"""
Estimate Sobol' indices for a function with multivariate output
===============================================================
"""
# %%
#
# In this example, we estimate the Sobol' indices of a function by sampling methods.
# This function has several outputs, which leads to the need of aggregated Sobol' indices.
#
# %%
# Introduction
# ------------
#
# In this example we quantify the sensitivity of a function's outputs to its inputs with Sobol' indices.
#
# The function we consider has 5 outputs. In this case, it may be convenient to consider each output separately.
# It may also be interesting to aggregate the sensitivity indices to get a global understanding of the sensitivity of the inputs to the average output.
# %%
# Define the model
# ----------------
# %%
import openturns as ot
import openturns.viewer
import openturns.viewer as viewer
ot.Log.Show(ot.Log.NONE)
# %%
# We define a linear model with 5 independent Gaussian inputs and 2 outputs.
inputDistribution = ot.Normal(5)
function = ot.SymbolicFunction(
["x0", "x1", "x2", "x3", "x4"],
["x0 + 4.0 * x1 ^ 2 + 3.0 * x2", "-7.0 * x2 - 4.0 * x3 + x4"],
)
# %%
# Estimate the Sobol' indices
# ---------------------------
# %%
# We first create a design of experiments with `SobolIndicesExperiment`.
# %%
size = 1000
sie = ot.SobolIndicesExperiment(inputDistribution, size)
inputDesign = sie.generate()
input_names = inputDistribution.getDescription()
inputDesign.setDescription(input_names)
print("Sample size: ", inputDesign.getSize())
# %%
# We see that 7000 function evaluations are required to estimate the first order and total Sobol' indices.
# Then we evaluate the outputs corresponding to this design of experiments.
# %%
outputDesign = function(inputDesign)
# %%
# Then we estimate the Sobol' indices with the `SaltelliSensitivityAlgorithm`.
# %%
sensitivityAnalysis = ot.SaltelliSensitivityAlgorithm(inputDesign, outputDesign, size)
# %%
# The `getFirstOrderIndices` and `getTotalOrderIndices` method respectively return estimates of first order and total Sobol' indices with a given output.
# Since these depend on the output marginal, the index of the output must
# be specified (the default is to return the index for the first output).
# %%
output_dimension = function.getOutputDimension()
for i in range(output_dimension):
print("Output #", i)
first_order = sensitivityAnalysis.getFirstOrderIndices(i)
total_order = sensitivityAnalysis.getTotalOrderIndices(i)
print(" First order indices: ", first_order)
print(" Total order indices: ", total_order)
agg_first_order = sensitivityAnalysis.getAggregatedFirstOrderIndices()
agg_total_order = sensitivityAnalysis.getAggregatedTotalOrderIndices()
print("Agg. first order indices: ", agg_first_order)
print("Agg. total order indices: ", agg_total_order)
# %%
# We see that:
#
# * `x1` has a rather large first order index on the first output, but a small index on the second output,
#
# * `x2` has a rather large first order index on the first output on both outputs,
#
# * the largest aggregated Sobol' index is `x2`,
#
# * `x0` and `x5` have Sobol' indices which are close to zero regardless of whether the indices are aggregated or not.
# %%
# The `draw` method produces the following graph. The vertical bars represent the 95% confidence intervals of the estimates.
# %%
graph = sensitivityAnalysis.draw()
view = viewer.View(graph)
# %%
# Since there are several outputs, the graph presents the aggregated Sobol' indices.
# The aggregated Sobol' indices indicate that the input variable which has the largest impact on the variability of the several outputs is `x2`.