Interpolation Problems for Random Fields from Observations in Perforated Plane

Interpolation Problems for Random Fields from Observations in Perforated Plane

The problem of estimation of linear functionals which depend on the unknown values of a homogeneous random field ξ(k, j) in the region K ⊂ Z² from observations of the sum ξ(k, j)+η(k, j) at points (k, j)  Z²\K is investigated. Formulas for calculating the mean square errors and the spectral char...

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Дата:2016
Автори: Moklyachuk, M.P., Shchestyuk, N.Yu., Florenko, A.S.
Формат: Стаття
Мова:English
Опубліковано: Інститут кібернетики ім. В.М. Глушкова НАН України 2016
Назва видання:Математичне та комп'ютерне моделювання. Серія: Технічні науки
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/133755
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Цитувати:Interpolation Problems for Random Fields from Observations in Perforated Plane / M.P. Moklyachuk, N.Yu. Shchestyuk, A.S. Florenko // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 83-97. — Бібліогр.: 19 назв. — англ.

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spelling irk-123456789-1337552018-06-06T03:03:58Z Interpolation Problems for Random Fields from Observations in Perforated Plane Moklyachuk, M.P. Shchestyuk, N.Yu. Florenko, A.S. The problem of estimation of linear functionals which depend on the unknown values of a homogeneous random field ξ(k, j) in the region K ⊂ Z² from observations of the sum ξ(k, j)+η(k, j) at points (k, j)  Z²\K is investigated. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case where the spectral densities are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given. Досліджується задача оцінювання лінійних функціоналів від невідомих значень однорідного випадкового поля ξ(k, j) для області K ⊂ Z² за спостереженями суми полів ξ(k, j)+η(k, j) в точках (k, j)  Z²\K. Знайдено формули для обчислення середньоквадра- тичної похибки та спектральної характеристики оптимальної лінійної оцінки функціола у випадку відомих спектральних щільностей полів. Запропоновано формули для визначення найменш сприятливої спектральної щільності та мінімаксної (робастної) спектральної характеристики у випадку, коли спектральна характеристика точно не відома, але клас спектральних характеристик, до якого належить спектральна щільність визначено. 2016 Article Interpolation Problems for Random Fields from Observations in Perforated Plane / M.P. Moklyachuk, N.Yu. Shchestyuk, A.S. Florenko // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 83-97. — Бібліогр.: 19 назв. — англ. 2308-5916 http://dspace.nbuv.gov.ua/handle/123456789/133755 519.21 en Математичне та комп'ютерне моделювання. Серія: Технічні науки Інститут кібернетики ім. В.М. Глушкова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The problem of estimation of linear functionals which depend on the unknown values of a homogeneous random field ξ(k, j) in the region K ⊂ Z² from observations of the sum ξ(k, j)+η(k, j) at points (k, j)  Z²\K is investigated. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case where the spectral densities are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.
format Article
author Moklyachuk, M.P.
Shchestyuk, N.Yu.
Florenko, A.S.
spellingShingle Moklyachuk, M.P.
Shchestyuk, N.Yu.
Florenko, A.S.
Interpolation Problems for Random Fields from Observations in Perforated Plane
Математичне та комп'ютерне моделювання. Серія: Технічні науки
author_facet Moklyachuk, M.P.
Shchestyuk, N.Yu.
Florenko, A.S.
author_sort Moklyachuk, M.P.
title Interpolation Problems for Random Fields from Observations in Perforated Plane
title_short Interpolation Problems for Random Fields from Observations in Perforated Plane
title_full Interpolation Problems for Random Fields from Observations in Perforated Plane
title_fullStr Interpolation Problems for Random Fields from Observations in Perforated Plane
title_full_unstemmed Interpolation Problems for Random Fields from Observations in Perforated Plane
title_sort interpolation problems for random fields from observations in perforated plane
publisher Інститут кібернетики ім. В.М. Глушкова НАН України
publishDate 2016
url http://dspace.nbuv.gov.ua/handle/123456789/133755
citation_txt Interpolation Problems for Random Fields from Observations in Perforated Plane / M.P. Moklyachuk, N.Yu. Shchestyuk, A.S. Florenko // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 83-97. — Бібліогр.: 19 назв. — англ.
