View Communities

Research Publications (Mathematical and Statistical Sciences)

Number of results to display per page
Items in this Collection
  1. The Irreducible Characters of 2 x 2 Unitary Matrix Groups Over Finite Fields [Download]

    Title: The Irreducible Characters of 2 x 2 Unitary Matrix Groups Over Finite Fields
    Creator: Campbell, John J.
    Description: In this work we will construct the table of irreducible characters for the group of unitary 2 x 2 matrices over a finite field. The table and the methods for its construction will show interesting connections to the table and methods of construction of the table of irreducible characters for the general linear group.
    Subjects: Group Representations
    Date Created: 2014/08/28
  2. A hybrid weighted interacting particle filter for multi-target tracking. [Download]

    Title: A hybrid weighted interacting particle filter for multi-target tracking.
    Creator: Ballantyne, David
    Description: A hybrid weighted interacting particle filter, the selectively resampling particle filter (SERP), is used to detect and track multiple ships maneuvering in a region of water. The ship trajectories exhibit nonlinear dynamics and interact in a nonlinear manner such that the ships do not collide. There is no prior knowledge on the number of ships in the region. The observations model a sensor tracking the ships from above the region, as in a low observable SAR or infrared problem. The SERP filter simulates particles to provide the approximated conditional distribution of the signal in the signal domain at a particular time, given the sequence of observations. After each observation, the hybrid filter uses selective resampling to move some particles with low weights to locations that have a higher likelihood of being correct, without resampling all particles or creating bias. Such a method is both easy to implement and highly computationally efficient. Quantitative results recording the capacity of the filter to determine the number of ships in the region and the location of each ship are presented. Thy hybrid filter is compared against an earlier particle filtering method.
    Subjects: non-linear filtering, SERP, hybrid filter, particle-based filtering, tracking, multiple target
    Date Created: 2003
  3. Practical applications of a branching particle-base filter. [Download]

    Title: Practical applications of a branching particle-base filter.
    Creator: Ballantyne, David
    Description: Particle-based nonlinear filters provide a mathematically optimal (in the limit) and sound method for solving a number of difficult filtering problems. However, there are a number of practical difficulties that can occur when applying particle-based filtering techniques to real world problems. These problems include highly directed signal dynamics highly definitive observations clipped observation data. Current approaches to solving these problems generally require increasing the number of particles, but to obtain a given level of performance the number of particles required may be extremely large. We propose a number of techniques to ameliorate these difficulties. We adopt the ideas of simulated annealing and add noise which is damped in time to the particle states when they are evolved or duplicated, and also add noise which is damped in time to the interpretation of the observations by the filter, to deal with signal dynamics and observation problems. We modify the method by which particles are duplicated to deal with different information flows into the system depending on the location of the particle and the information flow into the particle. We discuss the success we have had with these solutions on some of the problems of interest to Lockheed Martin and the MITACS-PINTS research center.
    Subjects: nonlinear, tracking, image processing
    Date Created: 2001
  4. Handling Target Obscuration through Markov Chain Observations. [Download]

    Title: Handling Target Obscuration through Markov Chain Observations.
    Creator: Kouritzin, Michael
    Description: 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.
    Subjects: particle filter, classical filter equation, target obscuration, tracking problem, target identification, random blockage
    Date Created: 2008
  5. A Strong Law of Large Numbers for Super-stable Processes. [Download]

    Title: A Strong Law of Large Numbers for Super-stable Processes.
    Creator: Kouritzin, Michael
    Description: Let ℓ be Lebesgue measure and X=(Xt,t≥0;Pμ) be a supercritical, super-stable process corresponding to the operator −(−Δ)α/2u+βu−ηu2 on Rd with constants β,η>0 and α∈(0,2]. Put View the MathML source, which for each smallθ is an a.s. convergent complex-valued martingale with limit View the MathML source say. We establish for any starting finite measure μ satisfying View the MathML source that View the MathML source-a.s. in a topology, termed the shallow topology, strictly stronger than the vague topology yet weaker than the weak topology, where cα>0 is a known constant. This result can be thought of as an extension to a class of superprocesses of Watanabe’s strong law of large numbers for branching Markov processes.
    Subjects: robability measures, super-Brownian motion, Fourier transform, vague convergence, strong law of large numbers, super-stable process
    Date Created: 2014
  6. Continuous and discrete space particle filters for predictions in acoustic positioning. [Download]

