On the Scale free laws of Urban Facilities

TL;DR

This paper models the spatial distribution of urban facilities using a double stochastic process with a power covariance function, effectively capturing various statistical properties of urban spatial patterns.

Contribution

It introduces a stochastic model that accurately fits multiple spatial statistics of urban facilities, extending beyond traditional methods.

Findings

The model fits the $K$ function and its derivative well.

It reproduces mean-variance relationships of event counts.

It captures higher-order spatial statistics like $H(t)$, $G(t)$, and $F(t)$.

Abstract

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 $F(t)$.