194,313 research outputs found
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
Boosted Beta regression.
Regression analysis with a bounded outcome is a common problem in applied statistics. Typical examples include regression models for percentage outcomes and the analysis of ratings that are measured on a bounded scale. In this paper, we consider beta regression, which is a generalization of logit models to situations where the response is continuous on the interval (0,1). Consequently, beta regression is a convenient tool for analyzing percentage responses. The classical approach to fit a beta regression model is to use maximum likelihood estimation with subsequent AIC-based variable selection. As an alternative to this established - yet unstable - approach, we propose a new estimation technique called boosted beta regression. With boosted beta regression estimation and variable selection can be carried out simultaneously in a highly efficient way. Additionally, both the mean and the variance of a percentage response can be modeled using flexible nonlinear covariate effects. As a consequence, the new method accounts for common problems such as overdispersion and non-binomial variance structures
Functional linear regression analysis for longitudinal data
We propose nonparametric methods for functional linear regression which are
designed for sparse longitudinal data, where both the predictor and response
are functions of a covariate such as time. Predictor and response processes
have smooth random trajectories, and the data consist of a small number of
noisy repeated measurements made at irregular times for a sample of subjects.
In longitudinal studies, the number of repeated measurements per subject is
often small and may be modeled as a discrete random number and, accordingly,
only a finite and asymptotically nonincreasing number of measurements are
available for each subject or experimental unit. We propose a functional
regression approach for this situation, using functional principal component
analysis, where we estimate the functional principal component scores through
conditional expectations. This allows the prediction of an unobserved response
trajectory from sparse measurements of a predictor trajectory. The resulting
technique is flexible and allows for different patterns regarding the timing of
the measurements obtained for predictor and response trajectories. Asymptotic
properties for a sample of subjects are investigated under mild conditions,
as , and we obtain consistent estimation for the regression
function. Besides convergence results for the components of functional linear
regression, such as the regression parameter function, we construct asymptotic
pointwise confidence bands for the predicted trajectories. A functional
coefficient of determination as a measure of the variance explained by the
functional regression model is introduced, extending the standard to the
functional case. The proposed methods are illustrated with a simulation study,
longitudinal primary biliary liver cirrhosis data and an analysis of the
longitudinal relationship between blood pressure and body mass index.Comment: Published at http://dx.doi.org/10.1214/009053605000000660 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Varying-coefficient functional linear regression
Functional linear regression analysis aims to model regression relations
which include a functional predictor. The analog of the regression parameter
vector or matrix in conventional multivariate or multiple-response linear
regression models is a regression parameter function in one or two arguments.
If, in addition, one has scalar predictors, as is often the case in
applications to longitudinal studies, the question arises how to incorporate
these into a functional regression model. We study a varying-coefficient
approach where the scalar covariates are modeled as additional arguments of the
regression parameter function. This extension of the functional linear
regression model is analogous to the extension of conventional linear
regression models to varying-coefficient models and shares its advantages, such
as increased flexibility; however, the details of this extension are more
challenging in the functional case. Our methodology combines smoothing methods
with regularization by truncation at a finite number of functional principal
components. A practical version is developed and is shown to perform better
than functional linear regression for longitudinal data. We investigate the
asymptotic properties of varying-coefficient functional linear regression and
establish consistency properties.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ231 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Regression with Distance Matrices
Data types that lie in metric spaces but not in vector spaces are difficult
to use within the usual regression setting, either as the response and/or a
predictor. We represent the information in these variables using distance
matrices which requires only the specification of a distance function. A
low-dimensional representation of such distance matrices can be obtained using
methods such as multidimensional scaling. Once these variables have been
represented as scores, an internal model linking the predictors and the
response can be developed using standard methods. We call scoring the
transformation from a new observation to a score while backscoring is a method
to represent a score as an observation in the data space. Both methods are
essential for prediction and explanation. We illustrate the methodology for
shape data, unregistered curve data and correlation matrices using motion
capture data from an experiment to study the motion of children with cleft lip.Comment: 18 pages, 7 figure
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