The Lévy State Space Model

Abstract

In this paper we introduce a new class of state space models based on shot-noise simulation representations of non-Gaussian Lévy-driven linear systems, represented as stochastic differential equations. In particular a conditionally Gaussian version of the models is proposed that is able to capture heavytailed non-Gaussianity while retaining tractability for inference procedures. We focus on a canonical class of such processes, the α-stable Lévy processes, which retain important properties such as self-similarity and heavy-tails, while emphasizing that broader classes of non-Gaussian Lévy processes may be handled by similar methodology. An important feature is that we are able to marginalise both the skewness and the scale parameters of these challenging models from posterior probability distributions. The models are posed in continuous time and so are able to deal with irregular data arrival times. Example modelling and inference procedures are provided using Rao-Blackwellised sequential Monte Carlo applied to a two-dimensional Langevin model, and this is tested on real exchange rate data.

Lévy State-Space Models for Tracking and Intent Prediction of Highly Maneuverable Objects

“…We also show how it can be applied to estimate the kinematic state and/or destination of a manoeuvring target, presenting several different example models and inference strategies. The proposed stable Lévy model is devised here as a conditionally Gaussian infinite series [20,29] that can be adapted to include the unknown intent of the tracked object via a mean reverting term. This conditionally Gaussian series structure is exact, but in practical applications requires the truncation of the infinite series and suitable approximation of the residual error terms.…”

“…Sharp rates of convergence for these approaches have been recently proposed, see [37] and references therein. A more general Lévy state-space model was detailed in [20], however not as a spatial process applicable to object tracking. Here, we propose for the first time multidimensional Lévy state-space models in the conditionally Gaussian series form for spatial tracking and intent prediction, coupled with an efficient sequential Bayesian inference procedure based on Rao-Blackwellised particle filtering.…”

“…On account of the typically isotropic properties in most tracking applications, we focus particularly on the symmetric stable Lévy process. Although skewed dynamic models may be appropriate in certain scenarios and can be included in our framework as in [20], we leave this as a topic for future work.…”

“…A special case is when α = 2, and then W (t) is the m-dimensional isotropic Brownian motion. We focus in this paper on stable Lévy models with isotropic noise, as is common in much of the tracking literature [30,41,7]; however we note that the more general skewed case in [20] could also be extended to the current multidimensional setting in future work. The isotropic premise supports the typical lack of information about directionality of the randomness of dynamics in a tracking scenario.…”

“…Here, a novel modelling approach for manoeuvring targets that can incorporate both kinematics and intent is proposed. The methods are a novel extension of continuous-time Lévy state-space models [5,20], driven by heavy-tailed α-stable Lévy processes [44,40]. These can better characterise sudden dramatic changes in the state induced by swift target manoeuvres, compared with the more commonly used Gaussian-driven dynamic model.…”