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''stochastic process in multi-dimensional time, stochastic process with a multi-dimensional parameter''
 
  
A [[Random function|random function]] defined on a set of points in a multi-dimensional space. Random fields are an important example of random functions (cf. [[Random element|Random element]]), which are often encountered in various applications. Some examples of random fields depending on three spatial coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773001.png" /> (as well as on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773002.png" />) are the fields of components of the velocity, of the pressure and of the temperature of a turbulent fluid flow (see [[#References|[1]]]). An example of a random field depending on two coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773004.png" /> is the height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773005.png" /> of a wavy sea surface, or the surface of any rough plate (see [[#References|[2]]]). In the investigation of global atmospheric processes on an Earth scale, the field of ground pressure and other meteorological characteristics are sometimes regarded as random fields on a sphere, etc.
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{| class="wikitable"
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!Copyright notice <!-- don't remove! -->
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|-
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| This article ''Random field'' was adapted from an original article by Mikhail Moklyachuk, which appeared in ''StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies''. The original article ([<nowiki>http://statprob.com/encyclopedia/RandomField6.html</nowiki> StatProb Source], Local Files: [[Media:random_field.pdf|pdf]] | [[Media:random_field.tex|tex]]) is copyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
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|-
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|}
  
The theory of random fields of a general form is almost identical with the general theory of random functions. One can only obtain more interesting concrete results for various special classes of random fields with additional properties that facilitate their study. One such class is that of homogeneous random fields (cf. [[Random field, homogeneous|Random field, homogeneous]]), defined on a homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773006.png" /> with a group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773007.png" /> and having the property that the probability distributions of the values of the field on an arbitrary finite group of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773008.png" />, or the mean value of the field and the second moments of its values on pairs of points, are invariant under the action of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r0773009.png" /> on their arguments. Homogeneous random fields on a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730011.png" /> or on the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730012.png" /> of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730013.png" /> with integral coordinates, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730014.png" /> is the group of all possible (or all integral) parallel translations, are natural generalizations of stationary stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]), and many results that hold for such processes carry over in a simple way. Of great interest for applications (in particular, for the mechanics of turbulence, see [[#References|[1]]]) are the so-called homogeneous isotropic random fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730017.png" /> is the group of isometric transformations of the corresponding space. An important feature of homogeneous random fields is the existence of a spectral decomposition of special form, both of the fields themselves and of their correlation functions (see, for example, [[#References|[3]]], [[#References|[4]]], ; see also [[Spectral decomposition of a random function|Spectral decomposition of a random function]]).
+
{{MSC|62M40|60G60}}
  
Another class of random fields that has attracted considerable attention is that of Markov random fields, defined in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730019.png" />. The condition for a random field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730020.png" /> of being Markov asserts, roughly speaking, that for a sufficiently large family of open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730021.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730022.png" />, fixing the values taken by the field in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730023.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730025.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730026.png" /> gives that the families of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730029.png" /> is the complement of the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730031.png" />, are mutually independent (or, in the case of the Markov property in the wide sense, mutually uncorrelated; see, for example, [[#References|[5]]]). This can be generalized to the concept of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730033.png" />-Markov random fields, where the above independence (or orthogonality) need only hold when the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730034.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730035.png" /> of arbitrary width is replaced by a special type of thickened boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730036.png" />. The theory of Markov random fields and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730038.png" />-Markov fields has a number of important applications in quantum field theory and statistical physics (see [[#References|[6]]], [[#References|[7]]]). Another class of random fields arising from problems of statistical physics is that of Gibbs random fields, whose probability distributions can be expressed as Gibbs distributions (cf. [[Gibbs distribution|Gibbs distribution]]; see e.g. [[#References|[7]]], [[#References|[8]]], [[#References|[10]]]). A convenient way of defining Gibbs random fields involves a family of conditional probability distributions of values of the field in a finite domain corresponding to all its fixed values outside this domain. It must be noted that it is often convenient to regard random fields on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730039.png" /> as a special case of generalized random fields, for which there may not exist values at one specified point, but the smoothened values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730040.png" /> can be interpreted as random linear functionals defined on some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730041.png" /> of smooth test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730042.png" />. Generalized random fields (and especially generalized Markov random fields) are widely used in physical applications. By considering within the limits of the theory of generalized random fields (cf. [[Random field, generalized|Random field, generalized]]) the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730043.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730044.png" /> are such that
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730045.png" /></td> </tr></table>
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<center>{Department of Probability Theory, Statistics and
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Actuarial Mathematics,}</center>
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 +
<center>{Kyiv National Taras
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Shevchenko University, Ukraine}</center>
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<center>{Email: mmp\@ univ.kiev.ua}</center>
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<!-- \vskip5mm -->
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Keywords and Phrases: Random field; Kolmogorov Existence Theorem;
 +
Gaussian random field; Wiener sheet; Brownian sheet; Poisson random
 +
field; Markov random field; Homogeneous random field; Isotropic
 +
random field; Spectral decomposition
 +
 
 +
<!-- \vskip5mm -->
 +
 
 +
''Random field'' $X(t)$ on $D\subset\mathbb R^n$ (i.e. $t\in D\subset\mathbb R^n$) is a function whose values are random variables for any $t\in D$.
 +
The dimension of the coordinate is usually in the range from one to four, but any $n>0$ is possible. A one-dimensional random field is usually called a stochastic process. The term 'random field' is used to stress that the dimension of the coordinate is higher than one. Random fields in two and three dimensions are encountered in a wide range of sciences and especially in the earth sciences such as hydrology, agriculture, and geology. Random fields where $t$ is a position in space-time are studied in turbulence theory and in meteorology.
 +
 
 +
Random field $X(t)$ is described by its finite-dimensional
 +
(cumulative) distributions
 +
$$F_{t_1,\dots,t_k}(x_1,\dots,x_k)=P\{X(t_1)<x_1,\dots,X(t_k)<x_k\},
 +
k=1,2,\dots$$
 +
The cumulative distribution functions are by
 +
definition left-continuous and nondecreasing. Two
 +
requirements on the finite-dimensional distributions must be
 +
satisfied. The symmetry condition
 +
$$F_{t_1,\dots,t_k}(x_1,\dots,x_k)=F_{t_{\pi1},\dots,t_{\pi
 +
k}}(x_{\pi1},\dots,x_{\pi k}),$$ $\pi$ is a permutation of the
 +
index set $\{1,\dots,k\}$. The compatibility condition
 +
$$F_{t_1,\dots,t_{k-1}}(x_1,\dots,x_{k-1})=
 +
F_{t_1,\dots,t_k}(x_1,\dots,x_{k-1},\infty).$$
 +
 
