Precise estimation of connections of metro passengers from Smart Card data

Abstract

The aim of this study is to estimate both the physical and schedule-based connections of metro passengers from their entry and exit times at the gates and the stations, a data set available from Smart Card transactions in a majority of train networks. By examining the Smart Card data, we will observe a set of transit behaviors of metro passengers, which is manifested by the time intervals that identifies the boarding, transferring, or alighting train at a station. The authenticity of the time intervals is ensured by separating a set of passengers whose trip has a unique connection that is predominantly better by all respects than any alternative connection. Since the connections of such passengers, known as reference passengers, can be readily determined and hence their gate times and stations can be used to derive reliable time intervals. To detect an unknown path of a passenger, the proposed method checks, for each alternative connection, if it admits a sequence of boarding, middle train(s), and alighting trains, whose time intervals are all consistent with the gate times and stations of the passenger, a necessary condition of a true connection. Tested on weekly 32 million trips, the proposed method detected unique connections satisfying the necessary condition, which are, therefore, most likely true physical and schedule-based connections in 92.6 and 83.4 %, respectively, of the cases.

Appendices

Appendix

Probability estimation of schedule-based connections

Suppose the current physical connection requires a single transfer, say, at Station $A$. The schedule-based connections on a physical connection can be represented by a time-expanded network as in Fig. 11.

Fig. 11

Two schedule-based connections can be consistent

The consistency check is initiated by finding consistent trains at both $O$ and $D$. By this assumption, there can be at most two trains, say $X_1$ and $X_2$, at $O$, whose time intervals contain the entry time, while at most one train, say $Y$, can be consistent with the exit time at $D$. If there are no such trains at either $O$ or $D$, the passenger did not use the physical connection.

If neither $X_1$ and $X_2$ can be connected to $Y$, in the sense that there is no relevant transfer reference passenger, we conclude that the passenger did not use the physical connection.

If there is only one such train, say $X_1$, whose connection to $Y$ can be verified by transfer reference passengers, then the schedule-based connection, $X_1-Y$ is confirmed as the unique connection of the passenger.

Finally, if there are two trains, say $X_1$ and $X_2$, from both of which we can find transfer reference passengers to $Y$ as in Fig. 11, we need to return both $X_1-Y$ and $X_2-Y$. It is a worst case in that the maximum number of schedule-based connections are confirmed as consistent connections.

The estimation, however, can be refined by a probability distribution over the two connections. In Fig. 11, we introduce some notations as follows:

• $p$: The fraction of the boarding reference passengers from the overlap of the two time intervals that boarded train $X_1$

• $1-p$: The fraction of the boarding reference passengers from the overlap of the two time intervals that boarded train not $X_1$ but $X_2$

• $1-q_1$: The fraction of the transfer reference passenger from $X_1$ to $Y$

• $q_2$: The fraction of the transfer reference passenger from $X_2$ to $Y$

It is not then difficult to show that

$$\begin{aligned} \begin{array}{rl} \Pr \left\{ {\text{Passenger}} {\text{ chose }}\; X_1-Y\right\} &{}= \frac{p (1-q_1)}{p (1-q_1) + (1-p) q_2},\\ \Pr \left\{ {\text{Passenger}} {\text{ chose} }\; X_2-Y\right\} &{}= \frac{(1-p) q_2}{p (1-q_1) + (1-p) q_2}. \end{array} \end{aligned}$$

(1)

Table 6 Numbers and lists of consistent schedule-based connection(s), the corresponding conditions, and the probability distributions for a single-transfer physical connection

Table 6 summarizes the numbers and list of consistent schedule-based connection(s), the corresponding conditions, and the probability distributions. If none of the conditions from Table 6 is satisfied, no schedule-based connection can be consistent with the quadruple of our passenger and hence the physical connection is rejected.

For a physical connection that requires two transfers, there may be up to 3 schedule-based connections consistent with a quadruple if the trip is not abnormally delayed. The previous arguments can be easily extended to such a case.