Heteroscedastic Gaussian processes for uncertainty modeling in large-scale crowdsourced traffic data

Accurately modeling traffic speeds is a fundamental part of efficient intelligent transportation systems. Nowadays, with the widespread deployment of GPS-enabled devices, it has become possible to crowdsource the collection of speed information to road users (e.g. through mobile applications or dedicated in-vehicle devices). Despite its rather wide spatial coverage, crowdsourced speed data also brings very important challenges, such as the highly variable measurement noise in the data due to a variety of driving behaviors and sample sizes. When not properly accounted for, this noise can severely compromise any application that relies on accurate traffic data. In this article, we propose the use of heteroscedastic Gaussian processes (HGP) to model the time-varying uncertainty in large-scale crowdsourced traffic data. Furthermore, we develop a HGP conditioned on sample size and traffic regime (SRC-HGP), which makes use of sample size information (probe vehicles per minute) as well as previous observed speeds, in order to more accurately model the uncertainty in observed speeds. Using 6 months of crowdsourced traffic data from Copenhagen, we empirically show that the proposed heteroscedastic models produce significantly better predictive distributions when compared to current state-of-the-art methods for both speed imputation and short-term forecasting tasks.Comment: 22 pages, Transportation Research Part C: Emerging Technologies (Elsevier

Increasing powers in a degenerate parabolic logistic equation

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$\partial_t u-\Delta u=a u-b(x) u^p \text{in} \Omega\times \R^+, u(0)=u_0, u(t)|_{\partial \Omega}=0$$ as $p\to +\infty$, where $\Omega$ is a bounded domain and $b(x)$ is a nonnegative function. We deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards we fully describe its long time behavior.Comment: 15 page

Explosive synchronization enhanced by time-delayed coupling

We study the emergence of synchronization in scale-free networks by considering the Kuramoto model of coupled phase oscillators. The natural frequencies of oscillators are assumed to be correlated with their degrees and a time delay is included in the system. This assumption allows enhancing the explosive transition to reach the synchronous state. We provide an analytical treatment developed in a star graph which reproduces results obtained in scale-free networks. Our findings have important implications in understanding the synchronization of complex networks, since the time delay is present in most systems due to the finite speed of the signal transmission over a distance.Comment: 5 pages, 7 figure