View source for Large deviations of Gaussian random functions

A '''random function''' &ndash; of either one variable (a random process), or two or more variables
(a [[Random field|random field]]) &ndash; is called '''Gaussian''' if every [[Finite-dimensional distribution|finite-dimensional distribution]] is a [[Multivariate normal distribution|multivariate normal distribution]]. Gaussian random fields on the [[Sphere|sphere]] are useful (for example) when analysing

* the anomalies in the cosmic microwave background radiation (see,<ref name="adler2000">Robert J. Adler, "On excursion sets, tube formulas and maxima of random fields", [https://dx.doi.org/10.1214/aoap/1019737664 The Annals of Applied Probability 2000, Vol. 10, No. 1, 1&ndash;74]. (Special invited paper.)</ref> pp.&nbsp;8&ndash;9);
* brain images obtained by [[Physics:Positron emission tomography|positron emission tomography]] (see,<ref name="adler2000" /> pp.&nbsp;9&ndash;10).

Sometimes, a value of a Gaussian random function deviates from its [[Expected value|expected value]] by several [[Standard deviation|standard deviation]]s. This is a '''large deviation'''. Though rare in a small domain (of space or/and time), large deviations may be quite usual in a large domain.

== Basic statement ==
Let <math>M</math> be the maximal value of a Gaussian random function <math>X</math> on the
(two-dimensional) sphere. Assume that the expected value of <math>X</math> is <math>0</math> (at every point of the sphere), and the standard deviation of <math>X</math> is <math>1</math> (at every point of the sphere). Then, for large <math>a>0</math>, <math>P(M>a)</math> is close to <math>C a \exp(-a^2/2) + 2P(\xi>a)</math>,
where <math>\xi</math> is distributed <math>N(0,1)</math> (the standard normal distribution), and <math>C</math> is a constant; it does not depend on <math>a</math>, but depends on the [[Correlation function|correlation function]] of <math>X</math> (see below). The relative error of the approximation decays exponentially for large <math>a</math>.

The constant <math>C</math> is easy to determine in the important special case described in terms of the [[Directional derivative|directional derivative]] of <math>X</math> at a given point (of the sphere) in a given direction (tangential to the sphere). The derivative is random, with zero expectation and some standard deviation. The latter may depend on the point and the direction. However, if it does not depend, then it is equal to <math>(\pi/2)^{1/4} C^{1/2}</math> (for the sphere of radius <math>1</math>).

The coefficient <math>2</math> before <math>P(\xi>a)</math> is in fact the [[Euler characteristic]] of the sphere (for the [[Torus|torus]] it vanishes).

It is assumed that <math>X</math> is twice continuously differentiable ([[Almost surely|almost surely]]), and reaches its maximum at a single point (almost surely).

== The clue: mean Euler characteristic ==
The clue to the theory sketched above is, Euler characteristic <math>\chi_a</math> of the [[Set (mathematics)|set]] <math>\{X>a\}</math> of all points <math>t</math> (of the sphere) such that <math>X(t)>a</math>. Its expected value (in other words, mean value) <math>E(\chi_a)</math> can be calculated explicitly:

:<math> E(\chi_a) = C a \exp(-a^2/2) + 2 P(\xi>a) </math>

(which is far from being trivial, and involves [[Poincaré–Hopf theorem]], [[Gauss–Bonnet theorem]], [[Rice's formula]] etc.).

The set <math>\{X>a\}</math> is the [[Empty set|empty set]] whenever <math>M<a</math>; in this case <math>\chi_a=0</math>. In the other case, when <math>M>a</math>, the set  <math>\{X>a\}</math> is non-empty; its Euler characteristic may take various values, depending on the topology of the set (the number of [[Connected space|connected components]], and possible holes in these components). However, if <math>a</math> is large and <math>M>a</math> then the set <math>\{X>a\}</math> is usually a small, slightly deformed disk or [[Ellipse|ellipse]] (which is easy to guess, but quite difficult to prove). Thus, its Euler characteristic <math>\chi_a</math> is usually equal to <math>1</math> (given that <math>M>a</math>). This is why <math> E(\chi_a)</math> is close to <math>P(M>a)</math>.

== See also ==
* [[Gaussian process]]
* [[Gaussian random field]]
* [[Large deviations theory]]

== Further reading ==
The basic statement given above is a simple special case of a much more general (and difficult) theory stated by Adler.<ref name="adler2000"/><ref name="AT2007">Robert J. Adler, Jonathan E. Taylor, "Random fields and geometry", Springer 2007. {{isbn|978-0-387-48112-8}}</ref><ref name="adler2008">Robert J. Adler, "Some new random field tools for spatial analysis", [https://arxiv.org/abs/0805.1031 arXiv:0805.1031].</ref> For a detailed presentation of this special case see Tsirelson's lectures.<ref>[http://www.tau.ac.il/~tsirel/Courses/Gauss2/syllabus.html Lectures of B. Tsirelson] (especially, Sect. 5).</ref>

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[[Category:Stochastic processes]]

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