Filtering Problems for Periodically Correlated Isotropic
Random Fields
Iryna Dubovets’ka
1,Oleksandr Masyutka
2,Mikhail Moklyachuk
1,∗1Department of Probability Theory, Statistics and Actuarial Mathematics,
Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine
2Department of Mathematics and Theoretical Radiophysics,
Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine
∗Corresponding Author: [email protected]
Copyright c⃝2014 Horizon Research Publishing All rights reserved.
Abstract
Spectral theory of isotropic random fields in Euclidean space developed by M. I. Yadrenko is exploited to find solution to the problem of optimal linear estimation of the functionalAζ= ∞
∑
j=0
∫
Sn
a(j, x)ζ(−j, x)mn(dx)
which depends on unknown values of a periodically cor-related (cyclostationary with period T) with respect to time isotropic on the sphere Sn in Euclidean space En
random fieldζ(j, x),j ∈ Z,x ∈ Sn. Estimates are based
on observations of the field ζ(j, x) + θ(j, x) at points (j, x), j = 0,−1,−2, . . ., x ∈ Sn, where θ(j, x) is an
uncorrelated withζ(j, x)periodically correlated with respect to time isotropic on the sphereSn random field. Formulas
for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional Aζ are obtained. The least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functionalAζ are determined for some special classes of spectral densities.
Keywords
Random Field, Filtering, Robust Estimate, Mean Square Error, Least Favorable Spectral Densities, Minimax Spectral Characteristic1
Introduction
Cosmological Principle (first coined by Einstein): the Uni-verse is, in the large, homogeneous and isotropic (Bartlett 1999). In the last two decades there has been some grow-ing interest in studygrow-ing the spatial–time data measured on the surface of a sphere. These data includes cosmic microwave background (CMB) anisotropies (Bartlett[1]; Hu and Dodel-son[2]; Kogo and Komatsu[3]), medical imaging (Kakarala [4]), global and land-based temperature data (Jones[5]), grav-itational and geomagnetic data, climate model (North and Cahalan[6]). Theory of isotropic random fields on a sphere has a long history. Some basic results and references can be
found in books by Yadrenko[7]and Yaglom[8]. Due to the expansive recent applications there are new books by Gae-tan and Guyon[9], Cressie and Wikle [10], Marinucci and Peccati[11] and several papers covering a number of prob-lems in general for spatial-time observations (see, for exam-ple, Subba Rao and Terdik[12], Terdik[13]).
Periodically correlated processes are those signals whose statistics vary almost periodically, and they are present in nu-merous physical and man-made processes. A comprehensive listing most of the existing references up to the year 2005 on periodically correlated processes and their applications was proposed by Serpedin at al. [14]. See also a review by An-toni[15]. For more details see a survey paper by Gardner[16] and book by Hurd and Miamee[17]. Note, that in the litera-ture, periodically correlated processes are named in multiple different ways such as cyclostationary, periodically nonsta-tionary or cyclic correlated processes.
The least square optimal estimation problems for period-ically correlated with respect to time isotropic on a sphere random fields are natural generalization of the linear extrap-olation, interpolation and filtering problems for stationary stochastic processes and homogeneous random fields. Ef-fective methods of solution of the linear extrapolation, inter-polation and filtering problems for stationary stochastic pro-cesses were developed by Kolmogorov[18], Wiener[19], Ya-glom[8]. The further results one can find in survey article by Kailath[20], books by Rozanov[21], Yadrenko [7], articles by Moklyachuk and Yadrenko [22].
they minimize the maximal value of the error. A survey of results in minimax (robust) methods of data processing can be found in the paper by Kassam and Poor[24]. The paper by Grenander[25] should be marked as the first one where the minimax approach to extrapolation problem for stationary processes was proposed. Franke[26]investigated the mini-max extrapolation problem for stationary sequences with the help of convex optimization methods. This approach makes it possible to find equations that determine the least favorable spectral densities for various classes of densities. For more details see, for example, book by Moklyachuk[27]. Meth-ods of solution the minimax-robust estimation problems for vector stationary sequences and processes were developed by Moklyachuk and Masyutka[28]. Luz and Moklyachuk [29] – [31] investigated the minimax estimation problems for lin-ear functionals which depend on unknown values of stochas-tic process with stationarynth increments. Dubovets’ka and Moklyachuk [32] – [35] investigated the minimax-robust es-timation problems (extrapolation, interpolation and filtering) for the linear functionals which depend on unknown values of periodically correlated stochastic processes. Methods of solution the minimax-robust estimation problems for time-homogeneous isotropic random fields on a sphere were devel-oped by Moklyachuk [36] – [41]. In the paper by Dubovetska at al. [42]investigation of minimax-robust estimation prob-lems for periodically correlated isotropic random fields was initiated.
In this article we considered the problem of least square optimal linear estimation of the functional
Aζ= ∞
∑
j=0
∫
Sn
a(j, x)ζ(−j, x)mn(dx)
which depends on unknown values of a periodically corre-lated (cyclostationary with period T) with respect to time isotropic on the sphere Sn in Euclidean space En random
field ζ(j, x), j ∈ Z, x ∈ Sn. Estimates are based on
observations of the field ζ(j, x) +θ(j, x) at points (j, x),
j = 0,−1,−2, . . ., x ∈ Sn, where θ(j, x)is an
uncorre-lated withζ(t, x)periodically correlated with respect to time isotropic on the sphere Sn random field. Formulas are
de-rived for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functionalAζin the case of spectral certainty where spectral densities of the fields are exactly known. Formulas are pro-posed that determine the least favourable spectral densities and the minimax-robust spectral characteristic of the optimal estimate of the functionalAζfor concrete classes of spectral densities under the condition that spectral densities are not exactly known, but classes D = DF ×DG of admissible
spectral densities are given.
