Benchmark the Borgonovo test function

import openturns as ot
import otbenchmark as otb
import openturns.viewer as otv

problem = otb.BorgonovoSensitivity()
print(problem)

name = Borgonovo
distribution = ComposedDistribution(Uniform(a = 0, b = 1), Uniform(a = 0, b = 1), Uniform(a = 0, b = 1), IndependentCopula(dimension = 3))
function = [x1,x2,x3]->[x1 * x2 + x3]
firstOrderIndices = [0.157895,0.157895,0.631579]
totalOrderIndices = [0.210526,0.210526,0.631579]

distribution = problem.getInputDistribution()
model = problem.getFunction()

Exact first and total order

exact_first_order = problem.getFirstOrderIndices()
exact_first_order

class=Point name=Unnamed dimension=3 values=[0.157895,0.157895,0.631579]

exact_total_order = problem.getTotalOrderIndices()
exact_total_order

class=Point name=Unnamed dimension=3 values=[0.210526,0.210526,0.631579]

Plot the function

Create X/Y data

ot.RandomGenerator.SetSeed(0)
size = 200
inputDesign = ot.MonteCarloExperiment(distribution, size).generate()
outputDesign = model(inputDesign)

dimension = distribution.getDimension()
full_sample = ot.Sample(size, 1 + dimension)
full_sample[:, range(dimension)] = inputDesign
full_sample[:, dimension] = outputDesign
full_description = list(inputDesign.getDescription())
full_description.append(outputDesign.getDescription()[0])
full_sample.setDescription(full_description)

marginal_distribution = ot.ComposedDistribution(
    [
        ot.KernelSmoothing().build(full_sample.getMarginal(i))
        for i in range(1 + dimension)
    ]
)
clouds = ot.VisualTest.DrawPairsMarginals(full_sample, marginal_distribution)
_ = otv.View(clouds, figure_kw={"figsize": (6.0, 6.0)})

output_distribution = ot.KernelSmoothing().build(outputDesign)
_ = otv.View(output_distribution.drawPDF())

Perform sensitivity analysis

Create X/Y data

ot.RandomGenerator.SetSeed(0)
size = 10000
inputDesign = ot.SobolIndicesExperiment(distribution, size).generate()
outputDesign = model(inputDesign)

Compute first order indices using the Saltelli estimator

sensitivityAnalysis = ot.SaltelliSensitivityAlgorithm(inputDesign, outputDesign, size)
computed_first_order = sensitivityAnalysis.getFirstOrderIndices()
computed_total_order = sensitivityAnalysis.getTotalOrderIndices()

Compare with exact results

print("Sample size : ", size)
# First order
# Compute absolute error (the LRE cannot be computed,
# because S can be zero)
print("Computed first order = ", computed_first_order)
print("Exact first order = ", exact_first_order)
# Total order
print("Computed total order = ", computed_total_order)
print("Exact total order = ", exact_total_order)

Sample size :  10000
Computed first order =  [0.166136,0.157381,0.623798]
Exact first order =  [0.157895,0.157895,0.631579]
Computed total order =  [0.207054,0.2159,0.623765]
Exact total order =  [0.210526,0.210526,0.631579]

_ = otv.View(sensitivityAnalysis.draw())

otv.View.ShowAll()