Ishigami function

The ishigami function is a non-linear, non-monotonic function that is commonly used to benchmark uncertainty and senstivity analysis methods.

\[f(x_1, x_2, x_3) = sin(x_1) + a \cdot sin^2(x_2) + b \cdot x_3^4 sin(x_1)\]

\[x_1, x_2, x_3 \sim \mathcal{U}(-\pi, \pi), \quad a, b\in \mathbb{R}\]

First order Sobol indices

\[S_1 = \frac{V_1}{\mathbb{V}[Y]}, \quad S_2 = \frac{V_2}{\mathbb{V}[Y]}, \quad S_3 = \frac{V_3}{\mathbb{V}[Y]} = 0,\]

\[V_1 = 0.5 (1 + \frac{b\pi^4}{5})^2, \quad V_2 = \frac{a^2}{8}, \quad V_3 = 0\]

\[\mathbb{V}[Y] = \frac{a^2}{8} + \frac{b\pi^4}{5} + \frac{b^2\pi^8}{18} + \frac{1}{2}\]

Total order Sobol indices

\[S_{T_1} = \frac{V_{T1}}{\mathbb{V}[Y]}, \quad S_{T_2} = \frac{V_{T2}}{\mathbb{V}[Y]}, \quad S_{T_3} = \frac{V_{T3}}{\mathbb{V}[Y]}\]

\[V_{T_1} = 0.5 (1 + \frac{b\pi^4}{5})^2 + \frac{8b^2\pi^8}{225}, \quad V_{T_2}= \frac{a^2}{8}, \quad V_{T_3} = \frac{8b^2\pi^8}{225}\]

\[\mathbb{V}[Y] = \frac{a^2}{8} + \frac{b\pi^4}{5} + \frac{b^2\pi^8}{18} + \frac{1}{2}\]

import numpy as np

from UQpy.run_model.RunModel import RunModel
from UQpy.run_model.model_execution.PythonModel import PythonModel
from UQpy.distributions import Uniform
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

model = PythonModel(
    model_script="local_ishigami.py",
    model_object_name="evaluate",
    var_names=[r"$X_1$", "$X_2$", "$X_3$"],
    delete_files=True,
    params=[7, 0.1],
)

runmodel_obj = RunModel(model=model)

# Define distribution object
dist_object = JointIndependent([Uniform(-np.pi, 2 * np.pi)] * 3)

Compute Sobol indices

SA = SobolSensitivity(runmodel_obj, dist_object)

SA.run(n_samples=100_000, n_bootstrap_samples=100)

First order Sobol indices

Expected first order Sobol indices:

\(S_1\) = 0.3139

\(S_2\) = 0.4424

\(S_3\) = 0.0

SA.first_order_indices

Total order Sobol indices

Expected total order Sobol indices:

\(S_{T_1}\) = 0.55758886

\(S_{T_2}\) = 0.44241114

\(S_{T_3}\) = 0.24368366

SA.total_order_indices

Confidence intervals for first order Sobol indices

SA.first_order_confidence_interval

Confidence intervals for total order Sobol indices

SA.total_order_confidence_interval

Plot the first order sensitivity indices

fig1, ax1 = plot_sensitivity_index(
    SA.first_order_indices[:, 0],
    confidence_interval=SA.first_order_confidence_interval,
    plot_title="First order Sobol indices",
    variable_names=["$X_1$", "$X_2$", "$X_3$"],
    color="C0",
)

Plot the first and total order sensitivity indices

fig2, ax2 = plot_index_comparison(
    SA.first_order_indices[:, 0],
    SA.total_order_indices[:, 0],
    confidence_interval_1=SA.first_order_confidence_interval,
    confidence_interval_2=SA.total_order_confidence_interval,
    label_1="First order Sobol indices",
    label_2="Total order Sobol indices",
    plot_title="First and Total order Sobol indices",
    variable_names=["$X_1$", "$X_2$", "$X_3$"],
)