Estimation of travel distances with the weighted ℓp norm: Some empirical results

The weighted ℓp or ℓk,p norm is used to model travel distances in road networks. The parameters k and p are computed from a sample of actual road distances taken from the geographical area of interest. During the computation, a goodness-of-fit criterion is computed and minimized when the best-fit values of k and p have been determined. Greatly improved results are obtained when the coordinate axes are rotated to account for any rectangular bias in the road network. In this paper, an examination of 17 diverse geographical areas shows that in every case, within a rotation arc of 90°, there are two points where the goodness-of-fit function is minimized. These two rotation points are approximately 45° apart; one results in a best-fit p value in the interval (1, 2) and the other results in a best-fit p value in the interval (2, ∞). This paper thus empirically confirms the double-bottom phenomenon which was hypothesized in a previous work. The superiority of the weighted ℓp norm as a distance predictor is also demonstrated.