On the directional bias of the ℓ_<i>bp</i>-norm

Abstract

<p>The Weighted Sum of Order <em>p</em>, the ℓ<em>_bp</em>-norm, is a generalization of the well-known ℓ<em>_p</em>-norm used in predicting distances in a transportation network. The properties of the directional bias function and the unit balls for the ℓ<em>_bp</em>-norm are of theoretical and practical interest. We investigate these properties and compare them with the properties of the ℓ<em>_p</em>-norm's directional bias function and the unit balls. We find that the ℓ<em>_bp</em>-norm is better at capturing the nonlinearity in a transportation network than the weighted ℓ<em>_p</em>-norm. It is also shown that, in contrast to the weighted ℓ<em>_p</em>-norm, where the optimal parameter <em>p</em> value is confined to the interval (1, 2), for the ℓ<em>_bp</em>-norm the parameter <em>p</em> can have an optimal value greater than 2.</p>