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.JointDistribution(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())

[0.00175769]
[3.70801e-06]

Quick summary of sensitivity analysis

sensitivityAnalysis = ot.FunctionalChaosSobolIndices(result)
sensitivityAnalysis

FunctionalChaosSobolIndices

• input dimension: 5
• output dimension: 1
• basis size: 181
• mean: [75.5283]
• std-dev: [30.9634]

Input	Variable	Sobol' index	Total index
0	rw	0.698383	0.726572
1	Hu	0.084703	0.095073
2	Hl	0.084747	0.095132
3	L	0.081092	0.092178
4	Kw	0.019748	0.022672

Index	Multi-index	Part of variance
1	[1,0,0,0,0]	0.623730
3	[0,0,1,0,0]	0.084747
2	[0,1,0,0,0]	0.084703
4	[0,0,0,1,0]	0.080217
5	[0,0,0,0,1]	0.019748
56	[6,0,0,0,0]	0.019191
31	[5,0,0,0,0]	0.018039
61	[7,0,0,0,0]	0.011068
6	[2,0,0,0,0]	0.010331

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):
        sij = sensitivityAnalysis.getSobolIndex([i, j])
        print(f"{input_names[i]} & {input_names[j]}: {sij:.4f}")

plt.show()

Hu & rw: 0.0088
Hl & rw: 0.0088
Hl & Hu: 0.0000
L & rw: 0.0083
L & Hu: 0.0012
L & Hl: 0.0011
Kw & rw: 0.0020
Kw & Hu: 0.0003
Kw & Hl: 0.0003
Kw & L: 0.0003