Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Flexible Modelling of Discrete Failure Time Including Time-Varying Smooth Effects

Discrete survival models have been extended in several ways. More flexible models are obtained by including time-varying coefficients and covariates which determine the hazard rate in an additive but not further specified form. In this paper a general model is considered which comprises both types of covariate effects. An additional extension is the incorporation of smooth interaction between time and covariates. Thus in the linear predictor smooth effects of covariates which may vary across time are allowed. It is shown how simple duration models produce artefacts which may be avoided by flexible models. For the general model which includes parametric terms, time-varying coefficients in parametric terms and time-varying smooth effects estimation procedures are derived which are based on the regularized expansion of smooth effects in basis functions

A Unified Framework of Constrained Regression

Generalized additive models (GAMs) play an important role in modeling and understanding complex relationships in modern applied statistics. They allow for flexible, data-driven estimation of covariate effects. Yet researchers often have a priori knowledge of certain effects, which might be monotonic or periodic (cyclic) or should fulfill boundary conditions. We propose a unified framework to incorporate these constraints for both univariate and bivariate effect estimates and for varying coefficients. As the framework is based on component-wise boosting methods, variables can be selected intrinsically, and effects can be estimated for a wide range of different distributional assumptions. Bootstrap confidence intervals for the effect estimates are derived to assess the models. We present three case studies from environmental sciences to illustrate the proposed seamless modeling framework. All discussed constrained effect estimates are implemented in the comprehensive R package mboost for model-based boosting.Comment: This is a preliminary version of the manuscript. The final publication is available at http://link.springer.com/article/10.1007/s11222-014-9520-

Variable Selection and Model Choice in Geoadditive Regression Models

Model choice and variable selection are issues of major concern in practical regression analyses. We propose a boosting procedure that facilitates both tasks in a class of complex geoadditive regression models comprising spatial effects, nonparametric effects of continuous covariates, interaction surfaces, random effects, and varying coefficient terms. The major modelling component are penalized splines and their bivariate tensor product extensions. All smooth model terms are represented as the sum of a parametric component and a remaining smooth component with one degree of freedom to obtain a fair comparison between all model terms. A generic representation of the geoadditive model allows to devise a general boosting algorithm that implements automatic model choice and variable selection. We demonstrate the versatility of our approach with two examples: a geoadditive Poisson regression model for species counts in habitat suitability analyses and a geoadditive logit model for the analysis of forest health

Generalized structured additive regression based on Bayesian P-splines

Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. For the first time, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX

Inference of time-varying regression models

We consider parameter estimation, hypothesis testing and variable selection for partially time-varying coefficient models. Our asymptotic theory has the useful feature that it can allow dependent, nonstationary error and covariate processes. With a two-stage method, the parametric component can be estimated with a $n^{1/2}$-convergence rate. A simulation-assisted hypothesis testing procedure is proposed for testing significance and parameter constancy. We further propose an information criterion that can consistently select the true set of significant predictors. Our method is applied to autoregressive models with time-varying coefficients. Simulation results and a real data application are provided.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1010 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org