summary a fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model the resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector the ideas of studentized residuals and leverage from regression analysis are generalized to state space models and the new algorithm is used to compute the various measures the results are also applied to obtain a new efficient algorithm for polynomial spline smoothing suppose observations are generated by a gaussian signal plus noise process with the signal described by a state space model such a model for the signal occurs often in practice for example when the signal is the output of a stochastic difference or differential equation this paper presents a new algorithm for signal extraction that is the computa tion of the conditional mean and variance of the signal given the observations the usual method for signal extraction is to use the kalman filter see for example anderson moore 1979 p 105 followed by a smoothing step using an algorithm such as the fixed interval smoothing algorithm anderson moore 1979 p 187 our approach also uses the kalman filter but for the smoothing step introduces a new set of recursions which are more efficient than those for the fixed interval smoothing algorithm as for the fixed interval smoothing algorithm recursions for estimating the signal and its variance are carried out separately so that considerable additional savings can be made if the signal estimate alone is required the development uses ideas introduced by ansley kohn 1987a a remarkable property of the new recursions is that they can be applied immediately

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

smoothing spline models with correlated random errors abstract spline smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model their performances depend greatly on the choice of smoothing parameters many methods of selecting smoothing parameters such as generalized maximum likelihood gml generalized cross validation gcv and unbiased risk ubr have been developed under the assumption of independent observations they tend to underestimate smoothing parameters when data are correlated in this article i assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters i extend the gml gcv and ubr methods to estimate the smoothing parameters and the correlation parameters simultaneously i also relate a smoothing spline model to three mixed effects models these relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction blup estimates and that the gml estimates of the smoothing parameters an

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

iterative solution of systems originating from integral equations and surface interpolation a simple procedure for generating diagonal dominance in certain linear systems is presented and the iterative solution of the transformed systems is discussed the method is presented for systems encountered in the numerical solution of fredholm equations of the first kind with singular kernels and for systems arising in the interpolation of surfaces by certain sets of radial functions in the second case the method provides new sets of basis functions generalizing in some sense the univariate b splines

the performance of cross validation and maximum likelihood estimators of spline smoothing parameters abstract an important aspect of nonparametric regression by spline smoothing is the estimation of the smoothing parameter in this article we report on an extensive simulation study that investigates the finite sample performance of generalized cross validation cross validation and marginal likelihood estimators of the smoothing parameter in splines of orders 2 and 3 the performance criterion for both the estimate of the function and its first derivative is measured by the square root of integrated squared error marginal likelihood using splines of degree 5 emerges as an attractive alternative to the other estimators in that it usually outperforms them and is also faster to compute

asymptotic variability analysis for a two stage tandem queue part ii the law of the iterated logarithm abstract a strong form of the law of the iterated logarithm lil is established for a two stage tandem queue the concerned lil with a stronger mathematical form is a later generalization of levy s lil and it quantifies the magnitude of asymptotic stochastic fluctuations of stochastic processes compensated by their deterministic fluid limits the lils are established in twelve cases covering three regimes the underloaded critically loaded and overloaded divided by traffic intensity of the two stages for five processes the queue length workload busy idle and departure processes all the lils are expressed into some simple analytic functions of the primitive data the first and second moments of the interarrival and service times the proofs are based on a fluid approximation and a strong approximation which approximate discrete performance processes with their expected mean values and reflected brownian motions respectively

smoothing splines with varying smoothing parameter this paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty the penalty is a function of the design points in order to accommodate local behaviour of the regression function we show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator and that the equivalent kernel is spatially dependent the equivalent kernels for traditional smoothing splines are a special case of this general solution with the aid of the green s function for a two point boundary value problem explicit forms of the asymptotic mean and variance are obtained for any interior point thus the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error simulation results and an application illustrate the performance of the proposed estimator copyright 2013 oxford university press

nonparametric input estimation in physiological systems problems methods and case studies input estimation from output data is an important problem in the analysis of physiological systems because many signals of interest are not directly accessible to measurement when the system is time invariant this problem is often referred to as deconvolution three representative physiological problems regarding hormone secretion insulin dynamics and hepatic glucose production are used to illustrate the major challenges ill conditioning confidence intervals assessment infrequent and nonuniform sampling nonnegativity constraints and computational efficiency the paper provides a critical overview of the existing techniques focusing on regularization theory and bayesian estimation in order to overcome some inadequacies of the existing methods some new results are derived in particular the connection between the maximum likelihood estimate of the regularization parameter and the notion of equivalent degree of freedom is studied moreover a fast svd based numerical algorithm is developed that includes the optimization of the regularization parameter and the computation of confidence intervals the proposed techniques are validated on a benchmark problem and are shown to provide effective solutions to the three physiological case studies

