State space modeling of time series.

*(English)*Zbl 0606.62102
Berlin etc.: Springer-Verlag. XI, 314 p. DM 120.00 (1987).

This is a book about linear systems and time series with a special emphasis on the construction of state space models from the observed data without the application of statistical techniques. Rather, the author advocates the use of the singular value decompositions and the associated theory with special balanced realizations to obtain models, where the crucial step of controlling the complexity or the dimensionality of the models is done by ”sound judgement”. This broad program is carried out quite thoroughly with a large number of numerical examples illustrating the details.

The first four chapters are preliminary ones, in which the elementary ideas of linear dynamic systems are introduced together with scattered practical tricks and devices about preprocessing observed time series. In the following six chapters the theory of multi-input/output linear systems is developed, and the algorithms and procedures for the various realizations are described. These chapters also include an adequate treatment of the theory of Kalman filters, while the discussion of the simpler and equally important ARMA predictors, where the role of the Riccati equation is played by the Cholesky factorization - not of the data sequence as described but of an input sequence - is only rudimentary.

Although we do not share the author’s enthusiasm of his general approach for modeling vector time series, we do think that the book fills a need in teaching the theory of multi-input/output systems, including the practicable aspects of the Adamjan-Arov-Krein theory.

The first four chapters are preliminary ones, in which the elementary ideas of linear dynamic systems are introduced together with scattered practical tricks and devices about preprocessing observed time series. In the following six chapters the theory of multi-input/output linear systems is developed, and the algorithms and procedures for the various realizations are described. These chapters also include an adequate treatment of the theory of Kalman filters, while the discussion of the simpler and equally important ARMA predictors, where the role of the Riccati equation is played by the Cholesky factorization - not of the data sequence as described but of an input sequence - is only rudimentary.

Although we do not share the author’s enthusiasm of his general approach for modeling vector time series, we do think that the book fills a need in teaching the theory of multi-input/output systems, including the practicable aspects of the Adamjan-Arov-Krein theory.

Reviewer: J.Rissanen

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

93B30 | System identification |

62M20 | Inference from stochastic processes and prediction |

91B84 | Economic time series analysis |

93E12 | Identification in stochastic control theory |