 +
''Kolmogorov Existence Theorem'' states: If a system of finite-dimensional distributions
 +
$F_{t_1,\dots,t_k}(x_1,\dots,x_k)$, $k=1,2,\dots$
 +
satisfies the symmetry and compatibility conditions, then there exists on some probability space a random field $X(t)$, $t\in D$, having $F_{t_1,\dots,t_k}(x_1,\dots,x_k)$, $k=1,2,\dots$
 +
as its finite-dimensional distributions.
 +
 
 +
The expectation (mean value) of a random field is by definition the Stieltjes integral
 +
$$
 +
m(t)=EX(t)=\int_{\mathbb R^1}xdF_t(x).
 +
$$
 +
The (auto-)covariance function is also expressed as the Stieltjes integral
 +
$$
 +
B(t,s)=E(X(t)X(s))-m(t)m(s)=\iint_{\mathbb R^2}xydF_{t,s}(x,y)-m(t)m(s),
 +
$$
 +
whereas the variance is $\sigma^2(t)=B(t,t)$.
 +
 
 +
 
 +
 
 +
''Gaussian random fields'' play an important role for several reasons: the specification of their finite-dimensional distributions is simple, they are reasonable models for many natural phenomenons, and their estimation and inference are simple.
 +
 
 +
A Gaussian random field is a random field where all the
 +
finite-dimensional distributions are multivariate normal
 +
distributions. Since multivariate normal distributions are
 +
completely specified by expectations and covariances, it suffices to
 +
specify $m(t)$ and $B(t, s)$ in such a way that the symmetry
 +
condition and the compatibility condition hold true. The
 +
expectation can be arbitrarily chosen, but the covariance function
 +
must be positive definite to ensure the existence of all
 +
finite-dimensional distributions (Adler and Taylor 2007; Piterbarg
 +
1996)
 +
 
 +
 
 +
 
 +
''Wiener sheet'' (''Brownian sheet'') is a Gaussian random field
 +
$W(t)$, $t=(t_1,t_2)\in\mathbb R_+^2$ with $EW(t)=0$ and correlation
 +
function $$ B(t,s)=E(X(t)X(s))=\min\{s_1,t_1\}\min\{s_2,t_2\}.$$
 +
Analogously, ($n$-parametric Wiener process) is a Gaussian random
 +
field $W(t)$, $t\in\mathbb R^n_+$ with $EW(t)=0$ and correlation
 +
function $ B(t,s)=\prod_{i=1}^n\min(s_i,t_i)$. Multiparametric
 +
Wiener process $W(t)$ has independent homogeneous increments.
 +
Generalized derivative of multiparametric Wiener process $W(t)$ is
 +
''Gaussian white noise process'' on $\mathbb R^n_+$ (Chung and
 +
Walsh 2005; Khoshnevisan 2002).
 +
 
 +
 
 +
 
 +
''Poisson random fields'' are also reasonable models for many
 +
natural phenomenon. A Poisson random field is an integer-valued
 +
(point) random field where the (random) amount of points which
 +
belong to a bounded set from the range of values of the field has a
 +
Poisson distribution and the random amounts of points which belong
 +
to nonoverlapping sets are mutually independent (Kerstan et al.
 +
1974). Point-valued random fields (Poisson random fields, Cox random
 +
fields, Poisson cluster random fields, Markov point random fields,
 +
homogeneous and isotropic point random fields, marked point random
 +
fields) are appropriate mathematical models for geostatistical data.
 +
A mathematically elegant approach to analysis of point-valued random fields
 +
(spatial point processes) is proposed by Noel A.C. Cressie (Cressie 1991).
 +
 
 +
 
 +
 
 +
''Markov random field'' $X(t)$, $t\in D\subset\mathbb R^n$, is a random function which has the Markov property with respect to a fixed system of ordered triples $(S_1,\Gamma,S_2)$ of nonoverlapping subsets from the domain of definition $D$.
 +
The Markov property means that for any measurable set $B$ from the range of values of the function $X(t)$ and every $t_0\in S_2$ the following equality holds true
 +
$$
 +
P\{X(t_0)\in B\vert X(t),t\in S_1\cup\Gamma\}=
 +
P\{X(t_0)\in B\vert X(t),t\in \Gamma\}.
 +
$$
 +
This means that the future $S_2$ does not depend on the past $S_1$
 +
when the present $\Gamma$ is given. Let, for example, $D=\mathbb
 +
R^n$, $\{\Gamma\}$ be a family of all spheres in $\mathbb R^n$,
 +
$S_1$ be the interior of $\Gamma$, $S_2$ be the exterior of
 +
$\Gamma$. A homogeneous and isotropic Gaussian random field $X(t)$,
 +
$t\in\mathbb R^n$, has the Markov property with respect to the
 +
ordered triples $(S_1,\Gamma,S_2)$ if and only if $X(t)=\xi$, where
 +
$\xi$ is a random variable. Nontrivial examples of homogeneous and
 +
isotropic Markov random fields can be constructed when consider the
 +
generalized random fields. Markov random fields are completely
 +
described in the class of homogeneous Gaussian random fields on
 +
$\mathbb Z^n$, in the class of multidimensional homogeneous
 +
generalized Gaussian random fields on the space $\mathbb
 +
C_0^{\infty}(\mathbb R^m)$ and the class of multidimensional
 +
homogeneous and isotropic generalized Gaussian random fields (Glimm
 +
and Jaffe 1981; Rozanov 1982; Yadrenko 1983).
 +
 
 +
 
 +
 
 +
''Gibbs random fields'' form a class of random fields that have
 +
extensive applications in solutions of problems in statistical
 +
physics. The distribution functions of these fields are determined
 +
by Gibbs distribution (Malyshev and Minlos 1985).
 +
 
 +
 
 +
 
 +
''Homogeneous random field'' in the strict sense is a real
 +
valued random function $X(t)$, $t\in\mathbb R^n$ (or $t\in\mathbb Z^n$), where
 +
all its finite-dimensional distributions are invariant under arbitrary translations, i.e.
 +
$$F_{t_1+s,\dots,t_k+s}(x_1,\dots,x_k)=F_{t_1,\dots,t_k}(x_1,\dots,x_k)\,\forall s\in\mathbb R^n.$$
 +
 
 +
 
 +
 
 +
''Homogeneous random field'' in the wide sense is a real
 +
valued random function $X(t)$, $t\in\mathbb R^n$ ($t\in\mathbb Z^n$), $E|X(t)|^2<+\infty$, where
 +
$EX(t)=m=\text{const}$ and the correlation function
 +
$EX(t)X(s)=B(t-s)$ depends on the difference $t-s$ of coordinates of points $t$ and $s$.
 +
 
