Advances in Dynamic Modeling and Computation for Count Data

A class of multi-process models is developed for collections of time indexed count data. Autocorrelation in counts is achieved with dynamic models for the natural parameter of the binomial distribution. In addition to modeling binomial time series, the framework includes dynamic models for multinomial and Poisson time series. Markov chain Monte Carlo (MCMC) and Po ́lya-Gamma data augmentation (Polson et al., 2013) are critical for fitting multi-process models of counts. To facilitate computation when the counts are high, a Gaussian approximation to the P ́olya- Gamma random variable is developed.

Three applied analyses are presented to explore the utility and versatility of the framework. The first analysis develops a model for complex dynamic behavior of themes in collections of text documents. Documents are modeled as a “bag of words”, and the multinomial distribution is used to characterize uncertainty in the vocabulary terms appearing in each document. State-space models for the natural parameters of the multinomial distribution induce autocorrelation in themes and their proportional representation in the corpus over time.

The second analysis develops a dynamic mixed membership model for Poisson counts. The model is applied to a collection of time series which record neuron level firing patterns in rhesus monkeys. The monkey is exposed to two sounds simultaneously, and Gaussian processes are used to smoothly model the time-varying rate at which the neuron’s firing pattern fluctuates between features associated with each sound in isolation.

The third analysis presents a switching dynamic generalized linear model for the time-varying home run totals of professional baseball players. The model endows each player with an age specific latent natural ability class and a performance enhancing drug (PED) use indicator. As players age, they randomly transition through a sequence of ability classes in a manner consistent with traditional aging patterns. When the performance of the player significantly deviates from the expected aging pattern, he is identified as a player whose performance is consistent with PED use.

All three models provide a mechanism for sharing information across related series locally in time. The models are fit with variations on the P ́olya-Gamma Gibbs sampler, MCMC convergence diagnostics are developed, and reproducible inference is emphasized throughout the dissertation.

Enhancing the Visualization of the Peripheral Retina with Wide Field-of-View Optical Coherence Tomography

The goal of my Ph.D. thesis is to enhance the visualization of the peripheral retina using wide-field optical coherence tomography (OCT) in a clinical setting.

OCT has gain widespread adoption in clinical ophthalmology due to its ability to visualize the diseases of the macula and central retina in three-dimensions, however, clinical OCT has a limited field-of-view of 300. There has been increasing interest to obtain high-resolution images outside of this narrow field-of-view, because three-dimensional imaging of the peripheral retina may prove to be important in the early detection of neurodegenerative diseases, such as Alzheimer's and dementia, and the monitoring of known ocular diseases, such as diabetic retinopathy, retinal vein occlusions, and choroid masses.

Before attempting to build a wide-field OCT system, we need to better understand the peripheral optics of the human eye. Shack-Hartmann wavefront sensors are commonly used tools for measuring the optical imperfections of the eye, but their acquisition speed is limited by their underlying camera hardware. The first aim of my thesis research is to create a fast method of ocular wavefront sensing such that we can measure the wavefront aberrations at numerous points across a wide visual field. In order to address aim one, we will develop a sparse Zernike reconstruction technique (SPARZER) that will enable Shack-Hartmann wavefront sensors to use as little as 1/10th of the data that would normally be required for an accurate wavefront reading. If less data needs to be acquired, then we can increase the speed at which wavefronts can be recorded.

For my second aim, we will create a sophisticated optical model that reproduces the measured aberrations of the human eye. If we know how the average eye's optics distort light, then we can engineer ophthalmic imaging systems that preemptively cancel inherent ocular aberrations. This invention will help the retinal imaging community to design systems that are capable of acquiring high resolution images across a wide visual field. The proposed model eye is also of interest to the field of vision science as it aids in the study of how anatomy affects visual performance in the peripheral retina.

Using the optical model from aim two, we will design and reduce to practice a clinical OCT system that is capable of imaging a large (800) field-of-view with enhanced visualization of the peripheral retina. A key aspect of this third and final aim is to make the imaging system compatible with standard clinical practices. To this end, we will incorporate sensorless adaptive optics in order to correct the inter- and intra- patient variability in ophthalmic aberrations. Sensorless adaptive optics will improve both the brightness (signal) and clarity (resolution) of features in the peripheral retina without affecting the size of the imaging system.

The proposed work should not only be a noteworthy contribution to the ophthalmic and engineering communities, but it should strengthen our existing collaborations with the Duke Eye Center by advancing their capability to diagnose pathologies of the peripheral retinal.