7,263 research outputs found
The Neural Particle Filter
The robust estimation of dynamically changing features, such as the position
of prey, is one of the hallmarks of perception. On an abstract, algorithmic
level, nonlinear Bayesian filtering, i.e. the estimation of temporally changing
signals based on the history of observations, provides a mathematical framework
for dynamic perception in real time. Since the general, nonlinear filtering
problem is analytically intractable, particle filters are considered among the
most powerful approaches to approximating the solution numerically. Yet, these
algorithms prevalently rely on importance weights, and thus it remains an
unresolved question how the brain could implement such an inference strategy
with a neuronal population. Here, we propose the Neural Particle Filter (NPF),
a weight-less particle filter that can be interpreted as the neuronal dynamics
of a recurrently connected neural network that receives feed-forward input from
sensory neurons and represents the posterior probability distribution in terms
of samples. Specifically, this algorithm bridges the gap between the
computational task of online state estimation and an implementation that allows
networks of neurons in the brain to perform nonlinear Bayesian filtering. The
model captures not only the properties of temporal and multisensory integration
according to Bayesian statistics, but also allows online learning with a
maximum likelihood approach. With an example from multisensory integration, we
demonstrate that the numerical performance of the model is adequate to account
for both filtering and identification problems. Due to the weightless approach,
our algorithm alleviates the 'curse of dimensionality' and thus outperforms
conventional, weighted particle filters in higher dimensions for a limited
number of particles
Data Assimilation: A Mathematical Introduction
These notes provide a systematic mathematical treatment of the subject of
data assimilation
Evaluating Data Assimilation Algorithms
Data assimilation leads naturally to a Bayesian formulation in which the
posterior probability distribution of the system state, given the observations,
plays a central conceptual role. The aim of this paper is to use this Bayesian
posterior probability distribution as a gold standard against which to evaluate
various commonly used data assimilation algorithms.
A key aspect of geophysical data assimilation is the high dimensionality and
low predictability of the computational model. With this in mind, yet with the
goal of allowing an explicit and accurate computation of the posterior
distribution, we study the 2D Navier-Stokes equations in a periodic geometry.
We compute the posterior probability distribution by state-of-the-art
statistical sampling techniques. The commonly used algorithms that we evaluate
against this accurate gold standard, as quantified by comparing the relative
error in reproducing its moments, are 4DVAR and a variety of sequential
filtering approximations based on 3DVAR and on extended and ensemble Kalman
filters.
The primary conclusions are that: (i) with appropriate parameter choices,
approximate filters can perform well in reproducing the mean of the desired
probability distribution; (ii) however they typically perform poorly when
attempting to reproduce the covariance; (iii) this poor performance is
compounded by the need to modify the covariance, in order to induce stability.
Thus, whilst filters can be a useful tool in predicting mean behavior, they
should be viewed with caution as predictors of uncertainty. These conclusions
are intrinsic to the algorithms and will not change if the model complexity is
increased, for example by employing a smaller viscosity, or by using a detailed
NWP model
Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation
Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper, the authors address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter
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