STATISTICAL METHODS FOR ADVANCED ECONOMETRIC MODELS WITH APPLICATIONS TO VEHICLE HOLDING AND SPEED QUANTILES DISTRIBUTION

This dissertation proposes statistical methods to formulate, estimate and apply complex transportation models. Two main problems are part of the analyses conducted and presented in this dissertation. The first method solves an econometric problem and is concerned with the joint estimation of models that contain both discrete and continuous decision variables. The use of ordered models along with a regression is proposed and their effectiveness is evaluated with respect to unordered models. Procedure to calculate and optimize the log-likelihood functions of both discrete-continuous approaches are derived, and difficulties associated with the estimation of unordered models explained. Numerical approximation methods based on the Genz algortithm are implemented in order to solve the multidimensional integral associated with the unordered modeling structure. The problems deriving from the lack of smoothness of the probit model around the maximum of the log-likelihood function, which makes the optimization and the calculation of standard deviations very difficult, are carefully analyzed. A methodology to perform out-of-sample validation in the context of a joint model is proposed. Comprehensive numerical experiments have been conducted on both simulated and real data. In particular, the discrete-continuous models are estimated and applied to vehicle ownership and use models on data extracted from the 2009 National Household Travel Survey. The second part of this work offers a comprehensive statistical analysis of free-flow speed distribution; the method is applied to data collected on a sample of roads in Italy. A linear mixed model that includes speed quantiles in its predictors is estimated. Results show that there is no road effect in the analysis of free-flow speeds, which is particularly important for model transferability. A very general framework to predict random effects with few observations and incomplete access to model covariates is formulated and applied to predict the distribution of free-flow speed quantiles. The speed distribution of most road sections is successfully predicted; jack-knife estimates are calculated and used to explain why some sections are poorly predicted. Eventually, this work contributes to the literature in transportation modeling by proposing econometric model formulations for discrete-continuous variables, more efficient methods for the calculation of multivariate normal probabilities, and random effects models for free-flow speed estimation that takes into account the survey design. All methods are rigorously validated on both real and simulated data.

MULTI-BAND BOSE HUBBARD MODELS AND EFFECTIVE THREE-BODY INTERACTIONS

Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.

Real-time analytics on large dynamic graphs

In today's fast-paced and interconnected digital world, the data generated by an increasing number of applications is being modeled as dynamic graphs. The graph structure encodes relationships among data items, while the structural changes to the graphs as well as the continuous stream of information produced by the entities in these graphs make them dynamic in nature. Examples include social networks where users post status updates, images, videos, etc.; phone call networks where nodes may send text messages or place phone calls; road traffic networks where the traffic behavior of the road segments changes constantly, and so on. There is a tremendous value in storing, managing, and analyzing such dynamic graphs and deriving meaningful insights in real-time. However, a majority of the work in graph analytics assumes a static setting, and there is a lack of systematic study of the various dynamic scenarios, the complexity they impose on the analysis tasks, and the challenges in building efficient systems that can support such tasks at a large scale. In this dissertation, I design a unified streaming graph data management framework, and develop prototype systems to support increasingly complex tasks on dynamic graphs. In the first part, I focus on the management and querying of distributed graph data. I develop a hybrid replication policy that monitors the read-write frequencies of the nodes to decide dynamically what data to replicate, and whether to do eager or lazy replication in order to minimize network communication and support low-latency querying. In the second part, I study parallel execution of continuous neighborhood-driven aggregates, where each node aggregates the information generated in its neighborhoods. I build my system around the notion of an aggregation overlay graph, a pre-compiled data structure that enables sharing of partial aggregates across different queries, and also allows partial pre-computation of the aggregates to minimize the query latencies and increase throughput. Finally, I extend the framework to support continuous detection and analysis of activity-based subgraphs, where subgraphs could be specified using both graph structure as well as activity conditions on the nodes. The query specification tasks in my system are expressed using a set of active structural primitives, which allows the query evaluator to use a set of novel optimization techniques, thereby achieving high throughput. Overall, in this dissertation, I define and investigate a set of novel tasks on dynamic graphs, design scalable optimization techniques, build prototype systems, and show the effectiveness of the proposed techniques through extensive evaluation using large-scale real and synthetic datasets.