pythonrandomscipy

Recommendations for Low Discrepancy (e.g. Sobol) quasi-random sequences in Python/SciPy?

I would like to use a quasi-random sequence, specifically Sobol, within a SciPy based simulation. Any recommendations on existing, efficient packages?

Solution

  • I would use OpenTURNS, which provides several low discrepancy sequences:

    Moreover, the sequence can be generated so that the marginals have arbitrary distribution. This is done with an probabilistic transformation, based on the inverse distribution function.

    In the following example, I generate a Sobol' sequence in 2 dimensions, based on the LowDiscrepancyExperiment class. The marginals are uniform in the [-1, 1] interval (which is the default Uniform distribution in OT). I suggest to use a sample size equal to a power of 2, because Sobol' sequence is based on base-2 integer decomposition. The generate method returns a ot.Sample.

    import openturns as ot
dim = 2
distribution = ot.ComposedDistribution([ot.Uniform()]*dim)
bounds = distribution.getRange()
sequence = ot.SobolSequence(dim)
samplesize = 2**5 # Sobol' sequences are in base 2
experiment = ot.LowDiscrepancyExperiment(sequence, distribution,
                                         samplesize, False)
sample = experiment.generate()
print(samplesize[:5])

    The previous sample has size 32. The first 5 elements are:

        y0      y1
0    0       0
1    0.5    -0.5
2   -0.5     0.5
3   -0.25   -0.25
4    0.75    0.75

    The Sobol' sequence in OT can generate an arbitrary sample size, in dimensions up to 1111.

    With a little more work, we may plot the design.

    import openturns.viewer as otv
fig = otv.PlotDesign(sample, bounds, 2**2, 2**1);
fig.set_size_inches(6, 6)

    which produces:

    A Sobol' sequence in 2 dimensions

    See how there is exactly 4 points in each elementary interval.

    If required, the sample can be easily converted into a Numpy array, which may better fits into your Scipy requirement:

    import numpy as np
array = np.array(sample)

    Other examples are provided at: http://openturns.github.io/openturns/latest/auto_reliability_sensitivity/index.html#design-of-experiments