Browsing by Author "Kowal, Daniel R"

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    Bayesian Approaches for Forecasting Count-Valued Time Series
    (2023-04-18) King, Brian; Kowal, Daniel R
    This thesis introduces novel Bayesian approaches to modeling and forecasting count-valued time series, particularly emphasizing the importance of distributional forecasting and proper, data-coherent uncertainty quantification. Time-ordered count data arise in many applications, including sales, finance, ecology, and epidemiology/public health. Such data exhibit a variety of complexities that make modeling difficult: they often inherit typical time series characteristics such as seasonality or high frequency, but also present numerous distributional features unique to the count setting, like zero-inflation, over-/under-dispersion, and heaping. Importantly, these count data features can rarely be summarized by a point estimate or even interval estimates. Instead, there is a pressing need for methods that can forecast entire distributions and allow decision makers access to uncertainty quantification built for the count data setting. In the first project, we introduce a broad class of multivariate state space models called the warped Dynamic Linear Model (warpDLM). Through a warping operation composed of a transformation function and rounding operator, the warpDLM connects count data to latent continuous data that can be modeled with a DLM. Thus, our model adapts the powerful existing methods for time series data in a way that allows for modeling the many complexities of count data. We develop conjugate inference for the warpDLM, which enables analytic and recursive updates, in turn facilitating the development of efficient algorithms for inference and forecasting, including Monte Carlo simulation for offline analysis and an optimal particle filter for online inference. The forecasting ability of this framework is demonstrated on simulated data as well as a real-data application of EMS calls regarding overdoses in the city of Cincinnati. In the second project, we take a different approach to the forecasting problem. Instead of training a model suited for count time series, we consider the scenario where several point forecasts for a count time series are available and explore how these could be combined to output a calibrated probabilistic forecast. To accomplish this task, we leverage a flexible Bayesian count regression model which (akin to the warpDLM) performs Simultaneous Transformation and Rounding (STAR) of a latent continuous regression model. The resulting forecast combination approach (called STARcast) can produce calibrated and sharp distributional forecasts, even with only a small collection of point forecasts as the input. The third project introduces a statistical software package, countSTAR, designed for both practitioners and researchers to utilize the methods proposed in this thesis, alongside many additional STAR models explored in other works. In addition to including functionality for warpDLM modeling, countSTAR features unified syntax, useful output, and detailed online documentation. This package has been accepted to CRAN, and thus is free and easily available to R users.
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    Efficient Bayesian Regression Methods for Dependent, Sparse Functional Data
    (2024-04-18) Sun, Thomas Y; Kowal, Daniel R
    Functional data analysis is widely useful for analysis of high resolution measurements over a continuous domain. However, Bayesian inference for functional regression models can be computationally challenging, especially in the presence of dependent or sparsely observed data. We provide novel Bayesian methods for functional regression across three specific areas. Existing algorithms for Bayesian inference with functional mixed models only provide either scalable computing or accurate approximations to the posterior distribution, but not both. We first introduce a new MCMC strategy for highly efficient and fully Bayesian regression with longitudinal functional data. Using a novel blocking structure paired with an orthogonalized basis reparametrization, our joint sampler optimizes efficiency for key parameters while preserving computational scalability. We surpass state-of-the-art algorithms for frequentist estimation and variational Bayes approximations while also providing accurate posterior uncertainty quantification. Next, we propose a fully Bayesian scalar-on-function regression model for sparse functional predictors measured with error. Estimation of sparsely-observed and noisy longitudinal data require careful consideration as to provide adequate uncertainty quantification and avoid overfitting, especially when used as covariates in regression models. We utilize functional factor models to parsimoniously represent the sparse curves while maintaining an efficient MCMC sampler that is stable under high missingness. We demonstrate the benefits of our modeling and algorithm design through simulations and applications on a bone mineral density study and actigraphy data. We also develop software to implement this approach and the longitudinal functional regression. Finally, we consider a challenging application of functional regression to study the dynamics between wastewater concentrations of SARS-CoV-2 and COVID-19 infection rates in the community. Wastewater-based surveillance yields a low-cost, noninvasive method for tracking disease transmissions and provides early warning signs of upcoming outbreaks. There is tremendous interest in understanding the exact dynamics between wastewater viral loads and infection rates in the population, but numerous complexities and dependency structures are present in the datasets. We propose a novel Bayesian functional concurrent regression model that accounts for both spatial and temporal correlations while estimating the dynamic effects and time lag between wastewater concentrations and positivity rates over time.