Additive function

We introduce the variance-based Sobol indices using an elementary example. For more details, refer [1].

\[f(x) = a \cdot X_1 + b \cdot X_2, \quad X_1, X_2 \sim \mathcal{N}(0, 1), \quad a,b \in \mathbb{R}\]

[1]

Saltelli A, T. (2008). Global sensitivity analysis: The primer. John Wiley.

import numpy as np

from UQpy.run_model.RunModel import RunModel
from UQpy.run_model.model_execution.PythonModel import PythonModel
from UQpy.distributions import Normal
from UQpy.distributions.collection.JointIndependent import JointIndependent
from UQpy.sensitivity.SobolSensitivity import SobolSensitivity
from UQpy.sensitivity.PostProcess import *

np.random.seed(123)

Define the model and input distributions

# Create Model object
a, b = 1, 2

model = PythonModel(
    model_script="local_additive.py",
    model_object_name="evaluate",
    var_names=[
        "X_1",
        "X_2",
    ],
    delete_files=True,
    params=[a, b],
)

runmodel_obj = RunModel(model=model)

# Define distribution object
dist_object = JointIndependent([Normal(0, 1)] * 2)

Compute Sobol indices

SA = SobolSensitivity(runmodel_obj, dist_object)

SA.run(n_samples=50_000)

First order Sobol indices

Expected first order Sobol indices:

\(\mathrm{S}_1 = \frac{a^2 \cdot \mathbb{V}[X_1]}{a^2 \cdot \mathbb{V}[X_1] + b^2 \cdot \mathbb{V}[X_2]} = \frac{1^2 \cdot 1}{1^2 \cdot 1 + 2^2 \cdot 1} = 0.2\)

\(\mathrm{S}_2 = \frac{b^2 \cdot \mathbb{V}[X_2]}{a^2 \cdot \mathbb{V}[X_1] + b^2 \cdot \mathbb{V}[X_2]} = \frac{2^2 \cdot 1}{1^2 \cdot 1 + 2^2 \cdot 1} = 0.8\)

SA.first_order_indices

Plot the first and total order sensitivity indices

fig1, ax1 = plot_index_comparison(
    SA.first_order_indices[:, 0],
    SA.total_order_indices[:, 0],
    label_1="First order Sobol indices",
    label_2="Total order Sobol indices",
    plot_title="First and Total order Sobol indices",
)