Latin Hypercube sampling

The LHS design is a statistical method for generating a quasi-random sampling distribution. It is among the most popular sampling techniques in computer experiments thanks to its simplicity and projection properties with high-dimensional problems. LHS is built as follows: we cut each dimension space, which represents a variable, into n sections where n is the number of sampling points, and we put only one point in each section.

The LHS method uses the pyDOE package (Design of Experiments for Python) [1]. Five criteria for the construction of LHS are implemented in SMT:

• Center the points within the sampling intervals.

• Maximize the minimum distance between points and place the point in a randomized location within its interval.

• Maximize the minimum distance between points and center the point within its interval.

• Minimize the maximum correlation coefficient.

• Optimize the design using the Enhanced Stochastic Evolutionary algorithm (ESE).

The four first criteria are the same than in pyDOE (for more details, see [1]). The last criterion, ESE, is implemented by the authors of SMT (more details about such method could be found in [2]).

[1] (1,2)

https://pydoe3.readthedocs.io/en/stable

[2]

Jin, R. and Chen, W. and Sudjianto, A. (2005), “An efficient algorithm for constructing optimal design of computer experiments.” Journal of Statistical Planning and Inference, 134:268-287.

Usage

import matplotlib.pyplot as plt
import numpy as np

from smt.sampling_methods import LHS

xlimits = np.array([[0.0, 4.0], [0.0, 3.0]])
sampling = LHS(xlimits=xlimits)

num = 50
x = sampling(num)

print(x.shape)

plt.plot(x[:, 0], x[:, 1], "o")
plt.xlabel("x")
plt.ylabel("y")
plt.show()

(50, 2)

Options

List of options

Option	Default	Acceptable values	Acceptable types	Description
xlimits	None	None	[‘ndarray’]	The interval of the domain in each dimension with shape nx x 2 (required)
criterion	c	[‘center’, ‘maximin’, ‘centermaximin’, ‘correlation’, ‘c’, ‘m’, ‘cm’, ‘corr’, ‘ese’]	[‘str’]	criterion used to construct the LHS design c, m, cm and corr are abbreviation of center, maximin, centermaximin and correlation, respectively
random_state	None	None	[‘NoneType’, ‘int’, ‘RandomState’]	Numpy RandomState object or seed number which controls random draws