Cluster-based estimation for sufficient dimension reduction

Sufficient dimension reduction (SDR) approaches such as ordinary least squares (OLS), sliced inverse regression, sliced average variance estimation and principal Hessian directions all require a linearity condition on the marginal distribution of the predictors X. This can be a significant limitation in practice. In this article we propose the use of clustering on the predictor space with the goal of breaking the nonlinearity among the predictors. We then estimate the dimension reduction subspace by combining estimates from individual clusters. OLS is used as a demonstration of the cluster-based estimation approach. Simulations have shown that the proposed method effectively mitigates the effects of nonlinearity; it thus has the potential to widen the applicative scope of existing SDR techniques.