Nonlinear Filters. Simon Haykin
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Название: Nonlinear Filters

Автор: Simon Haykin

Издательство: John Wiley & Sons Limited

Жанр: Программы

Серия:

isbn: 9781119078159

isbn:

СКАЧАТЬ k Baseline right-parenthesis plus bold w Subscript k Baseline comma EndLayout"/>

      the sequence of posterior information matrices, bold upper F Subscript k, for estimating state vectors, bold x Subscript k, can be computed as [55]:

      (4.22)bold upper F Subscript k plus 1 Baseline equals bold upper D Subscript k Superscript 22 Baseline minus bold upper D Subscript k Superscript 21 Baseline left-parenthesis bold upper F Subscript k Baseline plus bold upper D Subscript k Superscript 11 Baseline right-parenthesis Superscript negative 1 Baseline bold upper D Subscript k Superscript 12 Baseline comma

      where

      (4.23)bold upper D Subscript k Superscript 11 Baseline equals double-struck upper E left-bracket nabla Subscript bold x Sub Subscript k Subscript Baseline bold f Subscript k Baseline left-parenthesis bold x Subscript k Baseline right-parenthesis bold upper Q Subscript k Superscript negative 1 Baseline nabla Subscript bold x Sub Subscript k Subscript Superscript upper T Baseline bold f Subscript k Baseline left-parenthesis bold x Subscript k Baseline right-parenthesis right-bracket comma

      (4.24)bold upper D Subscript k Superscript 12 Baseline equals minus double-struck upper E left-bracket nabla Subscript bold x Sub Subscript k Subscript Baseline bold f Subscript k Baseline left-parenthesis bold x Subscript k Baseline right-parenthesis right-bracket bold upper Q Subscript k Superscript negative 1 Baseline comma

      (4.25)bold upper D Subscript k Superscript 21 Baseline equals left-parenthesis bold upper D Subscript k Superscript 12 Baseline right-parenthesis Superscript upper T Baseline comma

      (4.26)bold upper D Subscript k Superscript 22 Baseline equals double-struck upper E left-bracket nabla Subscript bold x Sub Subscript k plus 1 Subscript Baseline bold g Subscript k plus 1 Baseline left-parenthesis bold x Subscript k plus 1 Baseline right-parenthesis bold upper R Subscript k plus 1 Superscript negative 1 Baseline nabla Subscript bold x Sub Subscript k plus 1 Subscript Superscript upper T Baseline bold g Subscript k plus 1 Baseline left-parenthesis bold x Subscript k plus 1 Baseline right-parenthesis right-bracket plus bold upper Q Subscript k Superscript negative 1 Baseline comma

      where bold upper Q Subscript k and bold upper R Subscript k are the process and measurement noise covariance matrices, respectively.

      The general formulation of the optimal nonlinear Bayesian filtering leads to a computationally intractable problem; hence, the Bayesian solution is a conceptual solution. Settling for computationally tractable suboptimal solutions through deploying different approximation methods has led to a wide range of classic as well as machine learning‐based filtering algorithms. Such algorithms have their own advantages, restrictions, and domains of applicability. To assess and compare such filtering algorithms, several performance metrics can be used including entropy, Fisher information, and PCRLB. Furthermore, the Fisher information matrix is used to define the natural gradient, which is helpful in machine learning.

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