Minimax-robust filtering of functionals from periodically correlated random fields

Abstract: The problem of optimal estimation of linear functionals depending on unknown values of periodically correlated random field from observations of the field with noise is considered. Formulas for calculating mean square errors and spectral characteristics of optimal linear estimates of the functionals are derived in the case where spectral densities are exactly known. Formulas that determine least favourable spectral densities and minimax (robust) spectral characteristics are proposed in the case where spectral densities are not exactly known but a class of admissible spectral densities is given.


PUBLIC INTEREST STATEMENT
Cosmological Principle (first formulated by Einstein): the Universe is, in the large, homogeneous and isotropic. Last decades indicate growing interest to the spatio-temporal data measured on the surface of a sphere. These data include cosmic microwave background (CMB) anisotropies, medical imaging, global and land-based temperature data, gravitational and geomagnetic data, and climate model. Periodically correlated processes and fields are not homogeneous but have numerous properties similar to properties of stationary processes and fields. They describe appropriate models of numerous physical and man-made processes. In this article, we considered the problem of optimal estimation of functionals depending on unknown values of periodically correlated spatial temporal isotropic random fields from observations of the field with noise in the case of spectral certainty where spectral densities are known exactly as well as in the case of spectral uncertainty where spectral densities are not known exactly but a class of admissible spectral densities is given. Formulas that determine least favourable spectral densities and minimax (robust) spectral characteristics are derived.

Introduction
Cosmological Principle (first formulated by Einstein): the Universe is, in the large, homogeneous and isotropic (Bartlett, 1999). Last decades indicate growing interest to the spatio-temporal data measured on the surface of a sphere. These data include cosmic microwave background anisotropies (Adshead & Hu, 2012;Bartlett, 1999;Hu & Dodelson, 2002;Kogo & Komatsu, 2006;Okamoto & Hu, 2002), medical imaging (Kakarala, 2012), global and land-based temperature data (Jones, 1994;Subba Rao & Terdik, 2006), gravitational and geomagnetic data and climate model (North & Cahalan, 1981). Some basic results and references on the theory of isotropic random fields on a sphere can be found in the books by Yadrenko (1983) and Yaglom (1987). For more recent applications and results see new books by Cressie and Wikle (2011), Gaetan and Guyon (2010), Marinucci and Peccati (2011) and several papers covering a number of problems in general for spatial-time observations (Subba Rao & Terdik, 2012;Terdik, 2013).
Periodically correlated processes and fields are not homogeneous but have numerous properties similar to properties of stationary processes and fields. They describe appropriate models of numerous physical and man-made processes. A comprehensive list of the existing references up to the year 2005 on periodically correlated processes and their applications was proposed by Serpedin, Panduru, Sari, and Giannakis (2005). See also a review by Antoni (2009). For more details see a survey paper by Gardner (1994) and book by Hurd and Miamee (2007). Note that in the literature periodically correlated processes are named in multiple different ways such as cyclostationary, periodically nonstationary or cyclic correlated processes.
Among the current trends of the theory of stochastic processes and fields important is the direction which focuses on the problem of estimation of unknown values of random processes and fields. The problem of estimation of random processes and fields includes interpolation, extrapolation and filtering problems.
