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 usingnp.random.default_rng(seed)
. If seed is already aGenerator
instance, then the provided instance is used.
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.
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]])