Discrete Density Milestone

We think of a density function as being a formula rho(x,y). However, in real life applications, instead of a formula for rho(), we may only have density measurements at specify points, or we may have divided a polygon into subregions, and know a constant density value for each subregion.

Think about how the population density is defined over a state like Florida. It certainly varies from point to point, but we really don't have a formula for it, just values over large census blocks. How could we find the population "center of mass" of Florida? How could we do a Voronoi diagram of Florida, using a point in each of the current congressional districts as the generators? How could we do a CVT iteration for Florida which factored in the population density?

Topics:

• Keeping things simple, we will start by setting up a rectangujlar "state", and we will give it a set of 25 square counties, arranged in a 5x5 array. Each county will have a given population. We will convert the population arrays into probability arrays, then into a cumulative density function (CDF) array. Then we will see how this CDF array can be used so that a random number R between 0 and 1 can select a county uniformly with respect to population.
• Once we have selected a county, we need to select a person in the county. We will take this to mean that we need to select a point uniformly at random within the square that is the county. To do this requires some tedious and error prone calculation to turn a linear index into a row and column index, into a lower left (X,Y) corner and upper right (X,Y) corner and then into a random location.
• Now we can see how to sample 1,000 people at random from our state. We can make a plot of the locations of these people, and see if it corresponds to the original discrete density table we were given.