Superiority proof of "incoming layout" for full link flow observability under uncertainty

The full link flow observability problem is to identify the minimum set of links to be installed with sensors in a traffic network that allows the unique determination of all link flow volumes. In our previous work (Shao et al., 2016), we proposed a flow conservation system using turning ratios as prior information, and suggested that installing sensors on all exclusive incoming road links in the traffic network (called "incoming layout") can uniquely determine the flow information of all network links. However, the link flow observed by the sensor is inevitably subject to measurement errors, and there is also a risk that some deviation in prior information (i.e., turning ratios) will be propagated while extending flows over the whole network. Considering these two types of uncertainty, the "incoming layout" is not only a feasible solution, but in this study, has been proved to minimize the cumulative uncertainty in the process of inferring all link flows caused by the sensor measurement error and the deviation of prior information. Specifically, the superiority of the "incoming layout" is proved theoretically, including two cases. (i) Considering only the sensor measurement error, the error propagation theory is analytically expressed using the knowledge of linear algebra. The related error propagation matrix is found to be the key to help demonstrate that the cumulative uncertainty of the "incoming layout" is always smaller than that of the "general layout". (ii) Considering the sensor measurement error and the deviation of prior information, vectorization operator is introduced to quantify the effect of the prior information deviation on the accuracy of link flow inference, which is beneficial to prove the superiority of the "incoming layout" in minimizing the cumulative uncertainty of all link flows.