  The NGSIM trajectory data sets provide longitudinal and lateral positional
information for all vehicles in certain spatiotemporal regions. Velocity and
acceleration information cannot be extracted directly since the noise in the
NGSIM positional information is greatly increased by the necessary numerical
differentiations. We propose a smoothing algorithm for positions, velocities
and accelerations that can also be applied near the boundaries. The smoothing
time interval is estimated based on velocity time series and the variance of
the processed acceleration time series. The velocity information obtained in
this way is then applied to calculate the density function of the
two-dimensional distribution of velocity and inverse distance, and the density
of the distribution corresponding to the ``microscopic'' fundamental diagram.
Furthermore, it is used to calculate the distributions of time gaps and
times-to-collision, conditioned to several ranges of velocities and velocity
differences. By simulating virtual stationary detectors we show that the
probability for critical values of the times-to-collision is greatly
underestimated when estimated from single-vehicle data of stationary detectors.
Finally, we investigate the lane-changing process and formulate a quantitative
criterion for the duration of lane changes that is based on the trajectory
density in normalized coordinates. Remarkably, there is a very noisy but
significant velocity advantage in favor of the targeted lane that decreases
immediately before the change due to anticipatory accelerations.
