6,523 research outputs found
Multilevel Particle Filters for L\'evy-driven stochastic differential equations
We develop algorithms for computing expectations of the laws of models
associated to stochastic differential equations (SDEs) driven by pure L\'evy
processes. We consider filtering such processes and well as pricing of path
dependent options. We propose a multilevel particle filter (MLPF) to address
the computational issues involved in solving these continuum problems. We show
via numerical simulations and theoretical results that under suitable
assumptions of the discretization of the underlying driving L\'evy proccess,
our proposed method achieves optimal convergence rates. The cost to obtain MSE
scales like for our method, as compared with
the standard particle filter
Multilevel ensemble Kalman filtering for spatio-temporal processes
We design and analyse the performance of a multilevel ensemble Kalman filter
method (MLEnKF) for filtering settings where the underlying state-space model
is an infinite-dimensional spatio-temporal process. We consider underlying
models that needs to be simulated by numerical methods, with discretization in
both space and time. The multilevel Monte Carlo (MLMC) sampling strategy,
achieving variance reduction through pairwise coupling of ensemble particles on
neighboring resolutions, is used in the sample-moment step of MLEnKF to produce
an efficient hierarchical filtering method for spatio-temporal models. Under
sufficient regularity, MLEnKF is proven to be more efficient for weak
approximations than EnKF, asymptotically in the large-ensemble and
fine-numerical-resolution limit. Numerical examples support our theoretical
findings.Comment: Version 1: 39 pages, 4 figures.arXiv admin note: substantial text
overlap with arXiv:1608.08558 . Version 2 (this version): 52 pages, 6
figures. Revision primarily of the introduction and the numerical examples
sectio
Lookahead Strategies for Sequential Monte Carlo
Based on the principles of importance sampling and resampling, sequential
Monte Carlo (SMC) encompasses a large set of powerful techniques dealing with
complex stochastic dynamic systems. Many of these systems possess strong
memory, with which future information can help sharpen the inference about the
current state. By providing theoretical justification of several existing
algorithms and introducing several new ones, we study systematically how to
construct efficient SMC algorithms to take advantage of the "future"
information without creating a substantially high computational burden. The
main idea is to allow for lookahead in the Monte Carlo process so that future
information can be utilized in weighting and generating Monte Carlo samples, or
resampling from samples of the current state.Comment: Published in at http://dx.doi.org/10.1214/12-STS401 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Realtime Multilevel Crowd Tracking using Reciprocal Velocity Obstacles
We present a novel, realtime algorithm to compute the trajectory of each
pedestrian in moderately dense crowd scenes. Our formulation is based on an
adaptive particle filtering scheme that uses a multi-agent motion model based
on velocity-obstacles, and takes into account local interactions as well as
physical and personal constraints of each pedestrian. Our method dynamically
changes the number of particles allocated to each pedestrian based on different
confidence metrics. Additionally, we use a new high-definition crowd video
dataset, which is used to evaluate the performance of different pedestrian
tracking algorithms. This dataset consists of videos of indoor and outdoor
scenes, recorded at different locations with 30-80 pedestrians. We highlight
the performance benefits of our algorithm over prior techniques using this
dataset. In practice, our algorithm can compute trajectories of tens of
pedestrians on a multi-core desktop CPU at interactive rates (27-30 frames per
second). To the best of our knowledge, our approach is 4-5 times faster than
prior methods, which provide similar accuracy
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