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## Smoothing Parameter (smoothing + parameter)
## Selected Abstracts## Smoothing splines for trend estimation and prediction in time series ENVIRONMETRICS, Issue 3 2009Richard MortonAbstract We consider the use of generalized additive models with correlated errors for analysing trends in time series. The trend is represented as a smoothing spline so that it can be extrapolated. A method is proposed for choosing the smoothing parameter. It is based on the ability to predict a short term into the future. The choice not only addresses the purpose in hand, but also performs very well, and avoids the tendency to under-smooth or to interpolate the data that can occur with other data-driven methods used to choose the smoothing parameter. The method is applied to data from a chemical process and to stream salinity measurements. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Constraints on earthquake epicentres independent of seismic velocity models GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2004T. NicholsonSUMMARY We investigate the constraints that may be placed on earthquake epicentres without assuming a model for seismic wave speed variation within the Earth. This allows location improvements achieved using 1-D or 3-D models to be put into perspective. A simple, arrival order misfit criterion is proposed that may be used in standard location schemes. The arrival order misfit criterion does not use a seismic velocity model but simply assumes that the traveltime curve for a particular phase is monotonic with distance. Greater robustness is achieved by including a contribution from every possible pairing of stations and the effect of timing inconsistencies reduced by smoothing. An expression is found that relates the smoothing parameter to the number of observations. A typical event is studied in detail to demonstrate the properties of the misfit function. A pathological case is shown that illustrates that, like other location methods, the arrival order misfit is susceptible to poor station distribution. 25 ground truth and 5000 other teleseismically observed events are relocated and the arrival order locations compared to those found using a least-squares approach and a 1-D earth model. The arrival order misfit is found to be surprisingly accurate when more than 50 observations are used and may be useful in obtaining a model independent epicentre estimate in regions of poorly known velocity structure or the starting point for another location scheme. [source] ## A self-normalized approach to confidence interval construction in time series JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 3 2010Xiaofeng ShaoSummary., We propose a new method to construct confidence intervals for quantities that are associated with a stationary time series, which avoids direct estimation of the asymptotic variances. Unlike the existing tuning-parameter-dependent approaches, our method has the attractive convenience of being free of any user-chosen number or smoothing parameter. The interval is constructed on the basis of an asymptotically distribution-free self-normalized statistic, in which the normalizing matrix is computed by using recursive estimates. Under mild conditions, we establish the theoretical validity of our method for a broad class of statistics that are functionals of the empirical distribution of fixed or growing dimension. From a practical point of view, our method is conceptually simple, easy to implement and can be readily used by the practitioner. Monte Carlo simulations are conducted to compare the finite sample performance of the new method with those delivered by the normal approximation and the block bootstrap approach. [source] ## Nonparametric maximum likelihood estimation for shifted curves JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 2 2001Birgitte B. RønnThe analysis of a sample of curves can be done by self-modelling regression methods. Within this framework we follow the ideas of nonparametric maximum likelihood estimation known from event history analysis and the counting process set-up. We derive an infinite dimensional score equation and from there we suggest an algorithm to estimate the shape function for a simple shape invariant model. The nonparametric maximum likelihood estimator that we find turns out to be a Nadaraya,Watson-like estimator, but unlike in the usual kernel smoothing situation we do not need to select a bandwidth or even a kernel function, since the score equation automatically selects the shape and the smoothing parameter for the estimation. We apply the method to a sample of electrophoretic spectra to illustrate how it works. [source] ## On testing for multivariate ARCH effects in vector time series models THE CANADIAN JOURNAL OF STATISTICS, Issue 3 2003Pierre DuchesneAbstract Using a spectral approach, the authors propose tests to detect multivariate ARCH effects in the residuals from a multivariate regression model. The tests are based on a comparison, via a quadratic norm, between the uniform density and a kernel-based spectral density estimator of the squared residuals and cross products of residuals. The proposed tests are consistent under an arbitrary fixed alternative. The authors present a new application of the test due to Hosking (1980) which is seen to be a special case of their approach involving the truncated uniform kernel. However, they typically obtain more powerful procedures when using a different weighting. The authors consider especially the procedure of Robinson (1991) for choosing the smoothing parameter of the spectral density estimator. They also introduce a generalized version of the test for ARCH effects due to Ling & Li (1997). They investigate the finite-sample performance of their tests and compare them to existing tests including those of Ling & Li (1997) and the residual-based diagnostics of Tse (2002).Finally, they present a financial application. Adoptant une approche spectrale, les auteurs proposent des tests permettant de détecter des effets ARCH multivariés dans les résidus d'un modèle de régression multivarié. Leurs tests reposent sur une comparaison en norme quadratique