Explaining and Connecting Kriging with Gaussian Process Regression

Kriging and Gaussian Process Regression are statistical methods that allow predicting the outcome of a random process or a random field by using a sample of correlated observations. In other words, the random process or random field is partially observed, and by using a sample a prediction is made, pointwise or as a whole, where the latter can be thought as a reconstruction. In addition, the techniques permit to give a measure of uncertainty of the prediction. The methods have different origins. Kriging comes from geostatistics, a field which started to develop around 1950 oriented to mining valuation problems, whereas Gaussian Process Regression has gained popularity in the area of machine learning in the last decade of the previous century. In the literature, the methods are usually presented as being the same technique. However, beyond this affirmation, the techniques have yet not been compared on a thorough mathematical basis and neither explained why and under which conditions this affirmation holds. Furthermore, Kriging has many variants and this affirmation should be precised. In this paper, this gap is filled. It is shown, step by step how both methods are deduced from the first principles -- with a major focus on Kriging, the mathematical connection between them, and which Kriging variant corresponds to which Gaussian Process Regression set up. The three most widely used versions of Kriging are considered: Simple Kriging, Ordinary Kriging and Universal Kriging. It is found, that despite their closeness, the techniques are different in their approach and assumptions, in a similar way the Least Square method, the Best Linear Unbiased Estimator method and the Likelihood method in regression do. I hope this work deepen the understanding of the relation between Kriging and Gaussian Process Regression, and serves as a cohesive introductory resource for researchers.