series Математичне та комп'ютерне моделювання. Серія: Технічні науки
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AT shchestyuknyu interpolationproblemsforrandomfieldsfromobservationsinperforatedplane
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first_indexed 2025-07-09T19:34:37Z
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fulltext Серія: Технічні науки. Випуск 14 83 UDC 519.21 M. P. Moklyachuk*, D-r of Phys. and Mathem. Sci., Professor, N. Yu. Shchestyuk**, Cand. of Phys. and Mathem. Sci., A. S. Florenko**, Graduate Student * Taras Shevchenko National University of Kyiv, Kyiv, **National University «Kyiv-Mohila Academy», Kyiv INTERPOLATION PROBLEMS FOR RANDOM FIELDS FROM OBSERVATIONS IN PERFORATED PLANE The problem of estimation of linear functionals which depend on the unknown values of a homogeneous random field ( , )k j in the region 2K Z from observations of the sum ( , ) ( , )k j k j  at points   2, \k j Z K is investigated. Formulas for calculating the mean square errors and the spectral characteristics of the opti- mal linear estimate of functionals are derived in the case where the spectral densities are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spec- tral characteristics are proposed in the case where the spectral den- sities are not exactly known while a class of admissible spectral densities is given. Key words: random fields, estimation problem, minimax (ro- bust) spectral characteristic. Introduction. In engineering, geology, automatic control etc., one often has a number of data points, obtained by sampling which represents values of stochastic processes or random fields for a limited number of values of the independent variable. It is often required to interpolate (i.e. estimate) the value of processes or fields for an intermediate value of the independent variable. Methods of solution of linear estimation problems for stochastic processes and random fields were developed by A. N. Kol- mogorov [1], N. Wiener [2], A. M. Yaglom [3, 4], M. I. Yadrenko [5]. Traditional methods of solution of these problems are employed under the condition that spectral densities are known exactly. In practice, however, complete information on spectral densities is impossible in most cases. To solve the problem the parametric or nonparametric estimates of the un- known spectral densities are found or these densities are selected by other reasoning. Then the traditional estimation methods are applied provided that the estimated or selected densities are the true one. This procedure can result in a significant increasing of the value of the error as K. S. Vastola and H. V. Poor [6] have demonstrated with the help of some examples. This is a reason to search estimates which are optimal for all densities © M. P. Moklyachuk, N. Yu. Shchestyuk, A. S. Florenko, 2016 Математичне та комп’ютерне моделювання 84 from a certain class of the admissible spectral densities. These estimates are called minimax since they minimize the maximal value of the error. Many investigators have been interested in minimax extrapolation, inter- polation and filtering problems for stationary stochastic processes and ran- dom fields. A survey of results in minimax (robust) methods of data proc- essing can be found in the paper by S. A. Kassam and H. V. Poor [7]. M. P. Moklyachuk [8–11] proposed the minimax approach to extrapola- tion, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes and sequences. In papers by M. P. Moklyachuk and N. Yu. Shchestyuk [12–16], and M. P. Mokly- achuk and S. V. Tatarinov [17] problems of extrapolation, interpolation and filtering of functionals which depend on the unknown values