2
Spectral properties of periodically
correlated with respect to time
isotropic on a sphere random fields
Let Sn be the unit sphere in n-dimensional Euclidean
space En, let m
n(dx) be the Lebesgue measure on Sn,
let Sl
m(x), x ∈ Sn, m = 0,1, . . .; l = 1, . . . , h(m, n),
be orthonormal spherical harmonics of degree m, and let
h(m, n) = (2m+n−2)(m+n−3)!/((n−2)!m!)be the
num-ber of linearly independent orthonormal spherical harmonics of degreem(M¨uller[43]).
A mean-square continuous random field ζ(j, x),j ∈ Z,
x ∈ Sn, is called periodically correlated (cyclostationary
with periodT) with respect to time isotropic on the sphere
Snif
Eζ(j+T, x) = Eζ(j, x) = 0, E|ζ(j, x)|2<∞,
E
(
ζ(j+T, x)ζ(k+T, y)
)
=B(j, k,cos⟨x, y⟩),
wherecos⟨x, y⟩= (x, y)is the “angular” distance between points x, y ∈ Sn. Since it is isotropic on the sphere Sn
this random fieldζ(j, x)can be represented in the form (Ya-drenko[7], Yaglom[8])
ζ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Sml (x)ζml (j),
ζml (j) =
∫
Sn
ζ(j, x)Sml (x)mn(dx),
whereζml (j), j ∈ Z, m = 0,1, . . .; l = 1, . . . , h(m, n),
are mutually uncorrelated periodically correlated stochastic sequences with the correlation functionbζm(j, k) :
E
(
ζml (j+T)ζv u(k+T)
)
=δmuδlv bζm(j, k),
m, u= 0,1, . . .; l, v= 1, . . . , h(m, n); j, k∈Z.
The correlation function of the random fieldζ(j, x)can be represented as follows
B(j, k,cos⟨x, y⟩) =
= 1
ωn
∞
∑
m=0
h(m, n)C (n−2)/2
m (cos⟨x, y⟩)
Cm(n−2)/2(1)
bζm(j, k),
where ωn = (2π)n/2Γ(n/2), Cml (z) are the Gegenbauer
polynomials (M¨uller[43]). Stochastic sequencesζl
m(j),j ∈ Z, are periodically
cor-related with periodT if and only if there existT-variate sta-tionary sequences (Gladyshev[44], Makagon[45])
⃗
ξlm(j) ={ξlmk(j)}Tk=0−1, j∈Z,
such thatζml (j)can be represented in the form
ζml (j) =
T∑−1
k=0
e2πikj/Tξmkl (j), j ∈Z. (1)
The sequences⃗ξl
m(j) = {ξlmk(j)} T−1
k=0 are called generating sequences of the periodically correlated sequencesζl
m(j).
Denote by Fm(dλ), m = 0,1, . . . , the matrix spectral
measure (distribution) function of theT-variable vector sta-tionary sequence⃗ξl
m(j) ={ξmkl (j)} T−1
k=0, resulting from the Gladyshev representation. Denote by Φm(dλ) the matrix
spectral measure function of the T-variable vector stationary sequence
⃗
ζml (j) ={ζmkl (j)}Tk=0−1,
[⃗ζml (t)]j=ζml (jT+k), j ∈Z, k= 0,1, . . . , T −1
arising from the splitting into blocks of lengthT the univari-ate periodically correlunivari-ated sequenceζl
Spectral matricesΦm(dλ)andFm(dλ)are connected by
the formula
Φm(dλ) =T·V(λ)Fm(dλ/T)V−1(λ),
whereV(λ)is a unitaryT×Tmatrix whose(k, j)-th element is of the form
vkj(λ) =
1
√
Te
2πijk/T+ijλ/T, k, j= 0,1, . . . , T −1.
The invertibility and continuity of V(λ) for λ ∈ [−π, π) means that the relation can also be expressed as
Fm(dλ) =
1
T ·V
−1(T λ)Φ
m(T dλ)V(T λ).
The matrix spectral measuresFm(dλ)andΦm(dλ)are
mu-tually absolutely continuous. Consequently, if there exists the spectral density matrixFξ⃗
m(λ)of the T-variate stationary
sequence⃗ξml (t), then there exists the spectral density matrix
Fmζ⃗(λ)of theT-variate stationary sequence⃗ζml (t)obtained from the splitting into blocks of lengthT the univariate pe-riodically correlated sequenceζml (t), and these two density
matrices satisfy the relation
Fm⃗ζ(λ) =T·V(λ)Fmξ⃗(λ/T)V−1(λ).
We will suppose in the following text that matrix spec-tral measuresFm(dλ), m = 0,1, . . . ,of the generating
se-quences ⃗ξl
m(j) = {ξmkl (j)} T−1
k=0 are absolutely continuous with respect to the Lebesgue measure and form a sequence of spectral density matricesF(λ) = {Fmη⃗(λ) :m = 0,1, . . .}
called the spectral density of the fieldζ(j, x).
