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.

中文翻译：

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“传入布局”的优效性证明，在不确定性下实现完整的链路流可观察性


完整链路流可观测性问题是确定流量网络中要与传感器一起安装的最小链路集，以便唯一确定所有链路流量。在我们之前的工作（Shao et al.， 2016）中，我们提出了一个以转弯比为先验信息的流量守恒系统，并建议在交通网络中的所有专用进站道路链路（称为“进站布局”）上安装传感器可以唯一地确定所有网络链路的流向信息。然而，传感器观察到的链路流不可避免地会受到测量误差的影响，并且还存在先验信息（即转弯比）的一些偏差的风险，同时将流扩展到整个网络。考虑到这两种类型的不确定性，“来料布局”不仅是一个可行的解决方案，而且在这项研究中，已经证明可以最大限度地减少由传感器测量误差和先验信息偏差引起的推断所有链路流过程中的累积不确定性。具体来说，从理论上证明了 “incoming layout” 的优越性，包括两种情况。（i） 仅考虑传感器测量误差，利用线性代数知识对误差传播理论进行解析表示。相关的误差传播矩阵是帮助证明 “incoming layout” 的累积不确定性总是小于 “general layout” 的累积不确定性的关键。 （ii） 考虑到传感器测量误差和先验信息偏差，引入矢量化算子量化先验信息偏差对链路流推理精度的影响，有利于证明“传入布局”在最小化所有链路流的累积不确定性方面的优越性。