Rectangular Stratified Sampling

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

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

from UQpy.sampling.stratified_sampling.TrueStratifiedSampling import TrueStratifiedSampling
from UQpy.distributions import Exponential
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import expon
from UQpy.sampling.stratified_sampling.strata import RectangularStrata

Run TrueStratifiedSampling for 25 samples.

• 2 dimensions

• Five strata in each dimension

• Exponential distribution with location parameter = 1 and scale parameter = 1.

Create a distribution object.

marginals = [Exponential(loc=1., scale=1.), Exponential(loc=1., scale=1.)]

Create strata with equal volume

Create a strata object using RectangularStrata class.

strata = RectangularStrata(strata_number=[5, 5])

Generate samples using TrueStratifiedSampling class, one sample is generate inside each stratum.

x_sts = TrueStratifiedSampling(distributions=marginals,
                               strata_object=strata,
                               nsamples_per_stratum=1)

Plot the resulting stratified samples and the boundaries of the strata in the \(U(0,1)\) space.

fig = strata.plot_2d()
plt.title('Stratified Sample - U(0,1)')
plt.scatter(x_sts.samplesU01[:, 0], x_sts.samplesU01[:, 1], color='r')
plt.ylim(0, 1)
plt.xlim(0, 1)
plt.show()


print(x_sts.weights)

[0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04
 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04]

Plot the resulting stratified exponential samples and the boundaries of the strata in the exponential space.

fig, ax = plt.subplots()
plt.title('Stratified Sample - Exponential')
plt.scatter(x_sts.samples[:, 0], x_sts.samples[:, 1])
ax.set_yticks([1.0, expon.ppf(0.2, 1, 1), expon.ppf(0.4, 1, 1), expon.ppf(0.6, 1, 1), expon.ppf(0.8, 1, 1),
               expon.ppf(0.99, 1, 1)])
ax.set_xticks([1.0, expon.ppf(0.2, 1, 1), expon.ppf(0.4, 1, 1), expon.ppf(0.6, 1, 1), expon.ppf(0.8, 1, 1),
               expon.ppf(0.99, 1, 1)])
ax.yaxis.grid(True)
ax.xaxis.grid(True)
plt.ylim(1, expon.ppf(0.99, 1, 1))
plt.xlim(1, expon.ppf(0.99, 1, 1))
plt.show()

print(x_sts.samples)

[[1.06747969 1.1476857 ]
 [1.35088007 1.08848268]
 [1.70971223 1.16154741]
 [2.00944686 1.10808709]
 [3.20143124 1.10579233]
 [1.12696147 1.31286982]
 [1.35529156 1.50162259]
 [1.54746999 1.35553594]
 [2.57394918 1.46039875]
 [3.33062399 1.28660311]
 [1.03400835 1.71505305]
 [1.29919453 1.68834455]
 [1.74173834 1.6955551 ]
 [2.26988974 1.78513653]
 [4.01017631 1.62498802]
 [1.20050619 2.19993679]
 [1.43544144 2.05890816]
 [1.76529376 2.52148202]
 [2.23453661 2.44300227]
 [3.93930259 1.92154706]
 [1.05156352 2.70996901]
 [1.36170414 3.31801085]
 [1.59300373 3.52495366]
 [2.30208459 2.8156912 ]
 [5.15123197 2.65657644]]

Create stratification using seeds and widths

Strata object can be initiated by defining seeds and widths of the strata.

seeds = np.array([[0, 0], [0.4, 0], [0, 0.5], [0.4, 0.5]])
widths = np.array([[0.4, 0.5], [0.6, 0.5], [0.4, 0.5], [0.6, 0.5]])
strata_obj = RectangularStrata(seeds=seeds, widths=widths)

Generate samples using TrueStratifiedSampling class. User can control the number of samples generated inside each stratum. In this illustration, 10 samples are generated such that nsamples_per_stratum governs the number of samples.

sts_obj = TrueStratifiedSampling(distributions=marginals, strata_object=strata_obj, random_state=20)
sts_obj.run(nsamples_per_stratum=[1, 2, 3, 4])

Plot show the strata and samples generated in each stratum.

fig = strata_obj.plot_2d()
plt.title('Stratified Sample - U(0,1)')
plt.scatter(sts_obj.samplesU01[:, 0], sts_obj.samplesU01[:, 1], color='r')
plt.ylim(0, 1)
plt.xlim(0, 1)
plt.show()

Probability weights corresponding to each sample computed using Stratified sampling.

sts_obj.weights

array([0.2       , 0.15      , 0.15      , 0.06666667, 0.06666667,
       0.06666667, 0.075     , 0.075     , 0.075     , 0.075     ])

Create stratification using input file

Strata object can be defined using an input file, which contains the seeds and widths of each stratum.

strata_obj1 = RectangularStrata(input_file='strata.txt')

Generate samples inside each stratum using TrueStratifiedSampling class.

sts_obj1 = TrueStratifiedSampling(distributions=marginals, strata_object=strata_obj1, nsamples_per_stratum=1)

fig = strata_obj1.plot_2d()
plt.title('Stratified Sample - U(0,1)')
plt.scatter(sts_obj1.samplesU01[:, 0], sts_obj1.samplesU01[:, 1], color='r')
plt.ylim(0, 1)
plt.xlim(0, 1)
plt.show()

Proportional sampling

TrueStratifiedSampling class can generate samples proportional to volume of each stratum.

strata_obj.random_state = 24
sts_obj2 = TrueStratifiedSampling(distributions=marginals, strata_object=strata_obj)
sts_obj2.run(nsamples=10)

It can be noticed that new sample in each stratum is proportional to volume.

print('Volume: ', sts_obj2.strata_object.volume)
print('Number of samples in each stratum: ', sts_obj2.nsamples_per_stratum)


fig = strata_obj.plot_2d()
plt.title('Stratified Sample - U(0,1)')
plt.scatter(sts_obj.samplesU01[:, 0], sts_obj.samplesU01[:, 1], color='r')
plt.ylim(0, 1)
plt.xlim(0, 1)
plt.show()

Volume:  [0.2 0.3 0.2 0.3]
Number of samples in each stratum:  [2. 3. 2. 3.]