Kriging

Kriging is a sophisticated method of interpolation used to estimate unknown values at specific locations. In the field of statistics, it is also known as Gaussian process regression. This technique is essential for spatial analysis and computer experiments where data is incomplete. Instead of just connecting dots, kriging uses a mathematical model to predict values. It provides what is called the best linear unbiased prediction, or BLUP. This means the method seeks the most accurate guess possible without being systematically too high or too low.

The mechanism of kriging relies on computing a weighted average of known values. When predicting a value at an unsampled location, the method looks at the surrounding neighborhood. It assigns weights to these known points based on their proximity and their relationship to the target spot. These weights serve two vital functions. First, they reflect the structural proximity of the samples. Second, they provide a desegregation effect to prevent bias from sample clusters. By balancing these weights, the method achieves its goal of minimal variance.

Kriging operates through several distinct mathematical stages. It begins with a prior distribution over functions, specifically a Gaussian process. In this process, samples from a function are normally distributed. The relationship between any two samples is defined by a covariance function, also called a kernel. Once real values are observed at specific locations, the model combines this Gaussian prior with a Gaussian likelihood function. This results in a posterior distribution. This final distribution provides both a mean value for the prediction and a measure of covariance.

There are several different types of kriging methods tailored to specific data needs. Ordinary kriging is a common approach that assumes a constant but unknown mean within a local search neighborhood. Simple kriging is different because it assumes the mean is known across the entire domain. For data that follows a specific pattern, universal kriging uses a polynomial trend model. Other specialized versions include indicator kriging, which uses indicator functions to estimate transition probabilities. There is also co-kriging, which allows for the joint estimation of data from multiple different sources.

The history of this method is rooted in practical mining needs. It began with Danie G. Krige, a pioneer at the Witwatersrand reef complex in South Africa. Krige developed a way to plot distance-weighted average gold grades using only a few borehole samples. He wanted to estimate the most likely distribution of gold across the area. In 1960, the French mathematician Georges Matheron developed the formal theoretical basis for these ideas. Matheron's work was actually based on Krige's master's thesis. This connection is why the method carries Krige's name today.

Understanding the uncertainty of a prediction is a major significance of the kriging method. The estimation variance tells us how much we can trust a specific guess. This variance grows when the covariance between the samples and the point being estimated decreases. In simpler terms, as the known samples get farther away from the target, the estimation becomes less certain. Interestingly, the estimation variance does not depend on the actual values of the samples. Instead, it depends on the spatial configuration and the geometric relationship between the points.

Kriging connects deeply to several other mathematical and scientific fields. It is closely related to regression analysis, as both use assumptions on covariances to find estimators. While regression models focus on multiple observations in a multivariate data set, kriging is designed for a single realization of a random field. It can also be viewed as a spline in a reproducing kernel Hilbert space. Furthermore, kriging with polynomial trend surfaces is mathematically identical to generalized least squares polynomial curve fitting. This versatility makes it a cornerstone of modern geostatistics and spatial modeling.