15,577 research outputs found
Dynamic importance sampling for uniformly recurrent markov chains
Importance sampling is a variance reduction technique for efficient
estimation of rare-event probabilities by Monte Carlo. In standard importance
sampling schemes, the system is simulated using an a priori fixed change of
measure suggested by a large deviation lower bound analysis. Recent work,
however, has suggested that such schemes do not work well in many situations.
In this paper we consider dynamic importance sampling in the setting of
uniformly recurrent Markov chains. By ``dynamic'' we mean that in the course of
a single simulation, the change of measure can depend on the outcome of the
simulation up till that time. Based on a control-theoretic approach to large
deviations, the existence of asymptotically optimal dynamic schemes is
demonstrated in great generality. The implementation of the dynamic schemes is
carried out with the help of a limiting Bellman equation. Numerical examples
are presented to contrast the dynamic and standard schemes.Comment: Published at http://dx.doi.org/10.1214/105051604000001016 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Modeling the ecology and evolution of biodiversity: Biogeographical cradles, museums, and graves
Individual processes shaping geographical patterns of biodiversity are increasingly understood, but their complex interactions on broad spatial and temporal scales remain beyond the reach of analytical models and traditional experiments. To meet this challenge, we built a spatially explicit, mechanistic simulation model implementing adaptation, range shifts, fragmentation, speciation, dispersal, competition, and extinction, driven by modeled climates of the past 800,000 years in South America. Experimental topographic smoothing confirmed the impact of climate heterogeneity on diversification. The simulations identified regions and episodes of speciation (cradles), persistence (museums), and extinction (graves). Although the simulations had no target pattern and were not parameterized with empirical data, emerging richness maps closely resembled contemporary maps for major taxa, confirming powerful roles for evolution and diversification driven by topography and climate
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