Prediction problem for random fields on groups
The problem considered is the problem of optimal linear estimation of the functional Aξ = ∑↑∞↓j=0 ∫↓G a(g, j)ξ(g, j)dg which depends on the unknown values of a homogeneous random field ξ(g, j) on the group G × Z from observations of the field ξ(g, j) + η(g, j) for (g, j) belongs G×{−1,−2, . . .}, wher...
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Інститут математики НАН України
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| Цитувати: | Prediction problem for random fields on groups / M. Moklyachuk // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 148–162. — Бібліогр.: 20 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859754289728061440 |
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| author | Moklyachuk, M. |
| author_facet | Moklyachuk, M. |
| citation_txt | Prediction problem for random fields on groups / M. Moklyachuk // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 148–162. — Бібліогр.: 20 назв.— англ. |
| collection | DSpace DC |
| description | The problem considered is the problem of optimal linear estimation of the functional Aξ = ∑↑∞↓j=0 ∫↓G a(g, j)ξ(g, j)dg which depends on the unknown values of a homogeneous random field ξ(g, j) on the group G × Z from observations of the field ξ(g, j) + η(g, j) for (g, j) belongs G×{−1,−2, . . .}, where η(g, j) is an uncorrelated with ξ(g, j) homogeneous random field ξ(g, j) on the group G×Z. Formulas are proposed for calculation the mean square error and spectral characteristics of the optimal linear estimate in the case where spectral densities of the fields are known. The least favorable spectral densities and the minimax spectral characteristics of the optimal estimate of the functional are found for some classes of spectral densities.
|
| first_indexed | 2025-12-02T00:42:16Z |
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Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.148–162
MIKHAIL MOKLYACHUK
PREDICTION PROBLEM FOR RANDOM FIELDS
ON GROUPS
The problem considered is the problem of optimal linear estima-
tion of the functional Aξ =
∑∞
j=0
∫
G a(g, j)ξ(g, j)dg which depends
on the unknown values of a homogeneous random field ξ(g, j) on
the group G × Z from observations of the field ξ(g, j) + η(g, j) for
(g, j) ∈ G×{−1,−2, . . .}, where η(g, j) is an uncorrelated with ξ(g, j)
homogeneous random field ξ(g, j) on the group G×Z. Formulas are
proposed for calculation the mean square error and spectral characte-
ristics of the optimal linear estimate in the case where spectral den-
sities of the fields are known. The least favorable spectral densities
and the minimax spectral characteristics of the optimal estimate of
the functional are found for some classes of spectral densities.
1. Introduction
Traditional methods of solution of the linear extrapolation, interpolation
and filtering problems for stationary stochastic processes and homogeneous
random fields may be employed under the condition that spectral densities
of processes and fields are known exactly (see, for example, selected works
of A. N. Kolmogorov (1992), survey by T. Kailath (1974), Yu. A. Rozanov
(1990), N. Wiener (1966); A. M. Yaglom (1987),M. I. Yadrenko (1983)). In
practice, however, complete information on the spectral densities is impossi-
ble in most cases. To solve the problem one finds parametric or nonparame-
tric estimates of the unknown spectral densities or selects these densities by
other reasoning. Then applies the classical estimation method provided
that the estimated or selected densities are the true one. This procedure
can result in a significant increasing of the value of error as K. S. Vastola
and H. V. Poor (1983) have demonstrated with the help of some exam-
ples. This is a reason to search estimates which are optimal for all densities
Invited lecture.
2000 Mathematics Subject Classifications. 60G60, 62M20, 62M40, 93E10.
Key words and phrases. Random field, prediction, filtering, robust estimate, observa-
tions with noise, mean square error, least favorable spectral densities, minimax spectral
characteristic.
148
PREDICTION PROBLEM FOR RANDOM FIELDS 149
from a certain class of the admissible spectral densities. These estimates
are called minimax since 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 S. A. Kassam and H. V. Poor (1985). The paper by
Ulf Grenander (1957) should be marked as the first one where the minimax
approach to extrapolation problem for stationary processes was proposed.
J. Franke (1984, 1985, 1991), J. Franke and H. V. Poor (1984) investigated
the minimax extrapolation and filtering problems for stationary sequences
with the help of convex optimization methods. This approach makes it
possible to find equations that determine the least favorable spectral den-
sities for various classes of densities. In the papers by M. P. Moklyachuk
(1998, 2000, 2001, 2002) the minimax approach to extrapolation, interpola-
tion and filtering problems are investigated for functionals which depend on
the unknown values of stationary processes and random fields on a sphere.
