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Geometric Properties of Random Fields

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Random Fields for Spatial Data Modeling

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

Abstract

This chapter deals with the main concepts and mathematical tools that help to describe and quantify the shapes of random fields. The geometry of Gaussian random functions is to a large extent determined by the mean and the two-point correlation functions. The classical text on the geometry of random fields is the book written by Robert Adler [10]. The basic elements of random field geometry are contained in the technical report by Abrahamsen [3]. A more recent and mathematically advanced book by Taylor and Adler exposes the geometry of random field using the language of manifolds [11].

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

Pythagoras

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Notes

  1. 1.

    More generally, it can be shown that the Gaussian covariance function admits derivatives of all orders at ∥r∥ = 0.

  2. 2.

    In [135] R 2(1) = ξ 1ξ 2 is used instead of R. The results from [135] are equivalent to those presented herein following the transformation R 2(1)↦1∕R.

  3. 3.

    Note the slight difference of notation in Λi,j compared to (5.27). For the second-order moments the indices i, j are sufficient instead of the vector (p 1, …, p d). We also drop the superscript (2) since the order of the moments is obvious.

  4. 4.

    There is nothing magical about 5%; a different level, such as the distance at half-maximum could be selected instead.

  5. 5.

    A unimodal function is a function with a single peak.

  6. 6.

    A “hyper-cube” is used in d > 3 dimensions; in d = 1 the “hyper-cubes” correspond to line segments and in d = 2 to squares.

  7. 7.

    Condensed matter physicists call this less-than-perfect correlation quasi-long-range order, to distinguish it from “true” long-range order. The latter implies that all the system variables have the same value. A typical example is a ferromagnetic system in which all the magnetic moments are aligned in the same direction.

  8. 8.

    The exponents β refer to the equations (4.14) and (4.15) respectively.

  9. 9.

    In this case we should speak of a spectral function, since the spectral density is not well-defined.

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Hristopulos, D.T. (2020). Geometric Properties of 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_5

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