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Estimation of random fields. (English) Zbl 1026.62100
Theory Probab. Math. Stat. 66, 105-118 (2003) and Teor. Jmovirn. Mat. Stat. 66, 95-107 (2002).
In this paper some of the results are reviewed and some generalizations are mentioned from the author’s book, Random fields estimation theory. Harlow: Longman (1990; Zbl 0712.47042), and its expanded Russian edition from 1996. The paper deals with random fields estimation problems within the framework of covariance theory. No assumptions about distribution laws are made: the fields are not necessarily Gaussian or Markovian, the only information used is the covariance functions.
This work can be viewed as an extension of Wiener’s filtering theory. The statement of the problem is the same as in Wiener’s theory, but the author considers random functions of several variables, that is random fields, while Wiener (and many later researchers) studied estimation theory for stationary random processes, that is random functions of one variable. The analytical theory used by Wiener was the theory of Wiener-Hopf equations. Later Wiener’s theory was extended to the case of a finite interval of observations. A review of this theory with many references was proposed by T. Kailath [IEEE Trans. Inf. Theory 20, 146–181 (1974; Zbl 0307.93040)]; see also the collection of papers edited by T. Kailath, Linear last-squares estimation. Benchmark Papers in Electrical Engineering and Computer Science, V. 17 (1977).
Assume the random field is of the form $$u(x)=s(x)+n(x),\;x\in\mathbb R^r$$, where $$s(x),\;Es(x)=0,$$ is a signal and $$n(x),\;En(x)=0,$$ is a noise with known covariance functions $$Eu(x)u(y)=R(x,y)$$ and $$Eu(x)s(y)=f(x,y)$$. Let $$D\subset\mathbb R^r$$ be a bounded domain with a sufficiently smooth boundary $$\Gamma$$. Let the observations of the field $$u(x)$$ in the domain $$D$$ be given. The problem is to estimate the value $$As(x),\;x\in\mathbb R^r$$, where $$A$$ is a given operator and $$x$$ is a given point. The best possible linear estimate by the criterion of minimum variance of the error of the estimate is of the form $$\int_Dh(x,y)u(y)\,dy$$, where $$h(x,y)$$ is a distributional kernel determined by the equation $$\int_DR(z,y)h(x,y)u(y)\,dy=f(z,x)$$.
The basic topic of this article is the study of a class of such equations for which the analytical properties of the solutions $$h$$ can be obtained, numerical procedures for computing $$h$$ can be given, and properties of the operator $$R$$ can be studied. The answers to the above questions are given for the class of random fields whose covariance functions $$R(x,y)$$ are kernels of positive rational functions of self-adjoint elliptic operators in $$L_2(\mathbb R^r)$$. The class $$\mathcal R$$ of such kernels consists of kernels $R(x,y)= \int_{\Lambda}P(\lambda)Q^{-1}(\lambda)\Phi(x,y,\lambda)\,d\rho(\lambda),$ where $$\Lambda$$, $$d\rho(\lambda)$$, $$\Phi(x,y,\lambda)$$ are, respectively, the spectrum, spectral measure, and spectral kernel of an elliptic self-adjoint operator $$\mathcal L$$ in $$L_2(\mathbb R^r)$$ of order $$s$$, and $$P(\lambda)$$ and $$Q(\lambda)$$ are positive polynomials of degree $$p$$ and $$q$$, respectively.
Examples of one-dimensional kernels for various operators are discussed in the author’s book cited above, where a general theory of one-dimensional equations is given. A generalization of this class of kernels consists of kernels which satisfy the equation $$QR=P\delta(x-y)$$, where $$Q$$ and $$P$$ are elliptic, not necessarily self-adjoint or commuting formal differential operators.

##### MSC:
 62M40 Random fields; image analysis 60G60 Random fields 62M15 Inference from stochastic processes and spectral analysis 47N30 Applications of operator theory in probability theory and statistics 62M09 Non-Markovian processes: estimation 60G35 Signal detection and filtering (aspects of stochastic processes) 45A05 Linear integral equations 45H05 Integral equations with miscellaneous special kernels 45P05 Integral operators