Laurent Delsol

Advances on asymptotic normality in non-parametric functional time series analysis

We consider a stationary process and wish to predict future values from previous ones. Instead of considering the process in its discretized form, we choose to see it as a sample of dependent curves. Then, we cut the process into N successive curves. Obviously, the N curves are not independent. The prediction issue can be translated into a non-parametric functional regression problem from dependent functional variables. This paper aims to revisit and complete two recent works on this topic. This article extends recent literature and provides asymptotic law with explicit constants under α-mixing assumptions. Then we establish pointwise confidence bands for the regression function. To conclude, we present how our results behave on a simulation and on a real time series.

Structural test in regression on functional variables

Many papers deal with structural testing procedures in multivariate regression. More recently, various estimators have been proposed for regression models involving functional explanatory variables. Thanks to these new estimators, we propose a theoretical framework for structural testing procedures adapted to functional regression. The procedures introduced in this paper are innovative and make the link between former works on functional regression and others on structural testing procedures in multivariate regression. We prove asymptotic properties of the level and the power of our procedures under general assumptions that cover a large scope of possible applications: tests for no effect, linearity, dimension reduction, ...

Using Bagidis in nonparametric functional data analysis: Predicting from curves with sharp local features

Our goal is to predict a scalar value or a group membership from the discretized observation of curves with sharp local features that might vary both vertically and horizontally. To this aim, we propose to combine the use of the nonparametric functional regression estimator developed by Ferraty and Vieu (2006) [18] with the Bagidis semimetric developed by Timmermans and von Sachs (submitted for publication) [36] with a view of efficiently measuring dissimilarities between curves with sharp patterns. This association is revealed as powerful. Under quite general conditions, we first obtain an asymptotic expansion for the small ball probability indicating that Bagidis induces a fractal topology on the functional space. We then provide the rate of convergence of the nonparametric regression estimator in this case, as a function of the parameters of the Bagidis semimetric. We propose to optimize those parameters using a cross-validation procedure, and show the optimality of the selected vector. This last result has a larger scope and concerns the optimization of any vector parameter characterizing a semimetric used in this context. The performances of our methodology are assessed on simulated and real data examples. Results are shown to be superior to those obtained using competing semimetrics as soon as the variations of the significant sharp patterns in the curves have a horizontal component.

Structural Tests in Regression on Functional Variable

This work focuses on recent advances on the way general structural testing procedures can be constructed in regression on functional variable. Our test statistic is constructed from an estimator adapted to the specific model to be checked and uses recent advances concerning kernel smoothing methods for functional data. A general theoretical result states the asymptotic normality of our test statistic under the null hypothesis and its divergence under local alternatives. This result opens interesting prospects about tests for no-effect, for linearity, or for reduction dimension of the covariate. Bootstrap methods are then proposed to compute the threshold value of our test. Finally, we present some applications to spectrometric datasets and discuss interesting prospects for the future.

Bases Giving Distances. A New Semimetric and its Use for Nonparemetric Functional Data Analysis

The BAGIDIS semimetric is a highly adaptivewavelet-based semimetric. It is particularly suited for dealing with curves presenting horizontally- and verticallyvarying sharp local patterns. One can advantageously make use of this semimetric in the framework of nonparametric functional data analysis.

Nonparametric Regression on Functional Variable and Structural Tests

The aim of this talk is to highlight the usefulness of kernel methods in regression on functional variables. After reminding some asymptotic properties of the kernel estimator of the regression operator, we introduce a general framework to construct various innovative structural tests (no-e ect, linearity, single-index, ... ). Various bootstrap procedures are implemented on datasets in order to emphasize the pertinence of such structural testing methods.