Sobol

Concise implementation of the Sobol sequence for generating low-discrepancy quasi-random numbers in up to 1111 dimensions.

Note: scipy>=1.7 features a qmc module with a better Sobol implementation.

Install

pip install sobolsequence

Usage

import sobol

sobol.sample(dimension=3, n_points=8)
array([[0.5   , 0.5   , 0.5   ],
       [0.75  , 0.25  , 0.75  ],
       [0.25  , 0.75  , 0.25  ],
       [0.375 , 0.375 , 0.625 ],
       [0.875 , 0.875 , 0.125 ],
       [0.625 , 0.125 , 0.375 ],
       [0.125 , 0.625 , 0.875 ],
       [0.1875, 0.3125, 0.3125]])

Skip the first skip points:

X = sobol.sample(dimension=3, n_points=64, skip=1024)
plt.scatter(X[:, 0], X[:, 1]

Sample point by point using the underlying generator:

sob = sobol.generator(dimension=5)
for i in range(8):
    print(next(sob))

References

This implementation is based on the Python version by Corrado Chisari available here.

• Antonov, Saleev, USSR Computational Mathematics and Mathematical Physics, Volume 19, 1980, pages 252 - 256.
• Paul Bratley, Bennett Fox, Algorithm 659: Implementing Sobol’s Quasirandom Sequence Generator, ACM Transactions on Mathematical Software, Volume 14, Number 1, pages 88-100, 1988.
• Bennett Fox, Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators, ACM Transactions on Mathematical Software, Volume 12, Number 4, pages 362-376, 1986.
• Ilya Sobol, USSR Computational Mathematics and Mathematical Physics, Volume 16, pages 236-242, 1977.
• Ilya Sobol, Levitan, The Production of Points Uniformly Distributed in a Multidimensional Cube (in Russian), Preprint IPM Akad. Nauk SSSR, Number 40, Moscow 1976.

The direction numbers from Joe and Kuo are available here.

• Stephen Joe and Frances Kuo, Remark on Algorithm 659: Implementing Sobol’s quasirandom sequence generator, ACM Trans. Math. Softw. 29, 49-57 (2003), http://doi.acm.org/10.1145/641876.641879
• Stephen Joe and Frances Kuo, Constructing Sobol sequences with better two-dimensional projections, SIAM J. Sci. Comput. 30, 2635-2654 (2008), https://doi.org/10.1137/070709359