By Anders Lindquist
Maximizes reader insights into stochastic modeling, estimation, approach identity, and time sequence analysis
Reveals the ideas of stochastic nation house and country area modeling to unify the idea
Supports additional exploration via a unified and logically constant view of the subject
This ebook provides a treatise at the conception and modeling of second-order desk bound procedures, together with an exposition on chosen software components which are vital within the engineering and technologies. The foundational matters relating to desk bound approaches handled at the start of the e-book have an extended heritage, beginning within the Nineteen Forties with the paintings of Kolmogorov, Wiener, Cramér and his scholars, particularly Wold, and feature for the reason that been subtle and complemented via many others. difficulties in regards to the filtering and modeling of desk bound random indications and platforms have additionally been addressed and studied, fostered via the appearance of recent electronic pcs, because the primary paintings of R.E. Kalman within the early Sixties. The ebook deals a unified and logically constant view of the topic in keeping with basic rules from Hilbert house geometry and coordinate-free pondering. during this framework, the thoughts of stochastic country area and kingdom house modeling, in response to the proposal of the conditional independence of previous and destiny flows of the appropriate signs, are printed to be essentially unifying principles. The publication, in response to over 30 years of unique learn, represents a helpful contribution that might tell the fields of stochastic modeling, estimation, approach id, and time sequence research for many years to come back. It additionally offers the mathematical instruments had to grab and learn the buildings of algorithms in stochastic platforms conception.
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Extra info for Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
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.
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification by Anders Lindquist