In this article we considered the problem of estimation of the unknown
value of the functional Aξ =
∑∞
j=0
∫
G
a(g, j)ξ(g, j)dg which depends on
the unknown values of a homogeneous random field ξ(g, j) on the group
G × Z, where G is a compact Abelian group, from observations of the field
ξ(g, j)+η(g, j) for (g, j) ∈ G×Z
− = G×{−1,−2, . . .}, where η(g, j) is an un-
correlated with ξ(g, j) homogeneous random field on the group G×Z. For-
mulas are proposed for calculation the mean square error and spectral char-
acteristics of the optimal linear estimate of the unknown value of the func-
tional Aξ in the case where spectral densities f(λ) = {f (n)(λ) : n = 0, 1 . . .}
and g(λ) = {g(n)(λ) : n = 0, 1 . . .} of the fields are known. Formulas
are proposed that determine the least favorable spectral densities and the
minimax-robust spectral characteristic of the optimal estimate of the func-
tional Aξ for concrete classes D = DF × DG of spectral densities under
the condition that spectral densities f(λ), g(λ) are not known, but classes
D = Df × Dg of admissible spectral densities are given.
2. Homogeneous random fields on groups
Let G be a compact Abelian group and let Z be the group of all integers.
We consider random field ξ(g, j) on G × Z as a function on G × Z with
values in the Hilbert space L2(Ω,F ,P) of complex random variables with
finite second moment and zero mean. We will call a random field ξ(g, j)
homogeneous if
E{ξ(g, j)ξ(h, k)} = E{ξ(g−1g, j)ξ(g−1h, k)} = E{ξ(gg−1, j)ξ(hg−1, k)} =
= B(g−1h, j − k) = B(hg−1, j − k), ∀h, g ∈ G, ∀j ∈ Z.
In the case of commutative group G the correlation function has the prop-
erty:
B(h, j) = B(g−1hg, j), ∀h, g ∈ G, ∀j ∈ Z.
150 M. MOKLYACHUK
If the group G is compact it has countable nonequivalent finite dimensional
unitary representations. As a representation of the group G consider ho-
momorphism of the group G into the group of unitary matrices of finite
order
g → T (n)(g) = ‖T (n)
uv (g)‖, 1 ≤ u, v ≤ dn, n = 0, 1 . . .
T (n)(gh) = T (n)(g)T (n)(h), n = 0, 1 . . .
T (n)(g−1) = [T (n)(g)]−1 = [T (n)(g)]∗, n = 0, 1 . . ..
Elements T
(n)
uv (g), 1 ≤ u, v ≤ dn, n = 0, 1 . . . of matrices of such representa-
tion satisfy conditions
dn
∫
G
T (n)
uv (g)T (m)
u1v1
(g)dg = δn
mδu1
u δv1
v ,
where δn
m is the Kronecker symbol, and dg is an invariant unit measure on
G. The set of all matrix elements T
(n)
uv (g), 1 ≤ u, v ≤ dn, n = 0, 1 . . ., form
an orthogonal system in the space L2(G).
Let ξ(g, j) be a homogeneous random field on G × Z. Such a field may
be represented in the form of series
ξ(g, j) =
∞∑
n=0
dn∑
u,v=1
ξ(n)
uv (j)T (n)
uv (g), ξ(n)
uv (j) = dn
∫
G
ξ(g, j)T
(n)
uv (g)dg,
that converges in the mean square. Integral is also considered in the mean
square sense. It follows from the definition of homogeneous random field
ξ(g, j) that
Eξ(n)
uv (j)ξ
(m)
u1v1(k) = δn
mδu1
u δv1
v R(n)(j − k), 1 ≤ u, v ≤ dn, n = 0, 1 . . .,
where R(n)(j) are nonnegative numbers that satisfy the condition
∞∑
n=0
dnR(n)(j) < ∞.
The correlation function of the homogeneous random field ξ(g, j) on G×Z
may be represented in the form
B(g, j) =
∞∑
n=0
R(n)(j)χ(n)(g),
where χ(n)(g): = Tr[T (n)(g)] are characters of the group G.
3. Hilbert space projection method of estimation
Consider the problem of the mean square optimal linear estimation of
the functional
Aξ =
∞∑
j=0
∫
G
a(g, j)ξ(g, j)dg
PREDICTION PROBLEM FOR RANDOM FIELDS 151
which depends on the unknown values of a homogeneous random field ξ(g, j)
on the group G × Z from observations of the field ξ(g, j) + η(g, j) for
(g, j) ∈ G × {−1,−2, . . .}, where η(g, j) is an uncorrelated with ξ(g, j)
homogeneous random field ξ(g, j) on the group G × Z. It follows from the
matrix representation of the group G that such homogeneous random fields
can be represented in the form
ξ(g, j) =
∞∑
n=0
dn∑
u,v=1
ξ(n)
uv (j)T (n)
uv (g), ξ(n)
uv (j) = dn
∫
G
ξ(g, j)T
(n)
uv (g)dg,
η(g, j) =
∞∑
n=0
dn∑
u,v=1
η(n)
uv (j)T (n)
uv (g), η(n)
uv (j) = dn
∫
G
η(g, j)T
(n)
uv (g)dg,
where ‖T (n)
uv (g)‖ are matrix unitray representations of the group G, and
ξ
(n)
uv (j), η
(n)
uv (j), 1 ≤ u, v ≤ dn, n = 0, 1 . . . , j ∈ Z, are mutually orthogonal
stationary stochastic sequences. Homogeneous random fields ξ(g, j), η(g, j)
admit the spectral representations
ξ(g, j) =
∞∑
n=0
dn∑
u,v=1
∫ π
−π
eijλ(Zξ)
(n)
uv (dλ)T (n)
uv (g),
η(g, j) =
∞∑
n=0
dn∑
u,v=1
∫ π
−π
eijλ(Zη)
(n)
uv (dλ)T (n)
uv (g).
