19 research outputs found
Adaptive estimation in circular functional linear models
We consider the problem of estimating the slope parameter in circular
functional linear regression, where scalar responses Y1,...,Yn are modeled in
dependence of 1-periodic, second order stationary random functions X1,...,Xn.
We consider an orthogonal series estimator of the slope function, by replacing
the first m theoretical coefficients of its development in the trigonometric
basis by adequate estimators. Wepropose a model selection procedure for m in a
set of admissible values, by defining a contrast function minimized by our
estimator and a theoretical penalty function; this first step assumes the
degree of ill posedness to be known. Then we generalize the procedure to a
random set of admissible m's and a random penalty function. The resulting
estimator is completely data driven and reaches automatically what is known to
be the optimal minimax rate of convergence, in term of a general weighted
L2-risk. This means that we provide adaptive estimators of both the slope
function and its derivatives
On the effect of noisy observations of the regressor in a functional linear model
We consider the estimation of the slope function in functional linear regression, where a scalar response Y is modeled in dependence of a random function X, when Y and only a panel Z1,…., ZL of noisy observations of X are observable. Assuming an iid. sample of (Y; Z1,…, ZL) we derive in terms of both, the sample size and the panel size, a lower bound of a maximal weighted risk over certain ellipsoids of slope functions. We prove that a thresholded projection estimator can attain the lower bound up to a constant
Structural tests in regression on functional variable
International audienc
Robust, Adaptive Functional Regression in Functional Mixed Model Framework
Functional data are increasingly encountered in scientific studies, and their high dimensionality and
complexity lead to many analytical challenges. Various methods for functional data analysis have been
developed, including functional response regression methods that involve regression of a functional response on
univariate/multivariate predictors with nonparametrically represented functional coefficients.
In existing methods, however, the functional regression can be sensitive to outlying curves and outlying regions
of curves, so is not robust. In this paper, we introduce a new Bayesian method, robust functional mixed models (R-FMM),
for performing robust functional regression within the general functional mixed model framework, which includes multiple continuous or categorical predictors and random effect functions accommodating potential between-function correlation
induced by the experimental design. The underlying model involves a hierarchical scale mixture model
for the fixed effects, random effect and residual error functions. These modeling assumptions across curves result in robust
nonparametric estimators of the fixed and random effect functions which down-weight outlying curves and
regions of curves, and produce statistics that can be used to flag global and local outliers. These assumptions also lead to distributions across wavelet coefficients that have outstanding sparsity and adaptive shrinkage properties, with
great flexibility for the data to determine the sparsity and the heaviness of the tails.
Together with the down-weighting of outliers, these within-curve properties lead to fixed and random effect
function estimates that appear in our simulations to be remarkably adaptive in their ability to remove spurious features yet retain true features of the functions. We have developed general code to implement this fully Bayesian method that is automatic, requiring the user to only provide the functional data and design matrices. It is efficient enough to handle large data sets, and yields posterior samples of all model parameters that can be used to perform desired Bayesian estimation and inference. Although we present details for a specific implementation of the R-FMM using
specific distributional choices in the hierarchical model, 1D functions, and wavelet transforms, the method can be
applied more generally using other heavy-tailed distributions, higher dimensional functions
(e.g. images), and using other invertible transformations as alternatives to wavelets
Functional projection pursuit regression
International audienc