Generating Random Points from Arbitrary Distributions for 2D and Up

I have already covered how to generate random numbers from arbitrary distributions in the one-dimensional case. Here we look at a generalisation of that method that works for higher dimensions.

The basic trick, while easy to understand, is hard to put in words (without reverting to mathematical equations). For two dimensions, we divide the plane into slices. Each slice is a 1D distribution. We also calculate a distribution from summing the frequencies in each slice. The latter distribution gives us one coordinate, and the appropriate slice to use. The distribution of that slice then gives the second coordinate. All distributions are put into inverse accumulative response curves as was done to generate one-dimensional random numbers. (You should review that before implementing the 2D case).

In more dimensions, we also slice the space up into 1D distributions. Sums of these give us more distributions, which we can sum again, and again, until we reach a single distribution. This is used for the first coordinate, and to determine which distribution to use for the next coordinate. This goes on, until a 1D slice gives us the final coordinate. Again, all distributions are converted to inverse accumulative response curves.

If the above is unclear, I hope the detailed description below clears things up.

Generating Random Numbers with Arbitrary Distributions

For many applications, detailed statistical models are overkill. Instead, we can get away with a rough description of the distribution – not in mathematical formula form, but just as a graph with a few sample points.

For example, when trying to model the traffic around a school, you might know that the graph looks something like this:

The input is the number of minutes before the first bell rings, and the output the number of children dropped off at that time. You know that most kids are brought before the bell rings, and that the closer to the bell, the more kids are being brought every minute. Only a few kids are late.

This tutorial describes how to generate random numbers that can generate a distribution described by an arbitrary (piece-wise linear) curve, as the one above.