## 2 Methodology and Related Works
### Related Works  
A variety of sources have been used to estimate OD matrices in transportation systems. [PERSON] et al. [10] identified three models for destination estimation: (i) the probability model, which computes the alighting probability based on the traveled distance and the number of passengers on board, but without identifying pairs (board-alight) corresponding to the same passenger; (ii) the deep learning model, which requires boarding and alighting data for training, being more suitable to railway/subway systems where passengers are required to validate their cards both to enter and exit stations; and (iii) the trip chaining model. The most relevant works based on the trip chaining method, i.e., the one applied in this article, are reviewed next. The trip chaining model [11] infers destinations by looking at the history of trips of each cardholder. Two hypotheses are considered: the origin of a new trip is the destination of the previous one, and at the end of the day, users return to the origin of their first trip of the day. The proposed model was applied to the subway system of New York, where nearly 80% of riders use smart cards. The computed OD matrix was validated using station exit counts at different times of the day and using peak load passenger volume data and a trip assignment model. The authors estimated that 90% of destinations can be accurately inferred for a 78% share of the total number of subway users. [PERSON] et al. [12] proposed using the trip chaining model for estimating the destination of passengers boarding buses with smart cards, following a database programming approach. Trips for which chaining is not possible were compared with all other trips of the month for the same user to find similar trips with known destination. The experimental evaluation used real datasets from Gatineau, Quebec. The proposed approach allowed estimating the destination of 66% of the trips. However, the real estimation accuracy could not be assessed due to the lack of a second source of data (e.g., surveys) for comparison. [PERSON] et al. [13] applied the trip chaining method to infer bus passenger origin-destination from smart card transactions from London, UK. Results were compared against the passenger intercept survey performed every five to seven years for each bus route and includes the number of people boarding and alighting at each bus stop. The analysis showed that destinations could be estimated for nearly 57% of all trips. When compared to the survey, the difference on the estimated destinations were below 4% on the worst case. [PERSON] and [PERSON] [14] estimated OD matrices in the multimodal transportation system of Santiago, Chile, where passengers can use their smart cards to pay for tickets at metros, buses, and bus stations. The proposed approach was evaluated using smart card datasets corresponding to two different weeks, with over 35 million transactions each. The destination and time of alighting was estimated for over 80% of the transactions. Later [15], the authors validated the main assumptions of the model by comparing the estimated OD matrices with data from surveys and personal interviews to passengers. The authors concluded that the proposed model was highly reliable, accurately estimating 84.2% of the inferred destinations. [PERSON] et al. [16] analyzed one week of smart card data (628,479 transactions) from bus, train, and ferry networks of South East Queensland, Australia. The dataset contained both origin and destination records, since passengers are required to validate their smart cards when boarding and alighting. Therefore, the authors were able to study different variants of the trip chaining method and compare the resulting OD matrices against the real data from AFC records. Results showed that for nearly 88% of the passengers the last destination of the day was within a walkable distance of their first origin, thus validating one of the key assumptions of the trip chaining model.