By Peter Jan Van Leeuwen, Yuan Cheng, Sebastian Reich (auth.)
This e-book comprises overview articles on nonlinear info assimilation that care for heavily comparable issues yet have been written and will be learn independently. either contributions concentrate on so-called particle filters.
The first contribution through Jan van Leeuwen makes a speciality of the opportunity of concept densities. It discusses the problems with present-day particle filters and explorers new rules for thought densities to resolve them, converging to particle filters that paintings good in platforms of any size, final the contribution with a high-dimensional instance. the second one contribution by means of Cheng and Reich discusses a unified framework for ensemble-transform particle filters. this permits one to bridge winning ensemble Kalman filters with absolutely nonlinear particle filters, and permits a formal advent of localization in particle filters, which has been missing as much as now.
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This e-book comprises overview articles on nonlinear info assimilation that take care of heavily comparable issues yet have been written and will be learn independently. either contributions specialise in so-called particle filters. the 1st contribution by way of Jan van Leeuwen specializes in the potential for inspiration densities.
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Additional info for Nonlinear Data Assimilation
49) where we assumed that the weights are equal at time n m. These weights are called the ‘first-stage weights’ or the ‘simulation weights’. xki jxk i /. Note that ki connects the original particle i with its new position in state space, that of particle k from the resampling step at n m. 50) in which A is the normalisation factor. A resampling step can be done, but is not really necessary because the resampling is done at step 3. 1 Nonlinear Data Assimilation for high-dimensional systems resampling weighting at t=0 t=0 t=10 t=0 29 weighting t=10 Fig.
This last factor is related to the use of the proposed model instead of the original model. These equations form the basis for exploring proposal densities to find efficient particle filters. Finally, let us formulate an expression for the weights when multiple model time steps are present between observation times. Assume the model needs m time steps between observations. 61) jDn mC1 where we used the Markov property of the model. 64) So we find that the weights are multiplied by a p=q term each time step in which we use a modified model.
In the following we explore this idea in Bayes theorem, using the posterior pdf as the target pdf and the prior pdf as the proposal pdf. In later chapter we will explore more exotic proposals to increase the efficiency of the method. 33) which is just a number we can calculate directly as detailed in the previous section. Weighting the particles just means that their relative importance in the probability density changes. x/ are x itself, giving the mean of the pdf, and the squared deviation from the mean, giving the covariance.
Nonlinear Data Assimilation by Peter Jan Van Leeuwen, Yuan Cheng, Sebastian Reich (auth.)