Kernel inverse regression for random fields

In this paper, we propose a Dimension Reduction model for spatially dependent variables. Namely, we investigate a generalization of the Inverse Regression method under some mixing conditions. This method introduced by (Li, 1991) for i.i.d. data is based on the estimation of the matrix of covariance of the conditional expectation of the explanatory variable given the response variable. Here, we investigate the weak consistency of this estimate based on a kernel estimate of the Inverse Regression under strong mixing condition. Through some simulations, we show the difference of behavior between our method and its i.i.d. counterpart. We also, investigate applications of our method in spatial forecasting problems and confront its with some others whom make their proof through a real data application.