Differential geometry of a parametric family of invertible linear systems—Riemannian metric, dual affine connections, and divergence

A parametric model of systems is regarded as a geometric manifold imbedded in the enveloping manifold consisting of all the linear systems. The present paper aims at establishing a new geometrical method and framework for analyzing properties of manifolds of systems. A Riemannian metric and a pair of dual affine connections are introduced to a system manifold. They are calledα-connections. The duality of connections is a new concept in differential geometry. The manifold of all the linear systems isα-flat so that it admits natural and invariantα-divergence measures. Such geometric structures are useful for treating the problems of approximation, identification, and stochastic realization of systems. By using theα-divergences, we solve the problem of approximating a given system by one included in a model. For a sequence ofα-flat nesting models such as AR models and MA models, it is shown that the approximation errors are decomposed additively corresponding to each dimension of the model.

[1]  S. Amari,et al.  Geometrical theory of higher-order asymptotics of test, interval estimator and conditional inference , 1983, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[2]  Graeme Segal,et al.  The topology of spaces of rational functions , 1979 .

[3]  F. Götze.  Differential-geometrical methods in statistics. Lecture notes in statistics - A. Shun-ichi. , 1987 .

[4]  R. Kálmán.  Mathematical description of linear dynamical systems , 1963 .

[5]  Christopher I. Byrnes,et al.  Algebraic and Geometric Aspects of the Analysis of Feedback Systems , 1980 .

[6]  S. Amari.  Differential Geometry of Curved Exponential Families-Curvatures and Information Loss , 1982 .

[7]  J. Cadzow.  Maximum Entropy Spectral Analysis , 2006 .

[8]  J. Shore,et al.  Minimum cross-entropy spectral analysis of multiple signals , 1983 .

[9]  Robert Hermann,et al.  Applications of Algebraic Geometry to Systems Theory: The McMillan Degree and Kronecker Indices of Transfer Functions as Topological and Holomorphic System Invariants , 1978 .

[10]  甘利 俊一,et al.  Estimation of structural parameter in the presence of a large number of nuisance parameters , 1983 .

[11]  P. Bloomfield.  An exponential model for the spectrum of a scalar time series , 1973 .

[12]  E. J. Hannan,et al.  Estimating the dimension of a linear system , 1981 .

[13]  Manfred Deistler,et al.  Some properties of the parameterization of ARMA systems with unknown order , 1981 .

[14]  Roger W. Brockett,et al.  A scaling theory for linear systems , 1980 .

[15]  Michiel Hazewinkel.  Moduli and canonical forms for linear dynamical systems II: The topological case , 2005, Mathematical systems theory.

[16]  David F. Delchamps,et al.  Global structure of families of multivariable linear systems with an application to identification , 1985, Mathematical systems theory.

[17]  R. J. Bhansali,et al.  The inverse partial correlation function of a time series and its applications , 1983 .

[18]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[19]  E. Gilbert.  Controllability and Observability in Multivariable Control Systems , 1963 .

[20]  Michiel Hazewinkel,et al.  (Fine) moduli (spaces for linear systems : what are they and what are they good for , 1980 .

[21]  MEIR PACHTER,et al.  Some Geometric Questions in the Theory of Linear Systems , .

[22]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[23]  S. Amari.  A foundation of information geometry , 1983 .