Rates for branching particle approximations of continuous-discrete filters

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that $t\to X_t$ is a Markov process and we wish to calculate the measure-valued process $t\to\mu_t(\cdot)\doteq P\{X_t\in \cdot|\sigma\{Y_{t_k}, t_k\leq t\}\}$, where $t_k=k\epsilon$ and $Y_{t_k}$ is a distorted, corrupted, partial observation of $X_{t_k}$. Then, one constructs a particle system with observation-dependent branching and $n$ initial particles whose empirical measure at time $t$, $\mu_t^n$, closely approximates $\mu_t$. Each particle evolves independently of the other particles according to the law of the signal between observation times $t_k$, and branches with small probability at an observation time. For filtering problems where $\epsilon$ is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of $\epsilon$. We analyze the algorithm on L\'{e}vy-stable signals and give rates of convergence for $E^{1/2}\{\|\mu^n_t-\mu_t\|_{\gamma}^2\}$, where $\Vert\cdot\Vert_{\gamma}$ is a Sobolev norm, as well as related convergence results.Comment: Published at http://dx.doi.org/10.1214/105051605000000539 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Handling Target Obscuration through Markov Chain Observations

ABSTRACT Target Obscuration, including foliage or building obscuration of ground targets and landscape or horizon obscuration of airborne targets, plagues many real world filtering problems. In particular, ground moving target identification Doppler radar, mounted on a surveillance aircraft or unattended airborne vehicle, is used to detect motion consistent with targets of interest. However, these targets try to obscure themselves (at least partially) by, for example, traveling along the edge of a forest or around buildings. This has the effect of creating random blockages in the Doppler radar image that move dynamically and somewhat randomly through this image. Herein, we address tracking problems with target obscuration by building memory into the observations, eschewing the usual corrupted, distorted partial measurement assumptions of filtering in favor of dynamic Markov chain assumptions. In particular, we assume the observations are a Markov chain whose transition probabilities depend upon the signal. The state of the observation Markov chain attempts to depict the current obscuration and the Markov chain dynamics are used to handle the evolution of the partially obscured radar image. Modifications of the classical filtering equations that allow observation memory (in the form of a Markov chain) are given. We use particle filters to estimate the position of the moving targets. Moreover, positive proof-of-concept simulations are included

Rates for Branching Particle Approximations of Continuous-Discrete Filters

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that t → Xt is a Markov process and we wish to calculate the measure-valued process t →µt(∙) ≐ P(Xt ∉∙|σ{Ytk , tk ≤ t}), where tk = kε and Ytk is a distorted, corrupted, partial observation of Xtk. Then, one constructs a particle system with observation-dependent branching and n initial particles whose empirical measure at time t;µnt , closely approximates µt. Each particle evolves independently of the other particles according to the law of the signal between observation times tk, and branches with small probability at an observation time. For filtering problems where ε is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of ε. We analyze the algorithm on Lévy-stable signals and give rates of convergence for E1/2[||µnt - µt||2√], ||∙ ||√ is a Sobolev norm, as well as related convergence results