variational solution of penalized likelihood problems and smooth curve estimation usual nonparametric regression estimators often show many little wiggles which do not appear to be necessary for a good description of the data the new wp smoother is a maximum penalized likelihood mpl estimate with a novel roughness penalty it penalizes a relative change of curvature this leads to disjoint classes of functions each with a given number n w of inflection points for a wp estimate f x x w 1 x w nw exp h f x which is semiparametric with parameters w j and nonparametric part h f the main mathematical result is a convenient form of the characterizing differential equation for a very general class of mpl estimators

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

additive extensions to generalized estimating equation methods the methods underlying vector generalized additive models are extended to provide additive extensions to the generalized estimating equations approaches to multivariate regression problems of liang and zeger and the subsequent literature the methods are illustrated by two examples using correlated binary data concerning the presence or absence of several plant species at the same geographical site and chest pain in male workers from a workforce study with minor modification the extensions are shown to apply to longitudinal data

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

estimation filtering and smoothing in state space models with incompletely specified initial conditions on definit la vraisemblance pour un modele d espace d etats avec des conditions initiales incompletement specifiees par transformation des donnees pour eliminer la dependance sur des conditions non specifiees on etend cette approche pour obtenir des estimations sur les vecteurs d etat les predicteurs et les interpolateurs pour des observations manquantes

nonparametric spline regression with autoregressive moving average errors summary we estimate by spline nonparametric regression an unknown function observed with autocorrelated errors when the errors are modelled by an autoregressive moving average model unknown parameters are estimated by either maximum likelihood cross validation or generalized cross validation by expressing the problem in state space form we obtain 0 n algorithms to estimate the function and its derivatives and evaluate the marginal likelihood and cross validation functions the finite sample properties of the function estimates are evaluated by an extensive simulation study and examples are given

asymptotic variability analysis for a two stage tandem queue part i the functional law of the iterated logarithm abstract a functional law of the iterated logarithm lil and its corresponding lil are established for a two stage tandem queue the functional lil and the lil limits quantify the magnitude of asymptotic stochastic fluctuations of stochastic processes compensated by their deterministic fluid limits in two forms the functional and numerical respectively the earliest functional lil and lil concerned are both developed for brownian motion by volker strassen and levy respectively we establish the functional lils and their corresponding lils in twelve cases covering three regimes divided by the traffic intensity the underloaded critically loaded and overloaded for five processes the queue length workload busy idle and departure processes by the primitive data of the first and second moments of the interarrival and service times all the functional lils are expressed into some compact sets of continuous functions and all the corresponding lils are some analytic functions the proofs are based on the fluid approximation and the strong approximation of the queueing system with the fluid approximation characterizing the expected values of the performance functions and the strong approximation approximating discrete performance processes with reflected brownian motions a companion paper asymptotic variability analysis for a two stage tandem queue part ii the law of the iterated logarithm develops another different version of lil which is a later generalization of levy s lil and also quantifies the magnitude of the asymptotic stochastic fluctuations of the performance functions

a unified convergence theory for abstract multigrid or multilevel algorithms serial and parallel multigrid methods are analyzed in the style of standard iterative methods a basic error bound is derived in terms of residuals on neighboring levels the terms in this bound derive from the iterative methods used as smoothers on each level and the operators used to go from a level to the next coarser level this bound is correct whether the underlying operator is symmetric or nonsymmetric definite or indefinite and singular or nonsingular this paper allows any iterative method as a smoother or rougher in the multigrid cycle while standard multigrid error analysis typically assumes a specific multigrid cycle e g a v w or f cycle analysis for arbitrary multigrid cycles including adaptively chosen ones is provided this theory applies directly to aggregation disaggregation methods used to solve systems of linear equations

a signal extraction approach to modeling hormone time series with pulses and a changing baseline abstract hormones serve as regulating signals for many biological processes in recent years it was determined that many hormones are secreted in a pulsatile manner and that the pulsatile secretion pattern in addition to the absolute concentration level is important in regulating biological processes consequently it is necessary to characterize the latent secretion patterns from measurements of concentration levels the characterization is complicated by the presence of a biological circadian rhythm when hormone concentrations are plotted over time the resultant time series usually exhibits occasional short rises superimposed on a slowly changing baseline this is a result of a mixture of pulsatile secretions and a circadian rhythm in this article we present a signal extraction approach to model simultaneously a slowly changing component and a pulsatile component of a time series a smoothing spline is used to model the baseline and a multiprocess dynamic linear model is used to model the pulsati

a parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods a parallelizable interative procedure based on domain decomposition techniques is defined and analyzed for mixed finite element methods for elliptic equations with the analysis being presented for the decomposition of the domain into the individual elements associated with the mixed method or into larger subdomains applications to time dependent problems are indicated

two graphical displays for outlying and influential observations in regression summary the paper describes two procedures for detecting observations with outlying values either in the response variable or in the explanatory variables in multiple regression these procedures are presented as half normal plots with envelopes derived from simulation in order to avoid overinterpretation of the data analysis of a well known data set leads to the use of a data transformation a simple test for which is commended and to some comments on the relationship with robust regression the widespread availability of sophisticated computer software has made the fitting of multiple regression equations painless but the very ease of these procedures may cause insufficient care to be given to checking and scrutiny of the data transcription punching and data manipulation errors may lead to bad values both of the observations y and of the independent or explanatory variables x it is the purpose of the present paper to describe and exemplify two plots which provide checks against such bad values it is argued that these plots and a test for transformations should accompany any thorough regression analysis bad values of the response or outliers have long been detected by a variety of plots of residuals bad values of the explanatory variables are less easily detected even when they lead to an extreme point in the design space since the fitted equation will pass close to such an influential observation the residual may not be especially large even after allowance has been made for the small variance of the fitted value at such a point to detect such behaviour a quantity is needed which exhibits the dependence of the fitted model on each point or group of points discussions of such quantities are given by hoaglin welsch 1978 and more recently by belsley kuh welsch 1980 2 1 by cook weisberg 1980 and by pregibon 1981 the quantity used here is a modification of a statistic proposed by cook 1977 which is derived in 2 two half normal plots are presented in 3 as an aid to interpretation envelopes to the plots are generated by simulation examples of the plots are given in 4 both for simulated data and for the stack loss data given by brownlee 1965 p 454 transformations of this data set are discussed in 5 the paper concludes with some comments on the relationship with robust

bayesian confidence intervals for smoothing splines abstract the frequency properties of wahba s bayesian confidence intervals for smoothing splines are investigated by a large sample approximation and by a simulation study when the coverage probabilities for these pointwise confidence intervals are averaged across the observation points the average coverage probability acp should be close to the nominal level from a frequency point of view this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error these properties are independent of the bayesian assumptions used to derive this confidence procedure and they explain why the acp is accurate for functions that are much smoother than the sample paths prescribed by the prior this analysis accounts for the choice of the smoothing parameter bandwidth using cross validation in the case of natural splines an adaptive method for avo

artificial damping techniques for scalar waves in the frequency domain abstract an artificial damping algorithm for solving the helmholtz problem is considered when the imaginary part of the wave number is small the problem is known to be difficult to solve in this paper we propose an efficient artificial damping algorithm which can be viewed as a rational iteration each damped problem is solved incompletely by a nonoverlapping domain decomposition method convergence arguments are presented numerical results run on an ncube2 are presented to demonstrate the effectiveness of the method

semiparametric bayesian inference for time series with mixed spectra summary a bayesian analysis is presented of a time series which is the sum of a stationary component with a smooth spectral density and a deterministic component consisting of a linear combination of a trend and periodic terms the periodic terms may have known or unknown frequencies the advantage of our approach is that different features of the data such as the regression parameters the spectral density unknown frequencies and missing observations are combined in a hierarchical bayesian framework and estimated simultaneously a bayesian test to detect deterministic components in the data is also constructed by using an asymptotic approximation to the likelihood the computation is carried out efficiently using the markov chain monte carlo method in o mn operations where n is the sample size and m is the number of iterations we show empirically that our approach works well on real and simulated samples

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

necessary and sufficient conditions for the existence of a conjugate gradient method we characterize the class cg s of matrices a for which the linear system a bf x bf b can be solved by an s term conjugate gradient method we show that except for a few anomalies the class cg s consists of matrices a for which conjugate gradient methods are already known these matrices are the hermitian matrices a a and the matrices of the form ae i theta di b with b b

