Equivalences of Freight Trip Generation Functional Forms: Formulations, Approximations and Error Formalization Issues

The definition of freight intensity patterns, mainly on the form of freight trip or commodity generation, is essential in urban economics. Those patters are in general defined in a set of models that relate trips or commodity quantities to each individual activity size, in the form of employment mainly. Those models, which can have different functional forms, are defined at the level of each single establishment, but in some cases only aggregated zonal data is available, making it possible to define constant and linear models (since their formulations have a transitivity property) but not non-linear models directly, those last requiring the definition of individual employment for each establishment. This paper aims to overcome this limitation by proposing a forma aggregated formulation of four functional forms (constant, linear, logarithmic and potential) and defining, via mathematical transformation, equivalences based on quasi-arithmetic means, which are then approximated by the use of the arithmetic mean instead (which is calculable using aggregated data, where the total number of establishments and the total employment are known). The paper analyses those approximations and proposes a theoretical calculation of the maximum error those approximations can have, via the definition of the statistical limiting error as the limit of a percentage error calculated on those equivalences when the variability of data (then the standard deviation) is very important, covering 99.7% of the statistical distribution of this error. Results show that those errors are contained and using non-linear models, even with an approximated number of establishments, results on more accurate models than using linear forms when the most suitable model is non-linear.