As the impacts of climate change worsen in the coming decades, natural hazards are expected to increase in frequency and intensity, leading to increased loss and risk to human livelihood. The spatio-temporal statistical approaches developed and applied in this dissertation highlight the ways in which hazard data can be leveraged to understand loss trends, build forecasts, and study societal impacts of losses. Specifically, this work makes use of the Spatial Hazard Events and Losses Database which is an unparalleled source of loss data for the United States. The first portion of this dissertation develops accurate loss baselines that are crucial for mitigation planning, infrastructure investment, and risk communication. This is accomplished thorough a stationarity analysis of county level losses following a normalization procedure. A wide variety of studies employ loss data without addressing stationarity assumptions or the possibility for spurious regression. This work enables the statistically rigorous application of such loss time series to modeling applications. The second portion of this work develops a novel matrix variate dynamic factor model for spatio-temporal loss data stratified across multiple correlated hazards or perils. The developed model is employed to analyze and forecast losses from convective storms, which constitute some of the highest losses covered by insurers. Adopting factor-based approach, forecasts are achieved despite the complex and often unobserved underlying drivers of these losses. The developed methodology extends the literature on dynamic factor models to matrix variate time series. Specifically, a covariance structure is imposed that is well suited to spatio-temporal problems while significantly reducing model complexity. The model is fit via the EM algorithm and Kalman filter. The third and final part of this dissertation investigates the impact of compounding hazard events on state and regional migration in the United States. Any attempt to capture trends in climate related migration must account for the inherent uncertainties surrounding climate change, natural hazard occurrences, and socioeconomic factors. For this reason, I adopt a Bayesian modeling approach that enables the explicit estimation of the inherent uncertainty. This work can provide decision-makers with greater clarity regarding the extent of knowledge on climate trends.

It is well–established that physical phenomena occurring at the macroscale are the result of underlying molecular mechanisms that occur at the nanoscale. Understanding these mechanisms at the molecular level allows the development of semicrystalline polymers with tailored properties for different applications. Molecular Dynamics (MD) simulations offer significant insight into these mechanisms and their impact on various physical and mechanical properties. However, the temporostpatial limitations of all–atomistic (AA) MD simulations impede the investigation of phenomena with higher time– and length–scale. Coarse–grained (CG) MD simulations address the shortcomings of AAMD simulations by grouping atoms based on their chemical, structural, etc., aspects into larger particles, beads, and reducing the degrees offreedom of the atomistic system, allowing achievement of higher time– and length–scales. Among the approaches for generating CG models, the hybrid approach is capable of capturing the underlying mechanisms at the molecular level while replicating phenomena at temporospatial scales attainable by the CG model. In this dissertation, a novel hybrid method is developed for the systematic coarse–graining of semicrystalline polymers that uniquely blends the potential functions of both phases. The obtained blended potential not only faithfully reproduces the structural distributions of multiple phases simultaneously but also allows control over the dynamics of the obtained CG models employing a tunable parameter. Given that accelerated dynamics of the CG models hinder the investigation of phenomena in the crystal phase, such as α–α-relaxation, by utilizing the developed method, this phenomenon was successfully modeled for a semicrystalline polyethylene (PE) system with obtained values for the diffusion constant at room temperature and the activation energy in close agreement with experimental results. In a subsequent study, a family of potentials was developed for a sample semicrystalline polyethylene (PE) to investigate the impact of different potential functions on some physical properties, such as crystal diffusion and glass transition temperature, and their correlation with some mechanical properties obtained from uniaxial deformation.

The main objective of this work is to study novel stochastic modeling applications to cybersecurity aspects across three dimensions: Loss, attack, and detection. First, motivated by recent spatial stochastic models with cyber insurance applications, the first and second moments of the size of a typical cluster of bond percolation on finite graphs are studied. More precisely, having a finite graph where edges are independently open with the same probability $p$ and a vertex $x$ chosen uniformly at random, the goal is to find the first and second moments of the number of vertices in the cluster of open edges containing $x$. Exact expressions for the first and second moments of the size distribution of a bond percolation cluster on essential building blocks of hybrid graphs: the ring, the path, the random star, and regular graphs are derived. Upper bounds for the moments are obtained by using a coupling argument to compare the percolation model with branching processes when the graph is the random rooted tree with a given offspring distribution and a given finite radius. Second, the Petri Net modeling framework for performance analysis is well established; extensions provide enough flexibility to examine the behavior of a permissioned blockchain platform in the context of an ongoing cyberattack via simulation. The relationship between system performance and cyberattack configuration is analyzed. The simulations vary the blockchain's parameters and network structure, revealing the factors that contribute positively or negatively to a Sybil attack through the performance impact of the system. Lastly, the denoising diffusion probabilistic models (DDPM) ability for synthetic tabular data augmentation is studied. DDPMs surpass generative adversarial networks in improving computer vision classification tasks and image generation, for example, stable diffusion. Recent research and open-source implementations point to a strong quality of synthetic tabular data generation for classification and regression tasks. Unfortunately, the present state of literature concerning tabular data augmentation with DDPM for classification is lacking. Further, cyber datasets commonly have highly unbalanced distributions complicating training. Synthetic tabular data augmentation is investigated with cyber datasets and performance of well-known metrics in machine learning classification tasks improve with augmentation and balancing.

