Resistant Nonparametric Smoothing with S-PLUS

In this paper we introduce and illustrate the use of an S-PLUS set of functions to fit M-type smoothing splines with the smoothing parameter chosen by a robust criterion (either a robust version of cross-validation or a robust version of Mallows's Cp ). The main reference is: Cantoni, E. and Ronchetti, E. (2001). Resistant selection of the smoothing parameter for smoothing splines. Statistics and Computing, 11, 141-146.

Resistant Nonparametric Smoothing with S-PLUS

In this paper we introduce and illustrate the use of an S-PLUS set of functions to fit M-type smoothing splines with the smoothing parameter chosen by a robust criterion (either a robust version of cross-validation or a robust version of Mallows's Cp ). The main reference is: Cantoni, E. and Ronchetti, E. (2001). Resistant selection of the smoothing parameter for smoothing splines. Statistics and Computing, 11, 141-146

S-estimation for penalized regression splines.

This paper is about S-estimation for penalized regression splines. Penalized regression splines are one of the currently most used methods for smoothing noisy data. The estimation method used for fitting such a penalized regression spline model is mostly based on least squares methods, which are known to be sensitive to outlying observations. In real world applications, outliers are quite commonly observed. There are several robust estimation methods taking outlying observations into account. We define and study S-estimators for penalized regression spline models. Hereby we replace the least squares estimation method for penalized regression splines by a suitable S-estimation method. By keeping the modeling by means of splines and by keeping the penalty term, though using S-estimators instead of least squares estimators, we arrive at an estimation method that is both robust and flexible enough to capture non-linear trends in the data. Simulated data and a real data example are used to illustrate the effectiveness of the procedure.M-estimator; Penalized least squares method; Penalized regression spline; S-estimator; Smoothing parameter;

Robust estimation of mean and dispersion functions in extended generalized additive models.

Generalized Linear Models are a widely used method to obtain parametric estimates for the mean function. They have been further extended to allow the relationship between the mean function and the covariates to be more flexible via Generalized Additive Models. However the fixed variance structure can in many cases be too restrictive. The Extended Quasi-Likelihood (EQL) framework allows for estimation of both the mean and the dispersion/variance as functions of covariates. As for other maximum likelihood methods though, EQL estimates are not resistant to outliers: we need methods to obtain robust estimates for both the mean and the dispersion function. In this paper we obtain functional estimates for the mean and the dispersion that are both robust and smooth. The performance of the proposed method is illustrated via a simulation study and some real data examples.Dispersion; Generalized additive modelling; Mean regression function; M-estimation; P-splines; Robust estimation;

Robust Estimation of Mean and Dispersion Functions in Extended Generalized Additive Models

Generalized Linear Models are a widely used method to obtain parametric es- timates for the mean function. They have been further extended to allow the re- lationship between the mean function and the covariates to be more flexible via Generalized Additive Models. However the fixed variance structure can in many cases be too restrictive. The Extended Quasi-Likelihood (EQL) framework allows for estimation of both the mean and the dispersion/variance as functions of covari- ates. As for other maximum likelihood methods though, EQL estimates are not resistant to outliers: we need methods to obtain robust estimates for both the mean and the dispersion function. In this paper we obtain functional estimates for the mean and the dispersion that are both robust and smooth. The performance of the proposed method is illustrated via a simulation study and some real data examples.dispersion;generalized additive modelling;mean regression function;quasilikelihood;M-estimation;P-splines;robust estimation

Generalized additive modelling with implicit variable selection by likelihood based boosting

The use of generalized additive models in statistical data analysis suffers from the restriction to few explanatory variables and the problems of selection of smoothing parameters. Generalized additive model boosting circumvents these problems by means of stagewise fitting of weak learners. A fitting procedure is derived which works for all simple exponential family distributions, including binomial, Poisson and normal response variables. The procedure combines the selection of variables and the determination of the appropriate amount of smoothing. As weak learners penalized regression splines and the newly introduced penalized stumps are considered. Estimates of standard deviations and stopping criteria which are notorious problems in iterative procedures are based on an approximate hat matrix. The method is shown to outperform common procedures for the fitting of generalized additive models. In particular in high dimensional settings it is the only method that works properly

Variable Selection in Additive Models by Nonnegative Garrote

We adapt Breiman's (1995) nonnegative garrote method to perform variable selection in nonparametric additive models. The technique avoids methods of testing for which no reliable distributional theory is available. In addition it removes the need for a full search of all possible models, something which is computationally intensive, especially when the number of variables is moderate to high. The method has the advantages of being conceptually simple and computationally fast. It provides accurate predictions and is effective at identifying the variables generating the model. For illustration, we consider both a study of Boston housing prices as well as two simulation settings. In all cases our methods perform as well or better than available alternatives like the Component Selection and Smoothing Operator (COSSO).cross-validation, nonnegative garrote, nonparametric regression, shrinkage methods, variable selection

Detecting and handling outlying trajectories in irregularly sampled functional datasets

Outlying curves often occur in functional or longitudinal datasets, and can be very influential on parameter estimators and very hard to detect visually. In this article we introduce estimators of the mean and the principal components that are resistant to, and then can be used for detection of, outlying sample trajectories. The estimators are based on reduced-rank t-models and are specifically aimed at sparse and irregularly sampled functional data. The outlier-resistance properties of the estimators and their relative efficiency for noncontaminated data are studied theoretically and by simulation. Applications to the analysis of Internet traffic data and glycated hemoglobin levels in diabetic children are presented.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS257 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model

Modeling viral dynamics in HIV/AIDS studies has resulted in a deep understanding of pathogenesis of HIV infection from which novel antiviral treatment guidance and strategies have been derived. Viral dynamics models based on nonlinear differential equations have been proposed and well developed over the past few decades. However, it is quite challenging to use experimental or clinical data to estimate the unknown parameters (both constant and time-varying parameters) in complex nonlinear differential equation models. Therefore, investigators usually fix some parameter values, from the literature or by experience, to obtain only parameter estimates of interest from clinical or experimental data. However, when such prior information is not available, it is desirable to determine all the parameter estimates from data. In this paper we intend to combine the newly developed approaches, a multi-stage smoothing-based (MSSB) method and the spline-enhanced nonlinear least squares (SNLS) approach, to estimate all HIV viral dynamic parameters in a nonlinear differential equation model. In particular, to the best of our knowledge, this is the first attempt to propose a comparatively thorough procedure, accounting for both efficiency and accuracy, to rigorously estimate all key kinetic parameters in a nonlinear differential equation model of HIV dynamics from clinical data. These parameters include the proliferation rate and death rate of uninfected HIV-targeted cells, the average number of virions produced by an infected cell, and the infection rate which is related to the antiviral treatment effect and is time-varying. To validate the estimation methods, we verified the identifiability of the HIV viral dynamic model and performed simulation studies.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS290 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org