 +
Homogeneous random field $X(t)$, $t\in\mathbb R^n$, $EX(t)=0$, $E|X(t)|^2<+\infty$, and its correlation function $ B(t)=EX(t+s)X(s)$ admit the spectral representations
 +
$$ X(t)=\int\cdots\int\exp\left\{\sum_{k=1}^nt_k\lambda_k\right\}Z(d\lambda),
 +
$$
 +
$$ B(t)=\int\cdots\int\exp\left\{\sum_{k=1}^nt_k\lambda_k\right\}F(d\lambda),
 +
$$
 +
where $F(d\lambda)$ is a measure on the Borel $\sigma$-algebra $B_n$ of sets from $\mathbb R^n$, $Z(d\lambda)$ is an orthogonal random measure on $B_n$ such that
 +
$E(Z(S_1)Z(S_2))=F(S_1\cap S_2)$. The integration range is $\mathbb R^n$ in the case of continuous time random field $X(t)$, $t\in\mathbb R^n$ and $[-\pi,\pi]^n$ in the case of discrete time random field $X(t)$, $t\in\mathbb Z^n$.
 +
In the case where the spectral representation of the correlation function is of the form
 +
$$ B(t)=
 +
\int\cdots\int\exp\left\{\sum_{k=1}^nt_k\lambda_k\right\}f(\lambda)d\lambda,
 +
$$
 +
the function $f(\lambda)$ is called spectral density of the field $X(t)$.
 +
Based on these spectral representations we can prove, for example, the ''law of large numbers'' for random field $X(t)$:
 +
 
 +
The mean square limit
 +
$$
 +
\lim_{N\to\infty}\frac{1}{(2N+1)^n} \sum_{|t_i|\leq
 +
N,i=1,\dots,n}X(t)=Z\{0\}.
 +
$$
 +
This limit is equal to $EX(t)=0$ if and only if $E|Z\{0\}|^2=F\{0\}$. In the case where $F\{0\}=0$ and
 +
$$ \int_{-\pi}^{\pi}\cdots\int_{-\pi}^{\pi}
 +
\prod_{i=1}^n
 +
\log\left|\log\frac{1}{|\lambda_i|}\right|F(d\lambda)<+\infty,
 +
$$
 +
the ''strong law of large numbers'' holds true for the random field $X(t)$.
 +
 
 +
 
 +
 
 +
''Isotropic random field'' is a real
 +
valued random function $X(t)$, $t\in\mathbb R^n$, $E|X(t)|^2<+\infty$, where the expectation and the correlation function have properties:
 +
$EX(t)=EX(gt)$ and
 +
$EX(t)X(s)=EX(gt)X(gs)$ for all rotations $g$ around the origin of
 +
coordinates. An isotropic random field $X(t)$ admits the
 +
decomposition
 +
$$ X(t)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}
 +
X_m^l(r)S_m^l(\theta_1,\theta_2,\dots,\theta_{n-2},\varphi),
 +
$$
 +
where $(r,\theta_1,\theta_2,\dots,\theta_{n-2},\varphi)$ are spherical coordinates of the point $t\in\mathbb R^n$,
 +
$S_m^l(\theta_1,\theta_2,\dots,\theta_{n-2},\varphi)$ are spherical harmonics of the degree $m$, $h(m,n)$ is the amount of such harmonics,
 +
$X_m^l(r)$ are uncorrelated stochastic processes such that
 +
$E(X_m^l(r) X_{m_1}^{l_1}(s))=b_m(r,s)\delta_{m}^{m_1}\delta_l^{l_1}$, where
 +
$\delta_i^j$ is the Kronecker symbol, $b_m(r,s)$ is a sequence of positive definite kernels such that $\sum_{m=0}^{\infty}h(m,n)b_m(r,s)<+\infty$,
 +
$b_m(0,s)=0,m\not=0$.
 +
 
 +
Isotropic random field $X(t)$, $t\in\mathbb R^2$, on the plane admits the decomposition
 +
$$ X(r,\varphi)=\sum_{m=0}^{\infty}
 +
\left\{X_m^1(r)\cos(m\varphi)+X_m^2(r)\sin(m\varphi)\right\}.
 +
$$
 +
The class of isotropic random fields includes homogeneous and isotropic random fields, multiparametric Brownian motion processes.
 +
 
 +
 
 +
 
 +
''Homogeneous and isotropic random field'' is a real
 +
valued random function $X(t)$, $t\in\mathbb R^n$, $E|X(t)|^2<+\infty$, where the expectation $EX(t)=c=\text{const}$ and the correlation function
 +
$EX(t)X(s)=B(|t-s|)$ depends on the distance $|t-s|$ between points $t$ and $s$.
 +
Homogeneous and isotropic random field $X(t)$ and its correlation function $B(r)$ admit the spectral representations [7,8,9]
 +
$$
 +
X(t)=c_n\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}
 +
S_m^l(\theta_1,\theta_2,\dots,\theta_{n-2},\varphi)
 +
\int_0^{\infty}\frac{J_{m+(n-2)/2}(r\lambda)}{(r\lambda)^{(n-2)/2}}
 +
Z_m^l(d\lambda),
 +
$$
 +
$$ B(r)=
 +
\int_0^{\infty}Y_{n}(r\lambda)d\Phi(\lambda),
 +
$$
 +
where
 +
$$
 +
Y_n(x)=2^{(n-2)/2}\Gamma\left(\frac{n}{2}\right)\frac{J_{(n-2)/2}(x)}{x^{(n-2)/2}}
 +
$$
 +
is a spherical Bessel function, $\Phi(\lambda)$ is a bounded
 +
nondecreasing function called the spectral function the field
 +
$X(t)$, $Z_m^l(d\lambda)$ are random measures with orthogonal
 +
values such that
 +
$E(Z_m^l(S_1)Z_{m_1}^{l_1}(S_2))=\delta_m^{m_1}\delta_l^{l_1}\Phi(S_1\cap
 +
S_2)$, $c_n^2=2^{n-1}\Gamma(n/2)\pi^{n/2}$.
 +
 
 +
 
 +
Homogeneous and isotropic random field $X(t)$, $t\in\mathbb R^2$, on the plane admits the spectral representation
 +
$$ X(t,\varphi)=
 +
\sum_{m=0}^{\infty}
 +
\cos(m\varphi)Y_m(r\lambda)Z_m^1(d\lambda)+
 +
\sum_{m=1}^{\infty}
 +
\sin(m\varphi)Y_m(r\lambda)Z_m^2(d\lambda).
 +
$$
 +
These spectral decompositions of random fields form a power tool for solution of statistical problems for random fields such as extrapolation,
 +
interpolation, filtering, estimation of parameters of the distribution.
 +
 
