mrg32k3a

This package provides a Python implementation of the mrg32k3a pseudo-random number generator of L'Ecuyer (1999) and L'Ecuyer et al. (2002). It extends the implementation used in PyMOSO to handle streams, substreams, and subsubstreams. The generator's period of ~2¹⁹¹ is split into ~2⁵⁰ streams of length 2¹⁴¹, each containing 2⁴⁷ substreams of length 2⁹⁴, each containing 2⁴⁷ subsubstreams of length 2⁴⁷.

Details

The mrg32k3a module includes the MRG32k3a class and several useful functions for controlling the generators.

• The MRG32k3a class is a subclass of Python's random.Random class and therefore inherits easy-to-use methods for generating random variates. E.g., if rng is an instance of the MRG32k3a class, the command rng.normalvariate(mu=2, sigma=5) generates a normal random variate with mean 2 and standard deviation 5. Normal random variates are generated via inversion using the Beasley-Springer-Moro algorithm.
• The MRG32k3a class expands the suite of functions for random-variate generation available in random.Random to include lognormalvariate, mvnormalvariate, poissonvariate, gumbelvariate, binomialvariate. Additionally, the methods integer_random_vector_from_simplex and continuous_random_vector_from_simplex generate discrete and continuous vectors from a symmetric non-negative simplex.
• The advance_stream, advance_substream, and advance_subsubstream functions advance the generator to the start of the next stream, substream, or subsubstream, respectively. They make use of techniques for efficiently "jumping ahead," as outlined by L'Ecuyer (1990).
• The reset_stream, reset_substream, and reset_subsubstream functions reset the generator to the start of the current stream, substream, or subsubstream, respectively.

Installation

The mrg32k3a package is available to download through the Python Packaging Index (PyPI) and can be installed from the terminal with the following command:

python -m pip install mrg32k3a

Basic Example

After installing mrg32k3a, the package's main class (MRG32k3a) can be imported from the Python console (or in code):

from mrg32k3a.mrg32k3a import MRG32k3a

One can instantiate a random number generator set at a given stream, substream, and subsubstream triplet or seed. For example, the command

rng = MRG32k3a(s_ss_sss_index=[1, 2, 3])

creates a object of the MRG32k3a class called rng that it initialized at the start of subsubstream 3 of substream 2 of stream 1. If the argument s_ss_sss_index is not provided, the random number generator is initialized at stream-substream-subsubstream 0-0-0. (We adopt the Python convention of indexing from 0.) Alternatively, the command

rng = MRG32k3a(ref_seed=(12345, 12345, 12345, 12345, 12345, 12345))

initializes the random number generator at the state described by the length-6 tuple (12345, 12345, 12345, 12345, 12345, 12345). Streams, substreams, and subsubstreams are indexed using ref_seed as a point of reference.

After instantiating a random number generator, its methods can be invoked to generate (scalar or vector) random variates from a particular probability distribution. For example,

x = rng.normalvariate(mu=2, sigma=5)

returns a normally distributed random variate x with mean 2 and standard deviation 5.

Similarly,

x = rng.poissonvariate(lmdba=50)

returns a Poisson distributed random variate x with rate parameter (mean) 50.

Finally,

v = rng.integer_random_vector_from_simplex(n_elements=3, summation=10, with_zero=False))

returns a random length-3 vector v of positive integers summing to 10. The vector v is uniformly distributed over the set of such vectors.

Documentation

Full documentation for the mrg32k3a source code can be found here.

References

• Cooper, Kyle and Susan R. Hunter (2020). PyMOSO: Software for multi-objective simulation optimization with R-PERLE and R-MinRLE. INFORMS Journal on Computing 32(4): 1101-1108.
• L'Ecuyer, Pierre (1990). Random numbers for simulation. Communications of the ACM 33(10):85-97.
• L'Ecuyer, Pierre (1999). Good parameters and implementations for combined multiple recursive random number generators. Operations Research 47(1):159-164.
• L'Ecuyer, Pierre, Richard Simard, E Jack Chen, and W. David Kelton (2002). An object-oriented random number package with many long streams and substreams. Operations Research 50(6):1073-1075.