MINIMAX FILTERING OF LINEAR TRANSFORMS OF STATIONARY ...

T e o p . BeposTHOCT.

h

Theor. Probability and Math. Statist. No. 44, 1992

M aT eM . CTaTHCT.

Bun. 44, 1991

MINIMAX FILTERING OF LINEAR TRANSFORMS OF STATIONARY PROCESSES UDC 519.21

M. P. MOKLYACHUK This article considers the problem of linear mean-square optimal estima­ tion of the transform AS = / 0°° a(t)S(~t)dt of a stationary random process S{t) with density f{X) from observations of the process S(t) + rj(t) for t < 0 , where r\{t) is a stationary random process with density g(X) that is uncorrelated with SU) . Formulas are obtained for computing the mean-square error and the spectral characteristic of an optimal linear estimator of the transform AS . The least favorable spectral densities fo(X) e 2 f and £oW e and the minimax (robust) spectral characteristics of an optimal estimator of AS are found for various classes S f and 2 g of densities. A b s tra c t.

Suppose that the function a(t) determining the transform AS, satisfies the conditions (1)

If the spectral densities f{X) and g(A) are regular, then they adm it canonical fac­ torization [ 1 ]: (2 )

(3)

f(A) + g(A) = \d{iA)\2 = \b{iA)\ 2 , (4)

d(iA) = /

dte uXd t ,

b(iA) = I

bte ltXd t .

The density f(A)+g(A) adm its canonical factorization if one o f the densities /(A) or g(A) is regular. If g(A) is regular, then the mean-square error o f a linear estim ator AS with spectral characteristic h(A) = / 0°° hte~ ‘a d t can be computed by the formula 1991 Mathematics Subject Classification. Primary 62M20, 60G35, 60G10. © 1 9 9 2 A m erican M a th e m atica l Society 0 0 9 4 -9 0 0 0 /9 2 $ 1 .0 0 + $.2 5 p e r page

95

96

M. P. MOKLYACHUK

A ( h ; f , g ) = E\AS-AS\2 1 /*°°

=2

MW - *№I2( / W + sW) /tt

-^J™ JA{X)-h{X))A{A)g{k)dk oo

1

f° ° _

- 2n ] _ poo

1

poo

p oo

/>oo /»oo

A/)S(J) /

•J0 Jo /»oo /»oo_________ _

/

J/ 00

J

Vt-uWs-u d u d s d t

J —oo /»min(£,s)

(a(t) - h t)a(s) /

Jo Jo poo poo

+

dt- ud s- u d u d s dt

/>m in(/,ij /»min(/,,s)

/ (a ( 0 -

- /

MW|2s W d k

p m i n ( t , s ) ________

/ / (a(t) - ht)(a(s) - hs) J0 J 0 J —co -

f°°

M W - AWMWtf W d * + 2^ J _

Wt-uWs-udb

J —oo pm in(t,s)

/ a{t)a{s) /

Vt-uVs_u d u d s d t

J —oo

»/0 0

= (£)(a - h ) , a - h) - (»Fa, a - h) - (a - h , T'a) + (a, T 'a ), where A(A) = / 0°° a(t)e~‘a d t , (a, h) is the inner product in the space L 2 [0 , oo), and D and T' are the operators acting in L 2 [0 , oo) with kernels / minier ,S) _ d t- uds- ud u,

0in f, ( / , g) e 31 is equivalent to the following unconditional extremum problem on the whole space L x x L \ [6 ]: A ® (/, g) = - A ( h ( f o , g o ) ; f , g) + ô{ { f , g)\3>) -> in f, where S((t , g)\31) is the indicator function of the set 3 1 . The solution fo , go o f this problem is characterized by the condition 0 € dAs (f0 , go), where d A ^ ( f 0 , go) is the subdifferential o f the convex functional A ^ ( f , g ) . Let us consider the problem for the set o f densities &o,o = { ( / № , g W )

2^

/_ °° №

d A < P x; ^

£

g(A) dA < P2 J

MINIMAX FILTERING OF LINEAR TRANSFORMS

99

From the condition 0 € dAg,(f0 , go) for such a set we get that the least favorable densities satisfy the relations PO O

(14)

f o W + go(A) = ai\rg{X)\2 = a\ / (Cgb)(t)e-i a c Jo

(15)

/o(A) + go(A) = f ( A ) , y2(A) > 0 a.e. and y2 (A) = 0 for _/ô(A) < / 2 (A), 7 3 (A) < 0 a.e. and y3 (A) = 0 for g 0 (A) > gi(A ), 74 (A) > 0 a.e. and 7 4 (A) = 0 for g 0 (A) < 5 2 (A). If the density g (A) is given, then the least favorable density /o(A) € 2 j j has the form pO O

(21)

/ (Cg b)(t)e- ■itAd t J0

fo(A) = min < max < ai

- * ( A ) , / i ( A ) L / 2 (A)

But if the density /(A) is given, then the least favorable density go (A) € 2 f ^ has the form pO O

(2 2 )

/ Jo

g 0 (A) = m in I max { a 2

{Cf \>)(ty

— itX

dt

~fW,

5 1 (A)

