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Toy multioutput function
In this example, we demonstrate the computation of the Generalised Sobol indices using the toy example in [1].
\[\begin{split}Y = f (X_{1}, X_{2}) := \left(\begin{array}{c}
X_{1}+X_{2}+X_{1} X_{2} \\
2 X_{1}+3 X_{1} X_{2}+X_{2}
\end{array}\right)\end{split}\]
\[\text{case 1: } X_1, X_2 \sim \mathcal{N}(0, 1)\]
\[\text{case 2: } X_1, X_2 \sim \mathcal{U}(0, 1)\]
from UQpy.run_model.RunModel import RunModel
from UQpy.run_model.model_execution.PythonModel import PythonModel
from UQpy.distributions import Uniform, Normal
from UQpy.distributions.collection.JointIndependent import JointIndependent
from UQpy.sensitivity.GeneralisedSobolSensitivity import GeneralisedSobolSensitivity
from UQpy.sensitivity.PostProcess import *
np.random.seed(123)
Define the model and input distributions
# Create Model object
model = PythonModel(
model_script="local_multioutput.py",
model_object_name="evaluate",
var_names=[r"X_1$", r"X_2"],
delete_files=True,
)
runmodel_obj = RunModel(model=model)
# Define distribution object
dist_object_1 = JointIndependent([Normal(0, 1)] * 2)
Compute generalised Sobol indices
SA = GeneralisedSobolSensitivity(runmodel_obj, dist_object_1)
SA.run(
n_samples=20_000, confidence_level=0.95, n_bootstrap_samples=5_00
)
First order Generalised Sobol indices
Expected generalised Sobol indices:
Gaussian case
\(GS_1\) = 0.2941
\(GS_2\) = 0.1179
SA.generalized_first_order_indices
# **Plot the first order sensitivity indices**
fig1, ax1 = plot_sensitivity_index(
SA.generalized_first_order_indices[:, 0],
confidence_interval=SA.first_order_confidence_interval,
plot_title="First order Generalised Sobol indices",
color="C0",
)
SA.generalized_total_order_indices
# **Plot the first and total order sensitivity indices**
fig2, ax2 = plot_index_comparison(
SA.generalized_first_order_indices[:, 0],
SA.generalized_total_order_indices[:, 0],
confidence_interval_1=SA.first_order_confidence_interval,
confidence_interval_2=SA.total_order_confidence_interval,
label_1="First order",
label_2="Total order",
plot_title="First and Total order Generalised Sobol indices",
)
Compute generalised Sobol indices
dist_object_2 = JointIndependent([Uniform(0, 1)] * 2)
SA = GeneralisedSobolSensitivity(runmodel_obj, dist_object_2)
SA.run(
n_samples=20_000, confidence_level=0.95, n_bootstrap_samples=5_00
)
First order Generalised Sobol indices
Expected generalised Sobol indices:
Uniform case
\(GS_1\) = 0.6084
\(GS_2\) = 0.3566
SA.generalized_first_order_indices
# **Plot the first order sensitivity indices**
fig3, ax3 = plot_sensitivity_index(
SA.generalized_first_order_indices[:, 0],
confidence_interval=SA.first_order_confidence_interval,
plot_title="First order Generalised Sobol indices",
color="C0",
)
SA.total_order_confidence_interval
# **Plot the first and total order sensitivity indices**
fig4, ax4 = plot_index_comparison(
SA.generalized_first_order_indices[:, 0],
SA.total_order_confidence_interval[:, 0],
confidence_interval_1=SA.first_order_confidence_interval,
confidence_interval_2=SA.total_order_confidence_interval,
label_1="First order",
label_2="Total order",
plot_title="First and Total order Generalised Sobol indices",
)
Total running time of the script: ( 0 minutes 0.000 seconds)
