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Gaussian Random Fields

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Random Fields for Spatial Data Modeling

Part of the book series: Advances in Geographic Information Science ((AGIS))

Abstract

Gaussian random fields have a long history in science that dates back to the research of Andrey Kolmogorov and his group. Their investigation remains an active field of research with many applications in physics and engineering. The widespread appeal of Gaussian random fields is due to convenient mathematical simplifications that they enable, such as the decomposition of many-point correlation functions into products of two-point correlation functions. The simplifications achieved by Gaussian random fields are based on fact that the joint Gaussian probability density function is fully determined by the mean and the covariance function.

The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.

Albert Einstein

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Notes

  1. 1.

    The theorem essentially says that if we use a new set of integration variables, we need to take account of the Jacobian of the transformation from the old to the new variables.

  2. 2.

    As we have seen in (4.18), the autocorrelation function is constrained by ρ xx ≥−1∕d.

  3. 3.

    More precisely, the real-valued amplitude of the complex-valued wavefunction.

  4. 4.

    A clear explanation of the transformations from lattice space to the continuum is given in [122].

  5. 5.

    Summation is not implied over the indices i and j here.

  6. 6.

    The expectation operator acts over the degrees of freedom of the potential function V [X(s; ω)] which herein is denoted by V for short.

  7. 7.

    This includes products of k = 1, …, K moments of order m k < n such that \(\sum _{k=1}^{K}=n\).

  8. 8.

    The statistical physics definition of the free energy involves the temperature T and the Boltzmann constant k B, i.e., \(F = - k_{B} T\,\ln Z\). These factors do not play a role in determining the approximation of the partition function. They are also irrelevant if Z represents the partition function of a spatial random field instead of a system of particles at thermal equilibrium.

  9. 9.

    A symmetric N × N real matrix H is said to be positive definite if x ⊤Hx is non-negative for every non-zero column vector x of dimension N.

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  1. Technical University of Crete, Chania, Greece

    Dionissios T. Hristopulos

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Hristopulos, D.T. (2020). Gaussian Random Fields. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_6

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