Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності
We investigate the problem of linear estimation of unknown mathematical expectations based on observations of realizations of random matrix sequences. Constructive mathematical methods have been developed for finding linear guaranteed RMS estimates of unknown non-stationary parameters of average val...
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| author | Nakonechnyi, Oleksandr Kudin, Grygoriy Zinko, Petro Zinko, Taras |
| author_facet | Nakonechnyi, Oleksandr Kudin, Grygoriy Zinko, Petro Zinko, Taras |
| author_institution_txt_mv | [
{
"author": "Oleksandr Nakonechnyi",
"institution": "Київський національний університет імені Тараса Шевченка, Київ"
},
{
"author": "Grygoriy Kudin",
"institution": "Київський національний університет імені Тараса Шевченка, Київ"
},
{
"author": "Petro Zinko",
"institution": "Київський національний університет імені Тараса Шевченка, Київ"
},
{
"author": "Taras Zinko",
"institution": "Київський національний університет імені Тараса Шевченка, Київ"
}
] |
| author_sort | Nakonechnyi, Oleksandr |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2023-08-07T15:49:29Z |
| description | We investigate the problem of linear estimation of unknown mathematical expectations based on observations of realizations of random matrix sequences. Constructive mathematical methods have been developed for finding linear guaranteed RMS estimates of unknown non-stationary parameters of average values based on observations of realizations of random matrix sequences. It is shown that such guaranteed estimates are obtained either as solutions to boundary value problems for systems of linear differential equations or as solutions to the corresponding Cauchy problems. We establish the form and look for errors for the guaranteed RMS quasi-minimax estimates of the special forecast vector and parameters of unknown average values. In the presence of small perturbations of known matrices in the model of matrix observations, quasi-minimax RMS estimates are found, and their guaranteed RMS errors are obtained in the first approximation of the small parameter method. Two test examples for calculating the guaranteed root mean square estimates and their errors are given. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2023.2.07 |
| first_indexed | 2025-07-17T10:28:04Z |
| format | Article |
| fulltext |
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko, 2023
86 ISSN 1681–6048 System Research & Information Technologies, 2023, № 2
TIДC
МЕТОДИ ОПТИМІЗАЦІЇ, ОПТИМАЛЬНЕ
УПРАВЛІННЯ І ТЕОРІЯ ІГОР
UDC 519.711
DOI: 10.20535/SRIT.2308-8893.2023.2.07
GUARANTEED ROOT-MEAN-SQUARE ESTIMATES OF THE
FORECAST OF MATRIX OBSERVATIONS UNDER CONDITIONS
OF STATISTICAL UNCERTAINTY
O.G. NAKONECHNYI, G.I. KUDIN, P.M. ZINKO, T.P. ZINKO
Abstract. We investigate the problem of linear estimation of unknown mathematical
expectations based on observations of realizations of random matrix sequences.
Constructive mathematical methods have been developed for finding linear guaran-
teed RMS estimates of unknown non-stationary parameters of average values based
on observations of realizations of random matrix sequences. It is shown that such
guaranteed estimates are obtained either as solutions to boundary value problems for
systems of linear differential equations or as solutions to the corresponding Cauchy
problems. We establish the form and look for errors for the guaranteed RMS quasi-
minimax estimates of the special forecast vector and parameters of unknown aver-
age values. In the presence of small perturbations of known matrices in the model
of matrix observations, quasi-minimax RMS estimates are found, and their guaran-
teed RMS errors are obtained in the first approximation of the small parameter
method. Two test examples for calculating the guaranteed root mean square esti-
mates and their errors are given.
Keywords: matrix observations, linear estimations, guaranteed RMS estimates,
guaranteed RMS estimate errors, quasi-minimax guaranteed vector estimates, differ-
ence equation, small parameter method, matrix perturbation.
INTRODUCTION
This article examines estimates of unknown mathematical expectations based on
observations of realizations of random matrix sequences. Scientific publications
[1–14], in which estimates of distribution parameters were studied, are devoted to
the problems of matrix sequence statistics. We formulate and solve new problems
of estimating the mean values of random matrix sequences. Under the condition
that the mean values belong to sets of a special form, we have developed con-
structive algorithms for guaranteed root-mean-square estimates of the mean val-
ues. It is shown that such estimates can be obtained either as solutions of bound-
ary value problems for a system of linear differential equations, or as solutions of
the corresponding Cauchy problems. In the case of the dependence of the average
values on a small parameter, asymptotic distributions were obtained both for the
guaranteed estimates and for the guaranteed root mean square errors of such esti-
mates.
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 87
STATEMENT OF THE PROBLEM
We consider matrix observations of the form:
1,0 , )( NkkxY kkk , (1)
where ,1,0 ),()())(( 1 NkkxkAkx ss
m
sk
1,0 ,,1 ,)( NkmsHkA pns are known matrices;
pnH is the space of matrices pn dimensions;
Nkkxkxkx T
m ,0,))(),...,(()( 1 are unknown vectors, belonging to a
limited set
},1 )( :,0),({ 221
0
k
N
k qkfNkkxG
, 1,0 ),()1()( Nkkxkxkf
(to simplify the calculations, we assume that )0(x is known vector and, without
limiting the generality, we put 0)0( x );
1,0,2 Nkqk are known positive real numbers;
T is a transposition symbol;
1,0 , NkH pnk is a sequence of random matrices.
It is assumed that the average value of the random matrices 1,0 , Nkk
is equal to the null matrix, i.e. 0kE E( is a symbol of mathematical expecta-
tion), and correlation matrices 1,0 , NkHR nnk are known and determined
by relations
1,0 ,2,1 , ,,,, 2121 NkiHZZZRZZE pnikkk ,
where )(, T
ikik ZspZ is a scalar product of matrices.
