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49
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.
Selecting the Number of Knots For Penalized Splines
, 2000
"... Penalized splines, or Psplines, are regression splines fit by leastsquares with a roughness penaly. Psplines have much in common with smoothing splines, but the type of penalty used with a Pspline is somewhat more general than for a smoothing spline. Also, the number and location of the knots ..."
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Cited by 101 (10 self)
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Penalized splines, or Psplines, are regression splines fit by leastsquares with a roughness penaly. Psplines have much in common with smoothing splines, but the type of penalty used with a Pspline is somewhat more general than for a smoothing spline. Also, the number and location of the knots of a Pspline is not fixed as with a smoothing spline. Generally, the knots of a Pspline are at fixed quantiles of the independent variable and the only tuning parameter to choose is the number of knots. In this article, the effects of the number of knots on the performance of Psplines are studied. Two algorithms are proposed for the automatic selection of the number of knots. The myoptic algorithm stops when no improvement in the generalized cross validation statistic (GCV) is noticed with the last increase in the number of knots. The full search examines all candidates in a fixed sequence of possible numbers of knots and chooses the candidate that minimizes GCV. The myoptic algo...
Logspline density estimation for censored data
 Journal of Computational and Graphical Statistics
, 1992
"... Logspline density estimation is developed for data that may be right censored, left censored or interval censored. In solving the maximum likelihood equations, the NewtonRaphson method is augmented by occasional searches in the direction of steepest ascent. A fully automatic method, which may invo ..."
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Cited by 59 (17 self)
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Logspline density estimation is developed for data that may be right censored, left censored or interval censored. In solving the maximum likelihood equations, the NewtonRaphson method is augmented by occasional searches in the direction of steepest ascent. A fully automatic method, which may involve stepwise knot deletion and either AIC or BIC, is used to select the fin~l model. Also, a user interface based on S is described for obtaining estimates of the density function, distribution function and quantile function and for generating a random sample from the fitted distribution. AMS 1991 subject classifications. Primary 62G07; secondary 65D07. Key words and phrases. Polynomial splines, maximum likelihood, stepwise knot deletion, AIC, BIC, S, user interface. *This res<~ar(:h 1 Introduction. Consider data thought of as arising as a random sample from a on an open interval having an unknown density function. In the logspline method of density
Spatiallyadaptive penalties for spline fitting
 Australian and New Zealand Journal of Statistics
, 2000
"... We study spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. Our estimates are pth degree piecewise polynomials with p − 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty ..."
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Cited by 52 (7 self)
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We study spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. Our estimates are pth degree piecewise polynomials with p − 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the pth derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize GCV. This locallyadaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knotselection techniques for leastsquares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions,
Smoothing with Mixed Model Software
 Journal of Statistical Software
, 2004
"... Smoothing with mixed model software Smoothing methods that use basis functions with penalization can be formulated as fits in a mixed model framework. One of the major benefits is that software for mixed model analysis can be used for smoothing. We illustrate this for several smoothing models such a ..."
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Cited by 35 (2 self)
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Smoothing with mixed model software Smoothing methods that use basis functions with penalization can be formulated as fits in a mixed model framework. One of the major benefits is that software for mixed model analysis can be used for smoothing. We illustrate this for several smoothing models such as additive and varying coefficient models for both SPLUS and SAS software. Code for each of the illustrations is available on the Internet.
Density and Hazard Rate Estimation for Right Censored Data Using Wavelet Methods
, 1997
"... This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadi ..."
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Cited by 32 (4 self)
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This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators possess pointwise and global mean square consistency, obtain the best possible asymptotic MISE convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver...
Probabilistic CurveAligned Clustering and Prediction with Regression Mixture Models
 Ph.D. Dissertation, 2004. Laboratoire MAS
, 2004
"... in quality ..."
Spline adaptation in extended linear models
 Statistical Science
, 2002
"... Abstract. In many statistical applications, nonparametric modeling can provide insight into the features of a dataset that are not obtainable by other means. One successful approach involves the use of (univariate or multivariate) spline spaces. As a class, these methods have inherited much from cla ..."
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Cited by 19 (2 self)
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Abstract. In many statistical applications, nonparametric modeling can provide insight into the features of a dataset that are not obtainable by other means. One successful approach involves the use of (univariate or multivariate) spline spaces. As a class, these methods have inherited much from classical tools for parametric modeling. For example, stepwise variable selection with spline basis terms is a simple scheme for locating knots (breakpoints) in regions where the data exhibit strong, local features. Similarly, candidate knot con gurations (generated by this or some other search technique), are routinely evaluated with traditional selection criteria like AIC or BIC. In short, strategies typically applied in parametric model selection have proved useful in constructing exible, lowdimensional models for nonparametric problems. Until recently, greedy, stepwise procedures were most frequently suggested in the literature. Researchinto Bayesian variable selection, however, has given rise to a number of new splinebased methods that primarily rely on some form of Markov chain Monte Carlo to identify promising knot locations. In this paper, we consider various alternatives to greedy, deterministic schemes, and present aBayesian framework for studying adaptation in the context of an extended linear model (ELM). Our major test cases are Logspline density estimation and (bivariate) Triogram regression models. We selected these because they illustrate a number of computational and methodological issues concerning model adaptation that arise in ELMs.
Mixed modelbased hazard estimation
 Journal of Computational and Graphical Statistics
, 2002
"... We propose a new method for estimation of the hazard function from a set of censored failure time data, with a view to extending the general approach to more complicated models. The approach is based on a mixed model representation of penalized spline hazard estimators. One payoff is the automation ..."
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Cited by 16 (1 self)
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We propose a new method for estimation of the hazard function from a set of censored failure time data, with a view to extending the general approach to more complicated models. The approach is based on a mixed model representation of penalized spline hazard estimators. One payoff is the automation of the smoothing parameter choice through restricted maximum likelihood. Another is the option to use standard mixed model software for automatic hazard estimation. Key words: Nonparametric regression; Restricted maximum likelihood; Variance component; Survival analysis.
Penalized Likelihood Density Estimation: Direct CrossValidation and Scalable Approximation
 Statistica Sinica
, 2003
"... For smoothing parameter selection in penalized likelihood density estimation, a direct crossvalidation strategy is developed and its empirical performance explored. The strategy is as eective as the indirect crossvalidation developed earlier, but is much easier to implement in multivariate setti ..."
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Cited by 13 (7 self)
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For smoothing parameter selection in penalized likelihood density estimation, a direct crossvalidation strategy is developed and its empirical performance explored. The strategy is as eective as the indirect crossvalidation developed earlier, but is much easier to implement in multivariate settings. Also studied is the practical implementation of certain lowdimensional approximations of the estimate, with the dimension of the model space selected to achieve both asymptotic eciency and numerical scalability. The greatly reduced computational burden allows the routine use of the technique for the analysis of large data sets. Related practical issues concerning multivariate numerical integration are also briey addressed.