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 ()

from __future__ import print_function
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())

Out:

[0.00224141]
[8.8431e-09]

Quick summary of sensitivity analysis

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

Out:

 input dimension: 5
 output dimension: 1
 basis size: 44
 mean: [74.1358]
 std-dev: [28.7844]
------------------------------------------------------------
Index   | Multi-indice                  | Part of variance
------------------------------------------------------------
      1 | [1,0,0,0,0]                   | 0.654212
      3 | [0,0,1,0,0]                   | 0.0947941
      2 | [0,1,0,0,0]                   | 0.0946975
      4 | [0,0,0,1,0]                   | 0.0904842
      5 | [0,0,0,0,1]                   | 0.0221225
------------------------------------------------------------


------------------------------------------------------------
Component | Sobol index            | Sobol total index
------------------------------------------------------------
        0 | 0.662726               | 0.693362
        1 | 0.0946975              | 0.10585
        2 | 0.0947941              | 0.106069
        3 | 0.0914871              | 0.10387
        4 | 0.0221225              | 0.0253679
------------------------------------------------------------

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()

Out:

Hu & rw : 0.00939596043391626
Hl & rw : 0.0094957984784045
Hl & Hu : 0.0
L & rw : 0.00918479200468764
L & Hu : 0.0012912602896845825
L & Hl : 0.0013069732237138588
Kw & rw : 0.002220185764977052
Kw & Hu : 0.00031043066749530707
Kw & Hl : 0.0003119420529851785
Kw & L : 0.0003096614949037279