Sobol’ sensitivity indices from chaos

In this example we are going to compute global sensitivity indices from a functional chaos decomposition.

We study the Borehole function that models water flow through a borehole:

With parameters:

• : radius of borehole (m)

• : radius of influence (m)

• : transmissivity of upper aquifer ()

• : potentiometric head of upper aquifer (m)

• : transmissivity of lower aquifer ()

• : potentiometric head of lower aquifer (m)

• : length of borehole (m)

• : hydraulic conductivity of borehole ()

import openturns as ot
from operator import itemgetter
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)

borehole model

dimension = 8
input_names = ["rw", "r", "Tu", "Hu", "Tl", "Hl", "L", "Kw"]
model = ot.SymbolicFunction(
    input_names, ["(2*pi_*Tu*(Hu-Hl))/(ln(r/rw)*(1+(2*L*Tu)/(ln(r/rw)*rw^2*Kw)+Tu/Tl))"]
)
coll = [
    ot.Normal(0.1, 0.0161812),
    ot.LogNormal(7.71, 1.0056),
    ot.Uniform(63070.0, 115600.0),
    ot.Uniform(990.0, 1110.0),
    ot.Uniform(63.1, 116.0),
    ot.Uniform(700.0, 820.0),
    ot.Uniform(1120.0, 1680.0),
    ot.Uniform(9855.0, 12045.0),
]
distribution = ot.ComposedDistribution(coll)
distribution.setDescription(input_names)

Freeze r, Tu, Tl from model to go faster

selection = [1, 2, 4]
complement = ot.Indices(selection).complement(dimension)
distribution = distribution.getMarginal(complement)
model = ot.ParametricFunction(
    model, selection, distribution.getMarginal(selection).getMean()
)
input_names_copy = list(input_names)
input_names = itemgetter(*complement)(input_names)
dimension = len(complement)

design of experiment

size = 1000
X = distribution.getSample(size)
Y = model(X)

create a functional chaos model

algo = ot.FunctionalChaosAlgorithm(X, Y)
algo.run()
result = algo.getResult()
print(result.getResiduals())
print(result.getRelativeErrors())

[0.00195286]
[4.64697e-06]

Quick summary of sensitivity analysis

sensitivityAnalysis = ot.FunctionalChaosSobolIndices(result)
print(sensitivityAnalysis)

FunctionalChaosSobolIndices
- input dimension=5
- output dimension=1
- basis size=181
- mean=[76.0794]
- std-dev=[30.2678]

| Index | Multi-index   | Variance part |
|-------|---------------|---------------|
|     1 | [1,0,0,0,0]   | 0.662606      |
|     3 | [0,0,1,0,0]   | 0.0902125     |
|     2 | [0,1,0,0,0]   | 0.0901124     |
|     4 | [0,0,0,1,0]   | 0.0861668     |
|     5 | [0,0,0,0,1]   | 0.0209417     |

| Input | Name          | Sobol' index  | Total index   |
|-------|---------------|---------------|---------------|
|     0 | rw            | 0.677582      | 0.70826       |
|     1 | Hu            | 0.0901124     | 0.101381      |
|     2 | Hl            | 0.0902126     | 0.101371      |
|     3 | L             | 0.0871206     | 0.0992782     |
|     4 | Kw            | 0.0209417     | 0.0240504     |

draw Sobol’ indices

first_order = [sensitivityAnalysis.getSobolIndex(i) for i in range(dimension)]
total_order = [sensitivityAnalysis.getSobolTotalIndex(i) for i in range(dimension)]
graph = ot.SobolIndicesAlgorithm.DrawSobolIndices(input_names, first_order, total_order)
view = viewer.View(graph)

We saw that total order indices are close to first order, so the higher order indices must be all quite close to 0

for i in range(dimension):
    for j in range(i):
        print(
            input_names[i] + " & " + input_names[j],
            ":",
            sensitivityAnalysis.getSobolIndex([i, j]),
        )

plt.show()

Hu & rw : 0.009603147153069586
Hl & rw : 0.009486332516243895
Hl & Hu : 6.772494613216887e-08
L & rw : 0.009152262191425037
L & Hu : 0.0012275743968190782
L & Hl : 0.001230665255687928
Kw & rw : 0.0021338960376998777
Kw & Hu : 0.0002952362243352463
Kw & Hl : 0.0002981593839990994
Kw & L : 0.0002935613713562576