3
Method of filtering based on
factor-ization of spectral densities
Consider the problem of mean square optimal linear esti-mation of the unknown value of the functional
Aζ= ∞
∑
j=0
∫
Sn
a(j, x)ζ(−j, x)mn(dx)
which depends on unknown values of a periodically corre-lated with respect to time isotropic on the sphereSn in
Eu-clidean spaceEn random fieldζ(j, x),j ∈Z,x∈Sn.
Esti-mates are based on observations of the fieldζ(j, x) +θ(j, x) at points(j, x),j =−1,−2, . . .,x∈Sn, where the ‘noise’
fieldθ(j, x)is an uncorrelated withζ(t, x)mean-square con-tinuous periodically correlated with respect to time isotropic on the sphereSnrandom field which has the representation
θ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Slm(x)θlm(j) =
= ∞
∑
m=0
h(∑m,n)
l=1
Sml (x)
T∑−1
k=0
e2πikj/Tηmkl (j),
θml (j) =
∫
Sn
θ(j, x)Slm(x)mn(dx).
In this representation θl
m(j), j ∈ Z, m = 0,1, . . .; l =
1, . . . , h(m, n), are mutually uncorrelated periodically cor-related stochastic sequences with the correlation function
bθm(j, k):
E
(
θlm(j+T)θv u(k+T)
)
=δmuδlv bθm(j, k),
m, u= 0,1, . . .; l, v= 1, . . . , h(m, n);j, k∈Z,
and ⃗ηl
m(j) = {ηmkl (j)} T−1
k=0 are vector-valued stationary sequences generating the periodically correlated sequences
θl
m(j). We will suppose that the matrix spectral
mea-suresG⃗η
m(dλ), m = 0,1, . . . ,of the generating sequences
⃗ ηl
m(j) = {ηlmk(j)} T−1
k=0 are absolutely continuous with re-spect to the Lebesgue measure and form a sequence of spec-tral density matrices G(λ) = {G⃗η
m(λ) : m = 0,1, . . .}
called the spectral density of the fieldθ(j, x).
Assume that the function a(j, x) which determines the functional
Aζ= ∞
∑
j=0
∫
Sn
a(j, x)ζ(−j, x)mn(dx) =
= ∞
∑
m=0
h(∑m,n)
l=1 ∞
∑
j=0
alm(j)ζml (−j) (2)
has components
alm(j) =
∫
Sn
a(j, x)Sml (x)mn(dx)
which satisfy the following condition
∞
∑
m=0
h(∑m,n)
l=1 ∞
∑
j=0
alm(j)<∞. (3)
Condition (3) ensure convergence of the series representation (2) of the functionalAζ as well as finiteness of the second moment of the functional:E|Aζ|2<∞.
Making use the Gladyshev representation (1) we can write the functionalAζin the form
Aζ= ∞
∑
m=0
h(∑m,n)
l=1 ∞
∑
j=0
(⃗alm(j))⊤ξ⃗lm(−j),
⃗alm(j) = (alm0(j), alm1(j), . . . , alm(T−1)(j))⊤,
almk(j) =alm(j)e2πikj/T, k= 0,1, . . . , T −1,
whereξ⃗l
m(j) = {ξmkl (j)} T−1
k=0 are vector-valued stationary sequences generating the periodically correlated sequences
ζml (j).
Every linear estimate Aζc of the functionalAζ is deter-mined by spectral stochastic measures
(Z⃗ξ)lm(dλ) =
{
(Z⃗ξ)lmk(dλ)
}T−1
k=0
,
(Z⃗η)lm(dλ) =
{
(Z⃗η)lmk(dλ)
}T−1
k=0 ,
of the generating sequences ⃗ξl
m(j) = {ξmkl (j)} T−1
k=0 and
⃗ ηl
m(j) ={ηmkl (j)} T−1
k=0, and the spectral characteristic
hlm(λ) ={hlmk(λ)}Tk=0−1,
which is from the subspaceL−2(F +G)generated by func-tions
hlm(λ) = ∞
∑
j=0
⃗hl m(j)e−
ijλ, m= 0,1, . . .;l= 1, . . . , h(m, n),
in the spaceL2(F+G)of functions that satisfy condition
∞
∑
m=0
h(∑m,n)
l=1
∫ π
−π
(hlm(λ))⊤[Fm(λ) +Gm(λ)]hlm(λ)dλ <∞.
The estimateAζchas the spectral representation of the form
c
Aζ= ∞
∑
m=0
h(∑m,n)
l=1
∫ π
−π
(hlm(λ))⊤
[
(Zξ⃗)lm(dλ) + (Z⃗η)lm(dλ)
]
.
The mean square error ∆(h;F, G) = E|Aζ −Aζc|2 of the estimateAζc is determined by matrices of spectral densities
F(λ) ={Fm(λ) : m= 0,1. . .},G(λ) ={Gm(λ) : m=
0,1. . .}of the generating sequences⃗ξl
m(j) ={ξmkl (j)} T−1
k=0 and⃗ηml (j) = {ηlmk(j)}
T−1
k=0, and the spectral characteristic
h(λ)of the estimate
∆(h;F, G) = E|Aζ−Aζc|2=
= ∞
∑
m=0
h(∑m,n)
l=1 1 2π
∫ π
−π
{ [(
Alm(λ))⊤−(hlm(λ))⊤
]
Fm(λ)×
×[(Alm(λ)
)⊤
−(hlm(λ)
)⊤]∗}
dλ+
+ ∞
∑
m=0
h(∑m,n)
l=1 1 2π
∫ π
−π
(
hlm(λ))⊤Gm(λ)
((
hlm(λ))⊤
)∗
dλ,
(4)
Alm(λ) =
∞
∑
j=0
⃗alm(j)e−ijλ.
