Abstract
The kriging methodology can be applied to predict the value of a spatial variable at an unsampled location, from the available spatial data. Furthermore, additional information from secondary variables, correlated with the target one, can be included in the resulting predictor by using the cokriging techniques. The latter procedures require a previous specification of the multivariate dependence structure, difficult to characterize in practice in an appropriate way. To simplify this task, the current work introduces a nonparametric kernel approach for prediction, which satisfies good properties, such as asymptotic unbiasedness or the convergence to zero of the mean squared prediction error. The selection of the bandwidth parameters involved is also addressed, as well as the estimation of the remaining unknown terms in the kernel predictor. The performance of the new methodology is illustrated through numerical studies with simulated data, carried out in different scenarios. In addition, the proposed nonparametric approach is applied to predict the concentrations of a pollutant that represents a risk to human health, the cadmium, in the floodplain of the Meuse river (Netherlands), by incorporating the lead level as an auxiliary variable.
Highlights
A major challenge for the World Health Organization (WHO) is to protect human health
The current work deals with the construction of a prediction map that shows the concentrations of a pollutant, the cadmium, in the whole region of interest
This issue may be addressed by using the available data of the target variable (Cd) and incorporating information of a secondary correlated one (Pb), collected at various spatial sites. Problems of this kind are typically solved through the kriging or the cokriging techniques, depending on whether data of one or more variables are employed, respectively
Summary
A major challenge for the World Health Organization (WHO) is to protect human health. The incorporation of the resulting data can be advantageous and it provides a larger sample than the one derived from the principal variable, on which the cokriging approaches can be applied This methodology has been widely used to solve a variety of problems, specially to assess the contamination level by any pollutant, whose concentration can be influenced by the presence of other variables (chemical elements, atmospheric variables, etc.). By deriving function (5) with respect to each λiji and li , for each i and ji , as well as equaling the result to zero, an equation system is obtained, whose solution provides the values of the unknown parameters λiji They would be expressed in terms of the cross-covariograms Cii0 or cross-semivariograms γii0 , difficult to be characterized in practice [6].
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