The increase in the photovoltaic (PV) generation on distribution grids may cause reverse power flows and challenges such as service voltage violations and transformer overloading. To resolve these issues, utilities need situational awareness, e.g., PV-feeder mapping to identify the potential back-feeding feeders and meter-transformer mapping for transformer overloading. As circuit schematics are outdated, this work relies on data. In cases where the advanced metering infrastructure (AMI) data is unavailable, e.g., analog meters or bandwidth limitation, the dissertation proposes to use feeder measurements from utilities and solar panel measurements from solar companies to identify PV-feeder mapping. Several sequentially improved methods based on quantitative association rule mining (QARM) are proposed, where a lower bound for performance guarantee is also provided. However, binning data in QARM leads to information loss. So, bands are designed to replace bins for increased robustness. For cases where AMI data is available but solar PV data is unavailable, the AMI voltage data and location data are used for situational awareness, i.e., meter-transformer mapping, to resolve voltage violation and transformer overloading. A density-based clustering method is proposed that leverages AMI voltage data and geographical information to efficiently segment utility meters such that the segments comprise meters of few transformers only. Although it is helpful for utilities, it may not directly recover the meter-transformer connectivity, which requires transformer-wise segmentation. The proposed density-based method and other past methods ignore two common scenarios, e.g., having large distance between a meter and parent transformer or high similarity of a meter's consumption pattern to a non-parent transformer's meters. However, going from meter-meter can lead to the parent transformer group meters due to the usual observation that the similarity of intra-cluster meter voltages is usually stronger than the similarity of inter-cluster meter voltages. Therefore, performance guarantee is provided via spectral embedding with voltage data under reasonable assumption. Moreover, the assumption is partially relaxed using location data. It will benefit the utility in many ways, e.g., mitigating voltage violations by transformer tap settings and identifying overloaded transformers.

A leading crisis in the United States is the opioid use disorder (OUD) epidemic. Opioid overdose deaths have been increasing, with over 100,000 deaths due to overdose from April 2020 to April 2021. This dissertation presents two mathematical models to address illicit OUD (IOUD), treatment, and recovery within an epidemiological framework. In the first model, individuals remain in the recovery class unless they relapse. Due to the limited availability of specialty treatment facilities for individuals with OUD, a saturation treat- ment function was incorporated. The second model is an extension of the first, where a casual user class and its corresponding specialty treatment class were added. Using U.S. population data, the data was scaled to a population of 200,000 to find parameter estimates. While the first model used the heroin-only dataset, the second model used both the heroin and all-illicit opioids datasets. Backward bifurcation was found in the first IOUD model for realistic parameter values. Additionally, bistability was observed in the second IOUD model with the heroin-only dataset. This result implies that it would be beneficial to increase the availability of treatment. An alarming effect was discovered about the high overdose death rate: by 2038, the disease-free equilibrium would be the only stable equilibrium. This consequence is concerning because although the goal is for the epidemic to end, it would be preferable to end it through treatment rather than overdose. The IOUD model with a casual user class, its sensitivity results, and the comparison of parameters for both datasets, showed the importance of not overlooking the influence that casual users have in driving the all-illicit opioid epidemic. Casual users stay in the casual user class longer and are not going to treatment as quickly as the users of the heroin epidemic. Another result was that the users of the all-illicit opioids were going to the recovered class by means other than specialty treatment. However, the relapse rates for those individuals were much more significant than in the heroin-only epidemic. The results above from analyzing these models may inform health and policy officials, leading to more effective treatment options and prevention efforts.

For fifty years, inquiry has attempted to capture how groups of people experience microaggression phenomena through multiple methodological and analytic applications grounded in psychology-influenced frameworks. Yet, despite theoretical advancements, the phenomenon has met criticisms trivializing its existence, falsifiability, and social significance. Unpacking possible interactive factors of a microaggressive moment invites a revisitation of the known and unknown pragmatic conditions that may produce and influence its discomforting situational “content.” This study employs an intentional, game-theoretic methodology based on brief, publicly-recorded, everyday conversation segments. Conversation segments of social interactions provide a means to conduct a mathematically-solid, computationally-tractable analysis of explaining what is happening during encounters where disability microaggressions are likely the result of partial (non)cooperation between communicators. Such analysis extends the microaggression research program (MRP) by: (1) proposing theoretical consequences for conversational repair phenomena, algorithmic programming, and experimental designs in negotiation research; and (2) outlining practical approaches for preventing microaggressions with new communication pedagogy, anti-oppression/de-escalation training programs, and calculable, focus-oriented psychotherapy. It concludes with an invitation for scholars to “be” in ambiguity so that they may speculate possible trajectories for the study of microaggressions as a communicative phenomenon.

