IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-27, NO. 4,JULY 1981 zyx 446 1271 1281 ~291 C. R. Bishop and H. R. Warner, “A mathematical approach to medical diagnosis: Application to polycythemic states utilizing clini- cal findings with values continuously distributed,” Comput. Biomed. Res., vol. 2, pp. 486-493, 1969. - J. A. Bovle. W. R. Griee. D. A. Franklin. R. M. Harden. W. W. Buchanan, and E. M. MyGirr, “Construction of a model for com- puter-assisted diagnosis: Application to the problem of non-toxic goitre,” Quart. J. Med., N.S. 35, pp. 565-588, 1966. C. A. Nugent, H. R. Warner, J. T. Dunn, and F. H. Tyler, “ Probability theory in the diagnosis of Gushing’s syndrome,” J. Clin. Endocrinology vol. 24, pp. 621-627, 1964. [30] A. Reale, G. A. Maccacaro, E. Rocca, S. D‘Intino, P. A. Geoffre, A. Vestri, and M. Motolese, “ Computer diagnosis of congenital heart disease,” Comput. Biomed. Res., vol. 1, pp. 533-549, 1968. [3 l] C. R. Rao, “Inference in discriminant function coefficients,” Essays Probability and Statistics, R. C. Bose, et al., Eds. Chapel Hill, NC: University of North Carolina and Statistical Publishing Society, pp. 487-602, 1970. [32] W. W. Cooley and P. R. Lohnes, Multivariate Data Analysis, Chapter 9. New York: Wiley, 1971. [33] B. Busca and P. Diethelm, “ Discriminant analysis using information gain (DIG),” Computer program prepared for WHO. Schur Recursions, Error Formulas, and Convergence of Rational Estimators for Stationary Stochastic Sequences PATRICK DEWILDE, MEMBER, IEEE, AND HARRY DYM Abstracr- An exact and approximate realization theory for estimation and model filters of second-order stationary stochastic sequences is pre- sented. The properties of J-lossless matrices as a unifying framework are used. Necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrfx of a .7-lossless system are deduced. An extension of the so-called Schur algorithm yields an ap- proximate J-lossless realization based on partial past information about the process. The geometric properties of such partial realizations and their convergence are studied. Finally, connections with the Nevanlinna- Pick problem are made, and how the techniques presented constitute a generali- zation of many aspects of the Levinson-Szego theory of partial realiza- tions is shown. As a consequence generalized recursive formulas for reproducing kernels and Christoffel-Darboux formulas are obtained. In this paper the scalar case is considered. The matrix case will be considered in a separate publication. I. INTRODUCTION L ET x(t), t = 0, * 1; . ., denote a real stationary zero mean scalar stochastic sequence with covariance func- tion r(k) = E[x(t)x(t - k)] = &]-ne-i*eW(e’e) de and spectral density W. Manuscript received November 28, 1979. This paper was supported in part by the Mathematics Division of the Army Research office under Contract DAA G 29-77-C-0042. P. Dewilde is with the Department of Electrical Engineering, Delft University of Technology, Mekelweg 4, Delft, The Netherlands, H. Dym is with the Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel. A shaping (or modeling) filter for this stochastic process is a linear time-invariant causal filter which when driven by white noise produces an output with covariance equal to r(k). An innovations or estimation filter is the inverse of a shaping filter. It produces white noise when it is driven by the given stochastic sequence. In practice it is of interest to establish recursive procedures for computing approximate shaping and innovations filters and thus avoid having to recompute the whole filter each time the permitted com- plexity is increased: the additional improvement should be achieved by adding sections without changing the existing structure. The same philosophy underlies both the “ladder structure” used in modern digital filter design and the theory of orthonormal expansions: the initial coefficients 4 = (f, $I,) of f with respect to an orthonormal set of functions Gj, j = 1,. *. , n, in a Hilbert space with inner product (. , .) do not have to be recomputed as the size of the set changes. Levinson [l] established a recursive procedure for com- puting a sequence of approximate autoregressive (AR) shaping filters from the covariance data which, under mild conditions, converge to the exact shaping filter. The theory of these filters is intimately connected with the theory of polynomials on the unit circle which are orthogonal with respect to the spectral density of the underlying stochastic sequence. In the paper of Dewilde, Vieira, and Kailath [2] it was shown that the Levinson procedure could be viewed as a special case of an algorithm for coprime factorization. In this paper we exploit the greater flexibility afforded by 001 S-9448/81/0700-0446$00.75 0 198 1 IEEE