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the computational load increases linearly with |M|, these hybrid IPS approaches are intractable when |M| is very large This condition applies to the air traf c example (where |M| 1025 ) considered later in this chapter The idea is to improve the situation for very large |M| by developing a hybrid IPS approach not for { t , xt }, but for { t , ( t , xt )}, where { t } is some complementary K-valued process with |K| |M| In order to accomplish this, we group modes that have large differences in mode switching frequencies This de nes a partition {M , K}, that is, K M = M and M M = for = , and a K-valued aggregation mode process { t } as follows: t ( ) = , if t ( ) M (96)
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Because the evolution of the aggregation mode process { t } depends on the evolution of { t }, { t } may inherit rare mode switching from { t } In order to avoid these rare effects in the evolution of particles, we also de ne a K-valued Markov chain { t } with known non-rare transition rates, and use the transition rates of { t } to determine for each particle a new -value at some time step h later The particle weight is compensated with the corresponding importance switching ratio p +h | ,x , ( | , x, )/p +h | ( | ), where , x, denote the given ( , x , ) particle value, and denotes the value newly sampled for +h Next, the prediction of the new +h particle from the (x , ) particle values is done conditional on the newly sampled -value Theorem 1 provides a probabilistic characterization of such -conditional -prediction Theorem 1 ( -conditional -prediction) For an arbitrary stopping time , p +h |x , , +h ( |x, , ) = Proof Using Bayes yields: p +h |x , , +h ( |x, , ) = p +h | +h ( | )p +h |x , ( |x, ) M p +h | +h ( | )p +h |x , ( |x, ) 1M ( )p +h |x , ( |x, ) M 1M ( )p +h |x , ( |x, ) (97)
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Substituting p +h | +h ( | ) = 1M ( ) yields (97) The prediction of the x-part of the particle over time step h is done by drawing a sample from px +h |x , , +h ( |x, , ) In order to identify all particles that arrive at Qk before time T , the prediction over time step h has to be done up to T / h times After these prediction steps, there is no guarantee that for each K some minimum number of particles have arrived at Qk Hence, we resample the Qk -arrived particles such that we regain Np particles for each K In order to make this possible, in Theorem 2 we provide a characterization of the (conditional) probabilities p +h and px , | +h as a function of px , , for
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arbitrary stopping time and time step h This characterization allows us to sample a xed number of particles per aggregation mode K, and to sample for each particle a novel -value conditional on the aggregation mode value Theorem 2 (Hierarchical interaction) If p +h ( ) > 0 for an arbitrary stopping time , then px , | +h (dx, | ) =
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p +h |x , ( |x, )px , (dx, )/p +h ( ), p +h |x , ( |x, )px , (dx, )
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