This dissertation consists of three papers about opinion dynamics. The first paper is in collaboration with Prof. Lanchier while the other two papers are individual works. Two models are introduced and studied analytically: the Deffuant model and the Hegselmann-Krause~(HK) model. The main difference between the two models is that the Deffuant dynamics consists of pairwise interactions whereas the HK dynamics consists of group interactions. Translated into graph, each vertex stands for an agent in both models. In the Deffuant model, two graphs are combined: the social graph and the opinion graph. The social graph is assumed to be a general finite connected graph where each edge is interpreted as a social link, such as a friendship relationship, between two agents. At each time step, two social neighbors are randomly selected and interact if and only if their opinion distance does not exceed some confidence threshold, which results in the neighbors' opinions getting closer to each other. The main result about the Deffuant model is the derivation of a positive lower bound for the probability of consensus that is independent of the size and topology of the social graph but depends on the confidence threshold, the choice of the opinion space and the initial distribution. For the HK model, agent~$i$ updates its opinion~$x_i$ by taking the average opinion of its neighbors, defined as the set of agents with opinion at most~$\epsilon$ apart from~$x_i$. Here,~$\epsilon > 0$ is a confidence threshold. There are two types of HK models: the synchronous and the asynchronous HK models. In the former, all the agents update their opinion simultaneously at each time step, whereas in the latter, only one agent is selected uniformly at random to update its opinion at each time step. The mixed model is a variant of the HK model in which each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors. The main results of this dissertation about HK models show conditions under which the asymptotic stability holds or a consensus can be achieved, and give a positive lower bound for the probability of consensus and, in the one-dimensional case, an upper bound for the probability of consensus. I demonstrate the bounds for the probability of consensus on a unit cube and a unit interval.

Modeling human survivorship is a core area of research within the actuarial com

munity. With life insurance policies and annuity products as dominant financial

instruments which depend on future mortality rates, there is a risk that observed

human mortality experiences will differ from projected when they are sold. From an

insurer’s portfolio perspective, to curb this risk, it is imperative that models of hu

man survivorship are constantly being updated and equipped to accurately gauge and

forecast mortality rates. At present, the majority of actuarial research in mortality

modeling involves factor-based approaches which operate at a global scale, placing

little attention on the determinants and interpretable risk factors of mortality, specif

ically from a spatial perspective. With an abundance of research being performed

in the field of spatial statistics and greater accessibility to localized mortality data,

there is a clear opportunity to extend the existing body of mortality literature to

wards the spatial domain. It is the objective of this dissertation to introduce these

new statistical approaches to equip the field of actuarial science to include geographic

space into the mortality modeling context.

First, this dissertation evaluates the underlying spatial patterns of mortality across

the United States, and introduces a spatial filtering methodology to generate latent

spatial patterns which capture the essence of these mortality rates in space. Second,

local modeling techniques are illustrated, and a multiscale geographically weighted

regression (MGWR) model is generated to describe the variation of mortality rates

across space in an interpretable manner which allows for the investigation of the

presence of spatial variability in the determinants of mortality. Third, techniques for

updating traditional mortality models are introduced, culminating in the development

of a model which addresses the relationship between space, economic growth, and

mortality. It is through these applications that this dissertation demonstrates the

utility in updating actuarial mortality models from a spatial perspective.

The problem of modeling and controlling the distribution of a multi-agent system has recently evolved into an interdisciplinary effort. When the agent population is very large, i.e., at least on the order of hundreds of agents, it is important that techniques for analyzing and controlling the system scale well with the number of agents. One scalable approach to characterizing the behavior of a multi-agent system is possible when the agents' states evolve over time according to a Markov process. In this case, the density of agents over space and time is governed by a set of difference or differential equations known as a {\it mean-field model}, whose parameters determine the stochastic control policies of the individual agents. These models often have the advantage of being easier to analyze than the individual agent dynamics. Mean-field models have been used to describe the behavior of chemical reaction networks, biological collectives such as social insect colonies, and more recently, swarms of robots that, like natural swarms, consist of hundreds or thousands of agents that are individually limited in capability but can coordinate to achieve a particular collective goal.

This dissertation presents a control-theoretic analysis of mean-field models for which the agent dynamics are governed by either a continuous-time Markov chain on an arbitrary state space, or a discrete-time Markov chain on a continuous state space. Three main problems are investigated. First, the problem of stabilization is addressed, that is, the design of transition probabilities/rates of the Markov process (the agent control parameters) that make a target distribution, satisfying certain conditions, invariant. Such a control approach could be used to achieve desired multi-agent distributions for spatial coverage and task allocation. However, the convergence of the multi-agent distribution to the designed equilibrium does not imply the convergence of the individual agents to fixed states. To prevent the agents from continuing to transition between states once the target distribution is reached, and thus potentially waste energy, the second problem addressed within this dissertation is the construction of feedback control laws that prevent agents from transitioning once the equilibrium distribution is reached. The third problem addressed is the computation of optimized transition probabilities/rates that maximize the speed at which the system converges to the target distribution.