Predictions of complex systems with density-dependent interations using Density-functional Fluctuation Theory

Abstract

From swarms of insects to the migration of people, studies of collections of complex individuals are found across multiple disciplines. Accompanying the recent growth of data for such systems are methods for discovering trends hidden within large datasets. To join this effort, this thesis shall develop and apply a generic statistical framework, Density-functional Fluctuation Theory (DFFT), that can both quantify behaviors and predict responses of complex density-dependent systems using analysis of data alone. Specifically, we consider systems that are collections of individuals interacting with both their environment and neighboring individuals. These types of systems arise in fields from ecology to neighborhood-scale demography. One feature commonly shared in such studies is that data comes in the form of densities, e.g. the counts of individuals of a given type in an area. The challenge DFFT addresses is how to build a statistical framework for such data that is both descriptive and predictive while making minimal assumptions about the nature of the underlying interactions in a system. DFFT does so by means of a generic probabilistic model that separates the tendencies of individuals to be found in parts of a heterogeneous environment from the tendencies of individuals to group or not group with others of the same or different type. This thesis shall demonstrate the utility of this method on experimental, simulated, and extant data sets by developing predictions that are of use to each respective system. First, using an experimental collection of walking fruit flies, we measure fly-fly interactions to accurately predict the distribution of flies after the number of flies changes by two orders of magnitude. Second, using the classic Schelling model of residential segregation, we develop a multi-group segregation function to accurately predict how a simulated city responds at short time scales to changes in the population. Third, using US census population counts by race and ethnicity, we use our segregation function to predict and validate the probability of any neighborhood within the US to change its racial composition over a 10 year period. Each chapter is intended to stand on its own and so readers are encouraged to skip to the application that most interests them. This thesis, then, establishes DFFT as a useful tool towards the analysis and prediction of density-dependent complex systems.