Research Projects

Exploring the Maximum Likelihood Thresholds of Gaussian Graphical Models

Faculty Mentor: Dr. Elizabeth Gross

Abstract:

In this research, we investigate graphical models with a focus on maintaining the maximum likelihood threshold (MLT). Graphical models play a vital role in representing complex relationships between variables, such as gene regulatory networks in biology and microbiome data analysis. We aim to address the challenge of having a sufficient number of data points relative to the variables for a maximum likelihood estimate (MLE) to exist.

Our main hypothesis is that by leveraging combinatorics and algebraic statistics, we can adapt model selection methods to suit small datasets, where observations are fewer than variables. We aim to develop strategies and techniques that ensure the validity of maximum likelihood estimation in small-sample scenarios.

Our initial approach involves a comprehensive examination of various graph properties, such as planarity and the clique number, to understand how MLE changes with the graph's combinatorial structure. We plan to use these insights, along with other relevant properties, to maintain the maximum likelihood threshold in graphical models.

Our project focuses on small graphs to explore the space of graphs with a fixed maximum likelihood threshold. We will study the effects of simple operations like edge swapping, removal, and addition to determine if these actions maintain the fixed threshold. By enhancing our understanding of preserving the maximum likelihood threshold in graphs, we aim to contribute to network sampling, model selection, and statistical inference in the field of biological sciences.

Investigating Wildfires and Air Pollution: Implications for Health in Marginalized Communities

Abstract:

This study examines the relationship between wildfires, air pollution, and asthma prevalence, with a focus on marginalized communities. Wildfires release particulate matter (PM2.5), which significantly exacerbates asthma symptoms, disproportionately affecting those with limited healthcare access. The research highlights California’s vulnerability to wildfires and identifies how counties with lower GDP face higher asthma rates due to increased exposure to air pollution and fewer resources. Additionally, climate change and air quality are intertwined, with pollution contributing to and worsening climate change. Addressing these issues through comprehensive strategies is essential to protect public health, particularly for vulnerable populations.