Let’s introduce linear operators that act from space lR into space pnH :
,1,0,)( 1 NkzUzU iik
l
ikk
pnik HU , )( 1 lkkk UUU , Tlzzz ,,1 , ,1,0 ,,1 Nkli
and operators conjugated to them )()(*
kkk YU :
,),,,,()()( 1
* T
klkkkkkk YUYUYU
pnik HU , pnk HY , ,,1 li .1,0 Nk
It is necessary to evaluate the vector )(NVx , where .mlHV
Definition 1. A vector )(NVx of the form
cYUNVx kkk
N
k ))(( )( *1
0
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 88
,),,,,( 1
1
0 cYUYU T
klkkk
N
k
lRc
is called a linear estimate of a vector )(NVx .
Definition 2. The value
2
10
2 max),,(
NVxNVxEUU
G
N
is called the guaranteed root mean square (RMS) error of the linear estimate
)(NVx
.
SOLVING THE PROBLEMS OF LINEAR ESTIMATION OF THE FORECAST
OF MATRIX OBSERVATIONS.
І. Let’s introduce vectors , )( mRkz Nk ,0 , which are solutions of the differ-
ence equation:
, , )( ,0,1 ,))(()1()( * lT
kkk RaaVNzNkaUkzkz (2)
where 1,0,* Nkk are operators conjugated to k .
Denote by ),(kzi li ,1 the solutions of the difference equation (2) at iea ,
where liei ,1 , are the base vectors of space lR , and also enter the matrix Z :
; )(
,1, ljiijzZ ;))1(),1(( 21
0
k
jiN
kij qkzkzz .,1, lji (3)
The vectors ,1 kz i li ,1 finds from the difference equations:
,)( ),( )1()( )(i
iiii VNzkbkzkz (4)
where ,),(,,),()( 1
T
ikmik
i UkAUkAkb
T
imii VVV ),,( 1)( , .1,0 , ,1 Nkli
There is a formula
.1,0 ,,1 ),()1(
)1(
1
)(
NklijNbVkz i
kN
j
i
i (5)
Statement 1. Let GNkkx ,0),( , then the following equality holds:
.,)),(),((max),,( 1
1
0
22/1
1
10
2
ikikk
l
i
N
k
a
N UURacaZaUU
Proof. Fair equality:
2
*
1
0
2
)))((()(())(()()( ckxUNVxNVxNVxE kkk
N
k
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 89
2*1
0 )( kkk
N
k UE
),())))(((()(,))(,(max
2
*
1
01
cakxUaNxaV kkkk
N
k
T
a
1
1
0 , ikikk
l
i
N
k UUR
),()))((),(())(,(max
2
*
1
01
caaUkxNxaV kkk
N
k
T
a
+
., 1
1
0
ikikk
l
i
N
k UUR
Since
)))(),(())1(),1((())(),(())(,( 1
0 kxkzkxkzNxNzNxaV N
k
T
= )),(),1(())()),( )1((( 1
0
1
0 kfkzkxkzkz N
k
N
k
then, considering that ))(()()1( * aUkzkz kkk we get that
2
10
2 )()(max),,( NVxNVxEUU
G
N
=
1
1
0
21
0
1
,)),()(),1((maxmax ikikk
l
i
N
k
N
k
Ga
UURcakfkz
.,)),())1( ((max 1
1
0
22/1221
0
1
ikikk
l
i
N
kk
N
k
a
UURcaqkz
From the fact that equalities are fulfilled
), ()),(),())1( ),1((())1( ( 21
01,
221
0 aZaqeaeakzkzqkz k
jijiN
k
l
jik
N
k
,
we conclude that the statement 1 is correct.
Corollary 1. There is an equality:
,0ˆ,ˆ ,ˆ)ˆ()()(maxmin 1
1
0max
2
,
cUURZNVxNVxE ikikk
l
i
N
k
GcU
where )ˆ( , )( ˆ
,1, UzzzZZ ijijljiij , )ˆ(max Z is the maximum eigenvalue of
the matrix Ẑ , and pnik HU ˆ , ,,1 li 1,0 Nk are found from the condition:
).,,(minArg)ˆ;1,0,,1,ˆ( 10
2
1,0,,1,;
N
NkliUc
ik UUcNkliU
ik
Corollary 2. Let 1l . The estimation error )),(,( NxV where mRV is an
arbitrary vector, is as follows:
2
10
2 )()(max),,( NVxNVxEUU
G
N
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 90
,,)1( 2
11
1
0
221
0 cUURqkz kkk
N
kk
N
k
where Nkkz ,0, )( is a solution of the difference equation )(kz
),()1( 1
*
kk Ukz ,)( ,0,1 VNzNk for this case .0с̂
Statement 2. Let’s put the parameter 1l in the statement 1, then:
1) guaranteed root mean square estimate for )(1 NxV has the form:
) ˆ( )( 1
1
01
T
kk
N
k YUspNxV
;
2) the guaranteed root mean square error of the linear estimate )(1 NxV
has
the form:
)),((),...,( )1(1,110
2 VNpUU N
,
where 1,0 )),((1
NkkpRU kkk ;
kR is a pseudo-inverse operator;
Nkkp ,0),( are vectors that are determined from the system of equa-
tions:
,0)0( ,1,0 ),1()()1(
,)( ,0,1 ,ˆ)1()(
2
11
*
pNkkzqkpkp
VNzNkUkzkz
k
kk
(6)
Proof. Let’s define kU1
ˆ 1,0 Nk from conditions:
0)
~
,...,
~
( 011 00
2
NNUU
d
d , for 1,0 ,
~
Nkk .