Random measures (Zξ)
(n)
uv (Δ), (Zη)
(n)
uv (Δ) satisfy conditions
E(Zξ)
(n)
uv (Δ1)(Zξ)
(m)
u1v1
(Δ2) = δn
mδu1
u δv1
v F
(n)
ξ (Δ1 ∩ Δ2),
E(Zη)
(n)
uv (Δ1)(Zη)
(m)
u1v1
(Δ2) = δn
mδu1
u δv1
v F (n)
η (Δ1 ∩ Δ2),
1 ≤ u, v ≤ dn, n = 0, 1 . . ., Δi ∈ B ([−π, π ]) , i = 1, 2.
Let measures F
(n)
ξ (Δ), n = 0, 1 . . . have spectral densities f (n)(λ), n =
0, 1 . . ., and measures F
(n)
η (Δ), n = 0, 1 . . . have spectral densities g(n)(λ),
n = 0, 1 . . ..
If the spectral densities f (n)(λ), n = 0, 1 . . ., of the field ξ(g, j) on G×Z
admit the canonical factorizations [8,15]
f (n)(λ) = |d(n)(λ)|2, d(n)(λ) =
∞∑
j=0
d(n)(j)e−ijλ, (1)
the field ξ(g, j) can be represented as one sided moving average random
field
ξ(g, j) =
∞∑
n=0
dn∑
u,v=1
j∑
k=−∞
d(n)(j − k)ζ (n)
uv (k)T (n)
uv (g), (2)
152 M. MOKLYACHUK
where ζ
(n)
uv (k), 1 ≤ u, v ≤ dn, n = 0, 1 . . . , k ∈ Z, are mutually uncorrelated
stochastic sequences with orthonormal values (white noise).
Let M(f +g) be a set of n such that the minimality condition holds true
[1,2?]: ∫ π
−π
(f (n)(λ) + g(n)(λ))−1dλ < ∞.
A sequence ξ
(n)
uv (j) + η
(n)
uv (j) that has the spectral density f (n)(λ) + g(n)(λ)
which do not satisfies the minimality condition can be estimated with zero
mean square error. For fixed u, v: 1 ≤ u, v ≤ dn; n ∈ M(f + g) consider the
functional
A(n)
uv ξ =
∞∑
j=0
a(n)
uv (j)ξ(n)
uv (j)
from the sequence ξ
(n)
uv (j), where
a(n)
uv (j) = dn
∫
G
a(g, j)T
(n)
uv (g)dg.
We will suppose that the following conditions hold true
∞∑
n=0
dn∑
u,v=1
∞∑
j=0
|a(n)
uv (j)| < ∞,
∞∑
n=0
dn∑
u,v=1
∞∑
j=0
(j + 1)|a(n)
uv (j)|2 < ∞. (3)
Under these conditions the functional Aξ has the second moment and
operators A
(n)
uv , defined below are compact.
Let ξ
(n)
uv (j)+η
(n)
uv (j), 1 ≤ u, v ≤ dn; n ∈ M(f + g) be stationary sequence
from observations of which we find an estimate of the functional A
(n)
uv ξ.
For every u, v: 1 ≤ u, v ≤ dn; n ∈ M(f + g) denote by L2
(
f (n) + g(n)
)
the
Hilbert space of complex-valued functions on [−π, π ], which are integrable
in square with respect to the measure with the density f (n)(λ) + g(n)(λ),
denote by H− (
f (n) + g(n)
)
closed subspace of L2
(
f (n) + g(n)
)
generated by
functions {eikλ, k = −1,−2, . . .}. Let h
(
eiλ
)
= {h(n)
uv
(
eiλ
)
: 1 ≤ u, v ≤ dn;
n ∈ M(f + g)} be the spectral characteristic of the linear estimate Âξ of
the functional Aξ
Âξ =
∑
n∈M(f+g)
dn∑
u,v=1
∫ π
−π
h(n)
uv
(
eiλ
) (
(Zξ)
(n)
uv (dλ) + (Zη)
(n)
uv (dλ)
)
.
the mean square error of the estimate Âξ of the functional Aξ
Δ(h; f, g) = E|Aξ − Âξ|2 =
∑
n∈M(f+g)
dn∑
u,v=1
1
2π
∫ π
−π
[
|A(n)
uv
(
eiλ
)− h(n)
uv
(
eiλ
)|2f (n)(λ) + |h(n)
uv
(
eiλ
)|2g(n)(λ)
]
dλ.
