class scipy.stats.qmc.LatinHypercube(d, *, centered=False, seed=None)[source]

Latin hypercube sampling (LHS).

A Latin hypercube sample [1] generates \(n\) points in \([0,1)^{d}\). Each univariate marginal distribution is stratified, placing exactly one point in \([j/n, (j+1)/n)\) for \(j=0,1,...,n-1\). They are still applicable when \(n << d\). LHS is extremely effective on integrands that are nearly additive [2]. LHS on \(n\) points never has more variance than plain MC on \(n-1\) points [3]. There is a central limit theorem for LHS [4], but not necessarily for optimized LHS.


Dimension of the parameter space.

centeredbool, optional

Center the point within the multi-dimensional grid. Default is False.

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.



Mckay et al., “A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code”, Technometrics, 1979.


M. Stein, “Large sample properties of simulations using Latin hypercube sampling.” Technometrics 29, no. 2: 143-151, 1987.


A. B. Owen, “Monte Carlo variance of scrambled net quadrature.” SIAM Journal on Numerical Analysis 34, no. 5: 1884-1910, 1997


Loh, W.-L. “On Latin hypercube sampling.” The annals of statistics 24, no. 5: 2058-2080, 1996.


Generate samples from a Latin hypercube generator.

>>> from scipy.stats import qmc
>>> sampler = qmc.LatinHypercube(d=2)
>>> sample = sampler.random(n=5)
>>> sample
array([[0.1545328 , 0.53664833],  # random
       [0.84052691, 0.06474907],
       [0.52177809, 0.93343721],
       [0.68033825, 0.36265316],
       [0.26544879, 0.61163943]])

Compute the quality of the sample using the discrepancy criterion.

>>> qmc.discrepancy(sample)
0.019558034794794565  # random

Finally, samples can be scaled to bounds.

>>> l_bounds = [0, 2]
>>> u_bounds = [10, 5]
>>> qmc.scale(sample, l_bounds, u_bounds)
array([[1.54532796, 3.609945  ],  # random
       [8.40526909, 2.1942472 ],
       [5.2177809 , 4.80031164],
       [6.80338249, 3.08795949],
       [2.65448791, 3.83491828]])



Fast-forward the sequence by n positions.


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


Reset the engine to base state.