ENVIRONMETRICS Environmetrics 2010; 21: 365–381 Published online 3 July 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/env.1005 Nonlinear locally weighted kriging prediction for spatio-temporal environmental processes Olha Bodnar and Wolfgang Schmid ∗,† Department of Statistics, European University Viadrina, PO Box 1786, 15207 Frankfurt (Oder), Germany SUMMARY In the paper, a nonlinear interpolation procedure for the spatial prediction of an environmental process is proposed. The suggested interpolation is based on the locally weighted scatterplot smoothing method of Cleveland in 1979. This approach is applied to a nonlinear spatio-temporal model. In an empirical study, the PM10 concentration in the Berlin–Brandenburg region of Germany is considered. It is shown that the local approach permits a more structured interpolation of the air pollution. Copyright © 2009 John Wiley & Sons, Ltd. key words: nonlinear predictor; LOESS method; non-stationary spatio-temporal process; environmental statistics; Kalman filter 1. INTRODUCTION The prediction of the values of a spatio-temporal environmental process at sites, where no station of the monitoring network is available, is an important problem in environmental statistics. Instead of prediction we will also use the terminology interpolation because our aim is not to predict over time but over space. Usually, in that context the linear kriging predictor is used which is obtained by minimizing the mean squared error (MSE) (see, e.g., Zimmerman, 2006; Genton, 2007). Based on the linear kriging predictor Zimmerman (2006) discussed the problem of an optimal network design for spatial prediction and dealt with the problem, when the parameters of the spatio-temporal process are unknown and have to be estimated before the linear predictor is constructed. Genton (2007) considered the problem of separable approximations of space–time covariance matrices, which is used to reduce the dimensionality of the inverse covariance matrix used in the equation of the linear predictor. In the present paper, we propose a nonlinear predictor for spatial prediction which is based on the idea of the locally weighted scatterplot smoothing (LOESS) regression model (see, e.g., Cleveland, 1979). It is proposed to use a subset of stations of the monitoring network which are located nearest to the site, ∗ Correspondence to: W. Schmid, Department of Statistics, European University Viadrina, PO Box 1786, 15207 Frankfurt (Oder), Germany. † E-mail: schmid@euv-frankfurt-o.de Received 19 March 2009 Copyright © 2009 John Wiley & Sons, Ltd. Accepted 7 April 2009