of ho- mogeneous random fields were investigated. M. P. Moklyachuk and M. Sidey considerated the minimax approach to interpolation problems for functionals which depend on the unknown values of stationary sequences from observations with the missing intervals. In this paper we deal with the problem of estimation of the unknown values of a mean-square homogeneous random field from observations on the perforated plane. Optimal linear estimates. spectral densities are known. Let ( , )k j be a homogeneous random field on 2Z . It means that  , 0M k j  ,   2 ,M k j   and      , , ,B k n j m M k j n m    depends only on difference  ,k n j m  . The correlation function  ,B k n j m  of the homogeneous random field of discrete arguments has the following spectral representation ( )( , ) ( , ) ( , ) ( , )i k jB k n j m M k j n m e F d d                   , where ( , )F d d  is the spectral measure of the field on [ ; ) [ ; )      . If this measure is absolutely continuous with respect to Lebesgue measure, then the correlation function can be represented in the form ( )( , ) ( , ) ,i k jB k j e f d d                where ( , )f   is the spectral density of the field. Let the sum ( , ) ( , )k j k j  of the homogeneous random fields are observed in all points of perforated plane   2, \k j Z K . Perforations are small holes in a thin material or web. In our case the holes are rectangles Серія: Технічні науки. Випуск 14 85 x ym m . Assume that the number of rectangles on horizontal is xs and number of rectangles on vertical is ys (see picture 1). So we can formally define the set of perforations of the plane as   1 2 1 20 0 yx ss yx t t t t K m m      . Let 1 2 ... x x x s xm m m   . Denote ,x x xl = m + n y y yl = m + n , where ,xn yn are distances between the rectangles on horizontal and vertical coordinates correspondently. Fig. 1. The problem is to find the optimal in the mean square sense esti- mate KA   of the linear functional           21 1 2 1 2 1 11 1 , 0 0 , , , , y y yx x x x y s t l ms t l m K k l K t t k t l j t l A a k l k l a k j k j                           (1) which depend on the unknown values of the field  , ,k j ( , )k j K from observations of the sum ( , ) ( , ),k j k j  where   2, \k j Z K ,   1 2 1 20 0 yx ss yx t t t t K m m      . This problem is reduced to the optimization problem: 2 min .K KM A A      Математичне та комп’ютерне моделювання 86 Denote by 2 ( )L f f  the Hilbert space of complex valued functions on [ , ) [ , )      which are square integrated with respect to measure with the density ( , ) ( , )f f     . Denote by  2 KL f f    the subspace of  2L f f  , generated by functions  i u ve   ,   2, \u Z K  . Every linear estimate KA   of the unknown functional KA ξ  is of the form       , , , ,KA h Z d d Z d d                    (2) where  ,Z d d   and  ,Z d d   are orthogonal random measures of the fields ( , )u v and ( , )u v correspondingly, and ( , )h   is the spectral characteristic of the estimate KA   . We will suppose that the condition of «minimality» holds true     1 . , , d d f f                 (3) The spectral characteristic ( , )h   of the optimal linear estimate KA   is determined by the conditions [1]: 1) 2( , ) ( )Kh L f f     ; 2)   2( , ) ( , ) ( )K KA h L f f        . These conditions will be used to derive formulas for the mean square errors and spectral characteristic of the optimal linear estimate KA ξ  . From condition 2) we have       2( , ) ( , ) . , \i k j KA h e k j Z K         , where             21 1 2 1 2 11 , 0 0 , , , y y yx x x x y s t l ms t l m i k j i k j K k j K t t k t l j t l A a k j e a k j e                             . This condition can be represented