Aerospace Equipment

# Nonlinear and Non-Gaussian State-Space Modeling with Monte by Tanizaki H.

By Tanizaki H.

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Additional info for Nonlinear and Non-Gaussian State-Space Modeling with Monte Carlo Techniques: A Survey and Comparative Study

Example text

In procedure (i), typically, the smoothing estimates based on the extended Kalman ﬁlter are 33 taken for α0,t , t = 1, 2, · · · , T . αi,0 for i = 1, 2, · · · , N depend on the underlying assumption of α0 . That is, αi,0 for i = 1, 2, · · · , N are generated from Pα (α0 ) if α0 is stochastic and they are ﬁxed as α0 for all i if α0 is nonstochastic. Mean, Variance and Likelihood Function: Based on the random draws αi,t for i = 1, 2, · · · , N , evaluation of E(g(αt )|YT ) is simply obtained as the arithmetic average of g(αi,t ), i = 1, 2, · · · , N , which is represented by: E(g(αt )|YT ) ≈ 1 N −M N g(αi,t ).

Measurement equation) yt = Zt αt + dt + St t , αt = Tt αt−1 + ct + Rt ηt , (Transition equation) t ηt ∼N 0 Ht , 0 0 0 Qt , where Zt , dt , St , Tt , ct , Rt , Ht and Qt are assumed to be known for all time t = 1, 2, · · · , T . Deﬁne conditional mean and variance as ar|s ≡ E(αr |Ys ) and Σr|s ≡ Var(αr |Ys ) for (r, s) = (t + L, t), (t, t), (t, T ). Under the above setup, optimal prediction, ﬁltering and smoothing are represented as the standard linear recursive algorithms, which are easily derived from the ﬁrst- and second-moments of density functions (13) – (18).

Thus, the rejection sampling has the disadvantage that we cannot exactly predict computation time. To improve the rejection sampling procedure in the sense of computation time, we have the following strategies. One is that we may pick another j and/or i in procedure (i) and repeat procedures (ii) and (iii) again when the acceptance probability ω(·) is too small. Alternatively, we may switch random number generation from rejection sampling to the Metropolis-Hastings algorithm when ω(·) is too small (see Appendix B for the Metropolis-Hastings algorithm).