The spectral characteristich(F, G)of the least square op-timal linear estimate Aζc minimizes the value of the mean square error. We first apply the method based on factor-izations of matrices of spectral densities to find the spectral characteristich(F, G)and the mean square error of the least square optimal linear estimateAζc of the functionalAζ. For more relative results see articles by Moklyachuk [36] – [41] and books by Moklyachuk[27], Moklyachuk and Masyutka [28].
Suppose that matrices of spectral densities Fm(λ) and
Gm(λ)admit the canonical factorizations
Fm(λ) =φm(λ)(φm(λ))∗, (5)
φm(λ) =
∞
∑
k=0
φm(k)e−ikλ,
Gm(λ) =ψm(λ)(ψm(λ))∗, (6)
ψm(λ) =
∞
∑
k=0
ψm(k)e−ikλ,
Fm(λ) +Gm(λ) =dm(λ)(dm(λ))∗, (7)
dm(λ) =
∞
∑
k=0
dm(k)e−ikλ,
where φm(k), ψm(k), dm(k) are sequences of matrices of
T ×rdimension (ris the rank of the corresponding vector-valued stationary sequences).
These factorizations and the Parseval equality give us a possibility to represent the mean square error of the linear estimateAζc of the functionalAζin the form
∆(h;F, G) =EAζ−Aζc
2 =
= ∞
∑
m=0
h(∑m,n)
l=1 1 2π
∫ π
−π
[
Alm(λ)⊤Gm(λ)Alm(λ)+
+[Alm(λ)−hlm(λ)]⊤[Fm(λ) +Gm(λ)] [Alm(λ)−hlm(λ)]−
−[Alm(λ)−hlm(λ)]⊤Gm(λ)Alm(λ)−
−Alm(λ)⊤Gm(λ)[Alm(λ)−hlm(λ)]
]
dλ=
= ∞
∑
m=0
h(∑m,n)
l=1
[
Ψmalm
2
+Dm(alm−h l m)
2
−
−⟨Ψm(alm−h l
m),Ψmalm
⟩
−⟨Ψmalm,Ψm(alm−h l m)
⟩ ]
,
(8) where the following notions are used
Ψmalm
2 =
∞
∑
k=0
(Ψmalm)k
2
,
Dm(alm−h l m)
2 =
∞
∑
k=0
(Dm(alm−h l m)k
2
,
(Ψmalm)k= k
∑
j=0
ψm(k−j)⊤⃗alm(j),
(Dm(alm−hlm))k= k
∑
j=0
dm(k−j)⊤(⃗aml (j)−⃗hlm(j)),
⟨
Ψm(alm−hlm),Ψmalm
⟩
=
= ∞
∑
k=0
⟨
(Ψm(alm−h l
m))k,(Ψmalm)k
⟩
.
The spectral characteristic h(F, G) of the mean square optimal linear estimate Aζc of the functionalAζ minimizes the value of the mean square error. In the case where ma-trices of spectral densities Gm(λ) are regular and
matri-ces of spectral densities Gm(λ) and Fm(λ) + Gm(λ)
ad-mit the canonical factorizations (6), (7) we can find mini-mum of the obtained expression (8) of the mean square er-ror with respect tohl
mand represent the mean square error
∆(F, G) = ∆(h(F, G);F, G)of the optimal linear estimate
c
Aζof the functionalAζin the form
∆(F, G) = ∞
∑
m=0
h(∑m,n)
l=1
[
Ψmalm
2
−Bm∗Ψ∗mΨmalm
2] =
= ∞
∑
m=0
h(∑m,n)
l=1
[⟨
clm(G), alm⟩−Cml (G)b∗m2
]
Here
B∗mΨ∗mΨmalm
2 =
∞
∑
k=0
(Bm∗Ψ∗mΨmalm)k
2
,
(Bm∗Ψ∗mΨmalm)k =
∞
∑
j=0
bm(j)(Ψ∗mΨmalm)j+k,
⟨
clm(G), alm⟩= ∞
∑
k=0
⟨
clm(G)(k), ⃗alm(k)⟩,
clm(G)(k) = (Ψm∗Ψmalm)k =
∞
∑
j=0
ψm(j)(Ψmalm)j+k,
Cml (G)b∗m2= ∞
∑
k=0
(Cml (G)b∗m)k
2
,
(Cml (G)b∗m)k =
∞
∑
j=0
bm(j)clm(G)(j+k),
bm(λ) =
{
bij m(λ)
}j=1,T
i=1,r are matrix-valued functions which
satisfy equation
bm(λ)dm(λ) =E.
The spectral characteristich(F, G)of the mean square opti-mal linear estimateAζcof the functionalAζcan be calculated by the formula
hlm(F, G) =Aml (λ)−bm(λ)⊤(Cml (G)b∗m)(λ), (10)
(Cml (G)b∗m)(λ) =
∞
∑
k=0
(Cml (G)b∗m)ke−ikλ.