The mean square optimal estimation problems for periodically correlated with respect to time isotropic on sphere random fields are natural generalization of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes and homogeneous random fields. Effective methods of solution of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes were developed under the condition of certainty where spectral densities of processes and fields are known exactly [see, for example, selected works of Kolmogorov (1992), survey article by Kailath (1974), books by Rozanov (1967), Wiener (1966), Yaglom (1987), Yadrenko (1983) and articles by Yadrenko (1979, 1980)].
Particularly relevant in recent years is the problem of estimation of values of processes and fields under uncertainty where spectral densities of processes and fields are not known exactly. Such problems arise when considering problems of automatic control theory, coding and signal processing in radar and sonar, pattern recognition problems of speech signals and images.
The classical approach to the problems of interpolation, extrapolation and filtering of stochastic processes and random fields is based on the assumption that the spectral densities of processes and fields are known. In practice, however, complete information about the spectral density is impossible in most cases. To overcome this complication one finds parametric or nonparametric estimates of the unknown spectral densities or select these densities by other reasoning. Then applies the classical estimation method provided that the estimated or selected density is the true one. This procedure can result in a significant increasing of the value of error as Vastola and Poor (1983) have demonstrated with the help of some examples. This is a reason to search estimates which are optimal for all densities from a certain class of admissible spectral densities. These estimates are called minimax since they minimize the maximal value of the error of estimates. A survey of results in minimax (robust) methods of data processing can be found in the paper by Kassam and Poor (1985). The paper by Grenander (1957) should be marked as the first one where the minimax approach to extrapolation problem for stationary processes was proposed. For more details see, for example, books by Moklyachuk (2008), Moklyachuk and Masyutka (2012), Golichenko and Moklyachuk (2014).
In papers by Dubovets'ka, Masyutka, and , Dubovets'ka and Moklyachuk (2013Moklyachuk ( , 2014a, the minimax-robust estimation problems (extrapolation, interpolation and filtering) are investigated for the linear functionals which depend on unknown values of periodically correlated stochastic processes. Methods of solution of the minimax-robust estimation problems for time-homogeneous isotropic random fields on a sphere were developed by Moklyachuk (1994Moklyachuk ( , 1995Moklyachuk ( , 1996. In papers by Dubovets'ka, Masyutka, and Moklyachuk ( , 2015 results of investigation of minimax-robust estimation problems for periodically correlated isotropic random fields are described. In this article, we considered the problem of optimal linear estimation of the functional which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S n in Euclidean space n random field (j, x), j ∈ Z, x ∈ S n . Estimates are based on observations of the field (j, x) + (j, x) at points (j, x), j = −1, −2, …, x ∈ S n , where (j, x) is an uncorrelated with (t, x) periodically correlated with respect to time isotropic on the sphere S n random field. Formulas are derived for computing the value of the mean square error and the spectral characteristic of the optimal linear estimate of the functional A in the case of spectral certainty where spectral densities of the fields are known. Formulas are proposed that determine the least favourable spectral densities and minimax-robust spectral characteristic of the optimal estimate of the functional A in the case of spectral uncertainty where spectral densities are not known exactly, but classes D = D f × D g of admissible spectral densities are given.
We use the Kolmogorov (see, e.g. Kolmogorov, 1992) Hilbert space projection method based on properties of Fourier coefficients of the inverse to spectral density matrices. While in the paper by Dubovets'ka, Masyutka and Moklyachuk (2014a) the filtering problem is investigated with the help of the method based on factorization of spectral density matrices.

