A totally positive (TP) matrix is defined by the property that all its minors (determinants of square submatrices) are positive. The study of TP matrices has an extensive history, and has significant applications in various fields such as statistics, mathematical biology, combinatorics, dynamics, approximation theory, operator theory, and geometry. A central question in this field is the TP completion problem: given a partial TP matrix (with unspecified entries), can it be completed to form a TP matrix? While most 4x4 (and smaller) partial TP matrices can be completed, such matrices with unspecified entries at positions (1,4) and (4,1) present known incompletable cases. Building on prior research in quantum algebra, we applied methods including Cauchon’s Deleting Derivations Algorithm and the “path model” of quantum matrices to address this specific case. Our research identified the necessary conditions for completing a 4x4 partial TP matrix with unspecified (1,4) and (4,1) entries, and developed a step-by-step completion process. Additionally, we created a program that automates the path-finding process in our methods, adaptable to partial TP matrices of any size or pattern. Our work lays a foundation for future research on TP completion problems with similar patterns, as well as further deepening the connection between the fields of linear algebra and quantum algebra.

We study an inverse probability weighting (IPW) approach for estimating regression models with missing data. The study of such problems in the applied sciences has focused largely on multiple imputation techniques (i.e., simulated predictions). However, multiple imputation relies on a so-called imputation model for predicting the missing data. In types of data commonly used in applied studies, this model may be incorrectly specified and result in biased estimates and incorrect confidence intervals. Thus, we propose an IPW approach, which instead introduces a missingness model that is much easier to specify. We compare both methods in a simulation study, and show that the IPW approach provides unbiased estimates and better coverage rates than multiple imputation, when the imputation model is incorrectly specified. Thus the IPW approach provides a simpler and more flexible solution to missing data that is robust to violations of model assumptions, and confers more accurate parameter estimates and confidence intervals.

Accurate climate trend analysis is critical for understanding long-term climate change, but internal variability and instrumentation changes can introduce artificial signals into observational datasets. Therefore, a key focus within climate trend research is the homogenization of climate data, which involves correcting systematic errors in observational records. This project investigates the impact of a 2002 weather station instrumentation change in Phoenix, Arizona, on recorded humidity trends. Using ERA5 reanalysis and CONUS404 high-resolution model data, exploratory data analysis was conducted to compare specific humidity before and after the instrumentation change through time series and distribution analyses. Visualizations and distribution analyses reveal a consistent offset between the datasets ERA5 and CONUS404. Differences in humidity values after 2002 shifted towards more negative values relative to data before 2002, suggesting the presence of a potential breakpoint not readily visible in raw time series data. These findings highlight the limitations of traditional homogenization methods and suggest that incorporating dynamical adjustment to enhance the climate signal-to-noise ratio could improve the detection of undocumented inhomogeneities in climate records.

We formulate decentralized, cooperative multi-agent frameworks for the classical reinforcement learning bandit problem. In particular, we apply this framework to two specific settings, the stochastic cascading bandit and the adversarial bandit. In multi-agent cascading bandits, the stochastic reward in each round depends on the joint-ranking "cascade of actions" collectively taken by all agents. In adversarial multi-agent bandits, rewards for joint-actions may be given out arbitrarily and using knowledge on the agents' policies. The objective between agents is shared, but to make the coordination problem more challenging, we contend with variants of information asymmetry: action asymmetry, where the overall joint-action is unobservable to all agents but the feedback received is common; reward asymmetry, where the overall ranking is observable, but feedback received by each agent is i.i.d.; and that with both action and reward asymmetry. In the cascading setting, we develop optimal algorithms that achieve regret on or close to the order of O(log T) for all three settings. For the adversarial setting, we prove an impossibility theorem that a framework with no communication and simultaneous actions by agents is doomed to incur linear regret, and thus instead prove optimal algorithms for successive action frameworks.