Estimate the empty space function at a given distance:

Estimate the empty space function within a range of distances:

Visualize the result with ListPlot:

Empty space function of a cluster point process:

The empty space function for spatially random data:

Visualize the results:

The empty space function for hardcore data:

Estimate the values of the empty space function with given data:

Visualize the results:

On Monday, May 20, 2019, the Severe Prediction Center highlighted a small corridor from the northeastern part of the Texas Panhandle to central Oklahoma in a 45 percent probability of EF2–EF5 tornadoes to occur within 25 miles of a given location, which defines a value of EmptySpaceF:

Assuming uniform intensity over the region, use the Poisson point process as a tornado pattern model:

Define the Poisson point process:

Compute probabilities of a tornado within a given radius of a location:

Simulate a possible tornado spread over the state of Oklahoma:

The empty space function behaves like a CDF:

Visualize the results:

The empty space function for a PoissonPointProcess is equivalent to SurvivalFunction at 0 of PointCountDistribution on a disk of radius :

Define the point count distribution for the process on a ball of radius :

Compare its survival function at 0 with the empty space function for fixed dimensions:

In 3D:

In random dimension:

The empty space and the nearest neighbor functions of a PoissonPointProcess are identical:

In 1D they are both equivalent to the CDF of an ExponentialDistribution:

EmptySpaceF is often compared with NearestNeighborG, which estimates the probability of finding another point within distance r from a point in the point collection:

Visualize the result with ListPlot:

Compare the estimates between EmptySpaceF and NearestNeighborG for point data generated by HardcorePointProcess: