統計数学セミナー
Seminar on Probability and Statistics
Home : Archive [ 2003 to 04 ] [ 2004 to 05 ] [ 2005 to 06 ] [ 2006 to 07 ] [ 2007 to 08 ] [ 2008 to 09 ] [ 2009 to 10 ] [ 2010 to 11 ] [ 2011 to 12 ] [ 2012 to 13 ] [ 2013 to 14 ] [ 2014 to 15 ]
Previous Seminar : Next Seminar

Seminar on Probability and Statistics
Friday January 16 2015
Osaka J706 (Host) and Tokyo 052 (Web)
2:00-3:30 pm


A stable particle filter in high-dimensions


Ajay Jasra
National University of Singapore

Abstract

We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ with $d$ large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed as particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in $d$ for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space-time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed $d$, as do standard particle filters. Moreover, under a simple i.i.d. model structure, we show that in order to achieve some stability properties this new filter has cost $\mathcal{O}(nNd^2)$, where $n$ is the time parameter and $N$ is the number of Monte Carlo samples, that are fixed and independent of $d$. Similar results hold, under a more general structure than the i.i.d. one. Here we show that, under additional assumptions and with the same cost, the asymptotic variance of the relative estimate of the normalizing constant grows at most linearly in time and independently of the dimension. Our theoretical results are supported by numerical simulations. The results suggest that it is possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot.

This is joint work with: Alex Beskos (UCL), Dan Crisan (Imperial), Kengo Kamatani (Osaka) and Yan Zhou (NUS).




Previous Seminar : Next Seminar
Seminar on Probability and Statistics