5,707 research outputs found
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
Spike-and-Slab Priors for Function Selection in Structured Additive Regression Models
Structured additive regression provides a general framework for complex
Gaussian and non-Gaussian regression models, with predictors comprising
arbitrary combinations of nonlinear functions and surfaces, spatial effects,
varying coefficients, random effects and further regression terms. The large
flexibility of structured additive regression makes function selection a
challenging and important task, aiming at (1) selecting the relevant
covariates, (2) choosing an appropriate and parsimonious representation of the
impact of covariates on the predictor and (3) determining the required
interactions. We propose a spike-and-slab prior structure for function
selection that allows to include or exclude single coefficients as well as
blocks of coefficients representing specific model terms. A novel
multiplicative parameter expansion is required to obtain good mixing and
convergence properties in a Markov chain Monte Carlo simulation approach and is
shown to induce desirable shrinkage properties. In simulation studies and with
(real) benchmark classification data, we investigate sensitivity to
hyperparameter settings and compare performance to competitors. The flexibility
and applicability of our approach are demonstrated in an additive piecewise
exponential model with time-varying effects for right-censored survival times
of intensive care patients with sepsis. Geoadditive and additive mixed logit
model applications are discussed in an extensive appendix
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
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