By D. Dacunha-Castelle, H. Heyer, B. Roynette, P. L. Hennequin
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Additional resources for Ecole d'Ete de Probabilites de Saint-Flour VII. 1977
In Chap. 14 we consider the zero structure of minimal spectral factors and show that this leads to a geometric theory of zero dynamics and output-induced subspaces. This gives a geometric motivation for such phenomena as invariant directions in matrix Riccati equations. We also discuss the local structure of the geometry of the Riccati inequality. Chapter 15 deals with smoothing and interpolation. Smoothing is a basic applications area in systems and control which is particularly adapted to a stochastic realization approach.
Chapters 3 and 4 are intended to provide a reasonably coherent survey of parts of the theory of stationary stochastic processes relevant to the topics of this book, and some proofs have been omitted with reference to the literature. We cover among other things relevant topics in harmonic analysis of stationary processes, spectral representations, Wiener-Kolmogorov theory of ﬁltering and prediction, the Wold decomposition, spectral factorization, Toeplitz matrices and the Szegö formula. The reader is recommended to skim this material at a ﬁrst reading and return to it when needed.
A; B/. 23) is a generalization of the well-known Rayleigh quotient iteration of linear algebra. It can also be found in [121, p. 584]. v1 ; v2 ; v3 ; : : : / the canonical variables. These notions play an important role in various problems of model reduction, approximation and identiﬁcation of stochastic systems. See, in particular, Chap. 11 and the following chapters. To explain the role of canonical correlation analysis in statistics we shall consider two ﬁnite-dimensional subspaces A and B of zero-mean random variables of dimensions n and m, respectively.
Ecole d'Ete de Probabilites de Saint-Flour VII. 1977 by D. Dacunha-Castelle, H. Heyer, B. Roynette, P. L. Hennequin