Hervé Cardot

Functional linear model

In this paper, we study a regression model in which explanatory variables are sampling points of a continuous-time process. We propose an estimator of regression by means of a Functional Principal Component Analysis analogous to the one introduced by Bosq [(1991) NATO, ASI Series, pp. 509-529] in the case of Hilbertian AR processes. Both convergence in probability and almost sure convergence of this estimator are stated.

Autoregressive Forecasting of Some Functional Climatic Variations

Many variations such as the annual cycle in sea surface temperatures can be considered to be smooth functions and are appropriately described using methods from functional data analysis. This study defines a class of functional autoregressive (FAR) models which can be used as robust predictors for making forecasts of entire smooth functions in the future. The methods are illustrated and compared with pointwise predictors such as SARIMA by applying them to forecasting the entire annual cycle of climatological El Nino–Southern Oscillation (ENSO) time series one year ahead. Forecasts for the period 1987–1996 suggest that the FAR functional predictors show some promising skill, compared to traditional scalar SARIMA forecasts which perform poorly.

Testing Hypotheses in The Functional Linear Model

The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of ℝ and the response is scalar. The response is modelled as Y=Ψ(X)+ɛ, where Ψ is some linear continuous operator defined on the space of square integrable functions and valued in ℝ. The random input X is independent from the noise ɛ. In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of Ψ restricted to the Hilbert space generated by the random variable X. We introduce two test statistics based on the norm of the empirical cross‐covariance operator of (X,Y). The first test statistic relies on a χ2 approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X. The test procedures can be applied to check a given relationship between X and Y. The method is illustrated through a simulation study.

Nonparametric estimation of smoothed principal components analysis of sampled noisy functions

This study deals with the simultaneous nonparametric estimations of n curves or observations of a random process corrupted by noise in which sample paths belong to a finite dimension functional subspace. The estimation, by means of B-splines, leads to a new kind of functional principal components analysis. Asymptotic rates of convergence are given for the mean and the eigenelements of the empirical covariance operator. Heuristic arguments show that a well chosen smoothing parameter may improve the estimation of the subspace which contains the sample path of the process. Finally, simulations suggest that the estimation method studied here is advantageous when there are a small number of design points.

Smoothing splines estimators in functional linear regression with errors-in-variables

The total least squares method is generalized in the context of the functional linear model. A smoothing splines estimator of the functional coefficient of the model is first proposed without noise in the covariates and an asymptotic result for this estimator is obtained. Then, this estimator is adapted to the case where the covariates are noisy and an upper bound for the convergence speed is also derived. The estimation procedure is evaluated by means of simulations.

Simultaneous non-parametric regressions of unbalanced longitudinal data

Abstract The aim of this paper is to simultaneously estimate n curves corrupted by noise, this means several observations of a random process. The non-parametric estimation of the sampled paths leads to a new kind of functional principal components analysis which simultaneously takes into account a dimensionality and a smoothness constraint. Furthermore, the use of B-spline approximation to estimate the curves allows the study of unbalanced longitudinal data. The relationship between the choice of the smoothing parameter and that of dimensionality is discussed. A simulation study shows good behaviors of this proposed estimate compared to n independent smoothing splines under generalized cross-validation. Finally, the methodology of this paper is illustrated by its application to a real world data set.

Quantile regression when the covariates are functions

This article deals with a linear model of regression on quantiles when the explanatory variable takes values in some functional space and the response is scalar. We propose a spline estimator of the functional coefficient that minimizes a penalized L 1 type criterion. Then, we study the asymptotic behavior of this estimator. The penalization is of primary importance to get existence and convergence.

Thresholding projection estimators in functional linear models

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows us to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits us to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove that these estimators are minimax and rates of convergence are given for some particular cases.

Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm

With the progress of measurement apparatus and the development of automatic sensors it is not unusual anymore to get thousands of samples of observations taking values in high dimension spaces such as functional spaces. In such large samples of high dimensional data, outlying curves may not be uncommon and even a few individuals may corrupt simple statistical indicators such as the mean trajectory. We focus here on the estimation of the geometric median which is a direct generalization of the real median and has nice robustness properties. The geometric median being defined as the minimizer of a simple convex functional that is differentiable everywhere when the distribution has no atoms, it is possible to estimate it with online gradient algorithms. Such algorithms are very fast and can deal with large samples. Furthermore they also can be simply updated when the data arrive sequentially. We state the almost sure consistency and the L2 rates of convergence of the stochastic gradient estimator as well as the asymptotic normality of its averaged version. We get that the asymptotic distribution of the averaged version of the algorithm is the same as the classic estimators which are based on the minimization of the empirical loss function. The performances of our averaged sequential estimator, both in terms of computation speed and accuracy of the estimations, are evaluated with a small simulation study. Our approach is also illustrated on a sample of more 5000 individual television audiences measured every second over a period of 24 hours.

Testing for No Effect in Functional Linear Regression Models, Some Computational Approaches

Abstract The functional linear regression model is a regression model where the link between the response (a scalar) and the predictor (a random function) is expressed as an inner product between a functional coefficient and the predictor. Our aim is to test at first for no effect of the model, i.e., the nullity of the functional coefficient. A fully automatic permutation test based on the cross covariance operator of the predictor and the response is proposed. The model can be, in an obvious way, extended to the case of several functional predictors. When testing for no effect of some covariate on the response the permutation test is no longer valid. An alternative pseudo-likelihood ratio test statistic is then introduced. The procedure can be applied in some way to test partial nullity of a functional coefficient. All procedures are illustrated and compared by means of simulation studies.