  Recent transportation network studies on uncertainty and reliability call for
modeling the probabilistic O-D demand and probabilistic network flow. Making
the best use of day-to-day traffic data collected over many years, this paper
develops a novel theoretical framework for estimating the mean and
variance/covariance matrix of O-D demand considering the day-to-day variation
induced by travelers' independent route choices. It also estimates the
probability distributions of link/path flow and their travel cost where the
variance stems from three sources, O-D demand, route choice and unknown errors.
The framework estimates O-D demand mean and variance/covariance matrix
iteratively, also known as iterative generalized least squares (IGLS) in
statistics. Lasso regularization is employed to obtain sparse covariance matrix
for better interpretation and computational efficiency. Though the
probabilistic O-D estimation (ODE) works with a much larger solution space than
the deterministic ODE, we show that its estimator for O-D demand mean is no
worse than the best possible estimator by an error that reduces with the
increase in sample size. The probabilistic ODE is examined on two small
networks and two real-world large-scale networks. The solution converges
quickly under the IGLS framework. In all those experiments, the results of the
probabilistic ODE are compelling, satisfactory and computationally plausible.
Lasso regularization on the covariance matrix estimation leans to underestimate
most of variance/covariance entries. A proper Lasso penalty ensures a good
trade-off between bias and variance of the estimation.