de la densité spectrale uniforme et d'un estimateur à noyau de la densité spectrale des résidus carrés et des produits croisés des résidus. Ces tests sont convergents sous une contre-hypothèse fixe quelconque. Les auteurs présentent une nouvelle application du test de Hosking (1980) qui correspond dans leur approche au choix particulier d'un noyau uniforme tronqué. Cependant, l'emploi d'autres pondérations leur permet d'obtenir des test encore plus puissants. Les auteurs étudient notamment la procédure de Robinson (1991) pour le choix du paramètre de lissage de l'estimateur de la densité spectrale. Os proposent aussi une version généralisée du test pour effets ARCH de Ling & Li (1997). Ils examinent le comportement de leurs tests dans de petits échantillons par voie de simulation et les comparent aux tests de Ling & Li (1997) et aux diagnostiques de Tse (2002) fondés sur les résidus, us présentent en outre une application financière. [source] ## Adaptive tests of regression functions via multiscale generalized likelihood ratios THE CANADIAN JOURNAL OF STATISTICS, Issue 2 2003Chunming M. ZhangAbstract Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P-value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data. Moult applications des tests non paramétriques basés sur l'estimation de courbes font intervenir un paramètre de lissage. L'auteure propose un test adaptatif qui allie plusieurs tests du rapport de vraisemblances généralisés et rivalise de puissance avec le meilleur d'entre eux. Elle détermine la loi asymptotique conjointe des tests individuels et celle du test global sous l'hypothèse nulle. Elle montre aussi comment sélectionner facilement les paramètres de lissage du test global et propose deux méthodes de calcul approché de son seuil. Elle examine en outre le comportement du test proposé par voie de simulations et en illustre l'emploi dans un cas concret [source] ## Bayesian Inference in Semiparametric Mixed Models for Longitudinal Data BIOMETRICS, Issue 1 2010Yisheng LiSummary We consider Bayesian inference in semiparametric mixed models (SPMMs) for longitudinal data. SPMMs are a class of models that use a nonparametric function to model a time effect, a parametric function to model other covariate effects, and parametric or nonparametric random effects to account for the within-subject correlation. We model the nonparametric function using a Bayesian formulation of a cubic smoothing spline, and the random effect distribution using a normal distribution and alternatively a nonparametric Dirichlet process (DP) prior. When the random effect distribution is assumed to be normal, we propose a uniform shrinkage prior (USP) for the variance components and the smoothing parameter. When the random effect distribution is modeled nonparametrically, we use a DP prior with a normal base measure and propose a USP for the hyperparameters of the DP base measure. We argue that the commonly assumed DP prior implies a nonzero mean of the random effect distribution, even when a base measure with mean zero is specified. This implies weak identifiability for the fixed effects, and can therefore lead to biased estimators and poor inference for the regression coefficients and the spline estimator of the nonparametric function. We propose an adjustment using a postprocessing technique. We show that under mild conditions the posterior is proper under the proposed USP, a flat prior for the fixed effect parameters, and an improper prior for the residual variance. We illustrate the proposed approach using a longitudinal hormone dataset, and carry out extensive simulation studies to compare its finite sample performance with existing methods. [source] ## Smoothing Spline-Based Score Tests for Proportional Hazards Models BIOMETRICS, Issue 3 2006Jiang LinSummary We propose "score-type" tests for the proportional hazards assumption and for covariate effects in the Cox model using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate. The tests are based on the penalized partial likelihood and are derived by viewing the inverse of the smoothing parameter as a variance component and testing an equivalent null hypothesis that the variance component is zero. We show that the tests have a size close to the nominal level and good power against general alternatives, and we apply them to data from a cancer clinical trial. [source] ## Variable smoothing in Bayesian intrinsic autoregressions ENVIRONMETRICS, Issue 8 2007Mark J. BrewerAbstract We introduce an adapted form of the Markov random field (MRF) for Bayesian spatial smoothing with small-area data. This new scheme allows the amount of smoothing to vary in different parts of a map by employing area-specific smoothing parameters, related to the variance of the MRF. We take an empirical Bayes approach, using variance information from a standard MRF analysis to provide prior information for the smoothing parameters of the adapted MRF. The scheme is shown to produce proper posterior distributions for a broad class of models. We test our method on both simulated and real data sets, and for the simulated data sets, the new scheme is found to improve modelling of both slowly-varying levels of smoothness and discontinuities in the response surface. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Smoothing that does not blur: Effects of the anisotropic approach for evaluating diffusion tensor imaging data in the clinic JOURNAL OF MAGNETIC RESONANCE IMAGING, Issue 3 2010Marta Moraschi MSAbstract Purpose: To compare the effects of anisotropic and Gaussian smoothing on the outcomes of diffusion tensor imaging (DTI) voxel-based (VB) analyses in the clinic, in terms of signal-to-noise ratio (SNR) enhancement and directional information and boundary structures preservation. Materials and Methods: DTI data of 30 Alzheimer's disease (AD) patients and 30 matched control subjects were obtained at 3T. Fractional anisotropy (FA) maps with variable degrees and quality (Gaussian and anisotropic) of smoothing were created and compared with an unsmoothed dataset. The two smoothing approaches were evaluated in terms of SNR improvements, capability to separate differential effects between patients and controls by