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252
Flexible smoothing with Bsplines and penalties
 STATISTICAL SCIENCE
, 1996
"... Bsplines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots ..."
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Cited by 396 (6 self)
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Bsplines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent Bsplines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of Bsplines, their construction, and penalized likelihood is presented. We discuss properties of penalized Bsplines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.
Wavelet Thresholding via a Bayesian Approach
 J. R. STATIST. SOC. B
, 1996
"... We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. ..."
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Cited by 259 (32 self)
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We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might b...
Prediction With Gaussian Processes: From Linear Regression To Linear Prediction And Beyond
 Learning and Inference in Graphical Models
, 1997
"... The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. Th ..."
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Cited by 231 (4 self)
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The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. This leads in to a more general discussion of Gaussian processes in section 4. Section 5 deals with further issues, including hierarchical modelling and the setting of the parameters that control the Gaussian process, the covariance functions for neural network models and the use of Gaussian processes in classification problems. PREDICTION WITH GAUSSIAN PROCESSES: FROM LINEAR REGRESSION TO LINEAR PREDICTION AND BEYOND 2 1 Introduction In the last decade neural networks have been used to tackle regression and classification problems, with some notable successes. It has also been widely recognized that they form a part of a wide variety of nonlinear statistical techniques that can be used for...
A multivariate technique for multiply imputing missing values using a sequence of regression models. Survey Methodology 27
, 2001
"... This article describes and evaluates a procedure for imputing missing values for a relatively complex data structure when the data are missing at random. The imputations are obtained by fitting a sequence of regression models and drawing values from the corresponding predictive distributions. The ty ..."
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Cited by 169 (8 self)
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This article describes and evaluates a procedure for imputing missing values for a relatively complex data structure when the data are missing at random. The imputations are obtained by fitting a sequence of regression models and drawing values from the corresponding predictive distributions. The types of regression models used are linear, logistic, Poisson, generalized logit or a mixture of these depending on the type of variable being imputed. Two additional common features in the imputation process are incorporated: restriction to a relevant subpopulation for some variables and logical bounds or constraints for the imputed values. The restrictions involve subsetting the sample individuals that satisfy certain criteria while fitting the regression models. The bounds involve drawing values from a truncated predictive distribution. The development of this method was partly motivated by the analysis of two data sets which are used as illustrations. The sequential regression procedure is applied to perform multiple imputation analysis for the two applied problems. The sampling properties of inferences from multiply imputed data sets created using the sequential regression method are evaluated through simulated data sets. Key Words: Item nonresponse ; Missing at random ; Multiple imputation ; Nonignorable missing mechanism; Regression ; Sampling properties and simulations.
A unifying view of sparse approximate Gaussian process regression
 Journal of Machine Learning Research
, 2005
"... We provide a new unifying view, including all existing proper probabilistic sparse approximations for Gaussian process regression. Our approach relies on expressing the effective prior which the methods are using. This allows new insights to be gained, and highlights the relationship between existin ..."
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Cited by 155 (6 self)
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We provide a new unifying view, including all existing proper probabilistic sparse approximations for Gaussian process regression. Our approach relies on expressing the effective prior which the methods are using. This allows new insights to be gained, and highlights the relationship between existing methods. It also allows for a clear theoretically justified ranking of the closeness of the known approximations to the corresponding full GPs. Finally we point directly to designs of new better sparse approximations, combining the best of the existing strategies, within attractive computational constraints.
Smoothing Spline Models for the Analysis of Nested and Crossed Samples of Curves
 Journal of the American Statistical Association
, 1998
"... We introduce a class of models for an additive decomposition of groups of curves strati ed by crossed and nested factors, generalizing smoothing splines to such samples by associating them with a corresponding mixed e ects model. The models are also useful for imputation of missing data and explorat ..."
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Cited by 126 (1 self)
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We introduce a class of models for an additive decomposition of groups of curves strati ed by crossed and nested factors, generalizing smoothing splines to such samples by associating them with a corresponding mixed e ects model. The models are also useful for imputation of missing data and exploratory analysis of variance. We prove that the best linear unbiased predictors (BLUP) from the extended mixed e ects 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 e cient 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 e ects models. Our methods are applied to menstrual cycle data from studies of reproductive function that measure daily urinary progesterone; the sample of progesterone curves is strati ed by cycles nested within subjects nested within conceptive and nonconceptive groups.
Regularization of Wavelets Approximations
, 1999
"... this paper, weintroduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hardthresholding and s ..."
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Cited by 122 (16 self)
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this paper, weintroduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hardthresholding and softthresholding estimators of Donoho and Johnstone (1994) are specic members of nonlinear regularized wavelet estimators. They correspond to the lower and upper bound of a class of the penalized leastsquares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established, and are shown to be adaptively minimax. To eciently solve penalized leastsquares problems, Nonlinear Regularized Sobolev Interpolators (NRSI) are proposed as initial estimators, which are shown to have good sampling properties. The NRSI is further ameliorated by Regularized OneStep Estimators (ROSE), which are the onestep estimators of the penalized leastsquares problems using the NRSI as initial estimators. Two other approaches, the graduated nonconvexity algorithm and wavelet networks, are also introduced to handle penalized leastsquares problems. The newly introduced approaches are also illustrated by a few numerical examples. ####### ########## ## ########## ########### ## ############# ## ####### ######################### ##### ######## ##### ## ####### ######## ### ## ########## ########## ## ########### ########## ## ########### ### ######## ## ########## ### ### ####### ########## ## #### ##### ##### ########### ######### ######### ## ###...
Infinite Mixtures of Gaussian Process Experts
 In Advances in Neural Information Processing Systems 14
, 2001
"... We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using an inputdependent adaptation of the Dirichlet Process, we implement a gating network for an infinite number of Experts. Inference in this model may be do ..."
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Cited by 110 (6 self)
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We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using an inputdependent adaptation of the Dirichlet Process, we implement a gating network for an infinite number of Experts. Inference in this model may be done efficiently using a Markov Chain relying on Gibbs sampling. The model allows the effective covariance function to vary with the inputs, and may handle large datasets  thus potentially overcoming two of the biggest hurdles with GP models. Simulations show the viability of this approach.