in the form:                 , 2 , , , , 0, , \ . i k j KA f h f f e d d k j Z K                              From the indicated condition we conclude that               , , , , , , ,K KA f h f f C                Серія: Технічні науки. Випуск 14 87            21 1 2 1 2 1 11 1 , 0 0 , , , y y yx x x x y s t l ms t l m i k j i k j K k j K t t k t l j t l C c k j e c k j e                               , where ( , )c k j are unknown coefficients which we need to determine. From the derived relation we get the following formula for the spec- tral characteristic ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) K K A f C h f f f f                          . From condition 1) we have ( )( , ) 0i u vh e d d                for 2( , )u v K Z  , and we get that for any 2( , )u v K Z  :       2 21 1 1 2 1 2 1 2 1 2 21 1 2 1 2 1 1 1 11 1 1 1 0 0 0 0 1 11 1 0 0 , , , y y y y y yx x x x x x x y x y y y yx x x x y s t l m s t l ms t l m s t l m t t k t l j t l t t k t l j t l i ks t l ms t l m t t k t l j t l a k j a k j e c k j                                                                             0. u j v d d f f                          Let us represent the functions       , , , f f f          ,     1 , ,f f     in the Fourier series:               1 , , , , , , i p q pq p q i l t lt l t b e f f f d e f f                                      (4) and obtain the system for solving ,u vc , where ( , )k j K :                 , , , , 0, , . i u v i l t uv lt u v K l t i m n i p q i k j mn pq m n K p q a e d e C e b e e d d k j K                                                               Математичне та комп’ютерне моделювання 88     2 21 1 1 2 1 2 1 2 1 2 1 1 1 11 1 1 1 , , 0 0 0 0 , , . y y y y y yx x x x x x x y x y s t l m s t l ms t l m s t l m K K k u j v k u j v t t k t l j t l t t k t l j t l d a k j d c k j                                              From this system we derive equations for all ( , )k j K ; or in the matrix form K KD a B c   , where a  is a vector composed from coefficients determining KA  , c  is a vector composed from the unknown coefficients ( , )c k j , ( , )k j K , ,K KD B are operators, which are determined by matrices    , , , ,lt lt kj kjD d l k t j B b l k t j      (5) with elements that are the Fourier coefficients of the functions       , , , f f f          ,     1 , ,f f     correspondingly. So the un- known coefficients ( , )c k j can be computed by formula       1 ( , ) , , ,K K K k j c k j B D a k j K       (6) and the spectral characteristic of the optimal linear estimate KA   may be calculated by the formula                 21 1 2 1 2 1 1 11 1 ( ) ,0 0 , , ( , ) , , . , , y y yx x x y K s s t l mt l mx K K i k j K k jt t k t l j t l A f h f f B D a e f f                                                 (7) The mean square error is calculated by the formula                  2 2 2 2 ; , 1 , , , , 4 1 , , , 4 K K K h f f A Z d d hZ d d A Z d d hZ d d A h f h f d d                                                      where  ,K KA A   ,  ,h h   . Making use of the form (7) of the spectral characteristic we can get formula for calculating the mean square error of the optimal linear estimate KA   : Серія: Технічні науки. Випуск 14 89 ( ; , )h f f   (8)                        21 1 2 1 2 21 1 2 2 1 1 11 1 ( ) ,0 0 2 2 1 1 ( ) , 2 , , 1 , 4 , , , , 1 4 y y yx x x y y yx x y s s t l mt l mx K K i k j K K k jt t k t l j t l t l mt l K K i k j K K k jk t l j t l A f B D a e f d d f f A f B D a e                                                                                    1 2 2 1 1 1 0 0 2 , , , , y x s s mx t t f d d f f                                     Proposition 1. The