• Motivation
• Network Constrained Kernel Density Estimation (NetKDE)
• Network Constrained 2nd-Order Point Patterns Analysis
  • Network Constrained K-function
  • Network Constrained G-function

• Real world geography are heterogeneous and anisotropy.

• (a) an access point (the black circle) of a polygon to a network (the horizontal line segment),
• (b) a boundary line segment of a polygon shared with a network (the bold line segment)
• (c) an intersection point of two networks (the black circles),
• (d) a network intersecting an area (the bold line segment).

Typical network constrained events questionsare as follows:

• How can we obtain the catchment areas of parking lots in a downtown area including one-way streets, assuming that drivers access their nearest parking lots?
• Do coffee outlets tend to stand side-by-side alongside streets in a downtown area?
• Do Airbnb listings tend to locate near to MRT stations?
• Is the roadside retail rental price of a street segment similar to those of the adjacent street segments?
• How can we locate clusters of childcare centres within HDB towns?
• How can we estimate the density of traffic accidents and street crimes incidence, and how can we identify locations where the densities of those occurrence are high, referred to as black spots and hot spots?
• How can we spatially interpolate an unknown PM2.5 density at an arbitrary point on a road using known PM2.5 densities at observation points at CBD?

• With reference to (a), we will conclude that the spatial event is nonrandomly distributed points on a bounded plane,
• By constraining the points on a network, (b) reveals a randomly distributed patterns.

• To perform a NKDE, the events must be snapped on the network. The snapped events are shown here in green.

The simple method was proposed by Xie and Yan (2008). They defined the NetKDE with the following formula:

Ho: The observed spatial point events (i.e airbnb listings, coffee outlets, traffic accident locations etc) are uniformly distributed over a street network in a study area.

• The assumption of the binomial point process implies the hypothesis that objects represented by P (say, airbnb listings) are uniformly and independently distributed over the street network Lp.

• If this hypothesis is rejected, we may infer that the spatial point events are spatially interacting and dependent on each other; as a result, they may form nonuniform patterns.

In general, there are two types of nonuniform spatial point patterns on network, they are:

• clustering whereby the spatial point events tend to be close together (such as in Figure a), and
• repelling (also know as regular) whereby the spatial point event tend to keep away from each other (such as in Figure b).

Under the assumption of the binomial point process, network k-function K(t) (Atsuyuki Okabe and Ikuho Yarnada, 2001) is defined as:

For instance, the set A may be the set of Airbnb listings and the set B may be the set of MRT stations. We are concerned with whether the locations of MRT stations affect the distribution of Airbnb listing.

Ho: Airbnb listings are distributed according to the binomial point process.

• This assumption implies that Airbnb listings are uniformly and independently distributed over LT regardless of the locations of MRT stations.

• If the above hypothesis is rejected, we may infer that the locations of MRT stations affect the distribution of Airbnb listing.

It should be noted that no assumption is made with respect to the distribution of points B.

Okabe, Atsuyuki and Yarnada, Ikuho (2001) "The K-Function Method on a Network and Its Computational Implementation", Geographical Analysis, Vol. 33, No. 3, pp. 271-290

Abramson, Ian S. 1982. “On Bandwidth Variation in Kernel Estimates-a Square Root Law.” The Annals of Statistics, 1217–23.

Okabe, Atsuyuki, and Kokichi Sugihara. (2012) Spatial Analysis Along Networks: Statistical and Computational Methods. John Wiley & Sons.

Sugihara, Kokichi, Toshiaki Satoh, and Atsuyuki Okabe (2010) “Simple and Unbiased Kernel Function for Network Analysis.” In 2010 10th International Symposium on Communications and Information Technologies, 827–32. IEEE.

Xie, Zhixiao, and Jun Yan (2008) “Kernel Density Estimation of Traffic Accidents in a Network Space.” Computers, Environment and Urban Systems 32 (5): 396–406.