I have several areas of research, with an array of applications from chemistry and biology to anthropology.
Graphs and Networks:
Graph theory has many connections to statistics and recently there has been an explosion of research into modeling networks as well as using related ideas like graphical models to estimate covariance and understand relationships between variables. A large portion of my research work has been devoted to applications of graphs/networks in statistics with some applications to chemistry, particularly with protein conformational graphs. Related work with chemistry collaborators has led to several projects for students participating in the 4CBC Biomath program.
Nonparametric statistics are useful when the typical parametric conditions for common tests are not satisfied. A collaborator and I are currently working on a nonparametric test for interaction in the two-way ANOVA layout.
Dimension Reduction and Estimation:
Many modern data sets are characterized by high dimensionality - that is, the number of variables observed may be very large, especially when compared to the number of observations . For high-dimensional data, dimension estimation and reduction can be a valuable data analysis tool. Estimating the true ``intrinsic'' dimension of the data in p-dimensions can enable the use of dimension reduction methods and thus remove the curse of high-dimensionality. I have worked on projects that developed new dimension estimation techniques for various applications.
Data analysis techniques such as principal components analysis (PCA), classification by linear and quadratic discriminant analysis, and many more require an estimate of the covariance matrix or its inverse. However, it is known that the sample covariance performs poorly when p is large relative to n. Thus, alternative ways of estimating covariance are needed in high-dimensional situations. Alternative methods for estimating covariance can be divided into two classes depending on whether or not they require ordered variables. My work in this area has been developing a method to bridge the gap between the two classes of methods as well as discovering ways to uncover structure using covariance estimates.
Other Areas of Interest:
I have or have had projects in several other areas including:
Clustering and Classification - specifically cluster pruning, where a clustering solution is modified to deal with unusual points or possibly merged clusters
Longitudinal Data Analysis (applications to Anthropology) - analysis of survey data
Studying Probability in Classic Games (like Risk) - studying conquer odds with collaborators