PREDICTION PROBLEM FOR RANDOM FIELDS 153
The spectral characteristic h(f, g) = {h(n)
uv
(
f (n), g(n)
)
: 1 ≤ u, v ≤ dn;
n ∈ M(f + g)} of the mean square optimal linear estimate minimizes the
value of the mean square error
Δ(f, g) = Δ(h(f, g); f, g) =
=
∑
n∈M(f+g)
dn∑
u,v=1
min
h
(n)
uv ∈H−(f(n)+g(n))Δ(h(n)
uv ; f (n), g(n)).
With the help of the Hilbert space projection method proposed by A. N. Kol-
mogorov [8] we can find the following formulas for calculation the mean
square error Δ(f, g) = Δ(h(f, g); f, g) the spectral characteristic h(f, g) =
h
(n)
uv
(
f (n), g(n)
)
: 1 ≤ u, v ≤ dn; n ∈ M(f + g)} of the optimal linear estimate
of the functional Aξ
Δ(h(f, g); f, g) =
=
∑
n∈M(f+g)
dn∑
u,v=1
{
1
2π
∫ π
−π
|A(n)
uv
(
eiλ
)
g(n)(λ) + C
(n)
uv
(
eiλ
)|2
(g(n)(λ) + f (n)(λ))
2 f (n)(λ)dλ+
+
1
2π
∫ π
−π
|A(n)
uv
(
eiλ
)
f (n)(λ) − C
(n)
uv
(
eiλ
)|2
(g(n)(λ) + f (n)(λ))
2 g(n)(λ)dλ
}
=
=
∑
n∈M(f+g)
dn∑
u,v=1
[〈B(n)c(n)
uv , c(n)
uv 〉 + 〈R(n)a(n)
uv , a(n)
uv 〉
]
; (4)
h(n)
uv
(
f (n), g(n)
)
=
A
(n)
uv
(
eiλ
)
f (n)(λ) − C
(n)
uv
(
eiλ
)
f (n)(λ) + g(n)(λ)
=
= A(n)
uv
(
eiλ
) − A
(n)
uv
(
eiλ
)
g(n)(λ) + C
(n)
uv
(
eiλ
)
f (n)(λ) + g(n)(λ)
, (5)
1 ≤ u, v ≤ dn, n ∈ M(f + g).
A(n)
uv
(
eiλ
)
=
∞∑
j=0
a(n)
uv (j)eijλ, C(n)
uv
(
eiλ
)
=
∞∑
j=0
c(n)
uv (j)eijλ,
a(n)
uv =
(
a(n)
uv (0), a(n)
uv (1), . . .
)
, c(n)
uv =
(
c(n)
uv (0), c(n)
uv (1), . . .
)
,
c(n)
uv = (B(n))−1D(n)a(n)
uv , 〈c(n)
uv , a(n)
uv 〉 =
∞∑
j=0
c(n)
uv (j)a(n)
uv (j),
Here B(n),D(n),R(n) are operators in the space 2 generated by matrices
that are determined by the Fourier coefficients of the functions (f (n)(λ) +
154 M. MOKLYACHUK
g(n)(λ))−1, f (n)(λ)(f (n)(λ) + g(n)(λ))−1, f (n)(λ)g(n)(λ)(f (n)(λ) + g(n)(λ))−1
correspondingly
B(n)(k, j) =
1
2π
∫ π
−π
ei(j−k)λ 1
(f (n)(λ) + g(n)(λ)
dλ,
D(n)(k, j) =
1
2π
∫ π
−π
ei(j−k)λ f (n)(λ)
f (n)(λ) + g(n)(λ)
dλ,
R(n)(k, j) =
1
2π
∫ π
−π
ei(j−k)λ f (n)(λ)g(n)(λ)
f (n)(λ) + g(n)(λ)
dλ, k, j = 0, 1, . . .
The following statements holds true.
Theorem 1. Let ξ(g, j), η(g, j) be uncorrelated homogeneous random fields
on G×Z, which have spectral densities f(λ) = {f (n)(λ), n = 0, 1 . . .}, g(λ) =
{g(n)(λ), n = 0, 1 . . .}. If M(f +g) �= ∅ and conditions (3) are satisfied, then
the value of mean square error Δ(f, g) = Δ(h(f, g); f, g) and the spectral
characteristic h(f, g) = h
(n)
uv
(
f (n), g(n)
)
: 1 ≤ u, v ≤ dn; n ∈ M(f + g)} of
the optimal linear estimate of the functional Aξ from observations of the
field ξ(g, j) + η(g, j) for (g, j) ∈ G × {−1,−2, . . .} can be calculated by for-
mulas (4), (5).
