Random field

A random field , also a random field . random field , is required if you want to model random phenomena in space, e.g. B. the carbon dioxide content in the atmosphere in metropolitan areas, or the amount of precipitation in different regions of Germany.

Mathematical definition

A random field is a family of random quantities on a probability space . Where is a natural number . In the case one speaks of a stochastic process , then the time parameter plays the role and is usually referred to as. For a realized one there is then a realization , also a trajectory of the random field. Often one is interested in z. B. for the probability that a trajectory exceeds a certain value, see e.g. B. (Application example: flood protection) ${\displaystyle \{Y(x,\cdot )\}_{x\in R^{d}}}$ ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle d}$ ${\displaystyle d=1}$ ${\displaystyle x}$ ${\displaystyle t}$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle Y(x,\omega )=y(x)}$

Trend, covariance, stationarity, isotropy

The expected value function

${\displaystyle EY(x,\cdot )=m(x),x\in R^{d}}$

is called the trend and the second moment function

${\displaystyle E[Y(x,\cdot )-m(x)][Y(z,\cdot )-m(z)]=r(x,z)}$

referred to as the covariance function of the random field. Random fields for which trend and covariance function exist are called 2nd order random fields . Random fields with constant trend and shift-invariant covariance function, i.e. H.

${\displaystyle m(x)=m,\quad r(x,z)=r(x-z)}$

is called stationary in the broader sense . Stationarity in the narrower sense requires displacement invariance not only of the first two moments , but of all finite-dimensional distributions of the random field. Is the covariance function rotationally invariant, i. H.

${\displaystyle r(x,z)=r(\|x-z\|),\quad \|\cdot \|}$ Euclidean distance ,

then the random field is called isotropic .

Prediction of values ​​of the random field

If you have observed the random field at the locations with the results , you can use this to construct a prediction of the random field at an unobserved point . The best prediction that minimizes the mean square error is the conditional expectation of , given the observations , i. H. ${\displaystyle x_{1},...,x_{n}}$ ${\displaystyle y_{1},...,y_{n}}$ ${\displaystyle {\hat {Y}}(x)}$ ${\displaystyle x}$ ${\displaystyle Y(x,\cdot )}$ ${\displaystyle y_{1},...,y_{n}}$

${\displaystyle {\hat {Y}}(x)=E[Y(x,\cdot )|y_{1},...,y_{n}]}$.

This prediction cannot be calculated without further distribution assumptions. If the covariance function of the random field is known, however, the best linear prediction can be calculated with little effort .

Application in geostatistics

In geostatistics , instead of the covariance function, the semivariogram is used in an equivalent way

${\displaystyle V(x,z)={\frac {1}{2}}E[Y(x,\cdot )-Y(z,\cdot )]^{2}}$,

d. H. half the (semi) expected value of the quadratic difference is used. The best linear forecast is called Kriging in geostatistical terminology , see e.g. B. The dimension of the random field is here usually given in a natural way, e.g. B. for the surface temperature of a lake, for problems of the exploration of deposits in mining, for spatio-temporal phenomena in meteorology . ${\displaystyle Y(x)-Y(z)}$ ${\displaystyle d}$ ${\displaystyle d=2}$ ${\displaystyle d=3}$ ${\displaystyle d=4}$

literature

• R. Adler, J. Taylor: Random Fields and Geometry. Springer, New York 2007.
• N. Cressie: Statistics for Spatial Data. World Scientific, Singapore 2007.
• E. Vanmarcke: Random Fields: Analysis and Synthesis. World Scientific, Singapore 2010.

Individual evidence

1. ^ RJ Adler: The Geometry of Random Fields . Wiley, Chichester / New York / Toronto 1981.
2. ↑ JP Chiles, P. Delfiner: Geostatistics: Modeling Spatial Uncertainty . Wiley, New York 1999.