There is an equality:
011 00
2 )
~
,...,
~
(
2
1
NNUU
d
d =
kkk
N
kk
N
k URqkzkz
~
, ))1(~ ),1((
1
0
21
0 ,
where .1,0 ,0)(~ ),
~
()1(~)(~
* NkNzkzkz kk
If input the vectors NkRkp m ,0 ,)( , which are solutions of difference
equations
,0)0( ,1,0 ),1( )()1( 2 pNkkzqkpkp k
then we will get:
)))()1((),1(~(())1(~ ),1(( 1
0
21
0 kpkpkzqkzkz N
kk
N
k
))()),1(~)(~((1
0 kpkzkzN
k
.
~
)),(( ))(),
~
((
1
0
*1
0
kk
N
kkk
N
k kpkp
As a result, we get equality:
, 0
~
,)))((( 1
1
0
kkkk
N
k URkp
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 91
from which the representation derives
,1,0 )),((ˆ
1 NkkpRU kkk
which had to be shown.
Solution of the boundary value problem (6)
One of the options for solving the boundary value problem (6) is the possibility of
reducing it to the solution of the Cauchy problem for vectors Nkkp ,0 ),( .
This requires solving the Cauchy problem for the first equation of system (6).
Then, after substituting the result into the second equation of the system, solve the
Cauchy problem for the required vectors .,0 ),( Nkkp
It is also possible to use the homogeneity of the considered problem for the
required vectors Nkkp ,0 ),( . This requires the use of a base miei ,1 , of the
vector space mR . Expansions of vectors ),(kz Nkkp ,0 ),( in this basis have
the form:
),()( 1 kzxkz ii
m
i ,,0 ),()( 1 Nkkpxkp ii
m
i (7)
where the vectors miNkkzkp ii ,1 ,,0 ),( ),( are defined as solutions of m
boundary problems:
,,1,0)0( ,1,0 ),1()()1(
;)0( ,1,0 ),( )()1(
2 mipNkkzqkpkp
ezNkkpFkzkz
iikii
i
iikii
where
.
)()(
)()(
1
111
*
mk
T
mk
T
m
mk
T
k
T
kkkk
ARAspARAsp
ARAspARAsp
RF
Unknown coefficients mixi ,1 , in the expansions (7) are found as solu-
tions of the system of linear algebraic equations that ensure the fulfillment of the
boundary condition :)( VNz
.)(1 VxNz ii
m
i
According to the distribution of vectors Nkkp ,0 ),( (formula (7)), the
expressions for the matrices 1,0 )),((ˆ 1 NkkpRU kkk of the required es-
timate )(NVx
are obtained.
Another possibility of solving the boundary value problem (6) is to reduce it
to a difference boundary value problem of the second order with respect to vec-
tors Nkkp ,0 ),( and to find a general solution to the obtained problem. Arbi-
trary constants of the general solution are determined from the boundary condi-
tions of problem (6).
II. Let’s introduce vectors )(ˆ kp , ),(ˆ kx Nk ,0 , that are the solutions of
the system of difference equations:
.1,0 ,0)0(ˆ ),1(ˆ)(ˆ)1(ˆ
;0)(ˆ ))),(ˆ( ()1(ˆ)(ˆ
2
*
Nkxkpqkxkx
NpkxYRkpkp
k
kkkk (8)
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 92
Remark 1. If we take into account that the equality holds
),(ˆ))(),...,(()))(ˆ( ( 1
* kxFYRAspYRAspkxYR k
T
kk
T
mkk
T
kkkk
then we can find the solution of linear differential equation system (8) according
to the solution scheme of linear differential equation system (6).
Statement 3. The following equality holds
).),(ˆ()( VNxNVx (9)
Proof. For a guaranteed estimate, the following relations are fulfilled:
))(),(()),((,ˆ)( *1
0
1
0
1
0 kkk
N
kkkk
N
kkk
N
k YRkpYkpRYUNVx
. (10)
Let’s denote 1,0 ),1(ˆ)(ˆΔ Nkkpkpk . Then
)))(ˆ((Δ)( ** kxRYR kkkkkkk .
Hence
))))(ˆ((),(()Δ),(())(),(( ** kxRkpkpYRkp kkkkkkk .
Now we sum up both parts of the last equality:
)))(ˆ(),(()Δ),(())(),(( *
1
0
1
0
*
1
0
kxRkpkpYRkp kkk
N
k
k
N
k
kkk
N
k
(11)
and calculate the first term on the right-hand side:
))1(ˆ)(ˆ),(()Δ),((
1
0
1
0
kpkpkpkp
N
k
k
N
k
))1(),(ˆ)1(ˆ())()1(),1(ˆ(
1
0
1
0
kzkxkxkpkpkp
N
k
N
k
)),(ˆ())1()(),(ˆ(
1
0
VNxkzkzkx
N
k
))(ˆ))),(((()),(ˆ( *
1
0
kxkpRVNx kkk
N
k
))))(ˆ((),(()),(ˆ( *
1
0
kxRkpVNx kkk
N
k
. (12)
The required equality (9) follows from formulas (10)–(12).
