An Allometric Algorithm for Fractal-Based Cobb-Douglas Function of Geographical Systems

The generalized Cobb-Douglas production function has been derived from a general input-output relation based on fractality assumptions. It was proved to be a useful self-affine model for geographical analysis. However, the ordinary least square calculation is always an ineffectual method for the Cobb-Douglas modeling because of the multicollinearity in the logarithmic linear regression. In this paper, a novel approach is proposed to build the geographical Cobb-Douglas models. Combining the concept of allometric scaling with the linear regression technique, we obtain a simple algorithm that can be employed to estimate the parameters of the Cobb-Douglas function. As a case, the algorithm and models are applied to the public transportation of China’s cities, and the results validate the allometric algorithm. A conclusion can be drawn that the allometric analysis is an effective way of modeling geographical systems with the general Cobb-Douglas function. This study is significant for integrating the notions of allometry, fractals, and scaling into a new framework to form a quantitative methodology of spatial analysis.