Robust filtering of stochastic processes
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irk-123456789-44872011-03-20T03:29:07Z Robust filtering of stochastic processes Moklyachuk, M. Masyutka, A. 2007 Article Robust filtering of stochastic processes / M. Moklyachuk, A. Masyutka // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 166-181. — Бібліогр.: 22 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4487 en Інститут математики НАН України |
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Moklyachuk, M. Masyutka, A. Robust filtering of stochastic processes |
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Robust filtering of stochastic processes |
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Robust filtering of stochastic processes |
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Robust filtering of stochastic processes |
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Robust filtering of stochastic processes |
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Robust filtering of stochastic processes |
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robust filtering of stochastic processes |
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Robust filtering of stochastic processes / M. Moklyachuk, A. Masyutka // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 166-181. — Бібліогр.: 22 назв.— англ. |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.166-181
MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
ROBUST FILTERING OF STOCHASTIC
PROCESSES
The considered problem is estimation of the unknown value of the
functional A�ξ =
∫∞
0 �a(t)�ξ(−t)dt which depends on the unknown val-
ues of a multidimensional stationary stochastic process �ξ(t) based on
observations of the process �ξ(t) + �η(t) for t ≤ 0. Formulas are ob-
tained for calculation the mean square error and the spectral charac-
teristic of the optimal estimate of the functional under the condition
that the spectral density matrix F (λ) of the signal process �ξ(t) and
the spectral density matrix G(λ) of the noise process �η(t) are known.
The least favorable spectral densities and the minimax-robust spec-
tral characteristic of the optimal estimate of the functional A�ξ are
found for concrete classes D = DF × DG of spectral densities under
the condition that spectral density matrices F (λ) and G(λ) are not
known, but classes D = DF ×DG of admissible spectral densities are
given.
1. Introduction
Traditional methods of solution of the linear extrapolation, interpolation
and filtering problems for stationary stochastic processes may be employed
under the condition that spectral densities of processes 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. Ya-
glom (1987)). In practice, however, complete information on the spectral
densities is impossible in most cases. To solve the problem one finds para-
metric or nonparametric estimates of the unknown spectral densities or
selects these densities by other reasoning. Then applies the classical esti-
mation method provided that the estimated or selected density is the true
2000 Mathematics Subject Classifications. 60G10, 62M20, 60G35, 93E10, 93E11.
Key words and phrases. Stationary stochastic process, filtering, robust estimate,
observations with noise, mean square error, least favorable spectral densities, minimax-
robust spectral characteristic.
166
FILTERING OF STOCHASTIC PROCESSES 167
one. This procedure can result in a significant increasing of the value of er-
ror as K. S. Vastola and H. V. Poor (1983) have demonstrated with the help
of some examples. This is a reason to search estimates which are optimal for
all densities from a certain class of 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,
interpolation and filtering problems for stationary stochastic sequences. 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 Mikhail Mokly-
achuk (1994, 1997, 1998, 2000, 2001), Mikhail Moklyachuk and Aleksandr
Masyutka (2005, 2006) the minimax approach to extrapolation, interpola-
tion and filtering problems are investigated for functionals which depend on
the unknown values of stationary processes and sequences.
In this article we deal with the problem of estimation of the unknown
value of the functional A�ξ =
∫∞
0 �a(t)�ξ(−t)dt which depends on the unknown
values of a multidimensional stationary stochastic process �ξ(t) = {ξk(t)}T
k=1 ,
E�ξ(t) = 0, with the spectral density matrix F (λ) = {fij(λ)}T
i,j=1 based on
observations of the process �ξ(t)+�η(t) for t ≤ 0, where �η(t) = {ηk(t)}T
k=1 is an
uncorrelated with �ξ(t) multidimensional stationary stochastic process with
the spectral density matrix G(λ) = {gij(λ)}T
i,j=1. Formulas are proposed
that determine the least favorable spectral densities and the minimax-robust
spectral characteristic of the optimal estimate of the functional for concrete
classes D = DF ×DG of spectral densities under the condition that spectral
density matrices F (λ), G(λ) are not known, but classes D = DF × DG of
admissible spectral densities are given.
