Various design of experiments

The goal of this example is to present several design of experiments available in the library.

Distribution

import openturns as ot
import openturns.viewer as otv

Monte-Carlo sampling in 2D

dim = 2
X = [ot.Uniform()] * dim
distribution = ot.JointDistribution(X)
bounds = distribution.getRange()

sampleSize = 10
sample = distribution.getSample(sampleSize)

fig = otv.PlotDesign(sample, bounds)

We see that there a empty zones in the input space.

Monte-Carlo sampling in 3D

dim = 3
X = [ot.Uniform()] * dim
distribution = ot.JointDistribution(X)
bounds = distribution.getRange()

sampleSize = 10
sample = distribution.getSample(sampleSize)

fig = otv.PlotDesign(sample, bounds)
fig.set_size_inches(10, 10)

Latin Hypercube Sampling

distribution = ot.JointDistribution([ot.Uniform()] * 3)
samplesize = 5
experiment = ot.LHSExperiment(distribution, samplesize, False, False)
sample = experiment.generate()

In order to see the LHS property, we need to set the bounds.

bounds = distribution.getRange()

fig = otv.PlotDesign(sample, bounds)
fig.set_size_inches(10, 10)

We see that each column or row exactly contains one single point. This shows that a LHS design of experiments has good 1D projection properties, and, hence, is a good candidate for a space filling design.

Optimized LHS

distribution = ot.JointDistribution([ot.Uniform()] * 3)
samplesize = 10

bounds = distribution.getRange()

lhs = ot.LHSExperiment(distribution, samplesize)
lhs.setAlwaysShuffle(True)  # randomized
space_filling = ot.SpaceFillingC2()
temperatureProfile = ot.GeometricProfile(10.0, 0.95, 1000)
algo = ot.SimulatedAnnealingLHS(lhs, space_filling, temperatureProfile)
# optimal design
sample = algo.generate()

fig = otv.PlotDesign(sample, bounds)
fig.set_size_inches(10, 10)

We see that this LHS is optimized in the sense that it fills the space more evenly than a non-optimized does in general.

Sobol’ low discrepancy sequence

dim = 2
distribution = ot.JointDistribution([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()

samplesize

32

subdivisions = [2**2, 2**1]
fig = otv.PlotDesign(sample, bounds, subdivisions)
fig.set_size_inches(6, 6)

We have elementary intervals in 2 dimensions, each having a volume equal to 1/8. Since there are 32 points, the Sobol’ sequence is so that each elementary interval contains exactly 32/8 = 4 points. Notice that each elementary interval is closed on the left (or bottom) and open on the right (or top).

Halton low discrepancy sequence

dim = 2
distribution = ot.JointDistribution([ot.Uniform()] * dim)
bounds = distribution.getRange()

sequence = ot.HaltonSequence(dim)

Halton sequence uses prime numbers 2 and 3 in two dimensions.

samplesize = 2**2 * 3**2
experiment = ot.LowDiscrepancyExperiment(sequence, distribution, samplesize, False)
sample = experiment.generate()

samplesize

36

subdivisions = [2**2, 3]
fig = otv.PlotDesign(sample, bounds, subdivisions)
fig.set_size_inches(6, 6)

We have elementary intervals in 2 dimensions, each having a volume equal to 1/12. Since there are 36 points, the Halton sequence is so that each elementary interval contains exactly 36/12 = 3 points. Notice that each elementary interval is closed on the left (or bottom) and open on the right (or top).