Remark 2. The system of equations (8) can be obtained by solving the min-
imization problem of the function
))1(,),0(( NffJ
).( ))(())),((( 221
0
1
0 kfqkxYkxYR k
N
kkkkkk
N
k
III. Below we consider the case when the set G is in the space of possible
values ))(,),0(( Nxx , ,))(),...,(()( 1
T
m kxkxkx Nk ,0 is unbounded and has
the form:
}, )0( ,1 )()1( :))(,),0({( 221
0
m
k
N
k RxqkxkxNxxG
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 93
where 1,0,2 Nkqk are known positive real numbers.
It is necessary to determine the guaranteed root mean square error:
,)()(max),,(σ
2
10
2 NVxNVxEUU
G
N
where ,),,,,( )( 1
1
0 cYUYUNVx T
klkkk
N
k
lRc .
Let’s introduce vectors lRkz )( 1,0 Nk , that are solutions of differ-
ence equations
,)( ),()()1()( * aNzaUkzkz kkk Nk ,0 ,
and the set }. 0)0(:1,0 ,,1 ,{ zNkliUU ik
Statement 4.
1) If UNkliUik 1,0 ,,1, , then
2
2 )()(max)(σ NVxNVxEU
G
.,)),(),((max 1
1
1
22/1
1
ikikk
l
i
N
k
a
UURcaaZa (13)
2) If UNkliUik 1,0 ,,1, , then
.)()(max)(σ
2
2 NVxNVxEU
G
Proof.
1) If 1,0 ,,1, UNkliUik , then we obtain the formula (13) simi-
larly to the statement 1;
2) If ,UUik then there may exist a such that
0)0()0( aazz by ikU , . 1,0 ,,1 Nkli
Therefore, given unbounded of the set ,G we obtain the relation:
21
0
2 )),()0(),0())(),1(( (max)(σ caxzkfkzU N
k
G
ikikk
l
i
N
k UUR ,1
1
0 .
IV. We present a guaranteed linear RMS estimate of the scalar product
))( ,( Nxa according to matrix observations of the form
1,0 ,))(( NkkxY kkk ,
through the solutions of the Cauchy problem for linear differential equations.
Denote by ,...2 ,1 ,0 , kVk the sequence of linear operators of the form:
,)( 1
*
1 kkkkkkk PPRV
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 94
where matrices kP are solutions of difference equations:
,)( *
1
*
1 kkkkkkkkk QVPRVPP
,...2 ,1 ,0 ,0 , 1
2
kPIqQ mkk
Statement 5. The equality holds
),1( ˆ ,,ˆ ))( ,( 1
0
kzVUYUNxa kkkk
N
k
where Nkkz ,1),( are the solutions of difference equations:
.)( ,1,),1()()1()1()( aNzNkkzVIkzVkzkz kmk
Moreover
).,()))( ,())( ,((max 1
2 aaPNxaNxaE N
G
Proof. Let’s solve the problem of optimal system control:
0,1,)( ),()1()( * NkaNzUkzkz kk
with the criterion
kkk
N
kk
N
kN UURkzkzQUUJ ,))1( ),1((),...,( 1
0
1
010
by the method of dynamic programming.
Let’s introduce the Bellman function
,)1( ], ,))1(),1(([min)( 00
,...,0
xkzUURizizQxB iii
k
ii
k
i
UU
k
k
for which the Bellman equation holds
.0)( ], ,)()([min),()( 1
*
xBuuRuxxBxxQxB kkk
Hu
kk
pn
1,1 Nk
(by definition we consider that ).01
0
k
We find the function )(xBk in the form ). ,()( xxPxB kk Let’s choose the
matrices 1,1 , NkPk so that the Bellman equation holds true. After appro-
priate transformations (similarly as it is done, for example, in [15]) we obtain the
expressions for 1,0 , , ,ˆ NkPVU kkk .
It is obvious that .)))( ,())( ,(((max),()( 2
11
NxaNxaEaaPxB
G
NN
Statement 6. For a guaranteed linear RMS estimate of the scalar product the
following representation takes place
)),( ̂,())( ,( NxaNxa
where the vector )(ˆ Nx is a solution of difference equations
.0)0(ˆ ,1,0 )),(ˆ(( )( ̂ )1(ˆ * xNkkxYVkxkx kkk (14)
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 95
Proof. Since equalities are fulfilled:
,1,0 , )1(ˆ NkkzVU kk
then )),1((,ˆ *1
0
1
0 kk
N
kkk
N
k YVkzYU
.
Expressions take place:
;))(ˆ()(ˆ)1(ˆ ** kxVkxkxYV kkkk
))(ˆ)1(ˆ),1(()),1(( 1
0
*1
0 kxkxkzYVkz N
kkk
N
k
)));(ˆ(),1(( *1
0 kxVkz kk
N
k
))(ˆ,())(ˆ),()1(())(ˆ)1(ˆ),1(( 1
0
1
0 Nxakxkzkzkxkxkz N
k
N
k
))(ˆ)),1((())(ˆ,( *
1
0
kxkzVNxa kk
N
k
)))(ˆ(),1(())(ˆ,( *1
0 kxVkzNxa kk
N
k
.
From here we get the necessary equality.
Remark 3. The vector )(ˆ Nx is found as a solution of the linear difference
equation (14). It is possible to obtain the vector )(ˆ Nx even if the vectors
1,0 ),(ˆ )1(ˆ)( Nkkxkxkf are random and uncorrelated ,0)(( kEf
.1,0 ,)()( 2 NkqkfkEf k
T The given estimators are such that minimize the
root mean square error in the category of linear estimators.
