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scipy.stats.qmc.Halton#

class scipy.stats.qmc.Halton(d, *, scramble=True, optimization=None, 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.

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.

Added in version 1.10.0.

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

If seed is an int or None, a new numpy.random.Generator is created using np.random.default_rng(seed). If seed is already a Generator instance, then the provided 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(n)

Fast-forward the sequence by n positions.

integers(l_bounds, *[, u_bounds, n, ...])

Draw n integers from l_bounds (inclusive) to u_bounds (exclusive), or if endpoint=True, l_bounds (inclusive) to u_bounds (inclusive).

random([n, workers])

Draw n in the half-open interval [0, 1).

reset()

Reset the engine to base state.
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