Gamma deviates

gamma-dev implements a pseudorandom number generator for the Gamma distribution [1].

The following distributions are either special cases or derived easily from the gamma distribution:

• ✓ Beta: Package beta-dev.
• Exponential: is a special case of gamma. However, please use exp-dev as it is more efficient.
• ✖ Chi-square
• ✖ Erlang
• ✖ F-distribution

Usage

Gamma

For the gamma distribution, you need to instantiate a number generator with the set of parameters you want to use. We use the following parameters:

• a (alpha in [2]): shape parameter
• b (beta in [2]): scale parameter

Instantiation

make-gamma-gen a b => Instance of gamma-gen

(setf generator (make-gamma-gen a b))

Create a pseudorandom number generator for the Gamma distribution with shape parameter a and scale parameter b.

Drawing

draw gamma-gen => Pseudorandom number

(draw generator)

Draw a pseudorandom number using the generator instance.

Reading parameters back

You can read back the parameters of the generator with:

(a generator)
(b generator)

Beta

In very much the same way:

(setf generator (beta-dev:make-beta-gen alpha beta))
(beta-dev:draw generator) ;=> pseudorandom number
(beta-dev:alpha generator) ;=> alpha
(beta-dev:beta generator) ;=> beta

Implementation

We use the algorithm by Tanizaki [2]. It involves a rejection of uniform variates. We use the in-built random.

Histograms

Gamma

We generate histograms for the gamma distribution with some shape and scale parameters (see histograms).

This looks good enough when compared with the analytical PDFs plotted on Wikipedia [3].

Beta

Caveats

• There are some specializations etc. to perform for performance if that is what you need.
• The algorithm by Tanizaki [2] is, according to its own author, not the best algorithm performance-wise for all choices of parameters. An efficient gamma pseudorandom generator would need to incorporate several of these algorithms and switch depending on the parameters.

Test

Launch tests with:

(asdf:test-system "gamma-dev")

Includes a quick performance test.

Dependencies

• gamma-dev: None.
• gamma-dev/test:
  • rove

References

1. https://en.wikipedia.org/wiki/Gamma_distribution
2. H. Tanizaki, “A simple gamma random number generator for arbitrary shape parameters,” Economics Bulletin 3(7), 1–10 (2008).
3. https://en.wikipedia.org/wiki/File:Gamma_distribution_pdf.svg