In this dissertation, we consider three statistical problems unified by an underlying graph structure.

The first problem concerns graph scan statistics.Scan statistics can be used for early detection of disease outbreaks by testing large regions for faint but detectable abnormal measurements. Traditional scan statistic implementations, such as the SaTScan software, fuse neighboring measurements to obtain more powerful global null tests. In this project, we provide an algorithm for detecting and localizing elevated levels in graph data. We present methods and software that can utilize both spatial proximity and network information. We apply this methodology to an outbreak of porcine reproductive and respiratory syndrome (PRRS) in pig farms and find hotspots of the disease. Our method can provably control the family-wise error rate and control the false discovery rate while detecting and localizing outbreaks.

The second problem concerns distributed communication with memory constraints.Memory constraints are common in modern statistical systems like those occurring in wireless sensor networks. We consider a model system of sensors communicating in a tree structure under bit-rate constraints to solve a hypothesis testing problem. We obtain bounds on the hypothesis testing error rate for a central class of decision rules as functions of the bitrate, tree width, and depth. These bounds give fundamental tradeoffs on memory and communication in wireless sensor networks.

The final problem concerns random graph model fitting.Exponential random graph models (ERGMs) are a family of random graphs often used in social science due to their elegance of definition and flexibility of expression. Fitting the parameters of such models to data is quite challenging however, due to the complexity of the partition function and the need to count subgraphs. Traditional Markov chain Monte Carlo methods used for fitting suffer from poor initialization errors and the need to produce many graph samples. We study a pseudolikelihood based on the large deviations theory for ERGMs developed by Chatterjee and Diaconis and investigate its performance under noisy subgraph count estimates. The work in this project shows preliminary results that our proposed method may be viable in low edge density graph regimes.

The problem of estimating underlying signals is well studied that has ubiquitous applications, such as signal processing. In network and image analysis, the tasks of signal estimation usually involve denoising a matrix based on structural information. In the situations where the information structure could be modeled by a network, graph trend filtering is a useful estimator with penalizing the $\ell_1$ norm of discrete derivatives based on some graph structures. A special case of graph trend filtering, based on penalizing an incidence matrix of a graph, is called graph fused lasso. In this dissertation, we proposed power graph fused lasso, an approach to denoising a response matrix with a graph, where both rows and columns of the matrix correspond to the vertices in the graph.

There are two categories of graph fused lasso algorithms, solving the problem with a fixed hyperparameter and path algorithms solving for every hyperparameter. An overview of projected Newton-type method for graph fused lasso and ADMM-based graph trend filtering algorithms was provided. Ensemble graph fused lasso algorithms were proposed with some numerical results presented.

We further discussed graph trend filtering with Kullback-Leibler divergence loss and how to solve the problem with a Bregman ADMM-based algorithm. We demonstrated the desired property that the GTF reconstructed signal is piecewise polynomial with an application to Covid-19 test positive data. Two other applications were also presented.

This dissertation is a combination of two bodies of work in modern statistical inference. Thefirst work introduces and studies a novel conditional Independence test using neural networks. The second work is applied data analyses for the Healthy Davis Together (HDT) Program, for wastewater testing for Covid-19 infections and an analysis of the impact of the college reopening. We propose a neural Ising model to test conditional independence between binary random variables X and Y given a potentially complex random variable Z such as text or images. The method uses the score test statistic and employs a computationally efficient score-based bootstrap procedure [20] to generate the p-value. We extend the method to the multi-class X and Y by replacing the Ising model with a restricted Boltzmann machine. Empirical studies show that our model has high power against H1 and reliable type-I error control on both simulated and real-world data. We derive the asymptotic separability of the score-test statistics under the Ising model. On the applied side, we first summarize our collaboration with the wastewater team at Healthy Davis Together (HDT) initiative working on the wastewater monitoring project [36]. We provide a Bayesian Ct value imputation method via the EM-MCMC algorithm wrapped in a user-friendly API. The algorithm is able to produce Ct values matching the overall trend of the clinical data and has a stronger correlation with the clinical data when compared with existing methods [36]. The other data analysis project at HDT is measuring the impact of college reopening on the COVID19 outbreak level in their home county. The coronavirus disease 2019 (COVID-19) pandemic has dramatically impacted the 2020-2021 academic year in universities across the country, and conversely, college reopening has disrupted the course of the pandemic. We investigated COVID19 hotspot events in “college counties” which we defined as counties with at least 10% of its population composed of undergraduate students. We found that increments in cases could not be attributed to random chance by performing multiple hypothesis testing. Increments in confirmed cases among college counties from mid-August to mid-September were significantly higher than comparable non-college counties. After this period of reopening, hotspots of confirmed cases did not differ between counties, despite the college-town designation. Class setting (i.e., In-Person, Hybrid, Online) seemed to be associated with hotspot activity. We found no evidence to support an association between testing efforts and hotspots.