In the case where matrices of spectral densities Fm(λ)
are regular and matrices of spectral densities Fm(λ) and
Fm(λ) +Gm(λ)admit the canonical factorizations (5), (7)
the mean square error∆(F, G) = ∆(h(F, G);F, G)and the spectral characteristich(F, G)of the optimal linear estimate
c
Aζof the functionalAζcan be calculated by the formula
∆(F, G) = ∞
∑
m=0
h(∑m,n)
l=1
[⟨
clm(F), alm⟩−Cml (F)b∗m2
]
,
(11)
hlm(F, G) =bm(λ)⊤(Cml (F)b∗m)(λ), (12)
clm(F)(k) = (Φm∗ Φmalm)k=
∞
∑
j=0
φm(j)(Φmalm)j+k,
(Φmalm)k= k
∑
j=0
φm(k−j)⊤⃗alm(j),
(Cml (F)b∗m)(λ) = ∞
∑
k=0
(Cml (F)b∗m)ke−ikλ.
Let us summarize our results and present them in the form of a statement.
Theorem 1 Letζ(j, x),j ∈ Z,x ∈Sn andθ(j, x),j ∈ Z,
x ∈ Sn, be uncorrelated mean-square continuous
periodi-cally correlated with respect to time isotropic on the sphere Sn random fields, which have spectral densities F(λ) =
{Fm(λ) : m = 0,1. . .} and G(λ) = {Gm(λ) : m =
0,1. . .}. Let the functiona(j, x)which determine the func-tionalAζsatisfy condition (3). The value of the mean square error∆(F, G) = ∆(h(F, G);F, G)and the spectral charac-teristich(F, G)of the optimal linear estimateAζcof the func-tionalAζbased on observations of the fieldζ(j, x) +θ(j, x)
at points j = 0,−1,−2, . . . , x ∈ Sn, can be calculated
by formulas (9), (10) in the case where matrices of spec-tral densitiesGm(λ)andFm(λ) +Gm(λ)admit the
canon-ical factorizations (6), (7), and by formulas (11), (12) in the case where matrices of spectral densities Fm(λ) and
Fm(λ) +Gm(λ)admit the canonical factorizations (5), (7).
Example 3.1 Consider the problem of least square optimal linear estimation of the unknown value of the functional
A0ζ=
∫
Sn
a(x)ζ(0, x)mn(dx) =
∞
∑
m=0
h(∑m,n)
l=1
almζml (0),
which depends on unknown values ζ(0, x),x ∈ Sn, of the
random field ζ(j, x), j ∈ Z, x ∈ Sn, and are based on
observations of the field ζ(j, x) +θ(j, x) at points (j, x), j=−1,−2, . . .,x∈Sn.
Letζ(j, x)and θ(j, x), j ∈ Z,x ∈ Sn be uncorrelated
periodically correlated with respect to time isotropic on the sphereSnrandom fields, which have representations
ζ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Sml (x)ζml (j),
θ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Sml (x)θml (j),
where ζml (j), θlm(j), m = 0,1, . . . , l = 1, . . . , h(m, n), are mutually uncorrelated periodically correlated with pe-riodT = 2stochastic sequences of the form
ζml (j) =ξml 0(j) +eπijξlm1(j),
θml (j) =ηml 0(j) +eπijηlm1(j),
ξlm0(j),m= 0,1, . . . , l= 1, . . . , h(m, n), are uncorrelated
stationary Ornstein-Uhlenbeck stochastic sequences with the spectral densities
fm0(λ) =α2m·
5/4
2π|1−(1/2)e−iλ|2,
ξl
m1(j),m= 0,1, . . . , l= 1, . . . , h(m, n), are uncorrelated
withξl
m0(j)mutually uncorrelated stationary stochastic
se-quences with the spectral densities
fm2(λ) =α2m·
3 2π|1 +e
iλ|2,
ηl
m0(j) and ηml 1(j), m = 0,1, . . . , l = 1, . . . , h(m, n),
are mutually uncorrelated stationary white noise stochastic sequences with the spectral densities gmo(λ) = α2m·
3 2π,
gm1(λ) =α2m· π2. For all densities coefficientsα
2
mare such
that∑∞m=0h(m, n)α2m<∞.
It follows from the proposed relations that the optimal lin-ear estimateAd0ζof the functionalA0ζis calculated by the
formula
d
A0ζ= ∞
∑
m=0
h(∑m,n)
l=1
almαm
[
+ (1/3)(ξlm1(0) +ηlm1(0))+
+ (3/2)1/21367 3456
(
ξml 0(−1) +ηml 0(−1))+
+ (2/9)(ξml 1(−1) +ηlm1(−1))+
+ (3/2)1/21367 6912
(
ξml 0(−2) +ηml 0(−2))−
−(2/9)(ξml 1(−2) +ηlm1(−2))+
+2 ∞
∑
k=3
(−1)k−1 3k+1
(
ξml 1(−k) +ηlm1(−k)) ].
The value of the mean square error∆(F, G) = E|A0ζ−
d
A0ζ|2 of the optimal linear estimateAd0ζ of the functional
A0ζis calculated by formula
∆(F, G) = 1
ωn
∞
∑
m=0
h(∑m,n)
l=1
(
almαm
)2
·(0.596).
4
Minimax-robust method of filtering
The derived formulas may be employed in finding the mean square error ∆(F, G) = ∆(h(F, G);F, G) and the spectral characteristich(F, G)of the optimal linear estimate
c
Aζ of the functionalAζ based on observations of the field
ζ(j, x) +θ(j, x)at pointsj = 0,−1,−2, . . . , x∈ Sn,
un-der the condition that matrices of spectral densitiesF(λ) =
{Fm(λ) : m = 0,1. . .} and G(λ) = {Gm(λ) : m =
0,1. . .}of the fieldζ(j, x)and the fieldθ(j, x)are exactly known.
