Geometrically Intrinsic Nonlinear Recursive Filters I: Algorithms

The Geometrically Intrinsic Nonlinear Recursive Filter, or GI Filter, is designed to estimate an arbitrary continuous-time Markov diffusion process X subject to nonlinear discrete-time observations. The GI Filter is fundamentally different from the much-used Extended Kalman Filter (EKF), and its second-order variants, even in the simplest nonlinear case, in that: (i) It uses a quadratic function of a vector observation to update the state, instead of the linear function used by the EKF. (ii) It is based on deeper geometric principles, which make the GI Filter coordinate-invariant. This implies, for example, that if a linear system were subjected to a nonlinear transformation f of the state-space and analyzed using the GI Filter, the resulting state estimates and conditional variances would be the push-forward under f of the Kalman Filter estimates for the untransformed system - a property which is not shared by the EKF or its second-order variants. The noise covariance of X and the observation covariance themselves induce geometries on state space and observation space, respectively, and associated canonical connections. A sequel to this paper develops stochastic differential geometry results - based on "intrinsic location parameters", a notion derived from the heat flow of harmonic mappings - from which we derive the coordinate-free filter update formula. The present article presents the algorithm with reference to a specific example - the problem of tracking and intercepting a target, using sensors based on a moving missile. Computational experiments show that, when the observation function is highly nonlinear, there exist choices of the noise parameters at which the GI Filter significantly outperforms the EKF.Comment: 22 pages, 4 figure

Differential equation approximations for Markov chains

We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.Comment: Published in at http://dx.doi.org/10.1214/07-PS121 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Structure of large random hypergraphs

The theme of this paper is the derivation of analytic formulae for certain large combinatorial structures. The formulae are obtained via fluid limits of pure jump type Markov processes, established under simple conditions on the Laplace transforms of their Levy kernels. Furthermore, a related Gaussian approximation allows us to describe the randomness which may persist in the limit when certain parameters take critical values. Our method is quite general, but is applied here to vertex identifiability in random hypergraphs. A vertex v is identifiable in n steps if there is a hyperedge containing v all of whose other vertices are identifiable in fewer than n steps. We say that a hyperedge is identifiable if every one of its vertices is identifiable. Our analytic formulae describe the asymptotics of the number of identifiable vertices and the number of identifiable hyperedges for a Poisson random hypergraph on a set of N vertices, in the limit as N goes to infinity.Comment: Revised version with minor conceptual improvements and additional discussion. 32 pages, 5 figure