1,228 research outputs found
Sequential quasi-Monte Carlo: Introduction for Non-Experts, Dimension Reduction, Application to Partly Observed Diffusion Processes
SMC (Sequential Monte Carlo) is a class of Monte Carlo algorithms for
filtering and related sequential problems. Gerber and Chopin (2015) introduced
SQMC (Sequential quasi-Monte Carlo), a QMC version of SMC. This paper has two
objectives: (a) to introduce Sequential Monte Carlo to the QMC community, whose
members are usually less familiar with state-space models and particle
filtering; (b) to extend SQMC to the filtering of continuous-time state-space
models, where the latent process is a diffusion. A recurring point in the paper
will be the notion of dimension reduction, that is how to implement SQMC in
such a way that it provides good performance despite the high dimension of the
problem.Comment: To be published in the proceedings of MCMQMC 201
Online Smoothing for Diffusion Processes Observed with Noise
We introduce a methodology for online estimation of smoothing expectations
for a class of additive functionals, in the context of a rich family of
diffusion processes (that may include jumps) -- observed at discrete-time
instances. We overcome the unavailability of the transition density of the
underlying SDE by working on the augmented pathspace. The new method can be
applied, for instance, to carry out online parameter inference for the
designated class of models. Algorithms defined on the infinite-dimensional
pathspace have been developed in the last years mainly in the context of MCMC
techniques. There, the main benefit is the achievement of mesh-free mixing
times for the practical time-discretised algorithm used on a PC. Our own
methodology sets up the framework for infinite-dimensional online filtering --
an important positive practical consequence is the construct of estimates with
the variance that does not increase with decreasing mesh-size. Besides
regularity conditions, our method is, in principle, applicable under the weak
assumption -- relatively to restrictive conditions often required in the MCMC
or filtering literature of methods defined on pathspace -- that the SDE
covariance matrix is invertible
Linear and nonlinear filtering in mathematical finance: a review
Copyright @ The Authors 2010This paper presents a review of time series filtering and its applications in mathematical finance. A summary of results of recent empirical studies with market data are presented for yield curve modelling and stochastic volatility modelling. The paper also outlines different approaches to filtering of nonlinear time series
Neural decoding with visual attention using sequential Monte Carlo for leaky integrate-and-fire neurons
How the brain makes sense of a complicated environment is an important question, and a first step is to be able to reconstruct the stimulus that give rise to an observed brain response. Neural coding relates neurobiological observations to external stimuli using computational methods. Encoding refers to how a stimulus affects the neuronal output, and entails constructing a neural model and parameter estimation. Decoding refers to reconstruction of the stimulus that led to a given neuronal output. Existing decoding methods rarely explain neuronal responses to complicated stimuli in a principled way. Here we perform neural decoding for a mixture of multiple stimuli using the leaky integrate-and-fire model describing neural spike trains, under the visual attention hypothesis of probability mixing in which the neuron only attends to a single stimulus at any given time. We assume either a parallel or serial processing visual search mechanism when decoding multiple simultaneous neurons. We consider one or multiple stochastic stimuli following Ornstein-Uhlenbeck processes, and dynamic neuronal attention that switches following discrete Markov processes. To decode stimuli in such situations, we develop various sequential Monte Carlo particle methods in different settings. The likelihood of the observed spike trains is obtained through the first-passage time probabilities obtained by solving the Fokker-Planck equations. We show that the stochastic stimuli can be successfully decoded by sequential Monte Carlo, and different particle methods perform differently considering the number of observed spike trains, the number of stimuli, model complexity, etc. The proposed novel decoding methods, which analyze the neural data through psychological visual attention theories, provide new perspectives to understand the brain
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