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The mystery of Kriging: How does it achieve the best unbiased prediction?
Kriging method originated from geostatistics and has now been widely used in spatial analysis and computational experiments.This Gaussian process-based interpolation method aims to predict the values of unobserved locations by weighted averaging of known values.Among all kinds of prediction methods, the Kriging method stands out with its best linear unbiased prediction characteristics, making it an important tool for current research and application.
Kriging is not just an interpolation technique, it also involves in-depth understanding of stochastic processes.This allows analysts to make reasonable predictions in places where data does not exist and quantify relevant uncertainties.
The foundation of the Kriging method lies in the Gaussian process, where the samples of each point are distributed according to some covariance function.This means that the Kriging method not only takes into account the currently observed samples, but also makes predictions of future possible values for each potential unobserved location.The main theory of this technique was proposed in 1960 by French mathematician George Matelon, whose research was originally based on Danny Krieger's master's thesis, who conducted gold in the Witwattersland mine in South Africa. The distance-weighted average calculation of mining.
The Kriging method works by taking into account random variables around known data points and then computing predictions based on their spatial location.If the data considered have some degree of stability, a reasonable prediction of unknown values can be obtained.This assumption allows the Kriging method to set up a more efficient prediction model while obtaining uncertainty measures.
Through precise covariance function design, the kriging method can ensure the minimum mean square error of the prediction result, making it an extremely important tool in spatial inference.
Kriging can be regarded as a Bayesian optimization form.It starts with a prior distribution of the function, which is itself a Gaussian process.This means that for any two points, the process evaluates the covariance based on the spatial location of these two points.When new observations come in, combined with these observation data, a posterior distribution for any new location can be generated, which is also Gaussian distribution, which can be easily calculated from the observations and their variance.
When doing spatial inference, the core idea of Kriging is to use linear combinations to predict unobserved locations.These predicted values are based on a weighted average of known data, where the calculation of weights is intended to reflect structural proximity between the known values and the estimated location.More importantly, the design of the Kriging method also needs to avoid deviations caused by uneven sample distribution.
Not only that, the weights used in the Kriging method can also minimize the variance of the prediction, which ensures the robustness and accuracy of the prediction.
Kriging method can derive a variety of different prediction methods based on the random nature of the random field and the preset stability level.Classic kriging methods include ordinary kriging, simple kriging and general kriging, etc. These methods are applied to different assumptions under different circumstances.In ordinary kriging, it is assumed that the unknown mean is fixed within the search area, while the simple kriging further assumes that the mean is known for the overall range.The universal kriging rule takes into account the universal model of polynomial trends, thus providing more flexible prediction capabilities.
Various variants of the Kriging method are used in different fields, from natural resource exploration to environmental science, and even in urban planning.Taking geological exploration as an example, Kriging method can effectively convert sparse sampling points into comprehensive resource assessments, thereby helping enterprises make smarter investment decisions.Going deeper, Kriging method can also make more forward-looking predictions to help scientific researchers understand the potential changes in geological activities.
However, with the development of technology and the diversification of measurement methods, the challenges facing Kriging are also increasing.For example, how to process larger data sets and how to improve computing efficiency will become the direction of future research.
In future applications, how will Kriging combine emerging technologies and methods to further improve prediction accuracy?