 +
 
 +
''Estimation problems for random fields'' $X(t)$, $t\in\mathbb R^n$
 +
(estimation of the unknown mathematical expectation, estimation of the correlation function,
 +
estimation of regression parameters, extrapolation, interpolation, filtering, etc)
 +
are similar to the corresponding problems for stochastic processes
 +
(random fields of dimension 1). Complications usually are caused by the form
 +
of domain of points $\{t_j\}=D\subset\mathbb R^n $,
 +
where observations $\{X(t_j)\}$ are given and the dimension of the field.
 +
The complications are overcoming by considering specific domains of
 +
observations and particular classes of random fields.
 +
 
 +
Let in the domain $D\subset\mathbb R^n$ there are given observations of the
 +
random field
 +
$$
 +
X(t)=\sum_{i=1}^q\theta_ig_i(t)+Y(t),
 +
$$
 +
where $g_i(t),i=1,\dots,q,$ are known non-random functions, $\theta_i,i=1,\dots,q,$
 +
are unknown parameters, $Y(t)$ is a random field with $EY(t)=0$.
 +
The problem is to estimate the regression parameters $\theta_i,i=1,\dots,q$.
 +
This problem includes as a particular case $(q=1,g_1(t)=1)$ the problem of
 +
estimation of the unknown mathematical expectation. Linear unbiased least
 +
squares estimates of the regression parameters can be found by solving the
 +
corresponding linear algebraic equations or linear integral equations
 +
determined with the help of the correlation function.
 +
For the class of isotropic random fields formulas for estimates of
 +
the regression parameters are proposed by M. I. Yadrenko (Yadrenko 1983). For example,
 +
the estimate $\hat{\theta}$ of the unknown mathematical expectation ${\theta}$
 +
of an isotropic random field $X(t)=X(r,u)$ from observations on the sphere
 +
$S_n(r)=\{x\in\mathbb R^n,\|x\|=r\}$ is of the form
 +
$$
 +
\hat{\theta}=\frac{1}{\omega_n}\int_{S_n(r)}X(r,u)m_n(du), n\geq2,
 +
$$
 +
where $m_n(du)$ is the Lebesgue measure on the sphere $S_n(r)$,
 +
$\omega_n$ is the square of the surface of the sphere,
 +
$(r,u)$ are spherical coordinates of the point $t\in\mathbb R^n$.
 +
 
 +
 
 +
Consider the extrapolation problem. 1. Let observations of the
 +
mean-square continuous homogeneous and isotropic random field
 +
$X(t)$, $t\in\mathbb R^n$, are
 +
given on the sphere $S_n(r)=\{x\in\mathbb R^n,\|x\|=r\}$.
 +
The problem is to determine the optimal mean-square linear estimate
 +
$\hat{X}(s)$ of the unknown value $X(s)$, $s\not\in S_n(r)$, of the
 +
random field.
 +
It follows from the spectral representation of the field that this
 +
estimate is of the form
 +
$$
 +
\hat{X}(s)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}
 +
c_m^l(s)
 +
\int_0^{\infty}\frac{J_{m+(n-2)/2}(r\lambda)}{(r\lambda)^{(n-2)/2}}
 +
Z_m^l(d\lambda),
 +
$$
 +
where
 +
coefficients $c_m^l(s)$ are determined by a special algorithm (Yadrenko 1983).
 +
For practical purposes it is more convenient to have a formula where
 +
observations $X(t)$, $t\in S_n(r)$, are used directly.
 +
The composition theorem for spherical harmonics gives us this opportunity.
 +
We can write
 +
$$
 +
\hat{X}(s)=\int_{S_n(r)}c(s,t)X(t)dm_n(t),
 +
$$
 +
where the function $c(s,t)$ is determined by the spectral function
 +
$\Phi(\lambda)$ of the field $X(t)$ [6].
 +
 
 +
 
 +
2. Let an isotropic random field $X(t)$, $t=(r,u)\in\mathbb R^n$, is
 +
observed in the sphere $V_R=\{x\in\mathbb R^n,\|x\|\leq R\}$.
 +
The optimal liner estimate
 +
$\hat{X}(s)$ of the unknown value $X(s)$, $s=(\rho,v)\not\in V_R$, of the
 +
field has the form
 +
$$
 +
\hat{X}(s)=\int_{V_R}C(s,t)X(t)dm_n(t),
 +
$$
 +
$$
 +
C(s,t)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}
 +
c_m^l(r)S_m^l(u),
 +
$$
 +
where coefficients $c_m^l(r)$ are determined via special integral equations
 +
$$
 +
b_m(\rho,q)S_m^l(v)=\int_0^Rb_m(r,q)c_m^l(r)r^{n-1}dr,\quad\, m=0,1,\dots;\,\,l=1,2,\dots,h(m,n),\,
 +
q\in[0,R].
 +
$$
 +
If, for example, $X(t),t=(r,u),$ is an isotropic random field where
 +
$b_m(r,q)=a^{|m|}\exp\{-\beta|r-q|\}$,
 +
then it is easy to see that
 +
$\hat{X}(\rho,v)=\exp\{-\beta|\rho-R|\}X(R,v),\,v\in S_n$.
 +
 
 +
 
 +
 
 +
For methods of solutions of other estimation problems for random
 +
fields (extrapolation, interpolation, filtering, etc) see Cressie (1991), Grenander (1981),
 +
Moklyachuk (2008), Ramm (2005), Ripley (1981), Rozanov (1982),
 +
(Yadrenko 1983) and Yaglom (1987).
 +
 