    Title: Continuous and discrete space particle filters for predictions in acoustic positioning.
    Creator: Bauer, Will
    Description: Predicting the future state of a random dynamic signal based on corrupted, distorted, and partial observations is vital for proper real-time control of a system that includes time delay. Motivated by problems from Acoustic Positioning Research Inc., we consider the continual automated illumination of an object moving within a bounded domain, which requires object location prediction due to inherent mechanical and physical time lags associated with robotic lighting. Quality computational predictions demand high fidelity models for the coupled moving object signal and observation equipment pair. In our current problem, the signal represents the vector position, orientation, and velocity of a stage performer. Acoustic observations are formed by timing ultrasonic waves traveling from four perimeter speakers to a microphone attached to the performer. The goal is to schedule lighting movements that are coordinated with the performer by anticipating his/her future position based upon these observations using filtering theory. Particle system based methods have experienced rapid development and have become an essential technique of contemporary filtering strategies. Hitherto, researchers have largely focused on continuous state particle filters, ranging from traditional weighted particle filters to adaptive refining particle filters, readily able to perform path-space estimation and prediction. Herein, we compare the performance of a state-of-the-art refining particle filter to that of a novel discrete-space particle filter on the acoustic positioning problem. By discrete space particle filter we mean a Markov chain that counts particles in discretized cells of the signal state space in order to form an approximated unnormalized distribution of the signal state. For both filters mentioned above, we will examine issues like the mean time to localize a signal, the fidelity of filter estimates at various signal to noise ratios, computational costs, and the effect of signal fading; furthermore, we will provide visual demonstrations of filter performance.
    Subjects: particle system, target tracking, nonlinear filtering
    Date Created: 2002
  7. Particle filters for combined state and parameter estimation. [Download]

    Title: Particle filters for combined state and parameter estimation.
    Creator: Chan, Hubert
    Description: Filtering is a method of estimating the conditional probability distribution of a signal based upon a noisy, partial, corrupted sequence of observations of the signal. Particle filters are a method of filtering in which the conditional distribution of the signal state is approximated by the empirical measure of a large collection of particles, each evolving in the same probabilistic manner as the signal itself. In filtering, it is often assumed that we have a fixed model for the signal process. In this paper, we allow unknown parameters to appear in the signal model, and present an algorithm to estimate simultaneously both the parameters and the conditional distribution for the signal state using particle filters. This method is applicable to general nonlinear discrete-time stochastic systems and can be used with various types of particle filters. It is believed to produce asymptotically optimal estimates of the state and the true parameter values, provided reasonable initial parameter estimates are given and further estimates are constrained to be in the vicinity of the true parameters. We demonstrate this method in the context of search and rescue problem using two different particle filters and compare the effectiveness of the two filters to each other.
    Subjects: nonlinear filtering, target tracking, particle methods, system identification
    Date Created: 2001
  8. Computation of tail probability distributions via extrapolation methods and connection with rational and Padé approximants. [Download]

    Title: Computation of tail probability distributions via extrapolation methods and connection with rational and Padé approximants.
    Creator: Gaudreau, Philippe J.
    Description: Abstract. We use the recently developed algorithm for the G(1) n transformation to approximate tail probabilities of the normal distribution, the gamma distribution, the student’s t-distribution, the inverse Gaussian distribution, and Fisher’s F distribution. Using this algorithm, which can be computed recursively when using symbolic programming languages, we are able to compute these integrals to high predetermined accuracies. Previous to this contribution, the evaluation of these tail probabilities using the G(1) n transformation required symbolic computation of large determinants. With the use of our algorithm, the G(1) n transformation can be performed relatively easily to produce explicit approximations. After a brief theoretical study, a connection between the G(1) n transformation and rational and Pad´e approximants is established.
    Subjects: tails of probability distributions, rational and Pad´e approximants, extrapolation methods, G transformation, Slevinsky–Safouhi formulae
    Date Created: 2012
  9. A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter [Download]

    Title: A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter
    Creator: Kouritzin, Michael
    Description: Consider the following random ordinary differential equation: X˙ϵ(τ)=F(Xϵ(τ),τ/ϵ,ω)subject toXϵ(0)=x0, where {F(x,t,ω),t≥0} are stochastic processes indexed by x in Rd, and the dependence on x is sufficiently regular to ensure that the equation has a unique solution Xϵ(τ,ω) over the interval 0≤τ≤1 for each ϵ>0. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: x˙0(τ)=F¯¯¯(x0(τ))subject tox0(0)=x0, such that limϵ→0sup0≤τ≤1E|Xϵ(τ)−x0(τ)|=0. In this article we show that as ϵ→0 the random function (Xϵ(⋅)−x0(⋅))/2ϵloglogϵ−1−−−−−−−−−−√ almost surely converges to and clusters throughout a compact set K of C[0,1].
    Subjects: mixing processes, ordinary differential equation, laws of the iterated logarithm, central limit theorem
    Date Created: 1994
  10. On sonobuoy placement for submarine tracking. [Download]

    Title: On sonobuoy placement for submarine tracking.
    Creator: Kouritzin, Michael
    Description: This paper addresses the problem of detecting and tracking an unknown number of submarines in a body of water using a known number of moving sonobuoys. Indeed, we suppose there are N submarines collectively maneuvering as a weakly interacting stochastic dynamical system, where N is a random number, and we need to detect and track these submarines using M moving sonobuoys. These sonobuoys can only detect the superposition of all submarines through corrupted and delayed sonobuoy samples of the noise emitted from the collection of submarines. The signals from the sonobuoys are transmitted to a central base to analyze, where it is required to estimated how many submarines there are as well as their locations, headings, and velocities. The delays induced by the propagation of the submarine noise through the water mean that novel historical filtering methods need to be developed. We summarize these developments within and give initial results on a simplified example.
    Subjects: nonlinear filtering, sonobuoy, submarine, target tracking
    Date Created: 2005