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{{MSC|62M40|60G60}}


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<center>'''Random field'''</center>

<center>Mikhail Moklyachuk</center>

<center>Department of Probability Theory, Statistics and
Actuarial Mathematics</center>

<center>Kyiv National Taras
Shevchenko University, Ukraine</center>

<center>Email: mmp@univ.kiev.ua</center>

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 ({{Cite|Ad|Adler and Taylor 2007}}; {{Cite|Pit|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 $\mathbb R^n_+$ ({{Cite|Ch|Chung and
Walsh 2005}}; {{Cite|Khos|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 ({{Cite|Ker|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 ({{Cite|Cr|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 ({{Cite|Gl|Glimm
and Jaffe 1981}}; {{Cite|Roz|Rozanov 1982}}; {{Cite|Yadr|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 ({{Cite|Mal|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 ({{Cite|Yadr|Yadrenko 1983}})
$$
X(t)=c_n\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}
S_m^l(\theta_1,\theta_2,\dots,\theta_{n-2},\varphi)
\int_0^{\infty}\frac{J_{m+(n-2)/2}(r\lambda)}{(r\lambda)^{(n-2)/2}}
Z_m^l(d\lambda),
$$
$$ B(r)=
\int_0^{\infty}Y_{n}(r\lambda)d\Phi(\lambda),
$$
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 ({{Cite|Yadr|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 ({{Cite|Yadr|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 $X(t)$ ({{Cite|Yadr|Yadrenko 1983}}).


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 {{Cite|Cr|Cressie (1991)}}, {{Cite|Gren|Grenander (1981)}},
{{Cite|Mok|Moklyachuk (2008)}}, {{Cite|Ramm|Ramm (2005)}}, {{Cite|Ripl|Ripley (1981)}}, {{Cite|Roz|Rozanov (1982)}},
{{Cite|Yadr|Yadrenko (1983)}}, and {{Cite|Yagla|Yaglom (1987)}}.

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====References====
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|}


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Reprinted with permission from Lovric, Miodrag (2011), International
Encyclopedia of Statistical Science. Heidelberg: Springer Science
+Business Media, LLC

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Classification

AMS MSC:62M40;60G60

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[[Category:Statprob]]