 +
<!-- \vskip5mm -->
  
it is also possible to define the concepts of the locally homogeneous (and locally homogeneous and locally isotropic) random fields related to stochastic processes with stationary increments (cf. [[Stochastic process with stationary increments|Stochastic process with stationary increments]]); see [[#References|[10]]], . Such fields play an important role in the statistical theory of turbulence (see, for example, [[#References|[1]]], [[#References|[9]]]).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.S. Monin,  A.M. Yaglom,   "Statistical fluid mechanics" , '''1–2''' , M.I.T. (1971–1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.P. Khusu,  Yu.R. Vitenberg,  V.A. Pal'mov,   "Surface roughness (a probabilistic approach)" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.J. Hannan,   "Group representations and applied probability" , Methuen  (1965)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.I. Yadrenko,  "Spectral theory of random fields" , Optim. Software  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.A. Rozanov,   "Markov random fields" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B. Simon,   "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077300/r07730046.png" /> Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> K. Preston,   "Gibbs states on countable sets" , Cambridge Univ. Press  (1974)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> R.L. Dobrushin (ed.) Ya.G. Sinai (ed.) , ''Multi-component random systems'' , M. Dekker  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> I.M. Gel'fand,  N.Ya. Vilenkin,   "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> V.A. Malyshev,   R.A. Minlos,  "Gibbs random fields" , Kluwer  (1990)  (Translated from Russian)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|bookAd}}||valign="top"| Adler, Robert J. and Taylor, Jonathan E. (2007). ''Random fields and geometry''. Springer Monographs in Mathematics. New York, NY: Springer.
 +
|-
 +
|valign="top"|{{Ref|bookCh}}||valign="top"|  Chung, Kai Lai and Walsh, John B. (2005). ''Markov processes, Brownian motion, and time symmetry''. 2nd ed. Grundlehren der Mathematischen Wissenschaften 249. New York: Springer.  
 +
|-
 +
|valign="top"|{{Ref|bookCr}}||valign="top"| Cressie, Noel A.C. (1991). ''Statistics for spatial data''. Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons, Inc.
 +
|-
 +
|valign="top"|{{Ref|bookGl}}||valign="top"|  Glimm, James and Jaffe, Arthur. (1981). ''Quantum physics''. A functional integral point of view. New York - Heidelberg - Berlin: Springer-Verlag.
 +
|-
 +
|valign="top"|{{Ref|bookGren}}||valign="top"| Grenander, Ulf (1981). ''Abstract inference''. Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons.
 +
|-
 +
|valign="top"|{{Ref|bookKer}}||valign="top"| Kerstan, Johannes; Matthes, Klaus and Mecke, Joseph. (1974). ''Mathematische Lehrb&uuml;cher und Monographien''. II. Abt. Mathematische Monographien. Band XXVII. Berlin: Akademie-Verlag.
 +
|-
 +
|valign="top"|{{Ref|bookKhos}}||valign="top"| Khoshnevisan, Davar (2002). ''Multiparameter processes. An introduction to random fields''. Springer Monographs in Mathematics. New York, NY: Springer.  
 +
|-
 +
|valign="top"|{{Ref|bookMal}}||valign="top"| Malyshev, V. A. and Minlos, R. A. (1985). ''Stochastic Gibbs fields''. The method of cluster expansions. Moskva: "Nauka".
 +
|-
 +
|valign="top"|{{Ref|bookMok}}||valign="top"| Moklyachuk, M. P. (2008). ''Robust estimates of functionals of stochastic processes''. Vydavnycho-Poligrafichny\u \i Tsentr, Ky\"\i vsky\u\i Universytet, Ky\"\i v.
 +
|-
 +
|valign="top"|{{Ref|bookMona}}||valign="top"|  Monin, A. S. and Yaglom, A. M. (2007a). ''Statistical fluid mechanics: mechanics of turbulence''. Vol. I. Edited and with a preface by John L. Lumley. Mineola, NY: Dover Publications.  
 +
|-
 +
|valign="top"|{{Ref|bookMonb}}||valign="top"| Monin, A. S. and Yaglom, A. M. (2007b). ''Statistical fluid mechanics: mechanics of turbulence''. Vol. II. Edited and with a preface by John L. Lumley. Mineola, NY: Dover Publications.
 +
|-
 +
|valign="top"|{{Ref|bookPit}}||valign="top"| Piterbarg, V.I. (1996). ''Asymptotic methods in the theory of Gaussian processes and fields''. Translations of Mathematical Monographs. 148. Providence, RI: AMS.  
 +
|-
 +
|valign="top"|{{Ref|bookRamm}}||valign="top"| Ramm, A. G. (2005). ''Random fields estimation''. Hackensack, NJ: World Scientific.  
 +
|-
 +
|valign="top"|{{Ref|bookRipl}}||valign="top"| Ripley, B. D. (1981). ''Spatial statistics''. Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons, Inc.
 +
|-
 +
|valign="top"|{{Ref|bookRoz}}||valign="top"|  Rozanov, Yu. A. (1982). ''Markov random fields''. New York - Heidelberg - Berlin: Springer-Verlag.
 +
|-
 +
|valign="top"|{{Ref|bookYadr}}||valign="top"| Yadrenko, M. I. (1983). ''Spectral theory of random fields''. Translation Series in Mathematics and Engineering. New York: Optimization Software, Inc., Publications Division; New York-Heidelberg-Berlin: Springer-Verlag.  
 +
|-
 +
|valign="top"|{{Ref|bookYagla}}||valign="top"|  Yaglom, A. M. (1987a). ''Correlation theory of stationary and related random functions''. Volume I: Basic results. Springer Series in Statistics. New York etc.: Springer-Verlag.
 +
|-
 +
|valign="top"|{{Ref|bookYaglb}}||valign="top"| Yaglom, A. M. (1987b). ''Correlation theory of stationary and related random functions''. Volume II: Supplementary notes and references. Springer Series in Statistics. New York etc.: Springer-Verlag.
 +
|-
 +
|}
 +
 
 +
 
 +
<!-- \vskip5mm -->
  
 +
Reprinted with permission from Lovric, Miodrag (2011), International
 +
Encyclopedia of Statistical Science. Heidelberg: Springer Science
 +
+Business Media, LLC
  
 +
<!-- \vskip5mm -->
  
====Comments====
+
Classification
For Gibbs and Markov fields see also [[#References|[a2]]]–[[#References|[a3]]]. The estimation theory of random fields is discussed in [[#References|[a4]]]–[[#References|[a5]]]. For limit theorems concerning random fields cf. [[#References|[a5]]].
 
  
====References====
+
AMS MSC:62M40;60G60
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Adler,  "The geometry of random fields" , Wiley  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Albeverio,  J.E. Fenstad,  R. Høegh-Krohn,  T. Lindstrøm,  "Non standard methods in stochastic analysis and mathematical physics" , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.-O. Georgii,  "Gibbs measures and phase transitions" , de Gruyter  (1988)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,  "Random fields: estimation theory" , Longman &amp; Wiley  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.V. Ivanov,  N.N. Leonenko,  "Statistical analysis of random fields" , Kluwer  (1989)  (Translated from Russian)</TD></TR></table>
+
 
 +
<!-- \end{document} -->
 +
 
 +
<references />
 +
 
 +
[[Category:Statprob]]

Revision as of 20:20, 12 March 2016

Copyright notice
This article Random field was adapted from an original article by Mikhail Moklyachuk, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original article ([http://statprob.com/encyclopedia/RandomField6.html StatProb Source], Local Files: pdf | tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics, and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb.

