Nonlinear filtering via stochastic PDE projection on mixture manifolds in \(L^2\) direct metric

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• 作者：John Armstrong ; Damiano Brigo
• 关键词：Direct $$L^2$$ L 2 metric ; Exponential families ; Finite ; dimensional families of probability distributions ; Fisher information ; Hellinger distance ; Levy metric ; Mixture families ; Stochastic filtering ; Galerkin methods
• 刊名：Mathematics of Control, Signals, and Systems (MCSS)
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：28
• 期：1
• 全文大小：722 KB
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• 作者单位：John Armstrong (1)
  Damiano Brigo (2)
  
  1. Department of Mathematics, King鈥檚 College, London, UK
  2. Department of Mathematics, Imperial College, London, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Communications Engineering and Networks
  Control Engineering
  
• 出版者：Springer London
• ISSN：1435-568X

文摘

We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite-dimensional stochastic partial differential equation (SPDE), based on the direct \(L^2\) metric and on a family of normal mixtures. This results in a new finite-dimensional approximate filter based on the differential geometric approach to statistics. We compare this new filter to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and compare the \(L^2\) mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We discuss differences between projecting the SPDE for the normalized density, known as Kushner鈥揝tratonovich equation, and the SPDE for the unnormalized density known as Zakai equation. We prove that for a simple choice of the mixture manifold the \(L^2\) mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the \(L^2\) mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples. We leverage an algebraic ring structure by proving that in presence of a given structure in the system coefficients the key integrations needed to implement the new filter equations can be executed offline. Keywords Direct \(L^2\) metric Exponential families Finite-dimensional families of probability distributions Fisher information Hellinger distance Levy metric Mixture families Stochastic filtering Galerkin methods