Corollary 1. Let ξ(g, j), g ∈ G, j ∈ Z, be a homogeneous random field
on G × Z, which has spectral density f(λ) = {f (n)(λ), n = 0, 1 . . .}. If
M(f) �= ∅ and conditions (3) are satisfied, then the value of mean square
error Δ(f) = Δ(h(f); f) and the spectral characteristic h(f) = {h(n)
uv
(
f (n)
)
:
1 ≤ u, v ≤ dn; n ∈ M(f)} of the optimal linear estimate of the functional Aξ
from observations of the field ξ(g, j) for (g, j) ∈ G × {−1,−2, . . .} can be
calculated by formulas
Δ(h(f); f) =
∑
n∈M(f+g)
dn∑
u,v=1
1
2π
∫ π
−π
|C(n)
uv
(
eiλ
)|2(f (n)(λ))−1dλ =
=
∑
n∈M(f+g)
dn∑
u,v=1
〈 (
B(n)
)−1
a(n)
uv , a(n)
uv 〉 =
∑
n∈M(f+g)
dn∑
u,v=1
∥∥A(n)
uv d(n)
∥∥2
, (6)
h(n)
uv
(
f (n)
)
= A(n)
uv
(
eiλ
) − C(n)
uv
(
eiλ
) (
f (n)(λ)
)−1
=
= A(n)
uv
(
eiλ
) − (
A(n)
uv d(n)
)
(λ)
(
d(n)(λ)
)−1
, (7)
where
C(n)
uv
(
eiλ
)
=
∞∑
j=0
((
B(n)
)−1
a(n)
uv
)
(j)eijλ,
PREDICTION PROBLEM FOR RANDOM FIELDS 155
(
A(n)
uv d(n)
)
(λ) =
∞∑
j=0
(
A(n)
uv d(n)
)
(j)eijλ,
B(n),A
(n)
uv are operators in the space 2 generated by matrices
B(n)(k, j) =
1
2π
∫ π
−π
ei(j−k)λ
f (n)(λ)
dλ, A(n)
uv (k, j) = a(n)
uv (k + j), 0 ≤ k, j < ∞.
4. Minimax-robust method of estimation
The proposed formulas may be employed under the condition that spec-
tral densities f(λ) = {f (n)(λ), n = 0, 1 . . .}, g(λ) = {g(n)(λ), n = 0, 1 . . .}
of the field ξ(g, j) and the field η(g, j) are known. In the case where the
densities are not known exactly, but a set D = Df ×Dg of possible spectral
densities is given, the minimax (robust) approach to estimation of func-
tionals of the unknown values of homogeneous 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 f(λ), g(λ) from a given class Df × Dg simultaneously.
Definition 1. For a given class of spectral densities D = Df ×Dg spectral
densities f0(λ) ∈ Df , g0(λ) ∈ Dg are called least favorable for the optimal
linear estimate of the functional Aξ if the following relation holds true
Δ(f0, g0) = Δ(h(f0, g0); f0, g0) = max
(f,g)∈Df×Dg
Δ(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 functional
Aξ is 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
Δ(h; f, g) = max
(f,g)∈Df×Dg
Δ(h0; f, g).
Taking into account relations (3)–(7), we can conclude that the following
statements hold true.
Lemma 1. Spectral densities f0(λ) ∈ Df , g0(λ) ∈ Dg are least favorable
in Df × Dg for the optimal linear estimation of the functional Aξ from
observations of the field ξ(g, j) + η(g, j) for (g, j) ∈ G × {−1,−2, . . .} if
156 M. MOKLYACHUK
M(f0+g0) �= ∅ and the Fourier coefficients of functions (f
(n)
0 (λ)+g
(n)
0 (λ))−1,
f
(n)
0 (λ)(f
(n)
0 (λ) + g
(n)
0 (λ))−1, f
(n)
0 (λ)g
(n)
0 (λ)(f
(n)
0 (λ) + g
(n)
0 (λ))−1 determine
operators B
(n)
0 ,D
(n)
0 ,R
(n)
0 , which give a solution to the extremum problem
max
f,g∈Df×Dg
∑
n∈M(f+g)
dn∑
u,v=1
[
〈D(n)a(n)
uv ,
(
B(n)
)−1
D(n)a(n)
uv 〉 + 〈R(n)a(n)
uv , a(n)
uv 〉
]
=
∑
n∈M(f+g)
dn∑
u,v=1
[
〈D(n)
0 a(n)
uv ,
(
B
(n)
0
)−1
D
(n)
0 a(n)
uv 〉 + 〈R(n)
0 a(n)
uv , a(n)
uv 〉
]
. (8)
Minimax spectral characteristic h0 = h(f0, g0) is calculated by formula (5)
if the condition h0 = h(f0, g0) ∈ HD holds true.
Lemma 2. Spectral density f0(λ) ∈ Df , M(f0) �= ∅, is the least favorable in
Df for the optimal linear estimation of the functional Aξ from observations
of the field ξ(g, j) for (g, j) ∈ G × {−1,−2, . . .} if the Fourier coefficients
of functions (f
(n)
0 (λ))−1, n = 0, 1, . . . , determine operators B
(n)
0 , which give
a solution to the extremum problem
max
f∈Df
∑
n∈M(f+g)
dn∑
u,v=1
〈a(n)
uv ,
(
B(n)
)−1
a(n)
uv 〉 =
∑
n∈M(f+g)
dn∑
u,v=1
〈a(n)
uv ,
(
B
(n)
0
)−1
a(n)
uv 〉.