a standard VB analysis, and level of artifacts introduced by the preprocessing. Results: Gaussian smoothing regionally biased the FA values and introduced a high variability of results in clinical analysis, greatly dependent on the kernel size. On the contrary, anisotropic smoothing proved itself capable of enhancing the SNR of images and maintaining boundary structures, with only moderate dependence of results on smoothing parameters. Conclusion: Our study suggests that anisotropic smoothing is more suitable in DTI studies; however, regardless of technique, a moderate level of smoothing seems to be preferable considering the artifacts introduced by this manipulation. J. Magn. Reson. Imaging 2010;31:690,697. © 2010 Wiley-Liss, Inc. [source] ## Adaptive tests of regression functions via multiscale generalized likelihood ratios THE CANADIAN JOURNAL OF STATISTICS, Issue 2 2003Chunming M. ZhangAbstract Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P-value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data. Moult applications des tests non paramétriques basés sur l'estimation de courbes font intervenir un paramètre de lissage. L'auteure propose un test adaptatif qui allie plusieurs tests du rapport de vraisemblances généralisés et rivalise de puissance avec le meilleur d'entre eux. Elle détermine la loi asymptotique conjointe des tests individuels et celle du test global sous l'hypothèse nulle. Elle montre aussi comment sélectionner facilement les paramètres de lissage du test global et propose deux méthodes de calcul approché de son seuil. Elle examine en outre le comportement du test proposé par voie de simulations et en illustre l'emploi dans un cas concret [source] ## Varying Coefficient Model with Unknown Within-Subject Covariance for Analysis of Tumor Growth Curves BIOMETRICS, Issue 4 2008Robert T. KraftySummary In this article we develop a nonparametric estimation procedure for the varying coefficient model when the within-subject covariance is unknown. Extending the idea of iterative reweighted least squares to the functional setting, we iterate between estimating the coefficients conditional on the covariance and estimating the functional covariance conditional on the coefficients. Smoothing splines for correlated errors are used to estimate the functional coefficients with smoothing parameters selected via the generalized maximum likelihood. The covariance is nonparametrically estimated using a penalized estimator with smoothing parameters chosen via a Kullback,Leibler criterion. Empirical properties of the proposed method are demonstrated in simulations and the method is applied to the data collected from an ovarian tumor study in mice to analyze the effects of different chemotherapy treatments on the volumes of two classes of tumors. [source] ## Smooth Random Effects Distribution in a Linear Mixed Model BIOMETRICS, Issue 4 2004Wendimagegn GhideySummary A linear mixed model with a smooth random effects density is proposed. A similar approach to P -spline smoothing of Eilers and Marx (1996, Statistical Science11, 89,121) is applied to yield a more flexible estimate of the random effects density. Our approach differs from theirs in that the B -spline basis functions are replaced by approximating Gaussian densities. Fitting the model involves maximizing a penalized marginal likelihood. The best penalty parameters minimize Akaike's Information Criterion employing Gray's (1992, Journal of the American Statistical Association87, 942,951) results. Although our method is applicable to any dimensions of the random effects structure, in this article the two-dimensional case is explored. Our methodology is conceptually simple, and it is relatively easy to fit in practice and is applied to the cholesterol data first analyzed by Zhang and Davidian (2001, Biometrics57, 795,802). A simulation study shows that our approach yields almost unbiased estimates of the regression and the smoothing parameters in small sample settings. Consistency of the estimates is shown in a particular case. [source] ## Disease Mapping of Stage-Specific Cancer Incidence Data BIOMETRICS, Issue 3 2002Leonhard Knorr-HeldSummary. We propose two approaches for the spatial analysis of cancer incidence data with additional information on the stage of the disease at time of diagnosis. The two formulations are extensions of commonly used models for multicategorical response data on an ordinal scale. We include spatial and age-group effects in both formulations, which we estimate in a nonparametric smooth way. More specifically, we adopt a fully Bayesian approach based on Gaussian pairwise difference priors where additional smoothing parameters are treated as unknown as well. We argue that the methods are useful in monitoring the effectiveness of mass cancer screening and illustrate this through an application to data on cervical cancer in the former German Democratic Republic. The results suggest that there are large spatial differences in the stage proportions, which indicate spatial variability with respect to the introduction and effectiveness of Pap smear screening programs. [source] ## Estimation and Inference for a Spline-Enhanced Population Pharmacokinetic Model BIOMETRICS, Issue 3 2002Lang LiSummary. This article is motivated by an application where subjects were dosed three times with the same drug and the drug concentration profiles appeared to be the lowest after the third dose. One possible explanation is that the pharmacokinetic (PK) parameters vary over time. Therefore, we consider population PK models with time-varying PK parameters. These time-varying PK parameters are modeled by natural cubic spline functions in the ordinary differential equations. Mean parameters, variance components, and smoothing parameters are jointly estimated by maximizing the double penalized log likelihood. Mean functions and their derivatives are obtained by the numerical solution of ordinary differential equations. The interpretation of PK parameters in the model and its flexibility are discussed. The proposed methods are illustrated by application to the data that motivated this article. The model's performance is evaluated through simulation. [source] |