V. Definition 3. The vector
T
klkkk
N
k YUYUNVx ),ˆ,,,ˆ()( 1
1
0
,
which components are calculated according to formulas
1,0 ,,1 )),((ˆ NklikpRU ikkik ,
and )(kp i are vectors that are determined from the systems of difference equations
,0)0( ),1()()1(
,)( ),ˆ()1()(
2
*
iikii
iiikkii
pkzqkpkp
VNzUkzkz
1,0 ,,1 Nkli .
is called the quasi-minimax guaranteed estimation of the vector
T
klkkk
N
k YUYUNVx ),,,,()( 1
1
0
.
Statement 7. For the guaranteed root mean square error of quasi-minimax
estimates there is equality:
,ˆ,ˆ)ˆ()ˆ,,ˆ( 1
1
0max10
2
ikikk
l
i
N
kN UURZUU
where ,))1(ˆ),1(ˆ(ˆ ,)ˆ(ˆ 21
0,1,
kii
N
kijmjiij qkzkzzzZ mji ,1, , and ele-
ments ,)1(ˆ kzi mi ,1 are found as solutions of difference equations:
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 96
,)(ˆ ,1,1 ),(ˆ )1(ˆ)(ˆ )(iiiii VNzNkkbkzkz
,)ˆ),(,,ˆ),(()(ˆ
1
T
ikmiki UkAUkAkb
T
imii VVV ),,( 1)( , .1,1 ,,1 Nkli
Finding elements mjizij ,1,,ˆ of the matrix Ẑ is carried out according to
the algorithm for calculating elements of matrix Z (formulas (3)–(5)).
Quasi-minimax RMS estimates for small matrix perturbations.
Assume that the known matrices of model (1) have the form:
pnsss HkAkAkA )()()( )1()0( , 1,0 ,,1 Nkms ,
where 1R is small parameter; and the operators are as follows:
))(())(())(( )1()0()( kxkxkx kkk ,
where .1,0 ),()())(( ),()())(( )1(
1
)1()0(
1
)0( NkkxkAkxkxkAkx ss
m
skss
m
sk
We determine the effect of small perturbations of the matrices on the esti-
mates, as well as on their errors, using the results presented in statement 3.
Statement 8. Quasi-minimax guaranteed estimation of the vector )(NVx
within the framework of the first approximation of the small parameter method
has the form:
,),ˆ,...,,ˆ()( )()(
1
1
0
T
klkkk
N
k
YUYUNVx
where ,)(ˆˆˆ )1()0()(
pnikikik IoUUU
pnpn HI is the matrix, all elements of which are equal to one,
)),((ˆ )0()0()0( kpRU ikkik
1,0 ,,1 )),(())((ˆ )0()1()1()0()1( NklikpRkpRU ikkikkik ,
and vectors )()0( kp i , Nkli ,0 ,,1 are defined as solutions of boundary value
problems:
,)( ))),(( ()1( )(
)0()0(
)(
)0()0()0(
)(
)0(
)(
*
iiikkkii VNzkpRkzkz
,0)0( ),1()()1( )0()0(
)(
2)0(
)(
)0(
)(
iikii pkzqkpkp 1,0 ,,1 Nkli ,
–)))(( ()1()( )1(
)(
)0(
)0()1(
)(
)1(
)(
*
kpRkzkz ikkkii
,0 ))),(( ()))(( ( 1
)(
)0(
)(
)1(
0)0(
)(
)0(
)1( **
NzkpRkpR iikkkikkk
,0)0( ),1()1( )1(
)(
)1(
)(
2)1(
)(
1
)(
iikii pkzqkpkp 1,0 ,,1 Nkli .
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 97
Statement 9. There is the equality for the guaranteed root mean square error
of quasi-minimax estimates within the framework of the first approximation of
the small parameter method:
.ˆ,ˆ)ˆ()ˆ,,ˆ(σ )()(
1
1
0
)(
max
)(
1
)(
0
2
ikikk
l
i
N
kN UURZUU
Here ,)(ˆˆˆ )1()0()(
pnikikik IoUUU
1,0 ,,1 Nkli , and the expres-
sion for the matrix )(ˆ Z has the form
, )(ˆˆˆ )1()0()(
mmIoZZZ
where ;))1( ˆ),1(ˆ(ˆ ,)ˆ(ˆ 2)0()0(1
0
)0(
,,1,,
)0()0(
kji
N
kjimjiji qkzkzzzZ
.))1( ˆ),1(ˆ(2ˆ ,)ˆ(ˆ 2101
0
1
,,1,,
11
kji
N
kjimjiji qkzkzzzZ
The vectors ,1ˆ 0 kzi 1,0 ,,1 Nkmi are found as solutions of zero-
approximation difference equations:
;)( ),(ˆ )1(ˆ)(ˆ )0()0()0()0(
iiiii VNzkbkzkz (15)
;
)0()0(
,,
)0()0(
1
)0( ),( ),()(ˆ
T
ikmAiki UkUkAkb
T
imii VVV ),,( 1 , 1,0 ,,1 Nkli ,
and vectors 1,0 ,,1 ),1(ˆ 1 Nkmjkz j are found as solutions of first ap-
proximation difference equations:
;0)( ),(ˆ )1(ˆ)(ˆ 1111 Nzkbkzkz jjjj (16)
T
jkmjkj UkAUkAkb )ˆ),(,,ˆ),(()(ˆ )0()1()0()1(
1
)1(
1,0 ,,1,)ˆ),(,,ˆ),(( )1()0()1(0
1 NkljUkAUkA T
jkmjk .
Finding the solutions of differential equations (15), (16) is carried out ac-
cording to the algorithm for calculating elements of matrix ljiijzZ ,1,)( (formu-
las (3)–(5)).