(9)
Minimax spectral characteristic h0 = h(f0) is calculated by formula (7) if
the condition h(f0) ∈ HDf
holds true.
Lemma 3. Spectral density f0(λ) = {f (n)
0 (λ), n = 0, 1 . . .}, is the least
favorable in Df for the optimal linear estimation of the functional Aξ from
observations of the field ξ(g, j) for (g, j) ∈ G × {−1,−2, . . .} if sequences
d
(n)
0 = {d(n)
0 (j) : j = 0, 1, . . .}, that determine the canonical factorization
(1) of the density f0(λ), give a solution to the extremum problem
Δ(f) =
∑
n∈M(f+g)
dn∑
u,v=1
∥∥A(n)
uv d(n)
∥∥2 → sup, (10)
f(λ) =
⎧⎨⎩f (n)(λ) =
∣∣∣∣∣
∞∑
j=0
d(n)(j)e−ijλ
∣∣∣∣∣
2
, n = 0, 1 . . .
⎫⎬⎭ . (11)
The least favorable spectral densities f0(λ) ∈ Df , g0(λ) ∈ Dg and the
minimax (robust) spectral characteristic h(f0, g0) ∈ HD form a saddle point
of the function Δ(h; f, g) on the set HD ×D. The saddle point inequalities
PREDICTION PROBLEM FOR RANDOM FIELDS 157
hold when h0 = h(f0, g0), h(f0, g0) ∈ HD, and (f0, g0) is a solution to the
conditional extremum problem
Δ(h(f0, g0); f0, g0) = max
(f,g)∈Df×Dg
Δ(h(f0, g0); f, g), (12)
Δ(h(f0, g0); f, g) =
=
∑
n∈M(f+g)
dn∑
u,v=1
⎧⎪⎨⎪⎩ 1
2π
∫ π
−π
|A(n)
uv
(
eiλ
)
g
(n)
0 (λ) + C
(n)
uv
(
eiλ
)|2(
g
(n)
0 (λ) + f
(n)
0 (λ)
)2 f (n)(λ)dλ+
+
1
2π
∫ π
−π
|A(n)
uv
(
eiλ
)
f
(n)
0 (λ) − C
(n)
uv
(
eiλ
)|2(
g
(n)
0 (λ) + f
(n)
0 (λ)
)2 g(n)(λ)dλ
⎫⎪⎬⎪⎭ .
Conditional extremum problem (12) is equivalent to the unconditional extre-
mum problem
ΔD(f, g) = −Δ(h(f0, g0); f, g) + δ((f, g)|Df × Dg) → inf,
where δ((f, g)|Df × Dg) is the indicator function of the set Df × Dg. A
solution to this unconditional extremum problem is characterized by the
condition 0 ∈ ∂ΔD(f0, g0), where ∂ΔD(f, g) is the subdifferential of the
convex functional ΔD(f, g) [14].
5. Least favorable spectral densities in the class D0
f × Du
v .
Consider the problem for the set of spectral densities D0
f × Du
v , where
D0
f =
{
f(λ) :
1
2π
∞∑
n=0
dn
∫ π
−π
f (n)(λ)dλ ≤ P1
}
,
Du
v =
{
g(λ) : v(n)(λ) ≤ g(n)(λ) ≤ u(n)(λ);
1
2π
∞∑
n=0
dn
∫ π
−π
g(n)(λ)dλ ≤ P2
}
,
spectral densities v(λ) = {v(n)(λ), n = 0, 1 . . .}, u(λ) = {u(n)(λ), n =
0, 1 . . .} are known and fixed and densities u(n)(λ), n = 0, 1 . . . are bounded.
