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- All Subjects: Statistics
- All Subjects: spatial statistics
- Genre: Academic theses
- Creators: McCulloch, Robert

This article proposes a new information-based subdata selection (IBOSS) algorithm, Squared Scaled Distance Algorithm (SSDA). It is based on the invariance of the determinant of the information matrix under orthogonal transformations, especially rotations. Extensive simulation results show that the new IBOSS algorithm retains nice asymptotic properties of IBOSS and gives…

This article proposes a new information-based subdata selection (IBOSS) algorithm, Squared Scaled Distance Algorithm (SSDA). It is based on the invariance of the determinant of the information matrix under orthogonal transformations, especially rotations. Extensive simulation results show that the new IBOSS algorithm retains nice asymptotic properties of IBOSS and gives a larger determinant of the subdata information matrix. It has the same order of time complexity as the D-optimal IBOSS algorithm. However, it exploits the advantages of vectorized calculation avoiding for loops and is approximately 6 times as fast as the D-optimal IBOSS algorithm in R. The robustness of SSDA is studied from three aspects: nonorthogonality, including interaction terms and variable misspecification. A new accurate variable selection algorithm is proposed to help the implementation of IBOSS algorithms when a large number of variables are present with sparse important variables among them. Aggregating random subsample results, this variable selection algorithm is much more accurate than the LASSO method using full data. Since the time complexity is associated with the number of variables only, it is also very computationally efficient if the number of variables is fixed as n increases and not massively large. More importantly, using subsamples it solves the problem that full data cannot be stored in the memory when a data set is too large.

Bayesian Additive Regression Trees (BART) is a non-parametric Bayesian model

that often outperforms other popular predictive models in terms of out-of-sample error. This thesis studies a modified version of BART called Accelerated Bayesian Additive Regression Trees (XBART). The study consists of simulation and real data experiments comparing XBART to other leading…

Bayesian Additive Regression Trees (BART) is a non-parametric Bayesian model

that often outperforms other popular predictive models in terms of out-of-sample error. This thesis studies a modified version of BART called Accelerated Bayesian Additive Regression Trees (XBART). The study consists of simulation and real data experiments comparing XBART to other leading algorithms, including BART. The results show that XBART maintains BART’s predictive power while reducing its computation time. The thesis also describes the development of a Python package implementing XBART.

In this work, I present a Bayesian inference computational framework for the analysis of widefield microscopy data that addresses three challenges: (1) counting and localizing stationary fluorescent molecules; (2) inferring a spatially-dependent effective fluorescence profile that describes the spatially-varying rate at which fluorescent molecules emit subsequently-detected photons (due to different…

In this work, I present a Bayesian inference computational framework for the analysis of widefield microscopy data that addresses three challenges: (1) counting and localizing stationary fluorescent molecules; (2) inferring a spatially-dependent effective fluorescence profile that describes the spatially-varying rate at which fluorescent molecules emit subsequently-detected photons (due to different illumination intensities or different local environments); and (3) inferring the camera gain. My general theoretical framework utilizes the Bayesian nonparametric Gaussian and beta-Bernoulli processes with a Markov chain Monte Carlo sampling scheme, which I further specify and implement for Total Internal Reflection Fluorescence (TIRF) microscopy data, benchmarking the method on synthetic data. These three frameworks are self-contained, and can be used concurrently so that the fluorescence profile and emitter locations are both considered unknown and, under some conditions, learned simultaneously. The framework I present is flexible and may be adapted to accommodate the inference of other parameters, such as emission photophysical kinetics and the trajectories of moving molecules. My TIRF-specific implementation may find use in the study of structures on cell membranes, or in studying local sample properties that affect fluorescent molecule photon emission rates.