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

the signal extraction approach to nonlinear regression and spline smoothing abstract this article shows how to fit a smooth curve polynomial spline to pairs of data values yi xi prior specification of a parametric functional form for the curve is not required the resulting curve can be used to describe the pattern of the data and to predict unknown values of y given x both point and interval estimates are produced the method is easy to use and the computational requirements are modest even for large sample sizes our method is based on maximum likelihood estimation of a signal in noise model of the data we use the kalman filter to evaluate the likelihood function and achieve significant computational advantages over previous approaches to this problem

on bandwidth choice in nonparametric regression with both short and long range dependent errors we analyse methods based on the block bootstrap and leave out cross validation for choosing the bandwidth in nonparametric regression when errors have an almost arbitrarily long range of dependence a novel analytical device for modelling the dependence structure of errors is introduced this allows a concise theoretical description of the way in which the range of dependence affects optimal bandwidth choice it is shown that provided block length or leave out number respectively are chosen appropriately both techniques produce first order optimal bandwidths nevertheless the block bootstrap has far better empirical properties particularly under long range dependence

a signal extraction approach to the estimation of treatment and control curves abstract suppose that the response of a group of subjects which we call the control group is monitored over time at a given time after the start of the experiment the experimenter intervenes and applies a treatment to a subset of the group we propose a signal extraction approach to the simultaneous estimation of the treatment and control mean response curves and their rates of change imposing a smooth transition constraint between the two curves at the time of intervention by expressing the model in state space form we show how to compute the likelihood and the cross validation function efficiently in o n operations together with the response curve estimates and their standard errors we also show that the curve estimates are solutions to a penalized least squares problem a small simulation study to assess the performance of the generalized cross validation and marginal likelihood estimates of the smoothing parameter is carried out and the methodology in the article is applied to an experiment

strong approximation of extended renewal processes we develop a strong approximation approach to extended multidimensional renewal theory the consequences of this approximation are a bahadur kiefer type representation of the renewal process in terms of partial sums strassen and chung type laws of the iterated logarithm we also give a characterization of the renewal process by four classes of deterministic curves in the sense of revesz 1982 we generalize our results to the case of non independent and or nonidentically distributed random vectors

principal differential analysis data reduction by differential operators functional data are observations that are either themselves functions or are naturally representable as functions when these functions can be considered smooth it is natural to use their derivatives in exploring their variation principal differential analysis pda identifies a linear differential operator l w 0 i w 1 d w m 1 d m 1 d m that comes as close as possible to annihilating a sample of functions convenient procedures for estimating the m weighting functions w j are developed the estimated differential operator l is analogous to the projection operator used as the data annihilator in principal components analysis and thus can be viewed as a type of data reduction or exploration tool the corresponding linear differential equation may also have a useful substantive interpretation modelling and regularization features can also be incorporated into pda

a smoothness priors state space modeling of time series with trend and seasonality abstract a smoothness priors modeling of time series with trends and seasonalities is shown an observed time series is decomposed into local polynomial trend seasonal globally stationary autoregressive and observation error components each component is characterized by an unknown variance white noise perturbed difference equation constraint the constraints or bayesian smoothness priors are expressed in state space model form trading day factors are also incorporated in the model a kalman predictor yields the likelihood for the unknown variances hyperparameters likelihoods are computed for different constraint order models in different subsets of constraint equation model classes akaike s minimum aic procedure is used to select the best model fitted to the data within and between the alternative model classes smoothing is achieved by using a fixed interval smoother algorithm examples are shown

local nonlinear least squares using parametric information in nonparametric regression abstract we introduce a new nonparametric regression estimator that uses prior information on regression shape in the form of a parametric model in effect we nonparametrically encompass the parametric model we obtain estimates of the regression function and its derivatives along with local parameter estimates that can be interpreted from within the parametric model we establish the uniform consistency and derive the asymptotic distribution of the local parameter estimates and of the corresponding regression and derivative estimates for estimating the regression function our method has superior performance compared to the usual kernel estimators at or near the parametric model it is particularly well motivated for binary data using the probit or logit parametric model as a base we include an application to the horowitz 1993 journal of econometrics 58 49 70 transport choice dataset

semiparametric stochastic mixed models for longitudinal data abstract we consider inference for a semiparametric stochastic mixed model for longitudinal data this model uses parametric fixed effects to represent the covariate effects and an arbitrary smooth function to model the time effect and accounts for the within subject correlation using random effects and a stationary or nonstationary stochastic process we derive maximum penalized likelihood estimators of the regression coefficients and the nonparametric function the resulting estimator of the nonparametric function is a smoothing spline we propose and compare frequentist inference and bayesian inference on these model components we use restricted maximum likelihood to estimate the smoothing parameter and the variance components simultaneously we show that estimation of all model components of interest can proceed by fitting a modified linear mixed model we illustrate the proposed method by analyzing a hormone dataset and evaluate its performance through simulations