Network tomography. A transporation system in a big city, f. ex. a subway system

Question

Network tomography. A transporation system in a big city, f. ex. a subway system, consists of n edges between stations. Passengers enter the system through on station (departure station), travel by one or more edges until they exit the system at another station (arrival station). A ticket system keeps track of the trips passenger i takes and records the following information: si is the starting time of the trip, i.e. when the passenger enters the departure station, and is measurued in minutes that have passed since 6 AM, f is the end time of the trip (when the passenger exits the arrival station). Additionally, the ticket system keeps track of the order of the edges during the trip F. ex. s 128, f 144 and the order of edges (3, 7, 8, 10, 4) means that the passenger entered the departure station at 8:08 AM, left the arrival station at 8:24 AM, and traveled by edges 3, 7, 8, 10 and 4 (in this exact order). The total travel time is f s Suppose that the time that a trip takes is the sum of the time that it takes to travel by each edge. Let dj be equal to the time it takes to travel by edges i, i 1, n. We don't know the time, but we can assess it by using the information the ticket system records (si, f and the order of the edges) of passengers' trips m in the system. Suppose that m is a very large number We use least squares. The estimation, d, of the time it takes to travel each individual edge is acquired by minimizing the norm in the power of 2 of the travel time model's residual. In other words, we use d to minimized Rd cll for some mxn matrix R and m vector c. Describe R and c in your own words, i.e what their meaning is in the context above

Explanation / Answer

M in the above case means the number of passenger trips.

N in the above case means the number of edges through which the passengers travel.

So R in the above case means all the possible combinations of MxN i.e. number of passenger trips and number of edges through which the passenger travels. Which gives us all the possible combinations and helps to calculate every movement of the passengers.

c here means the vector for m which helps us to know the movement of number of passenger.

So when we find the least squares using these two values , it helps us to find the true and fair value of the total time taken by the passengers during travel.  

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