2. Hilbert space projection method of filtering
Let the vector function �a(t) which determines the functional A�ξ satisfies the
following technical conditions:
∞∫
0
T∑
k=1
|ak(t)| dt < ∞,
∞∫
0
t
T∑
k=1
|ak(t)|2dt < ∞. (1)
The process �ξ(t) + �η(t) admits the canonical moving average represen-
168 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
tation
�ξ(t) + �η(t) =
t∫
−∞
d(t − u) d�ε(u), (2)
if the spectral density matrix F (λ) + G(λ) = {fij(λ) + gij(λ)}T
i,j=1 of the
stationary stochastic process �ξ(t) + �η(t) admits the canonical factorization
F (λ) + G(λ) = d(λ)d∗(λ), d(λ) =
∞∫
0
d(u)e−iuλdu (3)
where d(u) = {dij(u)}j=1,m
i=1,T
, �ε(u) = {εk(u)}m
k=1 is a multidimensional sta-
tionary stochastic process with uncorrelated increments (see, for example,
Yu. A. Rozanov (1990)).
The spectral density matrices F (λ) and G(λ) admit the canonical fac-
torizations if
F (λ) = ϕ(λ)ϕ∗(λ), ϕ(λ) =
∞∫
0
ϕ(u)e−iuλdu, (4)
G(λ) = ψ(λ)ψ∗(λ), ψ(λ) =
∞∫
0
ψ(u)e−iuλdu, (5)
where ϕ(u) = {ϕij(u)}j=1,m
i=1,T
and ψ(u) = {ψij(u)}j=1,m
i=1,T
are matrix functions.
The value of the mean square error Δ(h, F, G) of a linear estimate Â�ξ of
the functional A�ξ with the spectral characteristic h(λ) =
∫∞
0
�h(t)e−itλdt is
determined by the formula
Δ(h; F, G) = E
∣∣∣A�ξ − Â�ξ
∣∣∣2 =
=
1
2π
∞∫
−∞
{
(A(λ) − h(λ))(F (λ) + G(λ))(A(λ) − h(λ))∗−
−(A(λ)−h(λ))G(λ)A∗(λ)−A(λ)G(λ)(A(λ)−h(λ))∗+A(λ)G(λ)A∗(λ)
}
dλ =
=
∞∫
0
∞∫
0
min(t,u)∫
−∞
{
(�a(t) −�h(t))d(t − x)d∗(u − x)(�a(u) −�h(u))∗−
−(�a(t)−�h(t))ψ(t−x)ψ∗(u−x)�a∗(u)−�a(t)ψ(t−x)ψ∗(u−x)(�a(u)−�h(u))∗+
+�a(t)ψ(t − x)ψ∗(u − x)�a∗(u)
}
dx du dt,
where A(λ) =
∫∞
0 �a(t)e−itλdt.
FILTERING OF STOCHASTIC PROCESSES 169
The spectral characteristic h(F, G) and the mean square error Δ(F, G) =
Δ(h(F, G); F, G) of the optimal estimate of the functional A�ξ in the case of
given spectral density matrices F (λ) and G(λ) are determined as solutions
to the extremum problem
Δ(F, G) = Δ(h(F, G); F, G) = min
h∈L−
2 (F+G)
Δ(h; F, G), (6)
where L−
2 (F + G) is the subspace of the space L2(F + G) generated by the
functions eitλδk, δk = {δkl}T
l=1 , k = 1, . . . , T, t < 0, δkk = 1, δkl = 0 for
k �= l.
If the spectral density matrices F (λ) and G(λ) admit the canonical fac-
torization (3), (5), then, as follows from the preceding formulas, the mean
square error of the optimal linear estimate can be calculated by the formula
Δ(F, G) = 〈cG, a〉 − ‖CGb∗‖2 , (7)
where
cG(t) =
∞∫
0
min(s,t)∫
−∞
�a(s)ψ(t − u)ψ∗(s − u)duds,
(CGb∗)(t) =
∞∫
0
cG(t + u)b∗(u)du,
〈cG, a〉 =
∞∫
0
T∑
k=1
(cG)k(t)ak(t) dt,
‖CGb∗‖2 =
∞∫
0
m∑
k=1
|(CGb∗)k(t)|2dt.
Here b(λ) = {bij(λ)}j=1,T
i=1,m
is such a matrix-valued function that b(λ) =
∞∫
0
b(t)e−itλdt, b(λ)d(λ) = Im, where Im is the identity matrix of order m.