Example 1. Let the matrix observations have the form:
; ,0,))(()( NkkxY kkk (17)
)),(())(())(( 10)( kxkxkx kkk (18)
where )()())(( ),())(( )1()1()0()0( kxkAkxkxAkx kk ,
Nk
k
kAIA ,0 ,
00
0
)( , )1(
2
)0(
, 1R is a small parameter;
1 )( Rkx , 1,0 Nk and belong to a bounded set G :
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 98
}; )()1( :,0),({ 2
1
21
0
qkxkxNkkxG N
k
,0 ,22 NkHk is a sequence of random matrices.
It is assumed that the average value of the random matrices 1,0 , Nkk
are equal to the zero matrix, and is the correlation matrices , 2
2
0 IqRk
1,0 Nk .
The guaranteed RMS estimate )(ˆ Nx has the form:
,,ˆ )(ˆ )(1
0
kk
N
k YUNx (19)
where 1,0 )),((ˆ )()(2
0
)( NkkpqU kk ;
Nkkp ,0),()( are values that are determined from the system of differ-
ence equations:
.0)0( ,1,0 ),1()()1(
;1)( ,1,1 ),()(),()1()(
)()(2
1
)()(
)()()()(2
0
)()(
pNkkzqkpkp
NzNkkpkAkAqkzkz
(20)
Applying the small parameter method for solving problems (19), (20), we in-
troduce the following notation:
1,0 ,)()1()0(ˆ
22
ˆ)(
NkIoUU kUkk
,
) ))(( ))((( , ))(()0( )0()1()1()0(2
01
)0()0(2
0 kpkpqUkpU kkkkqk
,
),()()()( ),()()()( )1()0()()1()0()( okzkzkzokpkpkp
where ),(0 kp Nk ,0 are the values of the zero approximation of the small
parameter method, which are defined as solutions of the boundary value problem:
,1,0 ,2
,0)0( ),1()()1(
,1)( ),( )1()(
2
0
)0()0(2
1
)0()0(
)0()0()0()0(
Nkq
pkzqkzkp
Nzkpkzkz
(21)
and Nkkp ,0 ),()1( are the values of the first approximation of the small pa-
rameter method, which are defined as solutions of the boundary value problem:
.1,0 ,0)0( ,)1()()1(
,0)( , )( )()1()(
)1()1(2
1
)1()1(
)1()0()1()1()1(
Nkpkzqkpkp
Nzkpkkpkzkz
(22)
Solution of the boundary value problem (21) of zero approximation
The boundary value problem (21) reduces to a boundary value problem for values
Nkkp ,0 ,)0( :
,0)0( ,0)()1()2()2( )0()0()0()0( pkpkpqkp
,)1()( 2
1
)0()0( qNpNp (23)
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 99
.2 ,2,0 2
0
2
1
qqqNk
The general solution of the corresponding homogeneous system has the
form:
.2/42 , )( 2
21
)0(
0
qqqCCkp kk
Taking into account the boundary conditions, the solution of the boundary
value problem (23) is as follows:
)).1()1/(( ,) ()( 122
111
)0(
0 NNkk qCCkp
Therefore, in the zero approximation of the small parameter method, expres-
sions are obtained:
. 1,0 , )()0( )0(2
0
NkkpU qk
(24)
The guaranteed RMS estimate in the zero approximation of the small pa-
rameter method has the form:
) (, )(ˆ )0(1
01
2
0
)0( kk
k
N
k YACqNx
, (25)
and the error of this estimate is as follows:
,)}({})](ˆ)([{max 2/1)0(2/12 NpNxNxE
G
(26)
where
)/1()1(
)/1(
)(
12
122
10
N
Nq
Np .
The representation of the estimation error in the zero approximation by for-
mula (26) allows one to notice a decrease in its value with an increase in the
quantity of observations, as well as to establish a limit value )()0( Np :
.
2
)211(
)(lim
2
0
2
12
1
)0( qq
qNp
N
Solving the boundary value problem (22) of the first approximation
The boundary value problem (22) is reduced to a boundary value problem for
values ,0 ),()1( Nkkp :
, ) ( )()1()2( )1()1()1( kkkkpkpkp (27)
2,0,0)1()( ,0)0( )1()1()1( NkNpNpp .
The partial solution of the inhomogeneous equation (27) is represented by
formulas with undefined coefficients:
,,0 , ) () ()( 32
2
132
2
1
)1(
part NkDkDkDBkBkBkp kk
which have the form:
,0 ), 1/()13( ), 1/( 3
22
12
2
1 BBBB
.0 , , 32211 DBDBD
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 100
The general solution of the inhomogeneous equation (27) has the form:
,,0 ),()( )( 2
2
121
)1( NkkBkBFFkp kkkk
where arbitrary constants are determined from boundary conditions (27).
The solution of the boundary value problem (27) is as follows:
,,0 , )(] )([)( 2
2
11
)1( NkkBkBFkp kk
where
)/1(1
)/1(
)()1(])1([
)1(
1
12
12
21211 N
N
NBNBNBNBF .
Thus, expressions are obtained for the corrections of the first approximation
of the small parameter method:
,1,0 ), )()()(()1(ˆ )0()1()1()0(2
0 NkkpkAkpAqUk
,),1(ˆ)(ˆ 1
0
)1(
kk
N
k YUNx
).(] )([)( 2
2
11
)1( NNNBNBFNp (28)
The guaranteed root mean square error in the first approximation of the
small parameter method is represented by the formula:
),()1()0()( 222 o
where ,
))/1(1(
))/1((
)1(
)0(
12
122
12
N
Nq
)())(()()1( 2
2
11
12 NNNBNBFNp .
Remark 4. It is worth noting that when using formula (28), it is necessary to
take into account the specific values of the model parameters of the observation
problem , , , , 10 qqq as well as the number of observations N , namely: order of
magnitude 12 a smaller than order of magnitude )0(2 .