Let the spectral densities f0(λ) ∈ D0
f , g0(λ) ∈ Du
v , M(f0 + g0) �= ∅ and let
the functions
h
(n)
f
(
f
(n)
0 , g
(n)
0
)
=
dn∑
u,v=1
|A(n)
uv
(
eiλ
)
g
(n)
0 (λ) + C
(n)
uv
(
eiλ
)|
g
(n)
0 (λ) + f
(n)
0 (λ)
(13)
158 M. MOKLYACHUK
h(n)
g
(
f
(n)
0 , g
(n)
0
)
=
dn∑
u,v=1
|A(n)
uv
(
eiλ
)
f
(n)
0 (λ) − C
(n)
uv
(
eiλ
)|
g
(n)
0 (λ) + f
(n)
0 (λ)
(14)
be bounded. The condition 0 ∈ ∂ΔD(f0, g0) is satisfied for D = D0
f ×Du
v if
components of the spectral densities f0(λ) = {f (n)
0 (λ), n = 0, 1 . . .}, g0(λ) =
{g(n)
0 (λ), n = 0, 1 . . .} satisfy equations
α
(n)
1
dn∑
u,v=1
|A(n)
uv
(
eiλ
)
g
(n)
0 (λ) + C(n)
uv
(
eiλ
)| =
(
g
(n)
0 (λ) + f
(n)
0 (λ)
)
, (15)
dn∑
u,v=1
|A(n)
uv
(
eiλ
)
f
(n)
0 (λ) − C(n)
uv
(
eiλ
)| =
=
(
g
(n)
0 (λ) + f
(n)
0 (λ)
)(
γ
(n)
1 (λ) + γ
(n)
2 (λ) + α
(n)
2
)
, (16)
n ∈ M(f0 + g0),
where α
(n)
1 ≥ 0, α
(n)
2 ≥ 0; γ
(n)
1 (λ) ≤ 0 and γ
(n)
1 (λ) = 0 if g
(n)
0 (λ) ≥ v(n)(λ);
γ
(n)
2 (λ) ≥ 0 and γ
(n)
2 (λ) = 0 if g
(n)
0 (λ) ≤ u(n)(λ), and conditions
1
2π
∞∑
n=0
dn
∫ π
−π
f
(n)
0 (λ)dλ = P1, (17)
1
2π
∞∑
n=0
dn
∫ π
−π
g
(n)
0 (λ)dλ = P2. (18)
The following statements hold true.
Theorem 2 Let spectral densities f0(λ) = {f (n)
0 (λ), n = 0, 1 . . .}, g0(λ) =
{g(n)
0 (λ), n = 0, 1 . . .} are from the set D0
f × Du
v and let the functions
h
(n)
f
(
f
(n)
0 , g
(n)
0
)
, h
(n)
g
(
f
(n)
0 , g
(n)
0
)
, m ∈ M(f0 + g0), determined by formu-
las (13), (14) be bounded. The spectral densities f0(λ) ∈ D0
f , g0(λ) ∈ Du
v
are the least favorable in the class D0
f ×Du
v for the optimal linear estimation
of the functional Aξ, if they satisfy equations (15), (16), conditions (17),
(18), and determine solution to the extremum problem (8). The minimax
spectral characteristic h0 = h(f0, g0) is calculated by formula (5).
Corollary 2. Let the spectral density f(λ) = {f (n)(λ), n = 0, 1 . . .} be
known, the spectral density g0(λ) = {g(n)
0 (λ), n = 0, 1 . . .} is from the class
Du
v , and let functions h
(n)
g
(
f
(n)
0 , g
(n)
0
)
, n ∈ M(f0+g0), determined by formu-
las (14) be bounded. The spectral density g0(λ) ∈ Du
v is the least favorable
PREDICTION PROBLEM FOR RANDOM FIELDS 159
in the class Du
v for the optimal linear estimation of the functional Aξ, if its
components satisfy equation
g
(n)
0 (λ) = max
{
v(n)(λ), min
{
u(n)(λ), α
(n)
2
dn∑
u,v=1
|C(n)
uv
(
eiλ
)|}}
,
condition (18), and densities (f(λ), g0(λ)) determine solution to the extre-
mum problem (8). The minimax spectral characteristic h0 = h(f, g0) is
calculated by formula (5).
Corollary 3. Let the spectral density g(λ) = {g(n)(λ), n = 0, 1 . . .} be
known, the spectral density f0(λ) = {f (n)
0 (λ), n = 0, 1 . . .} is from the class
D0
f and let the functions h
(n)
f
(
f
(n)
0 , g
(n)
0
)
, n ∈ M(f0 + g0), determined by
formulas (13) be bounded. The spectral density f0(λ) ∈ D0
f is the least
favorable in the class D0
f for the optimal linear estimation of the functional
Aξ, if its components satisfy equation
f
(n)
0 (λ) = max
{
0, α
(n)
1
dn∑
u,v=1
|A(n)
uv
(
eiλ
)
g
(n)
0 (λ) + C(n)
uv
(
eiλ
)| − g
(n)
0 (λ)
}
,
condition (17), and densities (f0(λ), g(λ)) determine solution to the extre-
mum problem (8). The minimax spectral characteristic h0 = h(f0, g) is
calculated by formula (5).