The spectral characteristic h(F, G) of the optimal estimate of the func-
tional A�ξ in this case can be calculated by the formula
h(F, G) = A(λ) − rG(λ)b (λ), rG(λ) =
∞∫
0
(CGb∗)(t)e−itλdt. (8)
If the spectral density matrices F (λ) and G(λ) admit the canonical factor-
ization (3), (4), then the mean square error and the spectral characteristic
of the optimal linear estimate can be calculated by the formulas
Δ(F, G) = 〈cF , a〉 − ‖CF b∗‖2 , (9)
170 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
h(F, G) = rF (λ)b(λ), rF (λ) =
∞∫
0
(CF b∗)(t)e−itλdt, (10)
where
cF (t) =
∞∫
0
min(s,t)∫
−∞
�a(s)ϕ(t − u)ϕ∗(s − u)du ds,
(CF b∗)(t) =
∞∫
0
cF (t + u)b∗(u) du.
The preceding reasonings show us that the following theorem holds true.
Theorem 1.1. Let �ξ(t) = {ξk(t)}T
k=1 , E�ξ(t) = 0 and �η(t) = {ηk(t)}T
k=1 ,
E�η(t) = 0, be uncorrelated multidimensional stationary stochastic processes
with the spectral density matrices F (λ) = {fij(λ)}T
i,j=1, G(λ) = {gij(λ)}T
i,j=1
and let condition (1) be satisfied. The mean-square error Δ(h(F ), F, G) of
the optimal linear estimate of the functional A�ξ =
∫∞
0 �a(t)�ξ(−t)dt which de-
pends on the unknown values of the process �ξ(t) based on observations of the
process �ξ(t) + �η(t) for t ≤ 0 can be calculated by formula (7) if the spectral
density matrices F (λ) and G(λ) admit the canonical factorization (3), (5)
(by formula (9) if the spectral density matrices F (λ) and G(λ) admit the
canonical factorization (3), (4)). The spectral characteristic h(F, G) of the
optimal linear estimate can be calculated by formula (8) (by formula (10)).
Remark. In the case where the process �ξ(t) + �η(t) is of the maximal
rank (m = T ), the matrix function b (λ) is the inverse matrix to the matrix
d(λ) : b(λ) = d−1(λ). In the case where the process �ξ(t)+�η(t) is of the rank
1 (m = 1), the matrix function b (λ) is a row matrix b (λ) = {bk(λ)}T
k=1
determined from the equation
∑T
k=1 bk(λ)dk(λ) = 1.
Example 1. Consider the problem of estimation of the value �a(0)�ξ(0) =
cξ1(0) + dξ2(0) based on observations of the process �ξ(t) + �η(t), t ≤ 0, in
the case where
F (λ) =
(
f(λ) f(λ)
f(λ) f(λ) + f1(λ)
)
, f(λ) =
P 2
1
|α1 + iλ|2 , f1(λ) =
P 2
2
|α2 + iλ|2 ;
G(λ) =
(
g(λ) g(λ)
g(λ) g(λ) + g1(λ)
)
, g(λ) = 0, g1(λ) =
P 2
3
|α3 + iλ|2 .
In this case F (λ) + G(λ) = d(λ)d∗(λ), F (λ) = ϕ(λ)ϕ∗(λ),
d(λ) =
( P1
α1+iλ
0
P1
α1+iλ
A β+iλ
(α2+iλ)(α3+iλ)
)
, A2 = P 2
2 + P 2
3 , β =
P 2
2 α2
3 + P 2
3 α2
2
A2
;
FILTERING OF STOCHASTIC PROCESSES 171
ϕ(λ) =
∞∫
0
ϕ(u)e−iuλdu, ϕ(u) =
(
P1e
−α1u 0
P1e
−α1u P2e
−α2u
)
.
The inverse matrix
b(λ) = d(λ)−1 =
( α1+iλ
P1
0
− 1
A
(α2+iλ)(α3+iλ)
β+iλ
1
A
(α2+iλ)(α3+iλ)
β+iλ
)
.
The spectral characteristic of the optimal estimate is of the form
h(λ) = (h1(λ), h2(λ)) = rF (λ)b(λ),
where
rF (λ) =
∞∫
0
(CF b∗)(t)e−itλdt, (CF b∗)(t) =
∞∫
0
cF (t + u)b∗(u)du.
Since
cF (t) = �a(0)
∞∫
0
ϕ(t + u)ϕ∗(u) du = (c1(t), c2(t)),
where
c1(t) =
[
(c + d)P 2
1
|α1 + iλ|2
]
t
, c2(t) = c1(t) +
[
dP 2
2
|α2 + iλ|2
]
t
are the Fourier transforms, we have
h1(λ) = (c + d) − dP 2
2
A2
(α2 + iλ)(α3 + iλ)
β + iλ
[
α3 − iλ
(β − iλ)(α2 + iλ)
]
−
,
where [f(λ)]− is a representation of the function as the Fourier integral
transform with respect to the negative powers of e−itλ, t ≥ 0. Taking into
account that [
α3 − iλ
(β − iλ)(α2 + iλ)
]
−
=
α2 + α3
(α2 + β)(α2 + iλ)
,
we will have
h1(λ) = (c + d) − dP 2
2 (α2 + α3)
A2(α2 + β)
α3 + iλ
β + iλ
.