2020 Mathematics Subject Classification: Primary: 62M40 Secondary: 60G60 [MSN][ZBL]


{Department of Probability Theory, Statistics and Actuarial Mathematics,}
{Kyiv National Taras Shevchenko University, Ukraine}
{Email: mmp\@ univ.kiev.ua}


Keywords and Phrases: Random field; Kolmogorov Existence Theorem; Gaussian random field; Wiener sheet; Brownian sheet; Poisson random field; Markov random field; Homogeneous random field; Isotropic random field; Spectral decomposition


Random field $X(t)$ on $D\subset\mathbb R^n$ (i.e. $t\in D\subset\mathbb R^n$) is a function whose values are random variables for any $t\in D$. The dimension of the coordinate is usually in the range from one to four, but any $n>0$ is possible. A one-dimensional random field is usually called a stochastic process. The term 'random field' is used to stress that the dimension of the coordinate is higher than one. Random fields in two and three dimensions are encountered in a wide range of sciences and especially in the earth sciences such as hydrology, agriculture, and geology. Random fields where $t$ is a position in space-time are studied in turbulence theory and in meteorology.

Random field $X(t)$ is described by its finite-dimensional (cumulative) distributions $$F_{t_1,\dots,t_k}(x_1,\dots,x_k)=P\{X(t_1)<x_1,\dots,X(t_k)<x_k\}, k=1,2,\dots$$ The cumulative distribution functions are by definition left-continuous and nondecreasing. Two requirements on the finite-dimensional distributions must be satisfied. The symmetry condition $$F_{t_1,\dots,t_k}(x_1,\dots,x_k)=F_{t_{\pi1},\dots,t_{\pi k}}(x_{\pi1},\dots,x_{\pi k}),$$ $\pi$ is a permutation of the index set $\{1,\dots,k\}$. The compatibility condition $$F_{t_1,\dots,t_{k-1}}(x_1,\dots,x_{k-1})= F_{t_1,\dots,t_k}(x_1,\dots,x_{k-1},\infty).$$

Kolmogorov Existence Theorem states: If a system of finite-dimensional distributions $F_{t_1,\dots,t_k}(x_1,\dots,x_k)$, $k=1,2,\dots$ satisfies the symmetry and compatibility conditions, then there exists on some probability space a random field $X(t)$, $t\in D$, having $F_{t_1,\dots,t_k}(x_1,\dots,x_k)$, $k=1,2,\dots$ as its finite-dimensional distributions.

The expectation (mean value) of a random field is by definition the Stieltjes integral $$ m(t)=EX(t)=\int_{\mathbb R^1}xdF_t(x). $$ The (auto-)covariance function is also expressed as the Stieltjes integral $$ B(t,s)=E(X(t)X(s))-m(t)m(s)=\iint_{\mathbb R^2}xydF_{t,s}(x,y)-m(t)m(s), $$ whereas the variance is $\sigma^2(t)=B(t,t)$.


Gaussian random fields play an important role for several reasons: the specification of their finite-dimensional distributions is simple, they are reasonable models for many natural phenomenons, and their estimation and inference are simple.

A Gaussian random field is a random field where all the finite-dimensional distributions are multivariate normal distributions. Since multivariate normal distributions are completely specified by expectations and covariances, it suffices to specify $m(t)$ and $B(t, s)$ in such a way that the symmetry condition and the compatibility condition hold true. The expectation can be arbitrarily chosen, but the covariance function must be positive definite to ensure the existence of all finite-dimensional distributions (Adler and Taylor 2007; Piterbarg 1996)


Wiener sheet (Brownian sheet) is a Gaussian random field $W(t)$, $t=(t_1,t_2)\in\mathbb R_+^2$ with $EW(t)=0$ and correlation function $$ B(t,s)=E(X(t)X(s))=\min\{s_1,t_1\}\min\{s_2,t_2\}.$$ Analogously, ($n$-parametric Wiener process) is a Gaussian random field $W(t)$, $t\in\mathbb R^n_+$ with $EW(t)=0$ and correlation function $ B(t,s)=\prod_{i=1}^n\min(s_i,t_i)$. Multiparametric Wiener process $W(t)$ has independent homogeneous increments. Generalized derivative of multiparametric Wiener process $W(t)$ is Gaussian white noise process on $\mathbb R^n_+$ (Chung and Walsh 2005; Khoshnevisan 2002).


Poisson random fields are also reasonable models for many natural phenomenon. A Poisson random field is an integer-valued (point) random field where the (random) amount of points which belong to a bounded set from the range of values of the field has a Poisson distribution and the random amounts of points which belong to nonoverlapping sets are mutually independent (Kerstan et al. 1974). Point-valued random fields (Poisson random fields, Cox random fields, Poisson cluster random fields, Markov point random fields, homogeneous and isotropic point random fields, marked point random fields) are appropriate mathematical models for geostatistical data. A mathematically elegant approach to analysis of point-valued random fields (spatial point processes) is proposed by Noel A.C. Cressie (Cressie 1991).


Markov random field $X(t)$, $t\in D\subset\mathbb R^n$, is a random function which has the Markov property with respect to a fixed system of ordered triples $(S_1,\Gamma,S_2)$ of nonoverlapping subsets from the domain of definition $D$. The Markov property means that for any measurable set $B$ from the range of values of the function $X(t)$ and every $t_0\in S_2$ the following equality holds true $$ P\{X(t_0)\in B\vert X(t),t\in S_1\cup\Gamma\}= P\{X(t_0)\in B\vert X(t),t\in \Gamma\}. $$ This means that the future $S_2$ does not depend on the past $S_1$ when the present $\Gamma$ is given. Let, for example, $D=\mathbb R^n$, $\{\Gamma\}$ be a family of all spheres in $\mathbb R^n$, $S_1$ be the interior of $\Gamma$, $S_2$ be the exterior of $\Gamma$. A homogeneous and isotropic Gaussian random field $X(t)$, $t\in\mathbb R^n$, has the Markov property with respect to the ordered triples $(S_1,\Gamma,S_2)$ if and only if $X(t)=\xi$, where $\xi$ is a random variable. Nontrivial examples of homogeneous and isotropic Markov random fields can be constructed when consider the generalized random fields. Markov random fields are completely described in the class of homogeneous Gaussian random fields on $\mathbb Z^n$, in the class of multidimensional homogeneous generalized Gaussian random fields on the space $\mathbb C_0^{\infty}(\mathbb R^m)$ and the class of multidimensional homogeneous and isotropic generalized Gaussian random fields (Glimm and Jaffe 1981; Rozanov 1982; Yadrenko 1983).


