A partitioned Single Functional Index Model

Abstract

Given a functional regression model with scalar response, the aim is to present a methodology in order to approximate in a semi-parametric way the unknown regression operator through a single index approach, but taking possible structural changes into account. Our paper presents this methodology and illustrates its behaviour both on simulated and real curves datasets. It appears, from an example of interest in spectrometry, that the method provides a nice exploratory tool both for analyzing structural changes in the spectrum and for visualizing the most informative directions, still keeping good predictive power. Even if the main objective of this work is to discuss applied issues of the method, asymptotic behaviour is shortly described.

Appendix: A few words on asymptotic behaviour

Cross-validation ideas have been firstly used in nonparametric framework for bandwidth selection in standard multivariate setting (see for instance Härdle and Marron 1985; Marron and Härdle 1986). Afterwards they have been extended for bandwidth selection in functional framework (see Rachdi and Vieu 2007), and more generally to other automatic parameter choices in functional data analysis, like choosing direction in functional single index modelling (see Ait-Saïdi et al. 2008), selecting the dimension in functional projection pursuit regression (see Ferraty et al. 2013) or structural-points in complex regression models (see Ferraty et al. 2011). This is why cross-validation has been used in our work both for fitting the model in (7) and for choosing the break-point in (8).

Observing that the partitioned model (4) can be equivalently written as

$$\begin{aligned} Y=\alpha +g_{1}\left( \int _{I}\overline{\theta }_{1}\left( t\right) X\left( t\right) dt\right) +g_{2}\left( \int _{I}\overline{\theta }_{2}\left( t\right) X\left( t\right) dt\right) +\mathcal {E} \end{aligned}$$

where

$$\begin{aligned} \overline{\theta }_{j}\left( t\right) =\left\{ \begin{array}{ll@{\quad }l} \theta _{j}\left( t\right) &{} &{} t\in I_{j} \\ 0 &{} &{} \text {otherwise} \end{array} \right. \end{aligned}$$

with $\overline{\theta }_{1}$ and $\overline{\theta }_{2}$ orthogonal by construction, PFSIM can be seen as some special case of the Functional Projection Pursuit Regression model developed recently in Ferraty et al. (2013). As a matter of conclusion, one could derive directly asymptotic results for the method proposed here just by straightforward adaptation of the proofs in the above mentioned paper. In particular one could get the following kinds of results:

• i. Asymptotic optimality (in terms of minimal quadratic prediction error) of the estimates of the directions $\theta _{j}$ obtained in (7), just from the proof of Theorem 5 in Ferraty et al. (2013);

• ii. Univariate rate of convergence of the estimates of the link functions $g_{j}$, just from the proof of Theorem 4 in Ferraty et al. (2013).

In the same spirit, by following the general methodology as presented in Ferraty et al. (2011) for structural parameter estimation, one could also get:

• iii. Consistency of the selected parameter $\widehat{\lambda }$ towards the value $\lambda _{0}\in \Lambda$ minimizing quadratic prediction errors;

• iv. Asymptotic optimality (inside of $\Lambda$) of the data-driven value $\widehat{\lambda }$.