Analogously
h2(λ) =
dP 2
2 (α2 + α3)
A2(α2 + β)
α3 + iλ
β + iλ
.
The value of the mean square error of the optimal estimate is calculated by
the formula
Δ(F, G) = 〈cF , a〉 − ‖CF b∗‖2 = 〈�cF (0),�a(0)〉 − ‖CF b∗‖2 .
172 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
Since
�cF (0) = �a(0)
∞∫
0
ϕ(u)ϕ∗(u)du,
we will have
〈�cF (0),�a(0)〉 =
(c + d)2P 2
1
2α1
+
d2P 2
2
2α2
.
The second summand
‖CF b∗‖2 = ‖(CF b∗)(t)‖2 ,
where
(CF b∗)(t) =
([
(c + d)P1
α1 + iλ
]
t
,
[
dP 2
2 (α3 − iλ)
A(α2 + iλ)(β − iλ)
]
t
)
=
=
(
(c + d)P1e
−α1t ,
dP 2
2
A
(
e−α1t +
α3 − β
α2 + β
e−α2t
))
.
Finally
Δ(F, G) =
d2P 2
2
2α2
− d2P 4
2
2α2A2
(
α2 + α3
α2 + β
)2
.
Example 2. Consider the problem of estimation of the value �a(t)�ξ(−t) =
cξ1(−t) + dξ2(−t) under conditions of Example 1. In this case cF (u) =
(c1(u), c2(u)), where
c1(u) =
[
(c + d)P 2
1 e−iλt
|α1 + iλ|2
]
u
, c2(u) = c1(u) +
[
dP 2
2 e−iλt
|α2 + iλ|2
]
u
.
We will have
h(λ) = ((c + d)e−iλt − q(λ) , q(λ)),
q(λ) =
dP 2
2
A2
(α2 + iλ)(α3 + iλ)
β + iλ
[
α3 − iλ
(α2 + iλ)(β − iλ)
e−iλt
]
−
=
=
dP 2
2
A2
α3 + iλ
β + iλ
(
e−iλt + eα2tα3 − β
α2 + β
)
.
The value of the mean square error of the optimal estimate is calculated by
the formula
Δ(F, G) = 〈�cF (t),�a(t)〉 − ‖(CF b∗)(u)‖2 =
d2P 2
2
2α2
− ‖(CF b∗)2(u)‖2 ,
where
(CF b∗)2(u) =
dP 2
2
A
[
1
α2 + iλ
(
e−iλt + eα2t α3 − β
α2 + β
)]
u
=
FILTERING OF STOCHASTIC PROCESSES 173
=
⎧⎨
⎩
dP 2
2 (α3−β)
A(α2+β)
eα2(t−u), 0 ≤ u < t ;
dP 2
2 (α2+α3)
A(α2+β)
eα2(t−u), u ≥ t .
The mean-square error of the optimal estimate can be calculated by the
formula
Δ(F, G) =
d2P 2
2
2α2
− d2P 4
2
2α2A2(α2 + β)2
((α3 − β)2(e2α2t − 1) + (α2 + α3)
2).
Example 3. Consider the problem of estimation of the value A1
�ξ =∫ 1
0 �a(t)�ξ(−t) dt. By using results of the previous examples we will get that
the spectral characteristic
h(λ) =
⎛
⎝ 1∫
0
(a1(t) + a2(t)e
−iλtdt − q(λ) , q(λ)
⎞
⎠ ,
where
q(λ) =
P 2
2 (α3 + iλ)
A2(β + iλ)
⎛
⎝ 1∫
0
a2(t)e
−iλtdt +
α3 − β
α2 + β
1∫
0
a2(t)e
α2tdt
⎞
⎠ .
The mean-square error of the optimal estimate can be calculated by the
formula
Δ(F, G) =
P 2
2
2α2
1∫
0
a2
2(t) dt − P 4
2
2α2A2(α2 + β)2
×
×
⎛
⎝(α3 − β)2
1∫
0
a2
2(t)(e
2α2t − 1) dt+(α2 + α3)
2
1∫
0
a2
2(t) dt
⎞
⎠ .
