14,030 research outputs found
A General Family of Penalties for Combining Differing Types of Penalties in Generalized Structured Models
Penalized estimation has become an established tool for regularization and model selection in regression models.
A variety of penalties with specific features are available
and effective algorithms for specific penalties have been proposed.
But not much is available to fit models that call for a combination of different penalties.
When modeling rent data, which will be considered as an example, various types of predictors call for a combination of a Ridge, a grouped Lasso and a Lasso-type penalty within one model.
Algorithms that can deal with such problems, are in demand.
We propose to approximate penalties that are (semi-)norms of scalar linear transformations of the coefficient vector in generalized structured models.
The penalty is very general such that the Lasso, the fused Lasso, the Ridge, the smoothly clipped absolute deviation penalty (SCAD), the elastic net and many more penalties are embedded.
The approximation allows to combine all these penalties within one model.
The computation is based on conventional penalized iteratively re-weighted least squares (PIRLS) algorithms and hence, easy to implement.
Moreover, new penalties can be incorporated quickly.
The approach is also extended to penalties with vector based arguments; that is, to penalties with norms of linear transformations of the coefficient vector.
Some illustrative examples and the model for the Munich rent data show promising results
Penalized Likelihood and Bayesian Function Selection in Regression Models
Challenging research in various fields has driven a wide range of
methodological advances in variable selection for regression models with
high-dimensional predictors. In comparison, selection of nonlinear functions in
models with additive predictors has been considered only more recently. Several
competing suggestions have been developed at about the same time and often do
not refer to each other. This article provides a state-of-the-art review on
function selection, focusing on penalized likelihood and Bayesian concepts,
relating various approaches to each other in a unified framework. In an
empirical comparison, also including boosting, we evaluate several methods
through applications to simulated and real data, thereby providing some
guidance on their performance in practice
Learning the Structure for Structured Sparsity
Structured sparsity has recently emerged in statistics, machine learning and
signal processing as a promising paradigm for learning in high-dimensional
settings. All existing methods for learning under the assumption of structured
sparsity rely on prior knowledge on how to weight (or how to penalize)
individual subsets of variables during the subset selection process, which is
not available in general. Inferring group weights from data is a key open
research problem in structured sparsity.In this paper, we propose a Bayesian
approach to the problem of group weight learning. We model the group weights as
hyperparameters of heavy-tailed priors on groups of variables and derive an
approximate inference scheme to infer these hyperparameters. We empirically
show that we are able to recover the model hyperparameters when the data are
generated from the model, and we demonstrate the utility of learning weights in
synthetic and real denoising problems
Structured penalties for functional linear models---partially empirical eigenvectors for regression
One of the challenges with functional data is incorporating spatial
structure, or local correlation, into the analysis. This structure is inherent
in the output from an increasing number of biomedical technologies, and a
functional linear model is often used to estimate the relationship between the
predictor functions and scalar responses. Common approaches to the ill-posed
problem of estimating a coefficient function typically involve two stages:
regularization and estimation. Regularization is usually done via dimension
reduction, projecting onto a predefined span of basis functions or a reduced
set of eigenvectors (principal components). In contrast, we present a unified
approach that directly incorporates spatial structure into the estimation
process by exploiting the joint eigenproperties of the predictors and a linear
penalty operator. In this sense, the components in the regression are
`partially empirical' and the framework is provided by the generalized singular
value decomposition (GSVD). The GSVD clarifies the penalized estimation process
and informs the choice of penalty by making explicit the joint influence of the
penalty and predictors on the bias, variance, and performance of the estimated
coefficient function. Laboratory spectroscopy data and simulations are used to
illustrate the concepts.Comment: 29 pages, 3 figures, 5 tables; typo/notational errors edited and
intro revised per journal review proces
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
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
A Framework for Unbiased Model Selection Based on Boosting
Variable selection and model choice are of major concern in many statistical applications, especially in high-dimensional regression models. Boosting is a convenient statistical method that combines model fitting with intrinsic model selection.
We investigate the impact of base-learner specification on the performance of boosting as a model selection procedure.
We show that variable selection may be biased if the covariates are of different nature.
Important examples are models combining continuous and categorical covariates, especially if the number of categories is large. In this case, least squares base-learners offer increased flexibility for the categorical covariate and lead to a preference even if the categorical covariate is non-informative.
Similar difficulties arise when comparing linear and nonlinear base-learners for a continuous covariate. The additional flexibility in the nonlinear base-learner again yields a preference of the more complex modeling alternative.
We investigate these problems from a theoretical perspective and suggest a framework for unbiased model selection based on a general class of penalized least squares base-learners.
Making all base-learners comparable in terms of their degrees of freedom strongly reduces the selection bias observed in naive boosting specifications. The importance of unbiased model selection is demonstrated in simulations and an application to forest health models
Bayesian Regularisation in Structured Additive Regression Models for Survival Data
During recent years, penalized likelihood approaches have attracted a lot of interest both in the area of semiparametric regression and for the regularization of high-dimensional regression models. In this paper, we introduce a Bayesian formulation that allows to combine both aspects into a joint regression model with a focus on hazard regression for survival times. While Bayesian penalized splines form the basis for estimating nonparametric and flexible time-varying effects, regularization of high-dimensional covariate vectors is based on scale mixture of normals priors. This class of priors allows to keep a (conditional) Gaussian prior for regression coefficients on the predictor stage of the model but introduces suitable mixture distributions for the Gaussian variance to achieve regularization. This scale mixture property allows to device general and adaptive Markov chain Monte Carlo simulation algorithms for fitting a variety of hazard regression models. In particular, unifying algorithms based on iteratively weighted least squares proposals can be employed both for regularization and penalized semiparametric function estimation. Since sampling based estimates do no longer have the variable selection property well-known for the Lasso in frequentist analyses, we additionally consider spike and slab priors that introduce a further mixing stage that allows to separate between influential and redundant parameters. We demonstrate the different shrinkage properties with three simulation settings and apply the methods to the PBC Liver dataset
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