Taylor's Power Law and its decomposition in urban facilities

As one of the few generalities in ecology, Taylor's power law admits a power function relationship $V=aM^{b}$ between the
variance $V$ and mean number $M$ of organisms in a quadrat. We examine the spatial distribution data of seven urban service facilities
in 37 major cities in China, and find that Taylor's law is validated among all types of facilities. Moreover, Taylor's law is robust
if we shift the observation window or vary the size of the quadrats. The exponent $b$ increases linearly with the logarithm of the
quadrat size, i.e., $b(s) = b_0 + A\log(s)$. Furthermore, the ANOVA test indicates that $b$ takes distinct values for different facilities
in different cities. We decompose $b$ into two different factors, a city-specific factor (CSF) and a facility-specific factor (FSF).
variations in $b$ can be explained to a large extent by the differences between cities and types of facilities. Facilities are more evenly distributed in larger and more developed cities. Competitive interchangeable facilities (e.g. pharmacy), with larger FSFs and smaller $b$s, are less aggregated than complimentary services (e.g., restaurants).