The extended possibilities of applying the small parameter method can be
seen in the following example for other small perturbations of the known matrices
in the model of the observation problem.
Example 2. Let the matrix observations have the form represented by for-
mulas (17), (18), but with other matrices of small perturbation:
.1,0 ,
00
01
1
NkakA k (29)
Let’s find the decomposition for the small parameter of the guaranteed value
estimate ),(Nx as well as its errors.
As in example 1, the guaranteed RMS estimate )(ˆ Nx has the form:
,,ˆ)(ˆ )(1
0
kk
N
k YUNx (30)
where 1,0 )),((ˆ )()(2
0
)( NkkpqU kk ; Nkkp ,0),()( are the values
that are determined from the system of difference equations:
Guaranteed root-mean-square estimates of the forecast of matrix observations under …
Системні дослідження та інформаційні технології, 2023, № 2 101
.00 ,1,0 ),1()()1(
,1)( ,1,1 ),()(),()1()(
)()(2
1
)()(
)()()()(2
0
)()(
pNkkzqkpkp
NzNkkpkAkAqkzkz
(31)
Applying the small parameter method to solve problems (30), (31) we obtain
a guaranteed root mean square estimate in the zero approximation and its error in
the formulas form (24)–(26).
First approximation corrections of the small parameter method for the
guaranteed RMS estimate require the definition of matrices 1ˆ
kU
) ))(( ))((( )0()1()1()0(2
0 kpkpq kk , where ),()0( kp Nk ,0 are the values of
the zero approximation of the small parameter method, and Nkkp ,0 ),()1( are
the values of the first approximation of the small parameter method, which are
defined as solutions of boundary value problems
,) ()()1()2( )1()1()1( kkkakpkpkp (32)
.2,0 ,0)1()( ,0)0( )1()1()1( NkNpNpp
The partial solution of the inhomogeneous equation (27) is represented by
formulas with undefined coefficients:
NkaB
a
Bkp k
k
,0 , )( 21
)1(
part
,
where these coefficients are calculated by formulas:
) 1(/ ), /( 2222
2
222
1 aaaBaaaB .
The general solution of the inhomogeneous equation (27) has the form:
,,0 ,) ( )( 2121
)1( NkaB
a
BFFkp k
k
kk
and arbitrary constants 21 , FF are determined from the boundary conditions (32):
,)( 2121 BBFF
.1
)1(
)1(
1
)1(
)(
)/11(
1
12
21122
a
aB
a
aBF NNN
N
Thus, expressions are obtained for the corrections of the first approximation
of the small parameter method:
,1,0 ,) )()()(()1(ˆ )0()1()1()0(2
0 NkkpkAkpAqUk
,),1(ˆ)(ˆ 1
0
)1(
kk
N
k YUNx
) ( )( 2121
)1( N
N
NN aB
a
BFFNp
.
The guaranteed root mean square error in the first approximation of the
small parameter method is represented by the formula:
,)()1()0()( 222 o
O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, T.P. Zinko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 102
where ,
))/1(1(
))/1((
)1(
)0(
12
122
12
N
Nq
.)()()1( 2121
)1(2 N
N
NN aB
a
BFFNp
Obviously, that at certain values of the parameter a in model (29), the de-
sired accuracy of the small parameter method can be achieved with a larger num-
ber of observations.
CONCLUSIONS
The article develops constructive mathematical methods for finding linear guaran-
teed root mean square estimates of unknown non-stationary parameters of average
values based on observations of realizations of a sequence of random matrices. It
is shown that, under certain conditions, such estimates are expressed in terms of
solutions of the boundary value problem for the system of difference equations.
Formulas are presented that allow obtaining recurrent estimates of unknown parame-
ters. In the case of the dependence of the average values on a small parameter, the
corresponding asymptotic formulas are given. Asymptotic distributions of linear
parameter estimates and their root mean square errors are given for partial cases.
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Received 20.01.2023
INFORMATION ON THE ARTICLE
Oleksandr G. Nakonechnyi, ORCID: 0000-0002-8705-3070, Taras Shevchenko
National University of Kyiv, Ukraine, e-mail: a.nakonechniy@gmail.com
Grygoriy I. Kudin, ORCID: 0000-0002-1322-4551, Taras Shevchenko National Univer-
sity of Kyiv, Ukraine, e-mail: gkudin@ukr.net
Petro N. Zinko, ORCID: 0000-0002-5111-4417, Taras Shevchenko National University
of Kyiv, Ukraine, e-mail: petro.zinko@gmail.com
Taras P. Zinko, ORCID: 0000-0003-1263-9293, Taras Shevchenko National University
of Kyiv, Ukraine, e-mail: taras.zinko@gmail.com
ГАРАНТОВАНІ СЕРЕДНЬОКВАДРАТИЧНІ ОЦІНКИ ПРОГНОЗУ
МАТРИЧНИХ СПОСТЕРЕЖЕНЬ В УМОВАХ СТАТИСТИЧНОЇ
НЕВИЗНАЧЕНОСТІ / О.Г. Наконечний, Г.І. Кудін, П.М. Зінько, Т.П. Зінько
Анотація. Досліджено задачу лінійного оцінювання невідомих математичних
сподівань за спостереженнями реалізацій випадкових матричних
послідовностей. Розроблено конструктивні математичні методи для знаход-
ження лінійних гарантованих середньоквадратичних оцінок невідомих
нестаціонарних параметрів середніх значень за спостереженнями реалізацій
послідовності випадкових матриць. Показано, що такі гарантовані оцінки
одержуються або як розв’язки крайових задач для систем лінійних різницевих
рівнянь, або як розв’язки відповідних задач Коші. Установлено вигляд по-
хибок для гарантованих середньоквадратичних квазімінімаксних оцінок
спеціального вектора прогнозу та параметрів невідомих середніх значень. За
наявності малих збурень відомих матриць у моделі матричних спостережень
знайдено квазімінімаксні середньоквадратичні оцінки і в першому наближенні
методу малого параметра отримано їх гарантовані середньоквадратичні по-
хибки. Наведено два тестові приклади обчислення гарантованих середньок-
вадратичних оцінок та їх похибок.