Spectral properties of periodically correlated random fields
Let S n be a unit sphere in the n-dimensional Euclidean space n , let m n (dx) be the Lebesgue measure on S n , and let be the orthonormal spherical harmonics of degree m (Müller, 1998).
A mean square continuous random field (j, x), j ∈ ℤ, x ∈ S n is called periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S n if where cos⟨x, y⟩ = (x, y) is the 'angular' distance between points x, y ∈ S n . This random field can be represented in the form where are mutually uncorrelated periodically correlated stochastic sequences with the correlation func- The correlation function of the random field (t, x) can be represented as follows: where n = (2 ) n∕2 Γ(n∕2) and C l m (z) are the Gegenbauer polynomials (Müller, 1998).
It follows from the Gladyshev (1961) (see also Makagon, 2011) results that the stochastic sequence This relation can also be expressed as and these two density matrices satisfy the relation

Hilbert space projection method of filtering
Consider the problem of mean square optimal linear estimation of the unknown value of the functional which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S n in Euclidean space E n random field (j, x), j ∈ Z, x ∈ S n . Estimates are based on observations of the field (j, is an uncorrelated with (j, x) periodically correlated with respect to time isotropic on the sphere S n random field which has the representation In this representation are mutually uncorrelated periodically correlated stochastic sequences with the correlation functions b m (j, k): k=0 are vector-valued stationary sequences generating the periodically correlated sequences l m (j).
Assume that the function a(j, x) which determines the functional which satisfy the following condition: Condition (Equation 2) ensures convergence of the series representation (Equation 1) of the functional A as well as finiteness of the second moment of the functional: E|A | 2 < ∞.
In this article, we use the Kolmogorov (1992) Hilbert space projection method based on properties of Fourier coefficients of the inverse to spectral density matrices. With the help of the proposed method we can find formulas for calculation the mean square error Δ(h;F, G) and the spectral characteristic h(F, G) of the optimal linear estimate Â of the functional A under the condition that spectral densities F m ( ), G m ( ) are known and satisfy the minimality condition (Equation 3). Following the method we found the optimal linear estimate Â as projection of A on the closed linear sub- are operators determined by the following block-matrices Let us summarize our results and present them in the form of a statement.
Theorem 3.1 Let the function a(j, x) which determines the functional A satisfy conditions (Equation 2). Let (j, x), (j, x) be uncorrelated periodically correlated with respect to time isotropic on the sphere S n random fields which have spectral densities F m ( ), G m ( ) that satisfy the minimality condition (Equation 3). The value of the mean square error Δ(h;F, G) and the spectral characteristic h(F, G) of the optimal linear estimate of the functional A based on observations of the field (j, x) + (j, x) for j ⩽ 0, x ∈ S n , can be calculated by formulas 5 and 6.

Minimax-robust method of filtering
The proposed formulas may be employed under the condition that spectral densities F m ( ) and G m ( ) of the fields (j, x), (j, x) are exactly known. In the case where the densities are not known exactly but a set D = D F × D G of possible spectral densities is given the minimax (robust) approach to estimation of functionals of the unknown values of random fields is reasonable. Instead of searching an estimate that is optimal for a given spectral densities we find an estimate that minimizes the mean square error for all spectral densities from given class simultaneously.
Definition 1 Spectral densities F 0 ( ), G 0 ( ) are called least favourable in a given class D for the optimal linear estimation of the functional A if It follows from the relationships (Equations 1-6) that the next theorem holds true.  G);F, G).
The form of the functional Δ(h(F 0 , G 0 );F, G) is convenient for application the Lagrange method of indefinite multipliers for finding solution to the problem (Equation 9). Making use the method of Lagrange multipliers and the form of subdifferentials of the indicator functions we describe relations that determine least favourable spectral densities in some special classes of spectral densities (see books by Moklyachuk, 2008;Moklyachuk and Masyutka, 2012;Golichenko and Moklyachuk, 2014for more details).

Least favourable spectral densities in the class
following pairs of sets of spectral densities that give restrictions on the first moment and describe the "strip" model of spectral densities are given. The first pair is The second pair of sets of admissible spectral densities is The third pair of sets of admissible spectral densities is The fourth pair of sets of admissible spectral densities is Here V m ( ), U m ( ) are given matrices of spectral densities, p, q, p k , q k , k = 1, T are given numbers, B 1 , B 2 , P, Q are given positive-definite Hermitian matrices.
Making use the method of Lagrange multipliers and the form of subdifferentials of the indicator functions of the sets D 0 and D U V (see Franke, 1985) we can conclude that the condition 0 For the first pair of sets of admissible spectral densities we have equations The following statement holds true.  (Equation 7), (Equation 14) and (Equation 15). The least favourable spectral densities F 0 m ( ), G 0 m ( ) in the fourth pair D 0 and D U V of spectral densities are determined by relations (Equation 7), (Equation 16) and (Equation 17). The minimax spectral characteristic of the optimal estimate of the functional A is calculated by formula 6.

Conclusions
In this paper, we study the filtering problem for functionals which depend on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S n in Euclidean space E n random field. The problem is considered in the case of spectral certainty where the matrices of spectral densities of random fields are known exactly and in the case of spectral uncertainty where matrices of spectral densities of random fields are not known exactly, but some restrictions on matrices are given which they must satisfy. We propose formulas for calculation the spectral characteristic and the mean square error of the optimal linear estimate of the functional which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S n random field (j, x) from observations of the field (j, x) + (j, x) at points (j, x), j ⩽ 0, x ∈ S n , where (j, x) is an uncorrelated with (j, x) periodically correlated with respect to time isotropic on the sphere S n random field provided that matrices of spectral densities F m ( ), G m ( ) of the vector-valued stationary sequences that generate the random fields (j, x), (j, x) are known exactly. We propose a representation of the mean square error in the form of linear functional in L 1 × L 1 with respect to spectral densities (F, G), which allows us to solve the corresponding conditional extremum problem and describe the minimax (robust) estimates of the functional. The least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional A are determined for some special classes of spectral densities.