.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "examples/uncertainty/sensitivity/plot_sobol.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here  to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_examples_uncertainty_sensitivity_plot_sobol.py: Sobol' analysis =============== .. GENERATED FROM PYTHON SOURCE LINES 25-33 .. code-block:: default import pprint from gemseo.algos.parameter_space import ParameterSpace from gemseo.api import create_discipline from gemseo.uncertainty.sensitivity.sobol.analysis import SobolAnalysis from matplotlib import pyplot as plt from numpy import pi .. GENERATED FROM PYTHON SOURCE LINES 34-42 In this example, we consider a function from :math:[-\pi,\pi]^3 to :math:\mathbb{R}^3: .. math:: (y_1,y_2)=\left(f(x_1,x_2,x_3),f(x_2,x_1,x_3)\right) where :math:f(a,b,c)=\sin(a)+7\sin(b)^2+0.1*c^4\sin(a) is the Ishigami function: .. GENERATED FROM PYTHON SOURCE LINES 42-51 .. code-block:: default expressions = { "y1": "sin(x1)+7*sin(x2)**2+0.1*x3**4*sin(x1)", "y2": "sin(x2)+7*sin(x1)**2+0.1*x3**4*sin(x2)", } discipline = create_discipline( "AnalyticDiscipline", expressions=expressions, name="Ishigami2" ) .. GENERATED FROM PYTHON SOURCE LINES 52-58 Then, we consider the case where the deterministic variables :math:x_1, :math:x_2 and :math:x_3 are replaced with the uncertain variables :math:X_1, :math:X_2 and :math:X_3. The latter are independent and identically distributed according to an uniform distribution between :math:-\pi and :math:\pi: .. GENERATED FROM PYTHON SOURCE LINES 58-64 .. code-block:: default space = ParameterSpace() for variable in ["x1", "x2", "x3"]: space.add_random_variable( variable, "OTUniformDistribution", minimum=-pi, maximum=pi ) .. GENERATED FROM PYTHON SOURCE LINES 65-70 From that, we would like to carry out a sensitivity analysis with the random outputs :math:Y_1=f(X_1,X_2,X_3) and :math:Y_2=f(X_2,X_1,X_3). For that, we can compute the correlation coefficients from a :class:.SobolAnalysis: .. GENERATED FROM PYTHON SOURCE LINES 70-74 .. code-block:: default sobol = SobolAnalysis([discipline], space, 100) sobol.main_method = "total" sobol.compute_indices() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none {'first': {'y1': [{'x1': array([0.19426511]), 'x2': array([0.21211854]), 'x3': array([0.01782884])}], 'y2': [{'x1': array([0.75387699]), 'x2': array([0.26516667]), 'x3': array([0.35758815])}]}, 'total': {'y1': [{'x1': array([0.87052721]), 'x2': array([0.37726189]), 'x3': array([0.29945874])}], 'y2': [{'x1': array([0.36991381]), 'x2': array([0.60974968]), 'x3': array([0.34776012])}]}} .. GENERATED FROM PYTHON SOURCE LINES 75-76 The resulting indices are the first and total order Sobol' indices: .. GENERATED FROM PYTHON SOURCE LINES 76-78 .. code-block:: default pprint.pprint(sobol.indices) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none {'first': {'y1': [{'x1': array([0.19426511]), 'x2': array([0.21211854]), 'x3': array([0.01782884])}], 'y2': [{'x1': array([0.75387699]), 'x2': array([0.26516667]), 'x3': array([0.35758815])}]}, 'total': {'y1': [{'x1': array([0.87052721]), 'x2': array([0.37726189]), 'x3': array([0.29945874])}], 'y2': [{'x1': array([0.36991381]), 'x2': array([0.60974968]), 'x3': array([0.34776012])}]}} .. GENERATED FROM PYTHON SOURCE LINES 79-80 They can also be accessed separately: .. GENERATED FROM PYTHON SOURCE LINES 80-83 .. code-block:: default pprint.pprint(sobol.first_order_indices) pprint.pprint(sobol.total_order_indices) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none {'y1': [{'x1': array([0.19426511]), 'x2': array([0.21211854]), 'x3': array([0.01782884])}], 'y2': [{'x1': array([0.75387699]), 'x2': array([0.26516667]), 'x3': array([0.35758815])}]} {'y1': [{'x1': array([0.87052721]), 'x2': array([0.37726189]), 'x3': array([0.29945874])}], 'y2': [{'x1': array([0.36991381]), 'x2': array([0.60974968]), 'x3': array([0.34776012])}]} .. GENERATED FROM PYTHON SOURCE LINES 84-86 The main indices corresponds to the Spearman correlation indices (this main method can be changed with :attr:.SobolAnalysis.main_method): .. GENERATED FROM PYTHON SOURCE LINES 86-90 .. code-block:: default pprint.pprint(sobol.main_indices) pprint.pprint(sobol.get_intervals()) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none {'y1': [{'x1': array([0.87052721]), 'x2': array([0.37726189]), 'x3': array([0.29945874])}], 'y2': [{'x1': array([0.36991381]), 'x2': array([0.60974968]), 'x3': array([0.34776012])}]} {'y1': [{'x1': array([-0.05887176, 0.57198378]), 'x2': array([0.0281138 , 0.66092945]), 'x3': array([-0.48592145, 0.41401005])}], 'y2': [{'x1': array([0.15165848, 2.21224422]), 'x2': array([-0.24741228, 1.07690768]), 'x3': array([-0.40957101, 2.01204153])}]} .. GENERATED FROM PYTHON SOURCE LINES 91-93 We can also sort the input parameters by decreasing order of influence: and observe that this ranking is not the same for both outputs: .. GENERATED FROM PYTHON SOURCE LINES 93-96 .. code-block:: default print(sobol.sort_parameters("y1")) print(sobol.sort_parameters("y2")) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none ['x1', 'x2', 'x3'] ['x2', 'x1', 'x3'] .. GENERATED FROM PYTHON SOURCE LINES 97-100 Lastly, we can use the method :meth:.SobolAnalysis.plot to visualize both first and total order Sobol' indices: .. GENERATED FROM PYTHON SOURCE LINES 100-104 .. code-block:: default sobol.plot("y1", save=False, show=False) sobol.plot("y2", save=False, show=False) # Workaround for HTML rendering, instead of show=True plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /examples/uncertainty/sensitivity/images/sphx_glr_plot_sobol_001.png :alt: Sobol indices for the output y1(0) :srcset: /examples/uncertainty/sensitivity/images/sphx_glr_plot_sobol_001.png :class: sphx-glr-multi-img * .. image-sg:: /examples/uncertainty/sensitivity/images/sphx_glr_plot_sobol_002.png :alt: Sobol indices for the output y2(0) :srcset: /examples/uncertainty/sensitivity/images/sphx_glr_plot_sobol_002.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.404 seconds) .. _sphx_glr_download_examples_uncertainty_sensitivity_plot_sobol.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: plot_sobol.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_sobol.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _