Krein factorization of covariance operators of 2-parameter random fields and application to the likelihood ratio

The covariance operator of a signal-plus-noise 0bserva:ion for a 2-parameter random field is factorized as products of Volterra type operators. As a result of this factorization we can express the determinant of this operator in terms of the system parameters. This determinant appears in the likelihood ratio that plays a significant role in identification and detection problems. Index Terms -Random fields, likelihood ratio, Krein factorization. I . INTRODUCTION ONSIDER a 2-parameter random field ( X , ) on some C compact set 3 in R2. Let {q} be the observation process of this field. For purposes of identification and signal detection, we are interested in determining the likelihood ratio of the model, which is the Radon-Nikodym derivative of the measure induced by the observation process with respect to the standard Gaussian measure. In 171, Shepp gives an expression for the likelihood ratio in the l-parameter case. This formula of Shepp involves the determinant term d e t ( I + R ) of the covariance operator of the observation process. In practice, however, this term is not readily computable. For 1-parameter two-point boundary value process models, Bagchi and Westdijk [31 succeeded in expressing this determinant in terms of the system parameters. The key was to use Krein factorization, which has already been applied earlier for this purpose in the Markov case [4]. In this paper we give an explicit Krein factorization formula for the operator ( I + R ) in the 2-parameter case when 9 is a rectangle. It turns out that we can factorize the operator in this case into a product of four “Volterra” type operators. An extension of our result to the multiparameter case is straightforward. For an abstract of this type of factorization, see Bromley and Kallianpur [6]. The likelihood ratio for the Gaussian signal-plus-noise model is derived next. The expression is very similar to that obtained by Shepp [7] in the I-parameter case. The determinant term in the likelihood ratio may be expressed in the system parameters using the Krein factorization of ( I + RI. In the 2-parameter situation, Wong and Zakai [81 gave an expression for the likelihood ratio using the 2-parameter Brownian motion model for the observation. This formula has to be modified in practice by a correction term when using real data. This is due to the fact that the observation Manuscript received November 8, 1989; revised April 20, 1990. The authors are with the Department of Applied Mathematics. University of Twente, P.O. Box 217. 7500 AE Enschede. The Netherlands. IEEE Log Number 9038859. noise can never be a true 2-parameter Brownian motion (in the integrated form). Using a direct white noise model for the observation noise, Bagchi [ l ] derived an expression for the likelihood ratio which differed from the Wong-Zakai formula. In analogy with the 1-parameter case, it may be inferred that this difference is precisely the correction term previously menlioned. The determinant term calculated in the present paper may give new possibilities for expressing the correction term. We do not pursue this point any further in this paper. 11. KREIN FACTORIZATION F R 2-PARAMETER COVARIANCE OPERATORS We first consider a 1-parameter stochastic process { X t ) in L2(Cl), t E [0, TI, taking values in R”, with Hilbert-Schmidt covariance operator R: L$[O, TI) 4 L;([O, TI), i.e., Rf(t) = / ( r r ( t , s)f(s)ds with n X n matrix valued kernel r ( t , s) = cov( X , , X , ) . Let {Y,}, t E [0, T ] be the measurement process Y/ = x, + n , (2.1) with {n , } , t E[O,T] , a Gaussian standard white noise independent of X . Then the observation process has covariance operator R , = I + R. Following a result of Krein [5, p. 1301 we can factorize (I + RI-’ as ( I + R)-I = ( I L*)( I L ) (2.2) where L is a Volterra operator on L;([O,T]) and L* is its adjoint, with L f ( t ) = / $ ( t , s)f(s)ds. Suppose further that L + L* is trace class. Then one can prove [4, p. 2321 that log(det ( I + R ) ) = t r ( L + L*) = JTt r I ( I , t ) d t , (2.3) where tr denotes the trace of an operator aind tr denotes the trace of a matrix. Our purpose is to extend this result to the 2-parameter case when the domain 9 is a rectangle. Let g = [ O , T , ] X [0, T,] , 2 = L ; ( g ) and consider a 2-parameter n-vector value random field ( X , } , t = ( I , , f 2 ) E 9. Let R: 3 + A? be the H-S covariance operator of the random field { X , } ; that is, Rf(t) = / ? r ( t , s ) f ( s ) d s (ds = ds, ds,), with r ( t , s) = cov(X,, X , ) continuous in 22 and /,/,(lr:t,s)lI’dsdt <m. Then we obtain the following factorization. 0 Theorem 2.1: ( I + R ) = ( I P l ) ( I P I ) ( I P T ) ( 1 P ; ) (2.4) 0018-9448/91/0100-0053$01.00