In many scientific studies, researchers are interested in geometric structures of the underlying density function. Common examples for these structures include local modes, ridges, and level sets. In this talk, I will focus on two geometric structures: density ridges and modal regression. Density ridges are curve-like structures characterizing high density regions. I will first describe statistical models for ridges and then discuss their asymptotic theory and methods for constructing confidence sets.
Modal regression is an alternative way to study conditional structure of the response variable given covariates. Instead of looking for conditional expectation, modal regression focuses on conditional local modes. I will present several useful statistical properties for modal regression, including asymptotic theory, confidence sets, prediction sets, and clustering.