22,339 research outputs found
Local Adaptive Grouped Regularization and its Oracle Properties for Varying Coefficient Regression
Varying coefficient regression is a flexible technique for modeling data
where the coefficients are functions of some effect-modifying parameter, often
time or location in a certain domain. While there are a number of methods for
variable selection in a varying coefficient regression model, the existing
methods are mostly for global selection, which includes or excludes each
covariate over the entire domain. Presented here is a new local adaptive
grouped regularization (LAGR) method for local variable selection in spatially
varying coefficient linear and generalized linear regression. LAGR selects the
covariates that are associated with the response at any point in space, and
simultaneously estimates the coefficients of those covariates by tailoring the
adaptive group Lasso toward a local regression model with locally linear
coefficient estimates. Oracle properties of the proposed method are established
under local linear regression and local generalized linear regression. The
finite sample properties of LAGR are assessed in a simulation study and for
illustration, the Boston housing price data set is analyzed.Comment: 30 pages, one technical appendix, two figure
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
Parametrization and penalties in spline models with an application to survival analysis
In this paper we show how a simple parametrization, built from the definition of cubic
splines, can aid in the implementation and interpretation of penalized spline models, whatever
configuration of knots we choose to use. We call this parametrization value-first derivative
parametrization. We perform Bayesian inference by exploring the natural link between quadratic
penalties and Gaussian priors. However, a full Bayesian analysis seems feasible only for some
penalty functionals. Alternatives include empirical Bayes methods involving model selection
type criteria. The proposed methodology is illustrated by an application to survival analysis
where the usual Cox model is extended to allow for time-varying regression coefficients
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
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