  Accurately monitoring road traffic state is crucial for various applications,
including travel time prediction, traffic control, and traffic safety. However,
the lack of sensors often results in incomplete traffic state data, making it
challenging to obtain reliable information for decision-making. This paper
proposes a novel method for imputing traffic state data using Gaussian
processes (GP) to address this issue. We propose a kernel rotation
re-parametrization scheme that transforms a standard isotropic GP kernel into
an anisotropic kernel, which can better model the congestion propagation in
traffic flow data. The model parameters can be estimated by statistical
inference using data from sparse probe vehicles or loop detectors. Moreover,
the rotated GP method provides statistical uncertainty quantification for the
imputed traffic state, making it more reliable. We also extend our approach to
a multi-output GP, which allows for simultaneously estimating the traffic state
for multiple lanes. We evaluate our method using real-world traffic data from
the Next Generation simulation (NGSIM) and HighD programs, along with simulated
data representing a traffic bottleneck scenario. Considering current and future
mixed traffic of connected vehicles (CVs) and human-driven vehicles (HVs), we
experiment with the traffic state estimation (TSE) scheme from 5% to 50%
available trajectories, mimicking different CV penetration rates in a mixed
traffic environment. We also test the traffic state estimation when traffic
flow information is obtained from loop detectors. The results demonstrate the
adaptability of our TSE method across different CV penetration rates and types
of detectors, achieving state-of-the-art accuracy in scenarios with sparse
observation rates.