Consider the problem of the optimal linear estimate of the functional Aξ
from observations of the field ξ(g, j) for (g, j) ∈ G × {−1,−2, . . .}. From
the condition 0 ∈ ∂ΔD(f0) for D = D0
f we find the following relations that
determine the least favorable spectral density f0(λ) = {f (n)
0 (λ), n = 0, 1 . . .}
from the class D0
f :
f
(n)
0 (λ) = α
(n)
1
dn∑
u,v=1
∣∣A(n)
uv d(n)(λ)
∣∣2 . (19)
To find the unknown α
(n)
1 , d
(n)
0 (j), n = 0, 1 . . . , j = 0, 1, 2, . . . we use
the factorization equations (1), extremum conditions (10), (11), and the
condition
‖d‖2 =
∞∑
n=0
dn
∞∑
j=0
|d(n)
0 (j)|2 = P1. (20)
For all solutions d(n) = {d(n)(j), j = 0, 1 . . .}, n = 0, 1, 2, . . . of the system
of equations
A(n)
uv d(n) = μ(n)
uv d(n), 1 ≤ u, v ≤ dn (21)
160 M. MOKLYACHUK
the following equality holds true
dn∑
u,v=1
∣∣∣∣∣
∞∑
j=0
(
A(n)
uv d(n)
)
(j)eijλ
∣∣∣∣∣
2
= α
(n)
1
∣∣∣∣∣
∞∑
j=0
d(n)(j) e−ijλ
∣∣∣∣∣
2
.
Denote by νP1 the maximum value of
∞∑
n=0
dn∑
u,v=1
∥∥A(n)
uv d(n)
∥∥2
=
∞∑
n=0
μ(n)
∥∥d(n)
∥∥2
, μ(n) =
dn∑
u,v=1
μ(n)
uv ,
where d(n) = {d(n)(j), j = 0, 1 . . .}, n = 0, 1, 2, . . . solutions to the system of
equations (21) that satisfy condition (20).
Denote by ν+P1 the maximum value of
∞∑
n=0
dn∑
u,v=1
∥∥A(n)
uv d(n)
∥∥2
under the condition that d(n) = {d(n)(j), j = 0, 1 . . .}, n = 0, 1, 2, . . ., gives
the canonical factorization (1) of the density (19) and satisfy condition (20).
If there exists a solution d(n0) = {d(n0)(j), j = 0, 1 . . .} to the system of
equations (21) for n = n0 such that dn0
∥∥d(n0)
∥∥2
= P1 and ν = ν+, then the
spectral density f0(λ) = {f (n)
0 (λ), n = 0, 1 . . .} with components
f
(n)
0 (λ) =
∣∣∣∣∣
∞∑
j=0
d(n0)(j) e−ijλ
∣∣∣∣∣
2
δn0
n , n = 0, 1 . . . (22)
of the moving average random field
ξ(g, j) =
dn0∑
u,v=1
j∑
k=−∞
d(n0)(j − k)ζ (n0)
uv (k)T (n0)
uv (g), (23)
where ζ
(n0)
uv (k), 1 ≤ u, v ≤ dn, are mutually uncorrelated stochastic sequen-
ces with orthonormal values (white noise), is the least favorable in the class
D0
f for the optimal linear estimation of the functional Aξ.
The following statement holds true.
Theorem 3. Spectral density f0(λ) = {f (n)
0 (λ), n = 0, 1 . . .} with compo-
nents (22) of the moving average random field (23) is the least favorable in
the class D0
f for the optimal linear estimation of the functional Aξ if there
exists a solution d(n0) = {d(n0)(j), j = 0, 1 . . .} to the system of equations
PREDICTION PROBLEM FOR RANDOM FIELDS 161
(21) for n = n0 such that dn0
∥∥d(n0)
∥∥2
= P1 and ν = ν+. If ν < ν+, then
the least favorable in the class D0
f for the optimal linear estimation of the
functional Aξ is determined by conditions (1), (10), (19), (20). The mini-
max spectral characteristic h0 = h(f0) is calculated by formula (7).
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv 01033, Ukraine
E-mail address: mmp@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4518 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-02T00:42:16Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Moklyachuk, M. 2009-11-24T15:31:52Z 2009-11-24T15:31:52Z 2007 Prediction problem for random fields on groups / M. Moklyachuk // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 148–162. — Бібліогр.: 20 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4518 The problem considered is the problem of optimal linear estimation of the functional Aξ = ∑↑∞↓j=0 ∫↓G a(g, j)ξ(g, j)dg which depends on the unknown values of a homogeneous random field ξ(g, j) on the group G × Z from observations of the field ξ(g, j) + η(g, j) for (g, j) belongs G×{−1,−2, . . .}, where η(g, j) is an uncorrelated with ξ(g, j) homogeneous random field ξ(g, j) on the group G×Z. Formulas are proposed for calculation the mean square error and spectral characteristics of the optimal linear estimate in the case where spectral densities of the fields are known. The least favorable spectral densities and the minimax spectral characteristics of the optimal estimate of the functional are found for some classes of spectral densities. en Інститут математики НАН України Prediction problem for random fields on groups Article published earlier |
| spellingShingle | Prediction problem for random fields on groups Moklyachuk, M. |
| title | Prediction problem for random fields on groups |
| title_full | Prediction problem for random fields on groups |
| title_fullStr | Prediction problem for random fields on groups |
| title_full_unstemmed | Prediction problem for random fields on groups |
| title_short | Prediction problem for random fields on groups |
| title_sort | prediction problem for random fields on groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4518 |
| work_keys_str_mv | AT moklyachukm predictionproblemforrandomfieldsongroups |