Gibbs random fields form a class of random fields that have extensive applications in solutions of problems in statistical physics. The distribution functions of these fields are determined by Gibbs distribution (Malyshev and Minlos 1985).


Homogeneous random field in the strict sense is a real valued random function $X(t)$, $t\in\mathbb R^n$ (or $t\in\mathbb Z^n$), where all its finite-dimensional distributions are invariant under arbitrary translations, i.e. $$F_{t_1+s,\dots,t_k+s}(x_1,\dots,x_k)=F_{t_1,\dots,t_k}(x_1,\dots,x_k)\,\forall s\in\mathbb R^n.$$


Homogeneous random field in the wide sense is a real valued random function $X(t)$, $t\in\mathbb R^n$ ($t\in\mathbb Z^n$), $E|X(t)|^2<+\infty$, where $EX(t)=m=\text{const}$ and the correlation function $EX(t)X(s)=B(t-s)$ depends on the difference $t-s$ of coordinates of points $t$ and $s$.

Homogeneous random field $X(t)$, $t\in\mathbb R^n$, $EX(t)=0$, $E|X(t)|^2<+\infty$, and its correlation function $ B(t)=EX(t+s)X(s)$ admit the spectral representations $$ X(t)=\int\cdots\int\exp\left\{\sum_{k=1}^nt_k\lambda_k\right\}Z(d\lambda), $$ $$ B(t)=\int\cdots\int\exp\left\{\sum_{k=1}^nt_k\lambda_k\right\}F(d\lambda), $$ where $F(d\lambda)$ is a measure on the Borel $\sigma$-algebra $B_n$ of sets from $\mathbb R^n$, $Z(d\lambda)$ is an orthogonal random measure on $B_n$ such that $E(Z(S_1)Z(S_2))=F(S_1\cap S_2)$. The integration range is $\mathbb R^n$ in the case of continuous time random field $X(t)$, $t\in\mathbb R^n$ and $[-\pi,\pi]^n$ in the case of discrete time random field $X(t)$, $t\in\mathbb Z^n$. In the case where the spectral representation of the correlation function is of the form $$ B(t)= \int\cdots\int\exp\left\{\sum_{k=1}^nt_k\lambda_k\right\}f(\lambda)d\lambda, $$ the function $f(\lambda)$ is called spectral density of the field $X(t)$. Based on these spectral representations we can prove, for example, the law of large numbers for random field $X(t)$:

The mean square limit $$ \lim_{N\to\infty}\frac{1}{(2N+1)^n} \sum_{|t_i|\leq N,i=1,\dots,n}X(t)=Z\{0\}. $$ This limit is equal to $EX(t)=0$ if and only if $E|Z\{0\}|^2=F\{0\}$. In the case where $F\{0\}=0$ and $$ \int_{-\pi}^{\pi}\cdots\int_{-\pi}^{\pi} \prod_{i=1}^n \log\left|\log\frac{1}{|\lambda_i|}\right|F(d\lambda)<+\infty, $$ the strong law of large numbers holds true for the random field $X(t)$.


Isotropic random field is a real valued random function $X(t)$, $t\in\mathbb R^n$, $E|X(t)|^2<+\infty$, where the expectation and the correlation function have properties: $EX(t)=EX(gt)$ and $EX(t)X(s)=EX(gt)X(gs)$ for all rotations $g$ around the origin of coordinates. An isotropic random field $X(t)$ admits the decomposition $$ X(t)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)} X_m^l(r)S_m^l(\theta_1,\theta_2,\dots,\theta_{n-2},\varphi), $$ where $(r,\theta_1,\theta_2,\dots,\theta_{n-2},\varphi)$ are spherical coordinates of the point $t\in\mathbb R^n$, $S_m^l(\theta_1,\theta_2,\dots,\theta_{n-2},\varphi)$ are spherical harmonics of the degree $m$, $h(m,n)$ is the amount of such harmonics, $X_m^l(r)$ are uncorrelated stochastic processes such that $E(X_m^l(r) X_{m_1}^{l_1}(s))=b_m(r,s)\delta_{m}^{m_1}\delta_l^{l_1}$, where $\delta_i^j$ is the Kronecker symbol, $b_m(r,s)$ is a sequence of positive definite kernels such that $\sum_{m=0}^{\infty}h(m,n)b_m(r,s)<+\infty$, $b_m(0,s)=0,m\not=0$.

Isotropic random field $X(t)$, $t\in\mathbb R^2$, on the plane admits the decomposition $$ X(r,\varphi)=\sum_{m=0}^{\infty} \left\{X_m^1(r)\cos(m\varphi)+X_m^2(r)\sin(m\varphi)\right\}. $$ The class of isotropic random fields includes homogeneous and isotropic random fields, multiparametric Brownian motion processes.


Homogeneous and isotropic random field is a real valued random function $X(t)$, $t\in\mathbb R^n$, $E|X(t)|^2<+\infty$, where the expectation $EX(t)=c=\text{const}$ and the correlation function $EX(t)X(s)=B(|t-s|)$ depends on the distance $|t-s|$ between points $t$ and $s$. Homogeneous and isotropic random field $X(t)$ and its correlation function $B(r)$ admit the spectral representations [7,8,9] $$ X(t)=c_n\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)} S_m^l(\theta_1,\theta_2,\dots,\theta_{n-2},\varphi) \int_0^{\infty}\frac{J_{m+(n-2)/2}(r\lambda)}{(r\lambda)^{(n-2)/2}} Z_m^l(d\lambda), $$ $$ B(r)= \int_0^{\infty}Y_{n}(r\lambda)d\Phi(\lambda), $$ where $$ Y_n(x)=2^{(n-2)/2}\Gamma\left(\frac{n}{2}\right)\frac{J_{(n-2)/2}(x)}{x^{(n-2)/2}} $$ is a spherical Bessel function, $\Phi(\lambda)$ is a bounded nondecreasing function called the spectral function the field $X(t)$, $Z_m^l(d\lambda)$ are random measures with orthogonal values such that $E(Z_m^l(S_1)Z_{m_1}^{l_1}(S_2))=\delta_m^{m_1}\delta_l^{l_1}\Phi(S_1\cap S_2)$, $c_n^2=2^{n-1}\Gamma(n/2)\pi^{n/2}$.


