Advances on nonparametric regression for functional variables

We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function $f$, consistent estimators of the mean embedding of a random variable $X$ lead to consistent estimators of the mean embedding of $f(X)$. For Mat\'ern kernels and sufficiently smooth functions we also provide rates of convergence. Our results extend to functions of multiple random variables. If the variables a...

Situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis. Predictors are often appropriately modeled as square integrable smooth random functions. Imposing minimal assumptions on the nature of the functional relationship, we aim to estimate the directional derivatives and gradients of the response with respect to the predictor functions. In statistical applications and data analysis, functional derivatives provide a quan...

When predicting scalar responses in the situation where the explanatory variables are functions, it is sometimes the case that some functional variables are related to responses linearly while other variables have more complicated relationships with the responses. In this paper, we propose a new semi-parametric model to take advantage of both parametric and nonparametric functional modeling. Asymptotic properties of the proposed estimators are established and finite sample be...

We present and describe the GPFDA package for R. The package provides flexible functionalities for dealing with Gaussian process regression (GPR) models for functional data. Multivariate functional data, functional data with multidimensional inputs, and nonseparable and/or nonstationary covariance structures can be modeled. In addition, the package fits functional regression models where the mean function depends on scalar and/or functional covariates and the covariance struc...

An extension of reproducing kernel Hilbert space (RKHS) theory provides a new framework for modeling functional regression models with functional responses. The approach only presumes a general nonlinear regression structure as opposed to previously studied linear regression models. Generalized cross-validation (GCV) is proposed for automatic smoothing parameter estimation. The new RKHS estimate is applied to both simulated and real data as illustrations.

In this paper we present a nonparametric method for extending functional regression methodology to the situation where more than one functional covariate is used to predict a functional response. Borrowing the idea from Kadri et al. (2010a), the method, which support mixed discrete and continuous explanatory variables, is based on estimating a function-valued function in reproducing kernel Hilbert spaces by virtue of positive operator-valued kernels.

This paper concerns the estimation of the regression function at a given point in nonparametric heteroscedastic models with Gaussian noise or with noise having unknown distribution. In the two cases an asymptotically efficient kernel estimator is constructed for the minimax absolute error risk.

New local linear estimators are proposed for a wide class of nonparametric regression models. The estimators are uniformly consistent regardless of satisfying traditional conditions of depen\-dence of design elements. The estimators are the solutions of a specially weighted least-squares method. The design can be fixed or random and does not need to meet classical regularity or independence conditions. As an application, several estimators are constructed for the mean of de...

Recent technological developments have enabled us to collect complex and high-dimensional data in many scientific fields, such as population health, meteorology, econometrics, geology, and psychology. It is common to encounter such datasets collected repeatedly over a continuum. Functional data, whose sample elements are functions in the graphical forms of curves, images, and shapes, characterize these data types. Functional data analysis techniques reduce the complex structu...

Suppose that $n$ statistical units are observed, each following the model $Y(x_j)=m(x_j)+ \epsilon(x_j),\, j=1,...,N,$ where $m$ is a regression function, $0 \leq x_1 <...<x_N \leq 1$ are observation times spaced according to a sampling density $f$, and $\epsilon$ is a continuous-time error process having mean zero and regular covariance function. Considering the local polynomial estimation of $m$ and its derivatives, we derive asymptotic expressions for the bias and variance...