In this dissertation, we apply deep learning tools and techniques to two problems: detecting strong gravitational lenses and modeling the effects of mobility on COVID-19 mortality. In Chapter 1, we give an overview of the problems we investigate and summarize some of the assumptions and results we made in our studies. In Chapter 2, we systematically study the effects of data augmentation, semi-supervised learning algorithms, and unsupervised learning algorithms in the context of training deep learning classifiers with only simulated lenses to detect real lenses. In Chapter 3, we conduct a descriptive data analysis of mobility in Davis during the COVID-19 pandemic. In Chapter 4, we study the causal effects of mobility on COVID-19 mortality across four counties in California, using neural nets to model the effects.

This dissertation aims to investigate three distinct problems. Firstly, it aims to enhance lung cancer diagnosis and survival predictions through the implementation of deep learning techniques and CT imaging. Secondly, it delves into understanding the differences in distortion patterns present in adversarial images generated by various attack methods. Lastly, it explores the application of the Minimum Description Length (MDL) principle for optimal threshold determination in microbiome community detection.

Supervised by Professor Thomas Lee and Professor James Sharpnack, Chapter \ref{ch:chap2} proposes the utilization of convolutional neural networks to model the intricate relationship between the risk of lung cancer and the morphology of the lungs depicted in CT images. Introducing a mini-batched loss extending the Cox proportional hazards model, this approach accommodates the non-convexity induced by neural networks, enabling training on large datasets. The combination of mini-batched loss and binary cross-entropy facilitates the prediction of both lung cancer occurrence and the risk of mortality. Results from simulations and real data experiments highlight the potential of this method to advance lung cancer diagnosis and treatment.

Supervised by Professor Thomas Lee, Chapter \ref{ch:chap4} discusses the application of the MDL principle in microbiome data analysis, particularly focusing on community detection methods. Addressing the challenge of subjective threshold selection in correlation-based techniques, MDL is employed to identify the optimal community structure by minimizing the subjectivity in choosing a cut-off for correlation strength. The chapter provides a detailed derivation of the MDL principle, discusses its consistency in threshold selection, and validates its effectiveness through simulations. A real data experiment involving microbiome data from the Great Lakes offers practical insights into the application of MDL in a real-world context.

Supervised by Professor Thomas Lee, Professor Yao Li, and Professor Cho-Jui Hsieh, Chapter \ref{ch:chap3} explores the vulnerability of deep neural networks to adversarial examples. Focusing on three common attack families – gradient-based, score-based, and decision-based – the research aims to recognize distinct types of adversarial examples. By identifying the information possessed by attackers, effective defense strategies can be developed. The study demonstrates that adversarial images from different attack families can be successfully identified with a simple model. Experiments on CIFAR10 and Tiny ImageNet reveal differences in distortion patterns between various attack types for both $L_2$ and $L_\infty$ norms.

Recent improvements in machine learning methods have significantly advanced many fields in- cluding computer vision and natural language processing. While the models have become increasingly complex, they have also become harder to interpret and be trusted.In my thesis, the focus is to improve or solve machine learning problems with clear guidance from statistics and design algorithms which can be supported or explained by statistics. We show statistics can be applied from various aspects and play different roles. In the first project, I use a heuristic search method to design a simply but effective neural network compression method, with a theoretical understanding of how the search works. In the second project, optimization based on Kullback-Leibler divergence is applied to fine-tune the output distribution of an image segmentation module to improve quality of segmentation. In the third project you will see how hypothesis testing can help measure the similarity between datasets, which is a crucial question for a better understanding for maching learning. Besides these applications on machine learning algorithms, there are also engineering challenges in real-life projects. For the fourth project, I summarized my contribution to Healthy Davis Together project, including the systems I built with full details, the logic behind those systems and a mobility variable I designed.

