James C. Spall

Xilei Zhao, James C. Spall

Real-time navigation services, such as Google Maps and Waze, are widely used in daily life. These services provide rich data resources in real-time traffic conditions and travel time predictions; however, they have not been fully applied in transportation modeling. This paper aims to use traffic data from Google Maps and applying cutting-edge technologies in maximum likelihood estimation to model traffic networks and travel time reliability. This paper integrates Google Maps travel time data for routes and traffic condition data for links to model the complexities of traffic networks. We then formulate the Fisher information matrix and apply the asymptotic normality to obtain the probability distribution of the travel time estimates for a random route within the network of interest. We also derive the travel time reliability by considering two levels of uncertainties, i.e., the uncertainty of the route's travel time and the uncertainty of its travel time estimates. The proposed method could provide a more realistic and precise travel time reliability estimate. The methodology is applied to a small network in the downtown Baltimore area, where we propose a link data collection strategy and provide empirical evidence to show data independence by following this strategy. We also show results for maximum likelihood estimates and travel time reliability measures for different routes within the network. Furthermore, we use the historical data from a different network to validate this approach, showing our method provides a more accurate and precise estimate compared to the sample mean of the empirical data.

Jingyi Zhu, Long Wang, James C. Spall

Stochastic approximation (SA) algorithms have been widely applied in minimization problems when the loss functions and/or the gradient information are only accessible through noisy evaluations. Stochastic gradient (SG) descent---a first-order algorithm and a workhorse of much machine learning---is perhaps the most famous form of SA. Among all SA algorithms, the second-order simultaneous perturbation stochastic approximation (2SPSA) and the second-order stochastic gradient (2SG) are particularly efficient in handling high-dimensional problems, covering both gradient-free and gradient-based scenarios. However, due to the necessary matrix operations, the per-iteration floating-point-operations (FLOPs) cost of the standard 2SPSA/2SG is $O(p^3)$, where $p$ is the dimension of the underlying parameter. Note that the $O(p^3)$ FLOPs cost is distinct from the classical SPSA-based per-iteration $O(1)$ cost in terms of the number of noisy function evaluations. In this work, we propose a technique to efficiently implement the 2SPSA/2SG algorithms via the symmetric indefinite matrix factorization and show that the FLOPs cost is reduced from $O(p^3)$ to $O(p^2)$. The formal almost sure convergence and rate of convergence for the newly proposed approach are directly inherited from the standard 2SPSA/2SG. The improvement in efficiency and numerical stability is demonstrated in two numerical studies.