spectral characteristic ( , )h   and the mean square error ( ; , )h f f  of the optimal linear estimate KA   of the func- tional KA  from observations of the field ( , ) ( , )u v u v  at points   2, \ ,u Z K    1 2 1 20 0 yx ss yx t t t t K m m      are calculated by formulas (7), (8). Corollary 1. The spectral characteristic ( , )h   and the mean square error ( )f of the optimal linear estimate KA   of the functional KA  from observations of the field ( , )u v at points   2, \u Z K  ,  1 2 1 20 0 yx ss yx t t t t K m m      are calculated by formulas     21 1 2 1 2 1 1 11 1 ( ) ,0 0 ( , ) ( , ) ( , ) y y yx x x x y K s s t l mt l m K K i k j K k jt t k t l j t l h A B D a e f                                   (9)     21 1 2 1 2 2 1 1 11 1 ( ) ,0 0 2 ( ) , 1 ( , )4 y y yx x x x y s s t l mt l m K K i k j K k jt t k t l j t l f d dB D a e f                                         (10) where KB is operator determined by the matrix with elements ( , )lt kjB b l k t j   that are the Fourier coefficients of the function 1 ( , )f   . Математичне та комп’ютерне моделювання 90 Example 1. (perforated plane with 3 holes). Suppose that we ob- serve the random field ( , )u v without noise at points   2, \u Z K  , where K is a union of 3 holes which are rectangles 3х2. Suppose that the dis- tance between the rectangles is equal to 3. So, 3xs  , 1ys  , 3xm  , 2ym  , 3xn  , 6x x xl m n   . Suppose that the spectral density of the field can be represented in the form 2 2 1 2 0 1 2( , ) ( ) ( ) / i if f f B e e          , 1 21, 1   , where 2 1 0 1( ) / if B e    , 2 2 0 2( ) / if B e    . The problem is to estimate the functional 1 1 1 6 22 1 ( , ) 0 6 0 ( , ) ( , ) ( , ) ( , ) t K k l K t k t j A a k l k l a k j k j                      2 1 8 1 14 1 0 0 6 0 12 0 ( , ) ( , ) ( , ) ( , ) ( , ) , ( , ). k j k j k j a k j k j a k j k j a k j k j               Let us extend the functions 1 ( , )f   in Fourier series:               2 2 0 2 2 0 2 2 2 2 2 ( ) ( ) 2 ( ) ( ) 1 / ( , ) (1/ ) (1/ ) 1 1 1 1 1 1 1 1 . i i i i i i i i i i i i i i f B e e B e e e e e e e e e e e e                                                                                 If we denote Fourier coefficients as        2 2 00 2 2 0,1 0, 1 1,0 1,0 1 1 , 1 , 1 , r G r r D r r A                     then ( ) ( ) ( ) ( ) 1 / ( , ) . i i i i i i i i f G De De Ae Ae e e e e                                     The unknown coefficients ( , )c k j , ( , )k j K can be computed from the equation Ka B c   , Серія: Технічні науки. Випуск 14 91 where   0,0 0,1 1,0 1,1 2,0 2,1 6,0 6,1 7,0 7,1 8,0 8,1 12,0 12,1 13,0 13,1 14,0 14,1 , , , , , , , , , , , , , , , , , a a a a a a a a a a a a a a a a a a a    0,0 0,1 1,0 1,1 2,0 2,1 12,0 12,1 13,0 13,1 14,0 14,1, , , , , ,... , , , , ,c c c c c c c c c c c c c  , KB is matrix from Fourier coefficients of 1 ( , )f   : 0 0 0 0 , 0 0 V V V           where 0 0 0 0 0 0 0 0 G D A D G A A G D A V A D G A A G D A D G                             . The spectral characteristic ( , )h   can be computed by (10) and fi- nally we get linear estimate of the functional KA  : ( , ) ( , )KA h Z d d                 2 2 2 2 2 1, 1, ,2 ,2 3, 3, , 1 , 1 5, 5, 1 0 1 0 1 j j i i j j i i j j j i j i j                             8 2 8 2 ,2 ,2 9, 9, , 1 , 1 11, 11, 6 1 6 1 i i j j i i j j i j i j                      14 2 14 ,2 ,2 15, 15, , 1 , 1 12 1 12 ,i i j j i i i j i                where 1, 1 0.0 1,0 0.0 0.1 1,1 0.1 0.0 1,2 0.1 0,2 0,1 1,1 , , , , , c Ac c Ac c c Dc c                             1,2 1,1 2,1 0,1 2,2 2,1 1,1 3,2 2,1 3, 1 2,0 , , , , Dc c c Dc c c c                     Математичне та комп’ютерне моделювання 92 3,1 2,1 2,0 3,0 2,0 2,1 2, 