Let �a(t) = (1 − t , e−at). In this case the spectral characteristic is of the
form
h(λ) =
(
1
iλ
+
1 − e−iλ
λ2
+
1 − e−(a+iλ)
a + iλ
− q(λ) , q(λ)
)
,
where
q(λ) =
P 2
2 (α3 + iλ)
A2(β + iλ)
(
1 − e−(a+iλ)
a + iλ
+
α3 − β
α2 + β
eα2−a − 1
α2 − a
)
.
The mean-square error of the optimal estimate can be calculated by the
formula
Δ(F, G) =
P 2
2
2α2
1 − e−2a
2a
− P 4
2
2α2A2(α2 + β)2
(
1 − e−2a
2a
(α2+β)(2α3+α2−β)+
174 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
+ (α3 − β)2 e2(α2−a) − 1
2(α2 − a)
)
.
Example 4. Consider the problem of estimation of the value A�ξ =
=
∫∞
0 �a(t)�ξ(−t) dt, where �a(t) = (e−at, e−bt), b > α2. In this case the
spectral characteristic is of the form
h(λ) =
(
1
iλ + a + b
− q(λ) , q(λ)
)
,
q(λ) =
P 2
2 (α3 + iλ)
A2(β + iλ)
(
1
iλ + b
+
α3 − β
(α2 + β)(b − α2)
)
.
The mean-square error of the optimal estimate can be calculated by the
formula
Δ(F, G) =
P 2
2
4bα2
− P 4
2
2α2A2(α2 + β)2
(
(α2 + α3)
2
2b
+
+(α3 − β)2
(
1
2(b − α2)
− 1
2b
))
.
3. Minimax-robust method of filtering
Formulas (l)-(10) may be used to determine the mean-square error and
the spectral characteristic of the optimal linear estimate of the functional
A�ξ when the spectral density matrices F (λ) and G(λ) of multidimensional
stationary stochastic processes �ξ(t), �η(t) are known. In the case where the
spectral density matrices are unknown, but a set D = DF × DG of ad-
missible spectral density matrices is given, the minimax-robust method of
estimation of the unknown values of the functional A�ξ is reasonable (see,
for example, the survey article by S. A. Kassam and H. V. Poor (1985)). By
means of this method it is possible to determine an estimate that minimizes
the mean-square error for all spectral density matrices F (λ), G(λ) from the
class D = DF × DG simultaneously.
Definition 3.1. Spectral density matrices F 0(λ), G0(λ) are called the least
favorable in the class D = DF × DG for the optimal linear filtering of the
functional A�ξ if the following relation holds true Δ(h(F 0, G0); F 0, G0) =
= max
(F,G)∈D
Δ(h(F, G); F, G) = max
F∈D
min
h∈L−
2 (F+G)
Δ(h, F, G).
Definition 3.2. A spectral characteristic h0(λ) of the optimal linear esti-
mate of the functional A�ξ is called the minimax-robust in the class D =
DF × DG if the conditions
h0(λ) ∈ HD =
⋂
(F,G)∈D
L−
2 (F + G),
FILTERING OF STOCHASTIC PROCESSES 175
min
h∈HD
max
(F,G)∈D
Δ(h; F, G) = max
(F,G)∈D
Δ(h0; F, G).
are satisfied.
Taking into account relations (1)–(10), it is possible to verify the follow-
ing propositions.
Proposition 3.1 The spectral density matrices F 0(λ) ∈ DF and G0(λ) ∈
DG are the least favorable in the class D = DF ×DG for the optimal linear
filtering of the functional A�ξ if the density matrices F 0(λ), G0(λ) admit the
canonical factorization (3)-(5) with functions d(u), 0 ≤ u ≤ ∞, ψ(u), 0 ≤
u ≤ ∞, ϕ(u), 0 ≤ u ≤ ∞, which are solutions to the conditional extremum
problem
Δ(F, G) = 〈cG, a〉 − ‖CGb∗‖2 → sup, (11)
G(λ) =
⎛
⎝ ∞∫
0
ψ(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
ψ(u)e−iuλdu
⎞
⎠
∗
∈ DG, (12)
F (λ) =
⎛
⎝
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠
∗
− G(λ)
⎞
⎠ ∈ DF . (13)
or the conditional extremum problem
Δ(F, G) = 〈cF , a〉 − ‖CF b∗‖2 → sup, (14)
F (λ) =
⎛
⎝ ∞∫
0
ϕ(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
ϕ(u)e−iuλdu
⎞
⎠
∗
∈ DF , (15)
G(λ) =
⎛
⎝
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠
∗
− F (λ)
⎞
⎠ ∈ DG. (16)
In the case where one of the spectral density matrices is fixed we have
conditional extremum problems with respect to the function b(u), 0 ≤ u ≤
∞. In this case the following lemmas hold true.