Ключові слова: матричні спостереження, лінійне оцінювання, гарантована
середньоквадратична оцінка, похибка гарантованої середньоквадратичної оці-
нки, квазімінімаксна гарантована оцінка вектора, різницеве рівняння, метод
малого параметра, збурення матриць.
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| id | journaliasakpiua-article-272758 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:04Z |
| publishDate | 2023 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/d8/4b8bf397486947c4d8800ae1800c48d8.pdf |
| spelling | journaliasakpiua-article-2727582023-08-07T15:49:29Z Guaranteed root-mean-square estimates of the forecast of matrix observations under conditions of statistical uncertainty Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності Nakonechnyi, Oleksandr Kudin, Grygoriy Zinko, Petro Zinko, Taras matrix observations linear estimations guaranteed RMS estimates guaranteed RMS estimate errors quasi-minimax guaranteed vector estimates difference equation small parameter method matrix perturbation матричні спостереження лінійне оцінювання гарантована середньоквадратична оцінка похибка гарантованої середньоквадратичної оцінки квазімінімаксна гарантована оцінка вектора різницеве рівняння метод малого параметра збурення матриць We investigate the problem of linear estimation of unknown mathematical expectations based on observations of realizations of random matrix sequences. Constructive mathematical methods have been developed for finding linear guaranteed RMS estimates of unknown non-stationary parameters of average values based on observations of realizations of random matrix sequences. It is shown that such guaranteed estimates are obtained either as solutions to boundary value problems for systems of linear differential equations or as solutions to the corresponding Cauchy problems. We establish the form and look for errors for the guaranteed RMS quasi-minimax estimates of the special forecast vector and parameters of unknown average values. In the presence of small perturbations of known matrices in the model of matrix observations, quasi-minimax RMS estimates are found, and their guaranteed RMS errors are obtained in the first approximation of the small parameter method. Two test examples for calculating the guaranteed root mean square estimates and their errors are given. Досліджено задачу лінійного оцінювання невідомих математичних сподівань за спостереженнями реалізацій випадкових матричних послідовностей. Розроблено конструктивні математичні методи для знаходження лінійних гарантованих середньоквадратичних оцінок невідомих нестаціонарних параметрів середніх значень за спостереженнями реалізацій послідовності випадкових матриць. Показано, що такі гарантовані оцінки одержуються або як розв’язки крайових задач для систем лінійних різницевих рівнянь, або як розв’язки відповідних задач Коші. Установлено вигляд похибок для гарантованих середньоквадратичних квазімінімаксних оцінок спеціального вектора прогнозу та параметрів невідомих середніх значень. За наявності малих збурень відомих матриць у моделі матричних спостережень знайдено квазімінімаксні середньоквадратичні оцінки і в першому наближенні методу малого параметра отримано їх гарантовані середньоквадратичні похибки. Наведено два тестові приклади обчислення гарантованих середньоквадратичних оцінок та їх похибок. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2023-06-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/272758 10.20535/SRIT.2308-8893.2023.2.07 System research and information technologies; No. 2 (2023); 86-103 Системные исследования и информационные технологии; № 2 (2023); 86-103 Системні дослідження та інформаційні технології; № 2 (2023); 86-103 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/272758/279554 |
| spellingShingle | матричні спостереження лінійне оцінювання гарантована середньоквадратична оцінка похибка гарантованої середньоквадратичної оцінки квазімінімаксна гарантована оцінка вектора різницеве рівняння метод малого параметра збурення матриць Nakonechnyi, Oleksandr Kudin, Grygoriy Zinko, Petro Zinko, Taras Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності |
| title | Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності |
| title_alt | Guaranteed root-mean-square estimates of the forecast of matrix observations under conditions of statistical uncertainty |
| title_full | Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності |
| title_fullStr | Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності |
| title_full_unstemmed | Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності |
| title_short | Гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності |
| title_sort | гарантовані середньоквадратичні оцінки прогнозу матричних спостережень в умовах статистичної невизначеності |
| topic | матричні спостереження лінійне оцінювання гарантована середньоквадратична оцінка похибка гарантованої середньоквадратичної оцінки квазімінімаксна гарантована оцінка вектора різницеве рівняння метод малого параметра збурення матриць |
| topic_facet | matrix observations linear estimations guaranteed RMS estimates guaranteed RMS estimate errors quasi-minimax guaranteed vector estimates difference equation small parameter method matrix perturbation матричні спостереження лінійне оцінювання гарантована середньоквадратична оцінка похибка гарантованої середньоквадратичної оцінки квазімінімаксна гарантована оцінка вектора різницеве рівняння метод малого параметра збурення матриць |
| url | https://journal.iasa.kpi.ua/article/view/272758 |
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