Homogeneous and isotropic random field $X(t)$, $t\in\mathbb R^2$, on the plane admits the spectral representation $$ X(t,\varphi)= \sum_{m=0}^{\infty} \cos(m\varphi)Y_m(r\lambda)Z_m^1(d\lambda)+ \sum_{m=1}^{\infty} \sin(m\varphi)Y_m(r\lambda)Z_m^2(d\lambda). $$ These spectral decompositions of random fields form a power tool for solution of statistical problems for random fields such as extrapolation, interpolation, filtering, estimation of parameters of the distribution.


Estimation problems for random fields $X(t)$, $t\in\mathbb R^n$ (estimation of the unknown mathematical expectation, estimation of the correlation function, estimation of regression parameters, extrapolation, interpolation, filtering, etc) are similar to the corresponding problems for stochastic processes (random fields of dimension 1). Complications usually are caused by the form of domain of points $\{t_j\}=D\subset\mathbb R^n $, where observations $\{X(t_j)\}$ are given and the dimension of the field. The complications are overcoming by considering specific domains of observations and particular classes of random fields.

Let in the domain $D\subset\mathbb R^n$ there are given observations of the random field $$ X(t)=\sum_{i=1}^q\theta_ig_i(t)+Y(t), $$ where $g_i(t),i=1,\dots,q,$ are known non-random functions, $\theta_i,i=1,\dots,q,$ are unknown parameters, $Y(t)$ is a random field with $EY(t)=0$. The problem is to estimate the regression parameters $\theta_i,i=1,\dots,q$. This problem includes as a particular case $(q=1,g_1(t)=1)$ the problem of estimation of the unknown mathematical expectation. Linear unbiased least squares estimates of the regression parameters can be found by solving the corresponding linear algebraic equations or linear integral equations determined with the help of the correlation function. For the class of isotropic random fields formulas for estimates of the regression parameters are proposed by M. I. Yadrenko (Yadrenko 1983). For example, the estimate $\hat{\theta}$ of the unknown mathematical expectation ${\theta}$ of an isotropic random field $X(t)=X(r,u)$ from observations on the sphere $S_n(r)=\{x\in\mathbb R^n,\|x\|=r\}$ is of the form $$ \hat{\theta}=\frac{1}{\omega_n}\int_{S_n(r)}X(r,u)m_n(du), n\geq2, $$ where $m_n(du)$ is the Lebesgue measure on the sphere $S_n(r)$, $\omega_n$ is the square of the surface of the sphere, $(r,u)$ are spherical coordinates of the point $t\in\mathbb R^n$.


Consider the extrapolation problem. 1. Let observations of the mean-square continuous homogeneous and isotropic random field $X(t)$, $t\in\mathbb R^n$, are given on the sphere $S_n(r)=\{x\in\mathbb R^n,\|x\|=r\}$. The problem is to determine the optimal mean-square linear estimate $\hat{X}(s)$ of the unknown value $X(s)$, $s\not\in S_n(r)$, of the random field. It follows from the spectral representation of the field that this estimate is of the form $$ \hat{X}(s)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)} c_m^l(s) \int_0^{\infty}\frac{J_{m+(n-2)/2}(r\lambda)}{(r\lambda)^{(n-2)/2}} Z_m^l(d\lambda), $$ where coefficients $c_m^l(s)$ are determined by a special algorithm (Yadrenko 1983). For practical purposes it is more convenient to have a formula where observations $X(t)$, $t\in S_n(r)$, are used directly. The composition theorem for spherical harmonics gives us this opportunity. We can write $$ \hat{X}(s)=\int_{S_n(r)}c(s,t)X(t)dm_n(t), $$ where the function $c(s,t)$ is determined by the spectral function $\Phi(\lambda)$ of the field $X(t)$ [6].


2. Let an isotropic random field $X(t)$, $t=(r,u)\in\mathbb R^n$, is observed in the sphere $V_R=\{x\in\mathbb R^n,\|x\|\leq R\}$. The optimal liner estimate $\hat{X}(s)$ of the unknown value $X(s)$, $s=(\rho,v)\not\in V_R$, of the field has the form $$ \hat{X}(s)=\int_{V_R}C(s,t)X(t)dm_n(t), $$ $$ C(s,t)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)} c_m^l(r)S_m^l(u), $$ where coefficients $c_m^l(r)$ are determined via special integral equations $$ b_m(\rho,q)S_m^l(v)=\int_0^Rb_m(r,q)c_m^l(r)r^{n-1}dr,\quad\, m=0,1,\dots;\,\,l=1,2,\dots,h(m,n),\, q\in[0,R]. $$ If, for example, $X(t),t=(r,u),$ is an isotropic random field where $b_m(r,q)=a^{|m|}\exp\{-\beta|r-q|\}$, then it is easy to see that $\hat{X}(\rho,v)=\exp\{-\beta|\rho-R|\}X(R,v),\,v\in S_n$.


For methods of solutions of other estimation problems for random fields (extrapolation, interpolation, filtering, etc) see Cressie (1991), Grenander (1981), Moklyachuk (2008), Ramm (2005), Ripley (1981), Rozanov (1982), (Yadrenko 1983) and Yaglom (1987).


References

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[bookMona] Monin, A. S. and Yaglom, A. M. (2007a). Statistical fluid mechanics: mechanics of turbulence. Vol. I. Edited and with a preface by John L. Lumley. Mineola, NY: Dover Publications.
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[bookPit] Piterbarg, V.I. (1996). Asymptotic methods in the theory of Gaussian processes and fields. Translations of Mathematical Monographs. 148. Providence, RI: AMS.
[bookRamm] Ramm, A. G. (2005). Random fields estimation. Hackensack, NJ: World Scientific.
[bookRipl] Ripley, B. D. (1981). Spatial statistics. Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons, Inc.
[bookRoz] Rozanov, Yu. A. (1982). Markov random fields. New York - Heidelberg - Berlin: Springer-Verlag.
[bookYadr] Yadrenko, M. I. (1983). Spectral theory of random fields. Translation Series in Mathematics and Engineering. New York: Optimization Software, Inc., Publications Division; New York-Heidelberg-Berlin: Springer-Verlag.
[bookYagla] Yaglom, A. M. (1987a). Correlation theory of stationary and related random functions. Volume I: Basic results. Springer Series in Statistics. New York etc.: Springer-Verlag.
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Reprinted with permission from Lovric, Miodrag (2011), International Encyclopedia of Statistical Science. Heidelberg: Springer Science +Business Media, LLC


Classification

AMS MSC:62M40;60G60


How to Cite This Entry:
Random field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Random_field&oldid=37788
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article