In the case where the densities are not known exactly, but a set D = DF ×DG of admissible 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 den-sitiesF(λ), G(λ)from given classDF×DGsimultaneously.
Definition 1 For a given class of spectral densities D =
DF ×DGspectral densitiesF0(λ)∈DFandG0(λ)∈DG
are called least favorable for the optimal linear estimation of the functionalAζif the following relation holds true
∆(h(F0, G0);F0, G0) = max (F,G)∈DF×DG
∆(h(F, G);F, G).
Definition 2 For a given class of spectral densities D =
DF ×DG the spectral characteristic h0(λ) of the optimal
linear estimate of the functionalAζis called minimax-robust if there are satisfied conditions
h0(λ)∈HD=∩(F,G)∈DF×DGL −
2(F+G),
min
h∈HD
max (F,G)∈DF×DG
∆(h;F, G) = max (F,G)∈DF×DG
∆(h0;F, G).
It follows from these Definitions and the above Theorem that the next Lemmas hold true.
Lema 1 Spectral densitiesF0(λ)∈ D
F andG0(λ)∈ DG
which admit the canonical factorizations (5) – (7) are least favorable in the classDF ×DGfor the optimal linear
filter-ing of the functionalAζif coefficients of the canonical factor-izations give a solution to the conditional extremum problem
∞
∑
m=0
h(∑m,n)
l=1
[⟨
clm(G), alm⟩−Cml (G)b∗m2
]
→sup, (13)
Gm(λ) =
[∞ ∑
k=0
ψm(k)e−ikλ
] [∞ ∑
k=0
ψm(k)e−ikλ
]∗
∈DG,
Fm(λ) =
[∞ ∑
k=0
dm(k)e−ikλ
] [∞ ∑
k=0
dm(k)e−ikλ
]∗
−
−
[∞ ∑
k=0
ψm(k)e−ikλ
] [∞ ∑
k=0
ψm(k)e−ikλ
]∗
∈DF,
or the conditional extremum problem
∞
∑
m=0
h(∑m,n)
l=1
[⟨
clm(F), alm⟩−Cml (F)b∗m2
]
→sup, (14)
Fm(λ) =
[∞ ∑
k=0
φm(k)e−ikλ
] [∞ ∑
k=0
φm(k)e−ikλ
]∗
∈DF,
Gm(λ) =
[∞ ∑
k=0
dm(k)e−ikλ
] [∞ ∑
k=0
dm(k)e−ikλ
]∗
−
−
[∞ ∑
k=0
φm(k)e−ikλ
] [∞ ∑
k=0
φm(k)e−ikλ
]∗
∈DG.
The minimax spectral characteristich0=h(F0, G0)is
cal-culated by formula (10) (or (12)) if the condition h0 =
h(F0, G0)∈HDholds true.
In the case where one of the densities is known the condi-tional extremum problems (13), (14) are extremum problems with respect to coefficientsbm(k)only.
Lema 2 Spectral density F0(λ) ∈ D
F which admits the
canonical factorizations (5), (7) with a given regular spec-tral densityG(λ)is least favorable in the classDF for the
optimal linear filtering of the functionalAζif
Fm0(λ)+Gm(λ) =
[∞ ∑
k=0
d0m(k)e−ikλ
] [∞ ∑
k=0
d0m(k)e−ikλ
]∗
.
Coefficients d0
m(k)are determined by decomposition of the
matrix functionb0
m(λ):
b0m(λ)d0m(λ) =E, b0m(λ) = ∞
∑
k=0
b0m(k)e−ikλ,
where coefficients b0
m(k)give a solution to the conditional
extremum problem
∞
∑
m=0
h(∑m,n)
l=1
Cml (G)b∗m2→inf, (15)
[∞ ∑
k=0
dm(k)e−ikλ
] [∞ ∑
k=0
dm(k)e−ikλ
]∗
−Gm(λ)∈DF.
The minimax spectral characteristich0=h(F0, G)is
calcu-lated by formula (12) if the conditionh0=h(F0, G)∈H
D
Lema 3 Spectral density G0(λ) ∈ D
G which admits the
canonical factorizations (6), (7) with a given regular spec-tral densityF(λ)is least favorable in the class DG for the
optimal linear filtering of the functionalAζif
Fm(λ)+G0m(λ) =
[∞ ∑
k=0
d0m(k)e−ikλ
] [∞ ∑
k=0
d0m(k)e−ikλ
]∗
.
Coefficientsd0
m(k)are determined by decomposition of the
matrix functionb0
m(λ):
b0m(λ)d0m(λ) =E, b0m(λ) = ∞
∑
k=0
b0m(k)e−ikλ,
where coefficients b0
m(k) give a solution to the conditional
extremum problem
∞
∑
m=0
h(∑m,n)
l=1
Cml (F)b∗m2→inf, (16)
[∞ ∑
k=0
dm(k)e−ikλ
] [∞ ∑
k=0
dm(k)e−ikλ
]∗
−Fm(λ)∈DG.
The minimax spectral characteristich0=h(F, G0)is
calcu-lated by formula (10) if the conditionh0 =h(F, G0)∈H
D
holds true.
