Russian Physics Journal, Vol. 38, No. 9, 1995

CONSISTENCY OF ADAPTIVE ESTIMATORS ON THE BASIS OF

CORRELATED OBSERVATIONS

G. A. Medvedev

Suppose that a set of sample values (observations) have the following structure:

rl

Yt = Y2,c~q~ (l) +~ t = q0" (t) c-k ~t, (1) K = I

where c is a column vector of unknown parameters the value of which is to be estimated in the process of identification; n,

the dimension of the vector c; c* = (c I . . . . . Cn), where* is the sign for the transpose; yt, observation of the output variable

at time t, t a discrete time (t = 1, 2 . . . . ); and ¢*(t) = (el(t) . . . . . Cn(t)) a known vector; ~t, additive noise with null

mathematical expectation M{( t } = 0 and correlation function M{(~(t} = p(K, t). The variance of ~t is denoted Pt. We also let

y*(N) = (Yl . . . . . YN) be an N-dimensional vector of observations; q,*(N) = (¢(1) ... ~(N)) a known matrix of dimension (n

x N); and (*(N) = ((1 . . . . . (n) , an N-dimensional noise vector.

Then the observations over time 1 _< t _< N is written in matrix form:

!l (N) = (I) (N) c ~- ~ (N). (2)

Here M{~(N)(*(N)} = R(N) will denote the noise correlation matrix. R(N) = Ilp( , t)II, >- 1, t _< N. We also let

r*(N) = (0(1, N) ... p ( N - 1, N)) be an (N - - D-dimensional correlation vector.

The most thorough investigation of the adaptive procedure of estimation carried out by means of the method of least

squares based on the use of independent observations is that of Albert and Sittler [1]. We will extend their results to the case

of dependent observations.

Theorem 1. I f the components of the noise vector in (2) are correlated, the adaptive estimator constructed by the

method of least squares has the form

c(N) = c ( N - - l ) + ~ (A')[1/,--r ~' (A')R-I (N- -1)y(N--1) - - --'t'~ (N) c (..V--I)].

Here ~(N) = ¢(N) - - q~* (N- 1 ) R - I ( N - Dr(N), and the influence factor 3,(N) is calculated by means of the formula

(3)

7 ( N ) =

a (N) , if a ( N ) -- 0, (4) tF* ( N ) a (N)

b (N) if a (N) = 0, (5) z~v + W* (N) b ( N ) '

where

and

Belorus State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 55-58,

September, 1995.

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a (..v) =A ( x - - 1 ) ~ (N)

b(N) = B(N--1 ,W(.,V)

are both n-dimensional vectors, while the (n x n)-dimensional matrices A and B satisfy definite recursion relations.

The computational complexity of adaptive estimators constructed by means of the method of least squares often is an

obstacle to their application. Estimators constructed after the pattern of stochastic approximation (called stochastic

approximation estimators), which for the case we are considering have the form

c(N) = c(N--1) + ~ (~.;) (~,,--~* (N)c (N- -1 ) ) .

prove to be far simpler.

For independent observations, influence factors selected simply as [2]

(6)

-; (.v) =cq: (N)/~';. (7)

ensure the convergence of (6).

The estimator (6) may be improved if the influence factor is selected in optimal fashion, i.e., to minimize the spur of

the matrix of variations. Suppose

(8) H(N) = M ( ( c ( N ) - - c ) ( c ( X ) - - c ) * } , h N = t r H ( N ) .

Theorem 2 [3]. If the components of the noise vector in [2] are correlated, the influence factor 3,(N) of the adaptive

estimator constructed as a stochastic approximation estimator, as in (6), and which minimizes h N is calculated by means of the

formula

H ( N - - ] ) ~ ( 3 " ) - - ) , ( N )

,~ - - ~* ( N ) / - / ( N -- 1) ? (N) - - 2?* (N/? . (N) '

(9) v(N) =

where X(N) = L ( N - 1)r(N) and the (n x N)-dimensional matrix L is determined recursively:

(lO) L (1) --- 7 (1) ,L (N) = : ( I - - ~,(A) ~F* (N~) L ( N - - 1)17 (N)~.

and X(1) = 0 if the observations are not correlated with the a priori estimators.

The variation matrix H(N) in this case is calculated recursively by means of the following relation:

H (N) = H ( N - - 1) -- ( H ( N - - 1) ~ ( N ) - - ) , ( N ) ) ( H ( N - - 1 ) ~ ( N ) - - ) , ( N ) ] * pN + ,~* (Af) H(?v - - 1) ~ (:V) -- 2"+ * (N) "~ (N) (11)

The existence of correlation between observations usually worsens the quality of the estimators. Therefore, situations

are possible in which correlation between observations does not enable us to construct a consistent estimator. Let us consider

this problem using as an example the scalar case of estimation of mathematical expectation. In this case n = 1, ~o(t) = 1, for

all t and

9 t = c + ~ , t ~ l . (12)

Let us first consider estimators constructed by means of the method of least squares. We let e denote a vector of the

corresponding dimension, consisting of ones, e = (1 1 ... 1). Let

c~=l , e t = l - - e R - I ( i - - 1 ) r ( t ) , t > l . (13)

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Then an estimator of the parameter c constructed by means of the method of least squares will, by (3), have the following form:

c t = c t - l q - "~ ( t) (y t - - r* ( t) t?, - t ( t - - 1 ) y ( t - - 1 ) - - e t c t - ~ ) . (14)

2 From (4) and (5), we find that a(1) = 1, a(t) = 0, t > 1, b(1) = 0, b(t) = bt_let , t > 1, and b 1 = Pl = ~1 and the following

recursion relation holds:

2 with a t as defined previously.

