Estimating origin-destination matrices with sparse seed matrices

Origin-destination matrices of traveller flows are a key ingredient to transport planning. In public transport planning, most agencies conduct origin-destination surveys to extract line-level origin-destination matrices. These matrices, however, only partially represent ridership, as only a fraction of travellers are surveyed. Hence, they need to be scaled to real ridership, typically by using automatic passenger counts (APC) and algorithms such as iterative proportional fitting (IPF). This procedure works well for busy lines, where seed matrices present few or no zeros (i.e., absence of observations for a given origin-destination pair), however it becomes less reliable on sparsely used lines, where seed matrices present a high percentage of structural and sampling zeros. It is currently unknown, up to which percentage of zeroes IPF can be reliably used, and how to handle zeroes more generally. In this paper, we apply IPF to simulated (ground truth) and real origin-destination seed matrices to quantify the reliability of IPF and to test different replacement values for zeroes. We work with matching data from automatic passenger counters and a large origin-destination survey with 26,000 + participants on 70 + public transport lines that was conducted in 2022 in Geneva, Switzerland. We find that the reliability of IPF measured by the estimation error exponentially correlates with the percentage of zeros in the seed matrix. We test replacement values of zeroes between 0 and 10 and find that 1 is the best replacement for sampling zeros in the seed matrix to minimize the estimation error and simultaneously improve convergence. Practitioners and academics can use these results to maintain the advantages of IPF in practice (computational lightweight, simplicity, implementation in most common software) yet improve its reliability on sparsely used lines.