# scipy.stats.qmc.Halton¶

class scipy.stats.qmc.Halton(d, *, scramble=True, seed=None)[source]

Halton sequence.

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-$$n$$ for its n-dimension.

Parameters
dint

Dimension of the parameter space.

scramblebool, optional

If True, use Owen scrambling. Otherwise no scrambling is done. Default is True.

seed{None, int, numpy.random.Generator}, optional

If seed is None the numpy.random.Generator singleton is used. If seed is an int, a new Generator instance is used, seeded with seed. If seed is already a Generator instance then that instance is used.

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 applicabiltiy to unbounded integrands.

References

1

Halton, “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals”, Numerische Mathematik, 1960.

2

A. B. Owen. “A randomized Halton algorithm in R”, arXiv:1706.02808, 2017.

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
array([[0.        , 0.        ],
[0.5       , 0.33333333],
[0.25      , 0.66666667],
[0.75      , 0.11111111],
[0.125     , 0.44444444]])


Compute the quality of the sample using the discrepancy criterion.

>>> qmc.discrepancy(sample)
0.088893711419753


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
array([[0.3125    , 0.37037037],
[0.8125    , 0.7037037 ],
[0.1875    , 0.14814815],
[0.6875    , 0.48148148],
[0.4375    , 0.81481481]])


Finally, samples can be scaled to bounds.

>>> l_bounds = [0, 2]
>>> u_bounds = [10, 5]
>>> qmc.scale(sample_continued, l_bounds, u_bounds)
array([[3.125     , 3.11111111],
[8.125     , 4.11111111],
[1.875     , 2.44444444],
[6.875     , 3.44444444],
[4.375     , 4.44444444]])


Methods

 Fast-forward the sequence by n positions. random([n, workers]) Draw n in the half-open interval [0, 1). Reset the engine to base state.