_ _ o~ bt 1 , " f t - b t ~ -k e[ bt-1

~t b ,_ 1 ( 1 5 ) , ' h = l .

Let us now estimate the variance of the estimator (14). Let ~/t = C t - - C. Then

~,a = ~J, ~t = (1 - - "h Et) ~ t - ~ + "h (~t - - r* ( t ) R -1 ( t - - 1 ) ~ ( t - - 1 ) ) . ( 1 6 )

Let us introduce the t-dimensional row vector q(t), determining it recursively by the following relation:

q ( 0 = ((1 - - ~'t ~t) q (t - - 1) - - "ft r* (t) R -1 (t - - 1) "~t), q (1) = 1. (17)

It is easily verified that ~t = q(t)~(t). Therefore, the variance of the estimator (14)

ht = D{ct} = M{~ ~t} = q (t) R (t) q* (t) . (18)

Using (17), it is possible to recast (18) in recursive form and calculate, by means of (18), the variance in explicit form:

t--1 [

h t = (1 -- "h ~t)~h, - , + "f~ ~ = "f~ n ~ + ~ , ~ ~ I-~ (1 - - "r, %)K (19) /c=i l= /¢+!

We denote

c~ ( 1 - - e R -1 (t - 1) r (t)) 2 (20)

~ p t - r * ( t ) R - ~ ( t - l ) r ( t )

From (15) it follows that

t

bt~l/(~t-l'Jf-U'):~I/K~= ( 2 1 )

and, further,

t i '--I f

n = l x = l h '=l

Using the latter result, we obtain the variance h t in the form

f

ht-=bt=l/Y~.u~. (23)

Theorem 3. An estimator constructed by means of the method of least squares will be consistent if and only if the t

series ~ u K diverges as t --, + oo. r = l

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In fact, if the series ( converges, then, as follows from (23), the variance will tend to a positive constant as t ~ oo

and the estimator will not be consistent.

Hence, certain useful corollaries may be established. Suppose that the noise is Markovian with variance 02 and t

parameter o(t, t + 1) = o < 1. Then u t = (1 - - o)/a2(1 + O) for all t. The series ~ u K diverges for all p < 1 and the K=I

variance of the estimator constructed by means of the method of least squares

h t : o 2 1 + p 1 *0 . 1 - - p t t~oo

Now suppose that the noise is uncorrelated. Then the limit ~ is a necessary condition that ensures that an estimator

constructed by means of the method of least squares will be consistent. The condition also constrains the growth of the variance t

of the noise over time: l i m ( ~ 1/pK) -1 =0. t'--> °° t¢=1

Let us now consider a stochastic approximation estimator (6) with influence factor Yt = a/t. As in (10), we introduce

the t-dimensional row vector l(t) by means of the following recursion relation:

/ ( 1 ) = a , l ( t ) = ( l - - a / t ) l ( t - - 1 ) a/ i ) , t > l . (24)

Then ~Tt = ct - - c = ~lt(a)~o + l(t)}(t), where

,~,t ( a) = ( l - - a / i ) ( 1 - - a / i - - 1 ) . ... . ( l - - a / 2 ) ( t - - a ) .

The variance of the estimator (6) has the form h t = ~r}t (oOh o + l(t)R(t)l*(t). Here h o is the variance of the a priori estimator. Using the recursion relation (24), we obtain

a ~ (2 2 h t = ( 1 - a ; ' t ) 2 h t _ , + 2 ( I - - a / t ) - / - l ( t - - 1 ) r ( t ) m - f i - O t =

t - I (25)

- - , : ~ , (a) ho + 7 = l = n + l

Here

0 t = T P t + 2 1 a

l ,~z = ~ (1 - - al l ) ~.

l - - 1

Theorem 4. Suppose that the following conditions are satisfied:

"(t = a / t , a > / 1 , i i m l ~ 9K = 0. t~oo t ~

(26)

In order for the estimator (18) to be consistent, it is sufficient that

Iim 1 . ~ - - ~ ( x - - a ) l ( 1 c - - I ) r ( ~ ' ) = O .

t - ~ t : (27)

The fact that, for a _> 1 and K < t, we have

l

l= t¢+1

is made essential use of in the proof of (27).

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Note that when a = 1, the estimator (6) turns into a sample mean, and the limit (27) into the well-known consistency

condition for a sample mean in the case of correlated observations:

lim 1 ~_~'~ ~ f-~o tS- 0 (l, ~c) = 0. K : 2 I = 1

REFERENCES

.

2.

3.

A. Albert and R. W. Sittler, J. SIAM Control. Series A, 3, No. 3,384-417 (1966).

Ya. Z. Tsypkin, Foundations of the Theory of Training Systems [in Russian], Nauka, Moscow (1970).

G. A. Medvedev, Avtomatika i Telemekhanika, No. 5, 110-116 (1974).

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