The least favorable spectral densitiesF0(λ),G0(λ) and the minimax (robust) spectral characteristic h0(λ) ∈ H
D
form a saddle point of the function ∆(h;F, G) on the set
HD ×D. The saddle point inequalities hold true ifh0 =
h(F0, G0)∈H
D, where(F0, G0)is a solution to the
condi-tional extremum problem
∆(h(F0, G0);F0, G0) = sup (F,G)∈D
∆(h(F0, G0);F, G),
(17) where
∆(h(F0, G0);F, G) =
= ∞
∑
m=0
h(∑m,n)
l=1 1 2π
∫ π
−π
{ [
(Cml (G)b∗m)(λ)]⊤b0(λ)Fm(λ)
(b0(λ))∗(Cl
m(G)b∗m)(λ)dλ+
[
(Cml (F)b∗m)(λ)
]⊤
b0(λ)Gm(λ)(b0(λ))∗(Cml (F)b∗m)(λ)
}
dλ,
The conditional extremum problem (17) is equivalent to the unconditional extremum problem
∆D(F, G) =−∆(h(F0, G0);F, G) +δ((F, G)|D)→inf,
(18) whereδ((F, G)|D)is the indicator function of the setD.
Solution to this unconditional extremum problem (18) is characterized by the condition 0 ∈ ∂∆D(F0, G0), where
∂∆D(F0, G0)is the subdifferential of the convex functional
∆D(F, G)at point(F0, G0).
With the help of condition0 ∈∂∆D(f0, g0)we can find
the least favorable spectral densities in some special classes of spectral densities (see books by Moklyachuk[27], Mokly-achuk and Masyutka[28]for more details).
5
Least favorable spectral densities in
the class
D
F0×
D
0GConsider the problem of the optimal linear estimation of the functionalAζbased on observations of the fieldζ(j, x) +
θ(j, x)at pointsj = 0,−1,−2, . . . , x∈Sn,under the
con-dition that matrices of spectral densitiesF(λ) = {Fm(λ) :
m = 0,1. . .}andG(λ) ={Gm(λ) : m= 0,1. . .}of the
fieldζ(j, x)and the fieldθ(j, x)are not known exactly, but the following pair of sets of possible spectral densities are given
DF0 =
{
F(λ)| 1 2πωn
∞
∑
m=0
h(m, n)
∫ π
−π
Fm(λ)dλ=P
}
,
DG0 =
{
G(λ)| 1 2πωn
∞
∑
m=0
h(m, n)
∫ π
−π
Gm(λ)dλ=Q
}
,
whereP andQare given positive definite matrices.
Making use the method of Lagrange multipliers and the form of setsD0
F andD
0
Gwe can conclude that the condition
0 ∈ ∂∆D(F0, G0)is satisfied forD = DF0 ×D
0
G if
com-ponents of the spectral densities F0(λ) = {F0
m(λ) : m =
0,1. . .}andG0(λ) = {G0
m(λ) : m= 0,1. . .}satisfy the
following pairs of equations
h(∑m,n)
l=1
(Cml (G0)b0m)(λ)(Cml (G0)b0m)(λ))∗=
=d0m(λ)⊤β⃗mβ⃗m∗d0m(λ), (19) h(∑m,n)
l=1
(Cml (F0)b0m)(λ)(Cml (F0)b0m)(λ))∗=
=d0m(λ)⊤⃗γm⃗γm∗d0m(λ). (20)
The unknown Lagrange multipliers β⃗m = {βmk, k =
0, . . . , T −1} and ⃗γm = {γmk, k = 0, . . . , T −1} are
calculated using the canonical factorization equations (5) – (7), conditions (13), (14) and restrictions which determine the corresponding sets of spectral densities.
Theorem 2 Let condition (3) be satisfied. Least favorable spectral densitiesF0(λ) ∈ D0
F, G0(λ) ∈ DG0 for the
op-timal linear estimation of the functionalAζ are determined relations (19), (20), (5) – (7), (13), (14).
In the case where matrices of spectral densitiesG(λ)are known the least favorable spectral densitiesF0(λ)∈D0
Fare
determined by relations (19), (6), (7), (15).
In the case where matrices of spectral densitiesF(λ)are known the least favorable spectral densitiesG0(λ)∈DG0 are determined by relations (20), (5), (7), (16).
The minimax spectral characteristic h0 = h(F0, G0)of
the optimal linear estimation of the functional is calculated by formulas (10), (12).
Example 5.1 Consider the problem of least square optimal linear estimation of the unknown value of the functional
A0ζ=
∫
Sn
a(x)ζ(0, x)mn(dx) =
∞
∑
m=0
h(∑m,n)
l=1
almζml (0),
which depends on unknown valuesζ(0, x), x ∈ Sn of the
observations of the field ζ(j, x) +θ(j, x) at points (j, x), j=−1,−2, . . .,x∈Sn.
Letζ(j, x)and θ(j, x), j ∈ Z,x ∈ Sn be uncorrelated
periodically correlated with respect to time isotropic on the sphereSnrandom fields, which have representations
ζ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Sml (x)ζml (j),
θ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Slm(x)θml (j),
whereζml (j),θml (j)are mutually uncorrelated periodically
correlated with periodT = 1stochastic sequences, spectral densitiesGm(λ) =α2m·|1−
√
2e−iλ|2,∑∞
m=0h(m, n)α 2
m<
∞are known and fixed, spectral densitiesFm(λ)are of the
formFm(λ) =α2m·F(λ), spectral densityF(λ)is unknown
and such that 21π∫−ππF(λ)dλ= 5.