Proposition 3.2 Let the spectral density matrix G(λ) ∈ DG be given. The
spectral density matrix F 0(λ) ∈ DF is the least favorable in the class DF
for the optimal linear filtering of the functional A�ξ if the density matrix
F 0(λ) + G(λ) admits the canonical factorization
F 0(λ) + G(λ) =
⎛
⎝ ∞∫
0
d0(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
d0(u)e−iuλdu
⎞
⎠
∗
.
Here d0(u), 0 ≤ u ≤ ∞, is determined by the function b0(u), 0 ≤ u ≤ ∞,
with the help of equation b0(λ)d0(λ) = Im, b0(λ) =
∫∞
0 b0(u)e−iuλdu, where
176 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
b0(u), 0 ≤ u ≤ ∞, and d0(u), 0 ≤ u ≤ ∞, gives a solution to the conditional
extremum problem
‖CGb∗‖2 → inf, (17)
F (λ) =
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠
∗
− G(λ) ∈ DF . (18)
Proposition 3.3 Let the spectral density matrix F (λ) ∈ DF be given. The
spectral density matrix G0(λ) ∈ DG is the least favorable in the class DG
for the optimal linear filtering of the functional A�ξ if the density matrix
F (λ) + G0(λ) admits the canonical factorization
F (λ) + G0(λ) =
⎛
⎝ ∞∫
0
d0(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
d0(u)e−iuλdu
⎞
⎠
∗
.
Here d0(u), 0 ≤ u ≤ ∞, is determined by the function b0(u), 0 ≤ u ≤ ∞,
with the help of equation b0(λ)d0(λ) = Im, b0(λ) =
∫∞
0 b0(u)e−iuλdu, where
b0(u), 0 ≤ u ≤ ∞, and d0(u), 0 ≤ u ≤ ∞, gives a solution to the conditional
extremum problem
‖CF b∗‖2 → inf, (19)
G(λ) =
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠ ·
⎛
⎝ ∞∫
0
d(u)e−iuλdu
⎞
⎠
∗
− F (λ) ∈ DG. (20)
The least favorable spectral density matrices F 0(λ) ∈ D and G0(λ) ∈ DG
and the minimax-robust spectral characteristic h0(λ) ∈ HD form a saddle
point of the function Δ(h; F, G) on the set HD × D. The saddle point
inequalities
Δ(h; F 0, G0) ≥ Δ(h0; F 0, G0) ≥ Δ(h0; F, G), ∀(F, G) ∈ D, ∀h ∈ HD
hold true if h0 = h(F 0, G0) ∈ HD and (F 0, G0) give a solution to the
conditional extremum problem
Δ(h(F 0, G0); F 0, G0) = sup
(F,G)∈D
Δ(h(F 0, G0); F, G), (21)
where
Δ(h(F 0, G0); F, G) =
1
2π
∞∫
−∞
rG(λ)b0(λ)F (λ)(b0(λ))∗(rG(λ))∗dλ+
+
1
2π
∞∫
−∞
rF (λ)b0(λ)G(λ)(b0(λ))∗(rF (λ))∗dλ.
FILTERING OF STOCHASTIC PROCESSES 177
The functions rF (λ), rG(λ) are calculated by formulas (8), (10) with F (λ) =
F 0(λ), G(λ) = G0(λ).
This conditional extremum problem is equivalent to the unconditional
extremum problem
ΔD(F, G) = −Δ(h(F 0, G0); F, G) + δ((F, G)|D) → inf, (22)
where δ((F, G)|D) is the indicator function of the set D = DF ×DG. A solu-
tion to this problem is determined by the condition 0 ∈ ∂ΔD(F 0, G0), where
∂ΔD(F 0, G0) is the subdifferential of the convex functional ∂ΔD(F, G) at
the point (F 0, G0).
4. Least favorable spectral densities in the class D0,0
Consider the problem of minimax filtering for the set of spectral density
matrices
D0,0 =
⎧⎨
⎩(F (λ), G(λ))| 1
2π
∞∫
−∞
F (λ)dλ = P1,
1
2π
∞∫
−∞
G(λ)dλ = P2
⎫⎬
⎭ .