1 2,0 1,0 0, 1 0,0 1,0 , , , , Ac c Ac c Dc c Dc c                      1, 1 5, 1 5,0 (1, 0) (2, 0) (0, 0), (6, 0), (6, 0) (6,1), Dc c c c Ac c                   5,2 5,1(6,1), (6,1) (6,0),c Ac c        6,2 7,2(6,1) (7,1), (7,1) (8,1) (6,1),Dc c Dc c c           8,2 8,1 7,1Dc c    , 9,2 8,1 9.0 8,0 8,1 9,1 8,1 8,0 9, 1 8,0, , , ,c Ac c Ac c c                 6, 1 7, 1 8, 1 (6,0) (7,0), (7,0) (8,0) (6,0), (8,0) (7,0) Dc c Dc c c Dc c                     11, 1 11,0 11,1 11,2 (12,0), (12,0) (12,1), (12,1) (12,0), (12,1), c Ac c Ac c c                    12,2 13,2 14,2 (12,1) (13,1), (13,1) (14,1) (12,1), (14,1) (13,1), Dc c Dc c c Dc c                  15,2 15,1 15,0 15, 1 (14,1), (14,1) (14,0), (6,1), (14,0), c Ac c c c                  12, 1 13, 1 14, 1 (12,0) (13,0), (13,0) (14,0) (12,0), (14,0) (13,0). Dc c Dc c c Dc c                     We can see that for estimating the functional were used only values in the neighbouring points (dots). It is naturally because we consider ran- dom field of the first order autoregression type for each argument. The mean square error ( ; , )h f f  may be calculated by the formula (9) after calculating vector c . Minimax (robust) approach to estimation problem. Formulas (1)– (10) can be applied to compute the spectral characteristic and the mean square error of the optimal linear estimate KA   of the functional KA  from observations of the field ( , ) ( , )u v u v  at points   2, \u Z K  ,   1 2 1 20 0 yx ss yx t t t t K m m      if the spectral densities f f  , f g  are ex- actly known. In the case where spectral densities f f  , f g  are not known exactly but sets  , f fD D D      of possible spectral densities Серія: Технічні науки. Випуск 14 93 are given we apply the minimax (robust) approach to estimate the func- tional KA  . With the help of this approach we can find an estimate that minimizes the mean square error for all spectral densities from a given class simultaneously. Such estimates are called minimax (robust). Definition 1.1. For a given class of spectral densities f gD D D  the spectral densities 0 0( , ) , ( , )f gf D g D     are called the least fa- vourable in f gD D D  for the optimal linear estimation of the func- tional KA  if        0 0 0 0 0 0 ( , ) , , ; , max , ; , f gf g D f g h f g f g h f g f g D       . Definition 1.2. For a given class of spectral densities f gD D D  the spectral characteristic  0 ,h   of the optimal linear estimation of the KA  is called minimax (robust) if the following conditions holds true  0 2 ( , ) , ( ), f g N D f g D h H L f g D          0 ( , ) ( , ) min max ( ; , ) max ( ; , ). Dh H f g D f g D h f g h f g       Spectral densities 0 0( , ), ( , )f g    are the least favourable in f gD D D  for the optimal linear estimation of the functional KA  , if Fourier coefficients (4), that correspond to the densities 0 0( , ), ( , )f g    determine operators 0 0,K KD B by matrices (5) which give a solution to the constrained optimization problem      0 0 0 0 0 0 ( , ) sup , ; , , ; , , f g D f g h f g f g h f g f g D     (11) where                         21 1 2 1 2 2 2 0 0 2 1 1 11 1 ( ) 0 ,0 0 2 2 0 0 1 1 ( ) 0 , 2 , ; , , , 1 , 4 ( , ) , , , 1 4 y y yx x x x y y y y s s t l mt l m K K i k j K K k jt t k t l j t l t l m K K i k j K K k jj t l h f g f g A g B D a e f d d f g A f B D a e                                                                             1 1 2 1 2 1 1 1 0 0 2 0 0 , ( , ) , yx x x x s s t l m t t k t l g d d f g                                    The constrained optimization problem (11) is equivalent to the un- constrained optimization problem Математичне та комп’ютерне моделювання 94        0 0, , ; , , | infD