A formula for EBK could probably be written, but it would be so complicated that no one would be able to understand it. To understand EBK, you need to understand the procedure that generates the predictions. There is more information in the documentation on Empirical Bayesian Kriging, but the general procedure is:
- Subset the input data.
- In each subset, estimate a semivariogram using maximum likelihood. This semivariogram will have a relatively simple formula and will serve as the seed for simulations later.
- Use the estimated semivariogram to simulate new values at each of the input point locations.
- Estimate a new semivariogram from the simulated data values.
- Simulate a new set of data from the seed semivariogram.
- Repeat steps 4 and 5 many times (by default, 100 times). This results in an empirical distribution of semivariograms in each subset.
- Use the empirical distribution in each subset as a Bayesian prior and calculate the posterior distribution.
- Finally, when making a prediction at a new location, identify the neighbors based on the search neighborhood and merge the posterior distributions from the subset associated with each neighbor. The distributions will be weighted proportionally by the number of neighbors from that subset.
The only difference for EBK Regression Prediction is that in step 2, you will estimate a coefficient for the explanatory variables that you provide in addition to the semivariogram parameters.
