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This is my dissertation for my PhD in Applied Mathematics at the University of Arizona. My thesis is about deblurring images in the Bayesian framework with spatially varying hyperparameters and mixed regularization types.
Size effects are an unavoidable nuisance in inertial navigation using sensors which are not co-located at the navigational point of interest. When estimating transforms between the navigation point and sensor locations, some trajectories preclude observation of all model parameters. Regularization is proposed to avoid over-fitting size-effects models. The result yields robust size effects compensation in other regions of flight.
Early detection of outbreaks caused by emergent pathogens, using epidemiological surveillance data i.e., daily case counts, is difficult. This is because the data tend to be noisy during the early epoch of the outbreak. In contrast, the spread-rate of the disease tends to be well-behaved, as it depends only on the mixing patterns of the population and the characteristics of the pathogen, neither of which behave erratically in space-time. In this report, we explore whether the spread-rate can be used for epidemiological surveillance, conditional on case count data. Estimating the spread-rate from case count data allows us to exploit exogenous information, e.g., incubation period distributions etc., which can considerably smooth out any erratic temporal behavior. Further, epidemiological dynamics are spatially correlated, and if case counts are available for multiple areal units e.g., counties, these correlations could potentially be used to suppress noise in the early epoch data. These exogenous information and structure are not exploited by conventional syndromic surveillance detectors to extract a well-behaved latent variable for monitoring purposes. The technical challenge lies in the estimation of the spread-rate field jointly over a collection of areal units; further, the spread-rate varies over time. We develop a method based on mean-field variational inference to approximately estimate the spread-rate field, using a Gaussian Random Field Model for spatial regularization. The method is tested on the estimation of spread-rate in the thirty-three counties of New Mexico and detect the arrival of the Fall 2020 COVID-19 wave in September 2020. We find that the method is scalable, but underestimates the uncertainty in the estimated spread-rate field. We detect the arrival of the Fall 2020 wave a week ahead of conventional syndromic surveillance algorithms, but our simplistic detection algorithm, based on simple anomaly detection, suffers from a high false positive rate, similar to conventional detectors.
The slides summarize recent advances in symbolic regression developed as part of the PrOMMiS project over the past year. It describes the use of various regularization metrics in the context of symbolic regression, and it includes computational experiments performed to build simple and accurate symbolic regression models.
We propose a new sparse Granger-causal learning framework for temporal event data. We focus on a specific class of point processes called the Hawkes process. We begin by pointing out that most of the existing sparse causal learning algorithms for the Hawkes process suffer from a singularity in maximum likelihood estimation. As a result, their sparse solutions can appear only as numerical artifacts. In this paper, we propose a mathematically well-defined sparse causal learning framework based on a cardinality-regularized Hawkes process, which remedies the pathological issues of existing approaches. We leverage the proposed algorithm for the task of instance-wise causal event analysis, where sparsity plays a critical role. We validate the proposed framework with two real use-cases, one from the power grid and the other from the cloud data center management domain.
Magnetometer for determination of angular motion of spinning body in regular precession
Regularities in excitation of short period pulses in middle latitudes of earth currents - geophysics
Generalization of space restricted three-body problem by spinor regularization
Perturbation theory for two-body problem, using regularized Levi-Civita variables
Regularization of relative many-body equations of celestial mechanics and relative three-body equations in case of binary collisions
Equations of motion and integrals of motion derived for Sundman regularization of three- body problem
Numerical and analytical methods for orbit computation in celestial mechanics during and beyond collision by introduction of regularized coordinates
Methods of regularization for computing orbits in celestial mechanics
Regularization of restricted three-body problem extended to case where three primaries of any mass revolve in circular orbits around common center of mass and fourth body of infinitesimal mass moves in their field
Independent variable changes in dynamical systems and applications to systems regularization, giving canonical and Lagrangian motion equations with tensor notation
Fixed point probability row vector of regular or ergodic transition matrices simplified using generalized matrix inversion theory
Trajectory optimization of space vehicle with continuous thrust based on regularized equations, comparing perturbation method for earth-Jupiter rendezvous transfer