Deriving the Upper Bound of the Number of Sensors Required to Know All Link Flows in a Traffic Network

Abstract

It is demonstrated that the minimum number of sensors required to know all link flows in a traffic network can be determined only if path information is available. However, not all paths need to be enumerated but, at most, a small subset defining the rank r w of the link-path incidence matrix W. If this rank for a reduced subset of paths is already m − n, where m and n are the number of links and noncentroid nodes, respectively, we can conclude that m − n sensors are sufficient. It is also shown that the formulas providing the dependent link flows in terms of the independent link flows can be obtained by the node-based or path-based approaches with the same results only when r w = m − n. Finally, an algorithm to obtain the small subsets of linearly independent path vectors is given. The methods are shown by a parallel network example and the Ciudad Real and Cuenca networks, for which the savings in link counts with respect to the m − n bound are larger than 16%. The corresponding savings in path enumeration are larger than 80%.Index Terms-Link flow estimation, optimal sensor location.

Data dependent input control for origin–destination demand estimation using observability analysis

“…However, feasibility requirements, such as non-negativity, have started to receive attention and been incorporated in a few recent link-flow observability studies by Castillo et al (2013Castillo et al ( , 2014a. Below, we show that, when the actual data of the input set are incorporated, there are four mutually exclusive and collectively exhaustive cases of the estimation ODR that all have a unique solution, but only some having a feasible solution.…”

Heterogeneous sensor location model for path reconstruction

“…This method was previously used in topological observability analysis in other applications, such as power systems ( Mori and Tsuzuki, 1991 ), water networks ( Rahal,1995;Gupta and Prasad, 20 0 0;Kumar et.al, 20 08 ), logistic chain networks ( Syarif et.al, 2002 ) and communication networks ( Sang-Moon et.al, 2005 ). Castillo et al (2013) focused on the upper bound of the number of passive sensors to determine all link flows with the partial link-path matrix consisting of only the set of linearly independent path vectors. More recently, a new concept of non-planar, hole-generated networks was introduced in the link-flow observability problem ( Castillo et al, 2014 ).…”

A State-of-the-Art Review of the Sensor Location, Flow Observability, Estimation, and Prediction Problems in Traffic Networks

“…Hu et al [74] solve the whole link observability problem (observing all link flows) assuming route information and using the concept of "reduced row echelon form" (RREF), a technique based on the well-known Gaussian elimination method, which emphasizes its algebraic character. Castillo et al [77,84] provide a pivoting technique to solve the same problem for partial and whole observability and extend the method to plate scanned data; this problem has also been treated by Castillo et al [72,77,84] and Ng [57] and consists of determining which subset of link flows can be calculated in terms of another subset of link flows but now using route information.…”

“…If route information is used, the bound given by node based approaches can be improved with savings reaching in some cases 16%, as demonstrated in Castillo et al [72]. However, since enumerating all routes is a difficult task, the first method was improved by considering linearly independent route vectors by Castillo et al [72]. More precisely, they show that only a subset of linearly independent routes is required and provide a method to select linearly independent route vectors.…”