`scipy.stats._qmc:Halton`

source: /scipy/stats/_qmc.py :1117

Signature

   def     Halton (    d  :  int | numpy.integer    ,  * ,     scramble  :  bool    =  True   ,    optimization  :  Literal['random-cd', 'lloyd'] | None    =  None   ,    rng  :  int | numpy.integer | numpy.random._generator.Generator | numpy.random.mtrand.RandomState | None    =  None   ,    seed    =  None     )   →  None

Members

Summary

Halton sequence.

Extended Summary

Pseudo-random number generator that generalize the Van der Corput sequence for multiple dimensions. The Halton sequence uses the base-two Van der Corput sequence for the first dimension, base-three for its second and base- $p$ for its $n$ -dimension, with $p$ the $n$ 'th prime.

Parameters

d : int

Dimension of the parameter space.

scramble : bool, optional

If True, use random scrambling from ^[2]. Otherwise no scrambling is done. Default is True.

optimization : {None, "random-cd", "lloyd"}, optional

Whether to use an optimization scheme to improve the quality after sampling. Note that this is a post-processing step that does not guarantee that all properties of the sample will be conserved. Default is None.

random-cd: random permutations of coordinates to lower the centered discrepancy. The best sample based on the centered discrepancy is constantly updated. Centered discrepancy-based sampling shows better space-filling robustness toward 2D and 3D subprojections compared to using other discrepancy measures.
lloyd: Perturb samples using a modified Lloyd-Max algorithm. The process converges to equally spaced samples.

rng : `numpy.random.Generator`, optional

Pseudorandom number generator state. When rng is None, a new numpy.random.Generator is created using entropy from the operating system. Types other than numpy.random.Generator are passed to numpy.random.default_rng to instantiate a Generator.

Notes

The Halton sequence has severe striping artifacts for even modestly large dimensions. These can be ameliorated by scrambling. Scrambling also supports replication-based error estimates and extends applicability to unbounded integrands.

Examples

Generate samples from a low discrepancy sequence of Halton.

from scipy.stats import qmc
sampler = qmc.Halton(d=2, scramble=False)
sample = sampler.random(n=5)

✓

sample

✗

Compute the quality of the sample using the discrepancy criterion.

qmc.discrepancy(sample)

✗

If some wants to continue an existing design, extra points can be obtained by calling again `random`. Alternatively, you can skip some points like:

_ = sampler.fast_forward(5)
sample_continued = sampler.random(n=5)

✓

sample_continued

✗

Finally, samples can be scaled to bounds.

l_bounds = [0, 2]
u_bounds = [10, 5]

✓

qmc.scale(sample_continued, l_bounds, u_bounds)