With the help of the Lagrange multipliers method we can find the follow-
ing relations that determine the least favorable spectral density matrices
(F 0(λ), G0(λ)) ∈ D0,0:
rG(λ)b0(λ)(b0(λ))∗(rG(λ))∗ = �α · �α∗,
rF (λ)b0(λ)(b0(λ))∗(rF (λ))∗ = �β · �β∗.
Here �α = (α1, . . . , αT )T , �β = (β1, . . . , βT )T are the Lagrange multipliers.
It follows from these relations that the least favorable density matrices are
such that
F 0(λ) + G0(λ) = �γ
⎛
⎝ ∞∫
0
(CGb∗)(t)e−itλdt
⎞
⎠ ·
⎛
⎝ ∞∫
0
(CGb∗)(t)e−itλdt
⎞
⎠
∗
�γ∗, (23)
F 0(λ) + G0(λ) = �δ
⎛
⎝ ∞∫
0
(CF b∗)(t)e−itλdt
⎞
⎠ ·
⎛
⎝ ∞∫
0
(CF b∗)(t)e−itλdt
⎞
⎠
∗
�δ∗. (24)
The unknown �β = (β1, . . . , βT )�, �δ = (δ1, . . . , δT )�, b = {b(u) : u ≥ 0} are
calculated with the help of the canonical factorization equations (3)-(5) of
the spectral density matrices F 0(λ), G0(λ), F 0(λ) + G0(λ) and conditions
1
2π
∞∫
−∞
F (λ)dλ = P1,
1
2π
∞∫
−∞
G(λ)dλ = P2. (25)
178 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
In the particular case where one of the spectral density matrices is fixed we
may use only one of these relations. If the spectral density matrix G(λ) is
given, then the least favorable density matrix F 0(λ) ∈ D0 is of the form
F 0(λ) =
max
⎧⎨
⎩�γ
⎛
⎝ ∞∫
0
(CGb∗)(t)e−itλdt
⎞
⎠
⎛
⎝ ∞∫
0
(CGb∗)(t)e−itλdt
⎞
⎠
∗
�γ∗ − G(λ), 0
⎫⎬
⎭ . (26)
If the spectral density matrix F (λ) is given, then the least favorable density
matrix G0(λ) ∈ D0 is of the form G0(λ) =
max
⎧⎨
⎩�δ
⎛
⎝ ∞∫
0
(CF b∗)(t)e−itλdt
⎞
⎠
⎛
⎝ ∞∫
0
(CF b∗)(t)e−itλdt
⎞
⎠
∗
�δ∗ − F (λ), 0
⎫⎬
⎭ . (27)
The unknown �γ,�δ, b(u), are calculated with the help of the canonical fac-
torization equations (3)-(5) of the spectral density matrices F 0(λ), G(λ),
F 0(λ) + G(λ) (or F (λ), G0(λ), F (λ) + G0(λ)) and conditions (25).
The preceding reasonings show us that the following theorem holds true.
Theorem 4.1. The least favorable density matrices F 0(λ), G0(λ) in the
class D0,0 for the optimal linear estimation of the functional A�ξ are de-
termined by relations (23), (24), (3)-(5), (11)–(16), (25). If the spectral
density matrix F (λ) (G(λ)) is given, then the least favorable density ma-
trix G0(λ) ∈ D0 (F 0(λ) ∈ D0) is determined by relations (26), (3)-(5),
(17), (18), (25) (or (27), (3)-(5), (19), (20), (25)). The minimax spectral
characteristic h(F ) of the optimal linear estimate of the functional A�ξ is
calculated by formulas (8), (10).