The primary objective in time series analysis is forecasting. Raw data often exhibits nonstationary behavior: trends, seasonal cycles, and heteroskedasticity. After data is transformed to a weakly stationary process, autoregressive moving average (ARMA) models may capture the remaining temporal dynamics to improve forecasting. Estimation of ARMA can be performed…

The primary objective in time series analysis is forecasting. Raw data often exhibits nonstationary behavior: trends, seasonal cycles, and heteroskedasticity. After data is transformed to a weakly stationary process, autoregressive moving average (ARMA) models may capture the remaining temporal dynamics to improve forecasting. Estimation of ARMA can be performed through regressing current values on previous realizations and proxy innovations. The classic paradigm fails when dynamics are nonlinear; in this case, parametric, regime-switching specifications model changes in level, ARMA dynamics, and volatility, using a finite number of latent states. If the states can be identified using past endogenous or exogenous information, a threshold autoregressive (TAR) or logistic smooth transition autoregressive (LSTAR) model may simplify complex nonlinear associations to conditional weakly stationary processes. For ARMA, TAR, and STAR, order parameters quantify the extent past information is associated with the future. Unfortunately, even if model orders are known a priori, the possibility of over-fitting can lead to sub-optimal forecasting performance. By intentionally overestimating these orders, a linear representation of the full model is exploited and Bayesian regularization can be used to achieve sparsity. Global-local shrinkage priors for AR, MA, and exogenous coefficients are adopted to pull posterior means toward 0 without over-shrinking relevant effects. This dissertation introduces, evaluates, and compares Bayesian techniques that automatically perform model selection and coefficient estimation of ARMA, TAR, and STAR models. Multiple Monte Carlo experiments illustrate the accuracy of these methods in finding the "true" data generating process. Practical applications demonstrate their efficacy in forecasting.

This dissertation investigates the classification of systemic lupus erythematosus (SLE) in the presence of non-SLE alternatives, while developing novel curve classification methodologies with wide ranging applications. Functional data representations of plasma thermogram measurements and the corresponding derivative curves provide predictors yet to be investigated for SLE identification. Functional…

This dissertation investigates the classification of systemic lupus erythematosus (SLE) in the presence of non-SLE alternatives, while developing novel curve classification methodologies with wide ranging applications. Functional data representations of plasma thermogram measurements and the corresponding derivative curves provide predictors yet to be investigated for SLE identification. Functional nonparametric classifiers form a methodological basis, which is used herein to develop a) the family of ESFuNC segment-wise curve classification algorithms and b) per-pixel ensembles based on logistic regression and fused-LASSO. The proposed methods achieve test set accuracy rates as high as 94.3%, while returning information about regions of the temperature domain that are critical for population discrimination. The undertaken analyses suggest that derivate-based information contributes significantly in improved classification performance relative to recently published studies on SLE plasma thermograms.

This dissertation develops versatile modeling tools to estimate causal effects when conditional unconfoundedness is not immediately satisfied. Chapter 2 provides a brief overview ofcommon techniques in causal inference, with a focus on models relevant to the data explored

in later chapters. The rest of the dissertation focuses on the development…

This dissertation develops versatile modeling tools to estimate causal effects when conditional unconfoundedness is not immediately satisfied. Chapter 2 provides a brief overview ofcommon techniques in causal inference, with a focus on models relevant to the data explored

in later chapters. The rest of the dissertation focuses on the development of novel “reduced

form” models which are designed to assess the particular challenges of different datasets.

Chapter 3 explores the question of whether or not forecasts of bankruptcy cause bankruptcy.

The question arises from the observation that companies issued going concern opinions were

more likely to go bankrupt in the following year, leading people to speculate that the opinions themselves caused the bankruptcy via a “self-fulfilling prophecy”. A Bayesian machine

learning sensitivity analysis is developed to answer this question. In exchange for additional

flexibility and fewer assumptions, this approach loses point identification of causal effects

and thus a sensitivity analysis is developed to study a wide range of plausible scenarios of

the causal effect of going concern opinions on bankruptcy. Reported in the simulations are

different performance metrics of the model in comparison with other popular methods and a

robust analysis of the sensitivity of the model to mis-specification. Results on empirical data

indicate that forecasts of bankruptcies likely do have a small causal effect.

Chapter 4 studies the effects of vaccination on COVID-19 mortality at the state level in

the United States. The dynamic nature of the pandemic complicates more straightforward

regression adjustments and invalidates many alternative models. The chapter comments on

the limitations of mechanistic approaches as well as traditional statistical methods to epidemiological data. Instead, a state space model is developed that allows the study of the

ever-changing dynamics of the pandemic’s progression. In the first stage, the model decomposes the observed mortality data into component surges, and later uses this information in

a semi-parametric regression model for causal analysis. Results are investigated thoroughly

for empirical justification and stress-tested in simulated settings.

