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

Fortran 4 program for two-impulse rendezvous analysis

Program determines if rendezvous in near space is possible, and performs an analysis to determine the approximate required values of the magnitude and direction of two thrust applications of the upper stage of a rocket firing. The analysis is performed by using ordinary Keplerian mechanics

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

On the X-ray Properties of OH Megamaser Sources: Chandra Snapshot Observations

We present Chandra snapshot observations for a sample of 7 sources selected from the Arecibo OH megamaser (OHM) survey at z~0.13-0.22 and with far-infrared luminosities in excess of 10^{11} L_sun. In contrast with the known H2O megamasers, which are mostly associated with powerful Active Galactic Nuclei (AGN), the situation is far less clear for OHMs, which have been poorly studied in the X-ray band thus far. All of the observed sources are X-ray weak, with only one OHM, IRAS FSC 03521+0028 (z=0.15), being detected by Chandra (with 5 counts). The results from this pilot program indicate that the X-ray emission, with luminosities of less than ~10^{42} erg/s, is consistent with that from star formation (as also suggested in some cases by the optical spectra) and low-luminosity AGN emission. If an AGN is present, its contribution to the broad-band emission of OHM galaxies is likely modest. Under reasonable assumptions about the intrinsic X-ray spectral shape, the observed count distribution from stacking analysis suggests absorption of ~10^{22} cm^{-2}.Comment: 8 pages, 3 figures, accepted for publication in MNRA

Searching for high-redshift centimeter-wave continuum, line and maser emission using the Square Kilometer Array

We discuss the detection of redshifted line and continuum emission at radio wavelengths using a Square Kilometer Array (SKA), specifically from low-excitation rotational molecular line transitions of CO and HCN (molecular lines), the recombination radiation from atomic transitions in almost-ionized hydrogen (radio recombination lines; RRLs), OH and water maser lines, as well as from synchrotron and free-free continuum radiation and HI 21-cm line radiation. The detection of radio lines with the SKA offers the prospect to determine the redshifts and thus exact luminosities for some of the most distant and optically faint star-forming galaxies and active galactic nuclei (AGN), even those galaxies that are either deeply enshrouded in interstellar dust or shining prior to the end of reionization. Moreover, it provides an opportunity to study the astrophysical conditions and resolved morphologies of the most active regions in galaxies during the most active phase of star formation at redshift z~2. A sufficiently powerful and adaptable SKA correlator will enable wide-field three-dimensional redshift surveys at chosen specific high redshifts, and will allow new probes of the evolution of large-scale structure (LSS) in the distribution of galaxies. The detection of molecular line radiation favours pushing the operating frequencies of SKA up to at least 26 GHz, and ideally to 40 GHz, while very high redshift maser emissions requires access to about 100 MHz. To search for LSS the widest possible instantaneous field of view would be advantageous.Comment: 12 pages, 2 figures. To appear in "Science with the Square Kilometer Array," eds. C. Carilli and S. Rawlings, New Astronomy Reviews (Elsevier: Amsterdam

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