Parameter estimators of random intersection graphs with thinned communities

This paper studies a statistical network model generated by a large number of randomly sized overlapping communities, where any pair of nodes sharing a community is linked with probability $q$ via the community. In the special case with $q=1$ the model reduces to a random intersection graph which is known to generate high levels of transitivity also in the sparse context. The parameter $q$ adds a degree of freedom and leads to a parsimonious and analytically tractable network model with tunable density, transitivity, and degree fluctuations. We prove that the parameters of this model can be consistently estimated in the large and sparse limiting regime using moment estimators based on partially observed densities of links, 2-stars, and triangles.Comment: 15 page

Estimation and Sparse Selection of Conditional Probability Models for Vector Time Series

Diverse scientific fields collect multiple time series data to investigate the dynamical behavior of complex systems: atmospheric and climate science, geophysics, neuroscience, epidemiology, ecology, and environmental science. Identifying patterns of mutual dependence among such data generates valuable knowledge that can be applied either for inferential or forecasting purposes. Vector autoregressive (VAR) processes provide a flexible class of statistical models for multiple time series that are easy to estimate using regression techniques. However, scaling to large data sets and extension to more general processes stretch the framework's capacity: due to the dense parametrization of VAR models, which have one parameter for every possible pairwise relationship between components (i.e., between each univariate time series in a collection), high-dimensional data generate difficulties associated with model selection and parametric regularization; and modeling more general processes requires data transformations that complicate inference, forecasting, and model interpretation. Fields such as epidemiology, neuroscience, and ecology generate high-dimensional time series of count vectors, which incur both sets of challenges at once. Autoregressive conditional probability models --- models in which the conditional means of a time series follow an autoregressive structure in the process history --- are natural generalizations that preserve ease of estimation and, in conjunction with selection methods in regression, promise to address challenges associated with modeling large multiple time series of count (and other discrete) data. This thesis focuses on developing empirical methodology for sparse selection of nonlinear VAR-type conditional Poisson models. Chapter 1 provides an overview of related existing work. Chapter 2 develops an empirical method for sparse selection in VAR models based on resampling methods. Chapter 3 presents a conditional probability generalization of the VAR model and analyzes the stability properties of Poisson generalized vector autoregressive (GVAR) processes. Chapter 4 combines the work of the preceding two chapters and develops a resampling-based method for sparse estimation of Poisson GVAR models. Finally, Chapter 5 summarizes key findings, challenges, and future work