4.2 A sliced inverse regression approach for block-wise evolving data streams

In this communication, we focus on data arriving sequentially by block in a stream. A semiparametric regression model involving a common EDR (Effective Dimension Reduc- tion) direction β is assumed in each block. Our goal is to estimate this direction at each arrival of a new block. A simple direct approach consists in pooling all the observed blocks and estimate the EDR direction by the SIR (Sliced Inverse Regression) method. But some disadvantages appear in practice such as the storage of the blocks and the running time for high dimensional data. To overcome these drawbacks, we propose an adaptive SIR es- timator of β based on the SIR approach for a stratified population developed by Chavent et al.(2011). The proposed approach is faster both from computational complexity and running time points of view, and provides data storage benefits. We show the consistency of our estimator at the root-n rate and give its asymptotic distribution. We propose an extension to multiple indices model. We also provide a graphical tool in order to detect if a drift occurs in the EDR direction or if some aberrant blocks appear in the data stream. In a simulation study, we illustrate the good numerical behavior of our estimator. One important advantage of this approach is its adaptability to changes in the underlying model.