Fundamental Limits of Missing Traffic Data Estimation in Urban Networks

Traffic data estimation plays an important role because traffic data often suffers from missing data problems, caused by a variety of reasons, i.e., temporary deployment of sensors, sensor malfunction and communication failure. Existing research on missing data estimation has mostly focused on using data-driven or model-driven models to estimate the missing data, and there is a lack of study on the achievable estimation accuracy and the conditions to achieve accurate missing data estimation. In this paper, we investigate the fundamental limits of missing traffic data estimation accuracy in urban networks using the spatial-temporal random effects model. We derive the squared flow error bound (SFEB) for the cases of the Fisher matrix being a singular and non-singular matrix, respectively. We show that the sufficient and necessary condition of the existence of an unbiased estimator is that the number of missing points is less than or equal to the rank of the Fisher matrix. For the case that no unbiased estimator can be found, we derive an inequality for the SPEB and show that the SFEB is readily determined by the covariance matrix of the unknown (missing) parameter vector, flow correlation between the unknown and the available data, and the sensor locations. Furthermore, we develop an optimal spatial-temporal Kriging estimator which is efficient in both cases where the causal relationship among available data points exists or does not exist. Our theoretical findings can be used to develop a sensor location optimization strategy to minimize the SFEB.