} , g 2 (A)

Theorem 2. The least favorable spectral densities /o(A) and g 0 (A) in the class 2 = 2 ^ x 2g* are determined by the relations (19), (20), (2)-(4), (9), (16). I f a regular density g (A) (or /(A )) is given, then the least favorable density fo(A) e 2jj(go(A) e 2 f f ) is determined by the relations (21), (2)-(4), (11), (16) (or (22), (2)-(4), (12), (16)). The m inim ax spectral characteristic o f an optimal estimator o f A£, is computed from the form ulas (7) and ( 8 ). From the relations ( 19)—(22) it is possible to find the form of least favorable spectral densities for the set 2 = 2 Bl x 2 Sl, ^

= |/(A )|/(A ) - (l - e 1 )«i(A) + e 1M(A), g(A)|g(A) = (1 - s 2)vi (A) + e2v(A),

f /

f(A)dA = Pi OO

g W dA — P2

2n -O O

where Mi (A) and v f A) are given densities. Such sets describe models of “econtam ination” o f random processes [2]. If in the relations (19)-(22) we set /i(A) = (1-«i)m i(A ), fi(A) = 0 0 , gi(A) = ( 1 - £ 2 )^ 1 (A), and g2(A) = 0 0 , then we get relations for determining the least favorable densities o f the set 2 Sl x 2 S2 : pO O

(23)

f o W + £o(A) — ot 1

/ Jo

(24)

fo(A) + go (A) =

/ (Cfb)(t)e Jo

(Cgb)(t)e

-itX

dt

OhW+i)

dt

( y 3( A) + i )

poo

(25)

0:2

/o(A) = max

(Cgb)(t)e {

-

I

dt

-1

.

-1

.

- g (A) , (1 - fii)«i(A)

f

poo

(26)

go (A) = max I a 2 / (Cf b)(t)e~M dt Jo

- f (A), (1 -

s 2) v i (A)

Here 71 (A) < 0 a.e. and 7 1 (A) = 0 for f 0(A) > (1 - s\)u\(A), 7 3 (A) = 0 for g 0 (A) > (1 - e2 )ui(A).

7 3 (A)

< 0 a.e. and

Theorem 3. The least favorable spectral densities f f X) and go(A) in the class 2 S{ x 2 &1 are determined by the relations (23), (24), (2)-(4), (19), (16). I f a regular density g(A) (or f(A)) is given, then the least favorable density f f X ) e D£t (go(A) e 2 £f) is determined by the relations (25), (2)-(4), (11), (16) (or (26), (2)-(4), (12), (16)).

101

MINIMAX FILTERING OF LINEAR TRANSFORMS

The m inim ax spectral characteristic o f an optimal estimator o f the transform At; is fo u n d from the form ulas (7) and ( 8 ). For the spectral densities in the set 2 = 2 ^ x 2 ^ , = [ m *£={*№

^

f~

I№

- A W \ d X < Bl} ,

^ £ j s № - g i ( A ) | < / A < e2}

,

where f ( A) and gi(A) are given bounded densities (the “e-neighborhood” model [2 ]), we get the following relations: pO O

(27)

fo(A) + go (A) — “

(28)

fo(A) + go(A) =

1

/

Jo

(Cgb)(t)e~ia dt

7 1 (A),

(Cfb)(t)e~‘a dt

72 W ,

poo

a2

/

Jo

where | 7 !(A)| < 1, yi(A) = sgn(/ 0 (A) - f ( A) ) for / 0 (A) f f ( A ) , and y2(A) < 1, and 7 2 (A) = sgn(g 0 (A) - gi (A)) for g0(A) f g, (A). Therefore, for a given regular density g (A) the least favorable density fo(A) e 2 e\ has the form 2

poo

/o(A) = max { c*i / (Cgb)(t)e~M dt Jo

(29)

1

-g(A),/i(A)

And for a given regular density /(A) the least favorable density go (A) e 2 } z has the form 2 Ï (30) ^o(A) = max < 0 2 | y (Cfb)(t)e ltX dt - / ( A ) , g , ( A ) The coefficients a 1 and a 2 o f the operators C f and C.g and the function bt are found from the factorization (2)-(4) and from the norm alization conditions (31)

1 2^

r°° / . \ f W - A W ] d A = Sl,

1 2^

J_

\ g W - g i W \ d A = e2 .

Theorem 4. The least favorable spectral densities fo(A) and go (A) in the class 2 ^ x 2 } 2 are found from the relations (27), (28), (2)-(4), (9), (31). I f a regular density g(A) (or /(A )) is given, then the least favorable density fo(T) e 2 j i(go(A) e 2 ^ ) is fo u n d from the relations (29), (2)-(4), (11), (31) (or (30), (2)-(4), (12), (31)). The m inim ax spectral characteristic o f an optimal estimator o f At; is computed from the form ulas (7) and ( 8 ). If the possible values of the densities belong to the set 2 = 2 2 x 2 }2 ,

={/№I^ 2

2

{^(A)

I№ J

- A W \ 2dA