Extrapolations of stable random fields

In this talk I’m going to speak about several methods of extrapolation of random fields used for prediction of field’s value at arbitrary point.
First part of the talk is devoted to a method, well-known in geostatistics, that is called kriging. It requires finiteness of the second moment of the field values. I will also mention an alternative to kriging- the quantile regression method.
In the second half of the talk, I will speak about three new extrapolation methods that are applied to random fields with heavy tail distributions, namely to stable random fields. These methods are all constructed by using of so-called covariation bracket, that plays the role of a dependence measure between stable random variables, i.e., it is an analogue for the covariance.
The first method minimizes L_p norm of the difference between the theoretical value and its predictor, the second method gives orthogonal (in the sense of the covariation bracket) projection to the linear subspace of the values at the extrapolation knots, the third method maximizes the dependence between the field and its linear predictor. We used numerical calculations to compare the methods, namely we calculated summary statistics for the deviation of the extrapolations build by using of each of the methods.