5. Least favorable spectral densities in the class DU
V × Dε
Consider the problem of minimax estimation of the functional A�ξ under
the condition that spectral density matrices (F (λ), G(λ)) of the multidi-
mensional stationary processes �ξ(t), �η(t) are from the set of spectral density
matrices DU
V × Dε, where
DU
V =
⎧⎨
⎩F (λ)
∣∣∣∣∣∣V (λ) ≤ F (λ) ≤ U(λ),
1
2π
∞∫
−∞
F (λ)dλ = P1
⎫⎬
⎭ ,
Dε =
⎧⎨
⎩G(λ)
∣∣∣∣∣∣G(λ) = (1 − ε)G1(λ) + εW (λ),
1
2π
∞∫
−∞
G(λ)dλ = P2
⎫⎬
⎭ ,
FILTERING OF STOCHASTIC PROCESSES 179
where V (λ), U(λ), G1(λ) are given fixed spectral density matrices, W (λ)
ia an unknown spectral density matrix, and expression B(λ) ≥ D(λ) means
that B(λ) − D(λ) ≥ 0 (positive definite matrix function). The set DU
V
describes the ‘band’ model of stochastic processes while the set Dε describes
the ε-contamination model of stochastic processes. For the set DU
V ×Dε from
the condition 0 ∈ ∂ΔD(F 0, G0) we can get the following relations which
determine the least favorable spectral density matrices
rG(λ)b0(λ)(b0(λ))∗(rG(λ))∗ = �α · �α∗ + Γ1(λ) + Γ2(λ); (28)
rF (λ)b0(λ)(b0(λ))∗(rF (λ))∗ = �β · �β∗ + Γ3(λ). (29)
The coefficients �α = (α1, . . . , αT )T , �β = (β1, . . . βT )T , the matrix function
b0(λ), the unknown functions rF (λ), rG(λ) are calculated with the help of
the canonical factorization equations (3)-(5) of the spectral density matrices
F 0(λ), G0(λ), F 0(λ) + G0(λ) and conditions
1
2π
∞∫
−∞
F (λ)dλ = P1,
1
2π
∞∫
−∞
G(λ)dλ = P2. (30)
The matrix functions Γ1(λ) ≥ 0, Γ2(λ) ≥ 0, Γ3(λ) ≥ 0 are determined by
the conditions
V (λ) ≤ F 0(λ) ≤ U(λ), G0(λ) = (1 − ε)G1(λ) + εW (λ), (31)
Γ1(λ) = 0 if F 0(λ) ≥ V (λ), Γ2(λ) = 0 if F 0(λ) ≤ U(λ), (32)
Γ3(λ) = 0 if G0(λ) ≥ (1 − ε)G1(λ). (33)
In the case where the spectral density matrix G(λ) that admits the canonical
factorization is given, the least favorable in the class D = DU
V spectral
density matrix F 0(λ) is of the form
F 0(λ) = min {U(λ), max {V (λ), �γ rG(λ)(rG(λ))∗�γ∗ − G(λ)}} . (34)
In the case where the spectral density matrix F (λ) that admits the canonical
factorization is given, the least favorable in the class Dε spectral density
matrix G0(λ) is of the form
G0(λ) = max
{
(1 − ε)G1(λ), �δ rF (λ)(rF (λ))∗�δ∗ − F (λ)
}
. (35)
In both cases �γ = (γ1, . . . , γT )T , �δ = (δ1, . . . , δT )T , b(u), rF (λ), rG(λ) are
determined by the factorization of the densities F 0(λ), G(λ), F 0(λ)+G(λ)
(or F (λ), G0(λ), F (λ) + G0(λ)).
The preceding reasonings show us that the following theorem holds true.
180 MIKHAIL MOKLYACHUK AND ALEKSANDR MASYUTKA
Theorem 5.1. The least favorable density matrices F 0(λ), G0(λ) in the
class DU
V × Dε for the optimal linear estimation of the functional A�ξ are
determined by relations (3)-(5), (11)–(16), (28)–(33). If the spectral den-
sity matrix F (λ) (G(λ)) is given, then the least favorable density matrix
G0(λ) ∈ D0 (F 0(λ) ∈ D0) is determined by relations (34), (3)-(5), (17),
(18), (30) (or (35), (3)-(5), (19), (20), (30)). The minimax spectral charac-
teristic h(F ) of the optimal linear estimate of the functional A�ξ is calculated
by formulas (8), (10).
6. Conclusions
We propose formulas for calculation the mean square errors and the spec-
tral characteristic of the optimal linear estimate of the unknown value of the
functional A�ξ =
∫∞
0 �a(t)�ξ(−t) dt which depends on the unknown values of
a multidimensional stationary stochastic process �ξ(t) based on observations
of the process �ξ(t) + �η(t) for t ≤ 0 under the condition that spectral den-
sity matrix F (λ) and spectral density matrix G(λ) of the noise process �η(t)
are known. Formulas are proposed that determine the least favorable spec-
tral densities and the minimax-robust spectral characteristic of the optimal
estimate of the functional for concrete classes D = DF × DG of spectral
densities under the condition that spectral density matrices F (λ) and G(λ)
are not known, but classes D = DF × DG of possible spectral densities are
given.
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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
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