f gf g h f g f g f g D D      , (12) where   , | f gf g D D  is the indicator function of the set .f gD D A solution  0 0,f g of the problem (12) is characterized by the condition  0 00 ,D f g , where  0 0,D f g is the subdifferential of the convex functional  ,D f g at point  0 0,f g . This condition gives us a possibility to determine the least favourable spectral densities for concrete classes of spectral densities. Least favorable densities in the class 1 22 2( , ) ( , )D D D      . Consider the problem for the class of spectral densities 1 22 2( , ) ( , )D D D      , where       1 2 2 1 12 1 ( , ) , | ( , , ) 4 D f f u d d                               ,       2 2 2 2 22 1 ( , ) , | ( , , ) 4 D g g u d d                               . Lema 1. Let  ,u   be a non-negative function for  ,     ,  ,     , let 0  and       2 2 2 1 , | ( , , ) 4 D f f u d d                             . Then the subdifferential of the indicator function  2|f D  can be represented in the form             2 02 0 2 2 0 02 1 0 , ( , , ) , 4 ( | ) 1 , ( , , ) , 4 f u d d f D f u d d                                              where      0 02 1 ( ) ( , , ) , . 4 f f u f d d                    This lemma is a corollary of results proved in paper. We can apply the condition  0 00 ,D f g for this class of spectral densities and have the following statement. Серія: Технічні науки. Випуск 14 95 Theorem 1. Let spectral densities 1 20 2 0 2( , ) , ( , )f D g D      satisfy condition (3) and suppose that functions  0 0,fh f g ,  0 0,gh f g , computed by the formula             0 0 0 0 0 , , , , , , K K f A g C h f g f g              , (13)             0 0 0 0 0 , , , , , , K K g A f C h f g f g              (14) are bounded. Spectral densities 1 20 2 0 2( , ) , ( , )f D g D      are the least favorable in the class 1 22 2D D  for the optimal linear estimation of the functional KA  , if they satisfy relations 1 2 2 2 1 2 1 2 2 2 11 1 1 1 0 0 0 00 0 0 0 0 0 1 1 (( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ( , ) ( , )) ( ), yx x x x ss t l m t l m K K K t t k t l j t l K B D a g A f g f g f u                                              1 2 2 2 1 2 1 2 2 2 11 1 1 1 0 0 0 00 0 0 0 0 0 2 2 (( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ( , ) ( , )) ( ) yx x x x ss t l m t l m K K K t t k t l j t l K B D a f A f g f g g u                                             and determine a solution of the extremum problem (11), where 1 2( ) 0, ( ) 0     can be computed by conditions     2 0 1 12 1 ( , , ) 4 f u d                 ,     2 0 2 22 1 ( , , ) . 4 g u d                 The minimax (robust) spectral characteristic  0 0,h f g may be cal- culated by formula (7). 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Sidesh // Statistics, Optimization & In- formation Computing. — 2015. — Vol. 3, № 3. — Р. 259–275. Досліджується задача оцінювання лінійних функціоналів від неві- домих значень однорідного випадкового поля ( , )k j для області 2K Z за спостереженями суми полів ( , ) ( , )k j k j  в точках   2, \k j Z K . Знайдено формули для обчислення середньоквадра- тичної похибки та спектральної характеристики оптимальної лінійної оцінки функціола у випадку відомих спектральних щільностей полів. Запропоновано формули для визначення найменш сприятливої спект- ральної щільності та мінімаксної (робастної) спектральної характери- стики у випадку, коли спектральна характеристика точно не відома, але клас спектральних характеристик, до якого належить спектральна щільність визначено. Ключові слова: випадкове поле, задача оцінювання, мінімаксна (робастна) спектральна характеристика. 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