Coefficients{b(0), b(1)}which gives a solution to the con-ditional extremum problem
(3b(0)−√2b(1))2+ 2b2(0)→min,
[
b2(0)−b2(1)]= 1 8.
determine the least favorable spectral densityFm0(λ)∈D0F
for the optimal linear estimation of the functionalA0ζ
Fm0(λ) =α2m
16 3 1− √ 2 2 e −iλ 2
−1−√2e−iλ
2
.
The minimax-robust linear estimateAd0ζof the functional
A0ζis calculated by the formula
d
A0ζ= ∞
∑
m=0
h(∑m,n)
l=1
almαm
[
(1/2)(ξml 0(0) +ηlm0(0))+
+ (1/4)(ξml0(−2) +ηlm0(−2)) ].
The value of the mean square error∆(F, G) = E|A0ζ−
d
A0ζ|2 of the optimal linear estimateAd0ζ of the functional
A0ζis calculated by formula
∆(F, G) = 1
ωn
∞
∑
m=0
h(∑m,n)
l=1
(
almαm
)2
·(2.5).
Example 5.2 Consider the problem of least square optimal linear estimation of the unknown value of the functional
A1ζ=
∫
Sn
[a(0, x)ζ(0, x) +a(1, x)ζ(−1, x)]mn(dx) =
= ∞
∑
m=0
h(∑m,n)
l=1
alm
[
7 2ζ
l
m(−1)−
√ 23 2 ζ l m(0) ]
which depends on unknown valuesζ(0, x),ζ(−1, x),x∈Sn,
of the random field ζ(j, x),j ∈ Z,x ∈ Sn and are based
on observations of the fieldζ(j, x) +θ(j, x)at points(j, x), j =−1,−2, . . .,x∈ Sn, and where the functiona(j, x)is
such thatalm(1) = 72a
l
m,alm(0) =−
√
23 2a
l m.
Let ζ(j, x)andθ(j, x),j ∈ Z,x ∈ Sn be uncorrelated
periodically correlated with respect to time isotropic on the sphereSnrandom fields, which have representations
ζ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Sml (x)ζml (j),
θ(j, x) = ∞
∑
m=0
h(∑m,n)
l=1
Sml (x)θlm(j),
whereζl
m(j),θlm(j)are mutually uncorrelated periodically
correlated with period T = 2 stochastic sequences, spec-tral densities Gm(λ) = αm2 ·I2, I2 is the identity matrix,
∑∞
m=0h(m, n)α 2
m < ∞, are given and fixed, spectral
den-sitiesFm(λ)are of the formFm(λ) = α2m·F(λ), spectral
density F(λ) is unknown and such that 21π∫−ππF(λ)dλ =
(
2 6
6 11
)
.
Coefficients{b(0), b(1)}which gives a solution to the con-ditional extremum problem
[ b(0) [√ 23 √ 2, √ 23 √ 2 ]⊤
−b(1)
[
7 2,−
7 2
]⊤]
×
[
b∗(0)
[√ 23 √ 2, √ 23 √ 2 ]
−b∗(1)
[
7 2,−
7 2 ]] + + [ b(0) [ 7 2,−
7 2
]⊤] [
b∗(0)
[
7 2,−
7 2
]]
→min,
[
I2+b(1)(I2+P)b∗(1)−b(0)(I2+P)b∗(0)
] = ( 0 0 0 0 ) .
determine the least favorable spectral density F m0(λ) ∈
D0
Ffor the optimal linear estimation of the functionalA1ζ
Fm0(λ) =α2m
{( 2 6 6 11 ) −178 √ 23 10569 ( 1 2 2 4 )
(e−2iλ +eiλ2)
}
.
The minimax-robust linear estimateAd1ζof the functional
A1ζis calculated by the formula
d
A1ζ= ∞
∑
m=0
h(∑m,n)
l=1
almαm
[
(−3.23)(ξml 0(0) +ηlm0(0))−
−(3.61)(ξml 1(0) +ηml 1(0))+(7.14)(ξml 0(−1) +ηlm0(−1))
−(6.74)(ξml 1(−1) +ηml 1(−1)) ].
The value of the mean square error∆(F, G) = E|A1ζ−
d
A1ζ|2 of the optimal linear estimate Ad1ζ of the functional
A1ζis calculated by formula
∆(F, G) = 1
ωn
∞
∑
m=0
h(∑m,n)
l=1
(
almαm
)2
·(23.37).
6
Conclusions
We propose a method of solution the problem of optimal linear estimation of functionals
Aζ= ∞ ∑ j=0 ∫ Sn
depending on unknown values of a periodically correlated with respect to time isotropic on the sphereSn in Euclidean
spaceEnrandom fieldζ(j, x),j∈Z,x∈Sn. Estimates are
based on observations of the fieldζ(j, x) +θ(j, x)at points
j = 0,−1,−2, . . .,x ∈ Sn, whereθ(j, x) is an
uncorre-lated withζ(j, x)periodically correlated with respect to time isotropic on the sphereSnrandom field. We propose a
repre-sentation of the mean square error in the form of linear func-tional in the spaceL1×L1with respect to spectral densities (F, G), which allows us to solve the corresponding condi-tional extremum problem and describe the minimax (robust) estimates of the functional.
Formulas for computing the value of the mean-square er-ror and the spectral characteristic of the optimal linear esti-mate of the functionalAζ are obtained. The least favorable spectral densities and the minimax (robust) spectral charac-teristics of the optimal estimates of the functionalAζare de-termined for some special classes of spectral densities.
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