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Stability of non-linear filter for deterministic dynamics

  • * Corresponding author: Amit Apte

    * Corresponding author: Amit Apte
The work of ASR was partially supported by Infosys Foundation Excellence Program of ICTS. AA acknowledges support from US Office of Naval Research under grant N00014-18-1-2204. Authors acknowledge the support of the Department of Atomic Energy, Government of India, under projects no.12-R & D-TFR-5.10-1100, and no.RTI4001
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  • This papers shows that nonlinear filter in the case of deterministic dynamics is stable with respect to the initial conditions under the conditions that observations are sufficiently rich, both in the context of continuous and discrete time filters. Earlier works on the stability of the nonlinear filters are in the context of stochastic dynamics and assume conditions like compact state space or time independent observation model, whereas we prove filter stability for deterministic dynamics with more general assumptions on the state space and observation process. We give several examples of systems that satisfy these assumptions. We also show that the asymptotic structure of the filtering distribution is related to the dynamical properties of the signal.

    Mathematics Subject Classification: Primary: 60G35, 93B07, 93E11, 62M20; Secondary: 93C10.


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  • Figure 1.  Dependence of $ \frac{D_N(x, y)}{\sum_{i = 0}^N \rho_{i\tau}} $ vs $ t = N\tau $ with $ \tau = 0.01 $ for $ 100 $ samples. We have $ t = N\tau $ with $ \tau = 0.01 $. We chose $ 100 $ different pairs of $ (x, y) $ for five different choices of $ \rho_t = 1000, \; t+1000, \; \log(t+1000); t^2+1000; t^3 + 1000 $. The initial conditions for the samples are randomly chosen from uniform distribution on $ [-10, 10]^{p} $ where the dimension $ p = 3 $ for Lorenz 63 model (left panel) and $ p = 36 $ for the Lorenz 96 model (right panel). The insets show the plots for large $ t $. (Note that the Lyapunov time scale for both these models is $ O(1) $.)

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