Kernel Inverse Regression for spatial random fields

TL;DR

This paper extends inverse regression for spatially dependent variables, providing consistent kernel-based estimates and a new spatial predictor as an alternative to traditional non-parametric methods.

Contribution

It introduces a dimension reduction model for spatial data using inverse regression under strong mixing conditions, with proven consistency and asymptotic properties.

Findings

Proposed a kernel-based inverse regression estimator for spatial data.

Proved weak and strong consistency of the estimator.

Developed a spatial predictor based on the dimension reduction approach.

Abstract

In this paper, we propose a dimension reduction model for spatially dependent variables. Namely, we investigate an extension of the \emph{inverse regression} method under strong mixing condition. This method is based on estimation of the matrix of covariance of the expectation of the explanatory given the dependent variable, called the \emph{inverse regression}. Then, we study, under strong mixing condition, the weak and strong consistency of this estimate, using a kernel estimate of the \emph{inverse regression}. We provide the asymptotic behaviour of this estimate. A spatial predictor based on this dimension reduction approach is also proposed. This latter appears as an alternative to the spatial non-parametric predictor.