Rectangular Refined Stratified Sampling - Random Refinement

In this example, the stratified sampling method is employed to generate samples from an exponential distribution using Voronoi stratification.

In this example, Stratified sampling is used to generate samples from Uniform probability distribution and sample are added using adaptive approach Refined Stratified Sampling.

Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to import the TrueStratifiedSampling and RefinedStratifiedSampling class from UQpy.

from UQpy.sampling.stratified_sampling.strata import RectangularStrata
from UQpy.sampling import TrueStratifiedSampling, RefinedStratifiedSampling, RandomRefinement
from UQpy.distributions import Uniform
import matplotlib.pyplot as plt
import matplotlib.patches as patches
import numpy as np

Create a distribution object.

marginals = [Uniform(loc=0., scale=1.), Uniform(loc=0., scale=1.)]

Create a strata object

strata = RectangularStrata(strata_number=[4, 4])

Run stratified sampling.

x = TrueStratifiedSampling(distributions=marginals, strata_object=strata, nsamples_per_stratum=1, random_state=1)

Using UQpy RefinedStratifiedSampling class to expand samples generated by TrueStratifiedSampling class. In this example, two new samples are generated using rectangular stratification .

refinement = RandomRefinement(strata=strata)
y = RefinedStratifiedSampling(stratified_sampling=x, refinement_algorithm=refinement,
                              nsamples=18, samples_per_iteration=2, random_state=2)

In the first figure shown below, samples generated from TrueStratifiedSampling class are plotted. Second figure, shows the new stratification corresponding to two generated samples using RefinedStratifiedSampling class.

fig1 = strata.plot_2d()
plt.plot(x.samples[:16, 0], x.samples[:16, 1], 'ro')
plt.xlim([0, 1])
plt.ylim([0, 1])
plt.show()

fig2 = strata.plot_2d()
plt.xlim([0, 1])
plt.ylim([0, 1])
plt.plot(y.samples[:16, 0], y.samples[:16, 1], 'ro')
plt.plot(y.samples[16:, 0], y.samples[16:, 1], 'yo')
plt.show()

Further, RefinedStratifiedSampling class is used to adaptively increase the sample size. In this example, samples are added in strata with maximum weights associated with it and strata’s are cut randomly along the maximum width.

y.run(nsamples=100)

In the figure shown below, all samples generated from TrueStratifiedSampling and RefinedStratifiedSampling class are plotted.

fig3 = strata.plot_2d()
plt.xlim([0, 1])
plt.ylim([0, 1])
plt.plot(y.samplesU01[:16, 0], y.samplesU01[:16, 1], 'ro')
plt.plot(y.samplesU01[16:, 0], y.samplesU01[16:, 1], 'go')
plt.show()