On the Scale free laws of Urban Facilities

We implement a double stochastic process as the mathematical model for the spatial point patterns of urban facilities. We find that the model with power covariance function can produce the best fit not only to $K$ function (whose derivative gives the radial distribution $\rho(t) = K'(t)/2\pi t$) but also to additional facts of spatial point patterns. These facts include the mean-variance relationship of number of events in a series of expanding bins, and other statistics beyond the first two orders, such as inter-event distribution function $H(t)$ and nearest neighbor distribution functions $G(t)$ and $F(t)$.