Abstract
Linear state-space systems, like ARMA systems, are models for stationary processes, more precisely for the class of stationary processes with rational spectral density. ARMA models and state-space models (with white noise as input) represent the same class of stationary processes. State-space systems became particularly popular through the work of Rudolf Kálmán. They contain an, in general, unobserved variable, the state, which contains all information from the past of the process that is relevant for the future. State-space systems lead to the Kalman filter discussed in this chapter. They are applied much more frequently in control theory, than the equivalent ARMA systems. Two key results in this chapter are the equivalence of controllability and observability with minimality and the description of equivalence classes of observationally equivalent minimal systems. Further, we provide construction for obtaining a state-space system from the Wold decomposition. In Sect. 7.4, we discuss the Kalman filter, which is an algorithm for estimating the unobserved state from observations and for predicting these observations. The Kalman filter is of significant importance for prediction and maximum likelihood estimation.
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Notes
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Rudolf Kálmán (1930–2016). Born in Hungary, active in the USA and Switzerland. Established modern systems theory. The Kalman filter named after him is one of the most widely used algorithms for prediction and filtering.
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Deistler, M., Scherrer, W. (2022). State-Space Systems. In: Time Series Models. Lecture Notes in Statistics, vol 224. Springer, Cham. https://doi.org/10.1007/978-3-031-13213-1_7
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DOI: https://doi.org/10.1007/978-3-031-13213-1_7
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