This dissertation covers several topics in machine learning and causal inference. First, the question of “feature selection,” a common byproduct of regularized machine learning methods, is investigated theoretically in the context of treatment effect estimation. This involves a detailed review and extension of frameworks for estimating causal effects and in-depth…

This dissertation covers several topics in machine learning and causal inference. First, the question of “feature selection,” a common byproduct of regularized machine learning methods, is investigated theoretically in the context of treatment effect estimation. This involves a detailed review and extension of frameworks for estimating causal effects and in-depth theoretical study. Next, various computational approaches to estimating causal effects with machine learning methods are compared with these theoretical desiderata in mind. Several improvements to current methods for causal machine learning are identified and compelling angles for further study are pinpointed. Finally, a common method used for “explaining” predictions of machine learning algorithms, SHAP, is evaluated critically through a statistical lens.

Tracking disease cases is an essential task in public health; however, tracking the number of cases of a disease may be difficult not every infection can be recorded by public health authorities. Notably, this may happen with whole country measles case reports, even such countries with robust registration systems.…

Tracking disease cases is an essential task in public health; however, tracking the number of cases of a disease may be difficult not every infection can be recorded by public health authorities. Notably, this may happen with whole country measles case reports, even such countries with robust registration systems. Eilertson et al. (2019) propose using a state-space model combined with maximum likelihood methods for estimating measles transmission. A Bayesian approach that uses particle Markov Chain Monte Carlo (pMCMC) is proposed to estimate the parameters of the non-linear state-space model developed in Eilertson et al. (2019) and similar previous studies. This dissertation illustrates the performance of this approach by calculating posterior estimates of the model parameters and predictions of the unobserved states in simulations and case studies. Also, Iteration Filtering (IF2) is used as a support method to verify the Bayesian estimation and to inform the selection of prior distributions. In the second half of the thesis, a birth-death process is proposed to model the unobserved population size of a disease vector. This model studies the effect of a disease vector population size on a second affected population. The second population follows a non-homogenous Poisson process when conditioned on the vector process with a transition rate given by a scaled version of the vector population. The observation model also measures a potential threshold event when the host species population size surpasses a certain level yielding a higher transmission rate. A maximum likelihood procedure is developed for this model, which combines particle filtering with the Minorize-Maximization (MM) algorithm and extends the work of Crawford et al. (2014).

Spatial regression is one of the central topics in spatial statistics. Based on the goals, interpretation or prediction, spatial regression models can be classified into two categories, linear mixed regression models and nonlinear regression models. This dissertation explored these models and their real world applications. New methods and models were…

Spatial regression is one of the central topics in spatial statistics. Based on the goals, interpretation or prediction, spatial regression models can be classified into two categories, linear mixed regression models and nonlinear regression models. This dissertation explored these models and their real world applications. New methods and models were proposed to overcome the challenges in practice. There are three major parts in the dissertation.

In the first part, nonlinear regression models were embedded into a multistage workflow to predict the spatial abundance of reef fish species in the Gulf of Mexico. There were two challenges, zero-inflated data and out of sample prediction. The methods and models in the workflow could effectively handle the zero-inflated sampling data without strong assumptions. Three strategies were proposed to solve the out of sample prediction problem. The results and discussions showed that the nonlinear prediction had the advantages of high accuracy, low bias and well-performed in multi-resolution.

In the second part, a two-stage spatial regression model was proposed for analyzing soil carbon stock (SOC) data. In the first stage, there was a spatial linear mixed model that captured the linear and stationary effects. In the second stage, a generalized additive model was used to explain the nonlinear and nonstationary effects. The results illustrated that the two-stage model had good interpretability in understanding the effect of covariates, meanwhile, it kept high prediction accuracy which is competitive to the popular machine learning models, like, random forest, xgboost and support vector machine.

A new nonlinear regression model, Gaussian process BART (Bayesian additive regression tree), was proposed in the third part. Combining advantages in both BART and Gaussian process, the model could capture the nonlinear effects of both observed and latent covariates. To develop the model, first, the traditional BART was generalized to accommodate correlated errors. Then, the failure of likelihood based Markov chain Monte Carlo (MCMC) in parameter estimating was discussed. Based on the idea of analysis of variation, back comparing and tuning range, were proposed to tackle this failure. Finally, effectiveness of the new model was examined by experiments on both simulation and real data.