The bandit problem models a sequential decision process between a player and an environment. In the beginning, the player has no prior knowledge about the environment, but can gain experience by sequentially pulling an arm out of a set of arms and receiving rewards from the environment. The goal is to maximize the cumulative rewards. In this dissertation, we focus on the stochastic contextual bandit problems, where each arm is associated with a feature vector and the observed reward of each pulled arm follows a stochastic process. The bandit problems enjoy solid theoretical foundations in the literature and have substantial applications in many real tasks such as recommender systems, online advertising, clinical trials, etc. However, there are still several challenges that prevent the success of bandit algorithms. To name a few, ensuring robustness, adaptive hyper-parameter selection, improving computational efficiency and adapting to non-stationarity.

We focus on resolving the underlying challenges of bandit problems, both theoretically and practically. In Chapter 1, we give a brief overview on bandit problems and the limitations in existing works. We propose a two-layer bandit algorithm in Chapter 2 to improve the robustness of stochastic linear contextual bandit algorithms under fully adaptive adversarial attacks. Based on a similar idea, we generalize the two-layer bandit structure to our proposed Syndicated Bandits framework to aid multiple hyper-parameters selection in online contextual bandit problems, where offline tuning is impossible. In Chapter 4 and 5, we address the computational bottlenecks of bandit problems and propose two novel algorithms, SGD-TS and Sampling-MIPS to improve computational efficiency in the estimation and prediction phases of contextual bandit algorithms. SGD-TS is the most efficient generalized linear bandit algorithm so far. In Chapter 6, we emphasize the limitations of the stationarity assumption in the classic bandit problems and propose a changepoint detection based bandit algorithm under a relaxed piecewise stationary bandit environment. We conclude this dissertation and mention some future directions in Chapter 7. All of our proposed algorithms and framework enjoy solid theoretical guarantees and proofs are presented in the Appendix. Extensive experiments in both simulations and real datasets validate the effectiveness of our proposed works.

Understanding the mobility pattern and network flow provides fundamental knowledge for decision making in transportation planning and operations. These insights play a critical role in travel demand modeling, traffic management and control, and processes in the development of robust and sustainable transportation systems. There has been a long history of developing estimation methods to better understand networked data. However, direct observations often suffer from several challenges: high-dimensionality, limited coverage, and low fidelity. Furthermore, existing literature tends to separate data with data-driven methods and domain knowledge with behavioral-based and physics-informed models.

This dissertation focuses on statistical learning on high-dimensional networked data in transportation. We consider signal from direct observations and the interdependence among various components in the network, under the regularization of the domain knowledge and structural information. The goal of the research is to develop a methodology to improve the quality, efficiency, and robustness of the estimation of high-dimensional networked data in transportation systems. Based on the interdependency of data from various network levels, the development of the statistical learning frameworks targets the following specific key objectives : 1) estimating unknown OD demand from observable link flow in the networks; 2) learning meaningful data representations in networks for critical information extraction and anomaly detection; and 3) statistical filtering for data on directed graphs.

In OD estimation, we present a modeling framework for OD demand estimation based on observed traffic flow data in a transportation network, from a fresh angle of stochastic programming. The proposed two-stage stochastic programming method is flexible for incorporating various design principles and risk preferences into domain knowledge regarding travel behavioral and physical rules. Besides, a benefit comes from the scenario representation, where the point estimate can be combined with estimation of the discrete approximation to the demand distribution. We demonstrate that under the proposed framework, well-established theories and methods for stochastic programming, including epi-convergence and scenario decomposition, can be exploited to advance the analytical and computational capability of the estimation model. The applicability and efficiency of the proposed method are illustrated via numerical examples based on highway and transit networks of various sizes.

In representation learning, we introduce a new perspective that the critical information of the data should reflect how the data is used in downstream applications, which carries a different research design philosophy adopted in most existing data representation learning methods. We propose an application-driven representation learning framework by incorporating information loss for the downstream application into the data encoding-decoding process. The proposed approach is formulated as a Stiefel manifold optimization problem. The effectiveness of the proposed framework is demonstrated through three case studies: transportation network performance assessment, vehicular emission estimation, and anomaly detection of travel demand. Experiments show that our proposed approach performs better than classical representation learning approaches, especially in applications involving complex network interdependence.

In graph signal filtering, we developed a Continuous-time Markov chain based filtering methods on directed graphs using nonparametric regression technique. Through bridging the stochastic process and algebraic graph theory, we utilize the transition matrix basis that is dependent on the connectivity of points across varied density regions. Compared with methods dedicated to undirected graph and spatial filtering methods, our approach is capable of capturing directional information flows and local asymmetric structure in data observations. The performance of our approach is evidenced through a series of synthetic and real-world case studies on network traffic flow. This work demonstrates the potential for incorporating heterogeneous structures for data defined on irregular domains.