A Direct Sampling Particle Filter from Approximate Conditional Density Function Supported on Constrained State Space

Constraints on the state vector must be taken into account in the state estimation problem. Recently, acceptance/rejection and projection methods are proposed in the particle filter framework for constraining the particles. A weighted least squares formulation is used for constraining samples in unscented and ensemble Kalman filters. In this paper, direct sampling from an approximate conditional probability density function (pdf) is proposed. It is obtained by approximating the a priori pdf as a Gaussian. The support of the conditional density is a subset of the intersection of two supports, the 3-sigma bounds of the priori Gaussian and the constrained state space. A direct sampling algorithm is proposed for handling linear and nonlinear equality and inequality constraints. The algorithm uses the constrained mode for nonlinear constraints

Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines MCMC with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM

Data assimilation in the low noise regime with application to the Kuroshio

On-line data assimilation techniques such as ensemble Kalman filters and particle filters lose accuracy dramatically when presented with an unlikely observation. Such an observation may be caused by an unusually large measurement error or reflect a rare fluctuation in the dynamics of the system. Over a long enough span of time it becomes likely that one or several of these events will occur. Often they are signatures of the most interesting features of the underlying system and their prediction becomes the primary focus of the data assimilation procedure. The Kuroshio or Black Current that runs along the eastern coast of Japan is an example of such a system. It undergoes infrequent but dramatic changes of state between a small meander during which the current remains close to the coast of Japan, and a large meander during which it bulges away from the coast. Because of the important role that the Kuroshio plays in distributing heat and salinity in the surrounding region, prediction of these transitions is of acute interest. Here we focus on a regime in which both the stochastic forcing on the system and the observational noise are small. In this setting large deviation theory can be used to understand why standard filtering methods fail and guide the design of the more effective data assimilation techniques. Motivated by our analysis we propose several data assimilation strategies capable of efficiently handling rare events such as the transitions of the Kuroshio. These techniques are tested on a model of the Kuroshio and shown to perform much better than standard filtering methods.Comment: 43 pages, 12 figure

Particle filtering for EEG source localization and constrained state spaces

Particle Filters (PFs) have a unique ability to perform asymptotically optimal estimation for non-linear and non-Gaussian state-space models. However, the numerical nature of PFs cause them to have major weakness in two important areas: (1) handling constraints on the state, and (2) dealing with high-dimensional states. In the first area, handling constraints within the PF framework is crucial in dynamical systems, which are often required to satisfy constraints that arise from basic physical laws or other considerations. The current trend in constrained particle filtering is to enforce the constraints on all particles of the PF. We show that this approach leads to more stringent conditions on the posterior density that can cause incorrect state estimates. We subsequently describe a novel algorithm that restricts the mean estimate without restricting the posterior pdf, thus providing a more accurate state estimate. In the second area, we tackle the curse of dimensionality, which causes the PF to require an exponential increase in computational complexity as the dimension of the state increases. The application of interest is localization of the brain neural generators that create the Electroencephalogram (EEG) signal. Specifically, we describe a state-space model that tracks the position and moments of multiple dynamic dipoles and apply the marginalized PF, which alleviates the curse of dimensionality for tracking multiple dynamic dipoles. This modified framework allows us to consider dynamic dipoles, which were historically considered time-invariant