.. _use-case-ishigami: The Ishigami function ===================== The Ishigami function of Ishigami & Homma (1990) is recurrent test case for sensitivity analysis methods and uncertainty. Let :math:a=7 and :math:b=0.1 (see Crestaux et al. (2007) and Marrel et al. (2009)). We consider the function .. math:: g(X_1,X_2,X_3) = \sin(X_1)+a \sin (X_2)^2 + b X_3^4 \sin(X_1) for any :math:X_1,X_2,X_3\in[-\pi,\pi] We assume that the random variables :math:X_1,X_2,X_3 are independent and have the uniform marginal distribution in the interval from :math:-\pi to :math:\pi: .. math:: X_1,X_2,X_3\sim \mathcal{U}(-\pi,\pi). Analysis -------- The expectation and the variance of :math:Y are .. math:: E(Y) = \frac{a}{2} and .. math:: V(Y) = \frac{1}{2} + \frac{a^2}{8} + \frac{b^2 \pi^8}{18} + \frac{b\pi^4}{5}. The Sobol' decomposition variances are .. math:: V_1 = \frac{1}{2} \left(1 + b\frac{\pi^4}{5} \right)^2, \qquad V_2 = \frac{a^2}{8}, \qquad V_{1,3} = b^2 \pi^8 \frac{8}{225} and :math:V_3=V_{1,2} = V_{2,3}=V_{1,2,3} = 0. This leads to the following first order Sobol' indices: .. math:: S_1 = \frac{V_1}{V(Y)}, \qquad S_2 = \frac{V_2}{V(Y)}, \qquad S_3 = 0, and the following total order indices: .. math:: ST_1 = \frac{V_1+V_{1,3}}{V(Y)}, \qquad ST_2 = S_2, \qquad S_3 = \frac{V_{1,3}}{V(Y)}. The third variable :math:X_3 has no effect at first order (because :math:X_3^4 it is multiplied by :math:\sin(X_1)), but has a total effet because of the interactions with :math:X_1. On the other hand, the second variable :math:X_2 has no interactions which implies that the first order indice is equal to the total order indice for this input variable. References ---------- * Ishigami, T., & Homma, T. (1990, December). An importance quantification technique in uncertainty analysis for computer models. In Uncertainty Modeling and Analysis, 1990. Proceedings., First International Symposium on (pp. 398-403). IEEE. * Sobol', I. M., & Levitan, Y. L. (1999). On the use of variance reducing multipliers in Monte Carlo computations of a global sensitivity index. Computer Physics Communications, 117(1), 52-61. * Saltelli, A., Chan, K., & Scott, E. M. (Eds.). (2000). Sensitivity analysis (Vol. 134). New York: Wiley. * Crestaux, T., Martinez, J.-M., Le Maitre, O., & Lafitte, O. (2007). Polynomial chaos expansion for uncertainties quantification and sensitivity analysis. SAMO 2007, http://samo2007.chem.elte.hu/lectures/Crestaux.pdf. Load the use case ----------------- We can load this classical model from the use cases module as follows : .. code-block:: python >>> from openturns.usecases import ishigami_function as ishigami_function >>> # Load the Ishigami use case >>> im = ishigami_function.IshigamiModel() API documentation ----------------- See :class:~openturns.usecases.ishigami_function.IshigamiModel. Examples based on this use case ------------------------------- .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/polynomial_chaos_metamodel/images/thumb/sphx_glr_plot_chaos_ishigami_thumb.png :alt: :ref:sphx_glr_auto_meta_modeling_polynomial_chaos_metamodel_plot_chaos_ishigami.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/polynomial_chaos_metamodel/plot_chaos_ishigami .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/sensitivity_analysis/images/thumb/sphx_glr_plot_sensitivity_fast_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_sensitivity_analysis_plot_sensitivity_fast.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/sensitivity_analysis/plot_sensitivity_fast .. raw:: html
.. only:: html .. figure:: /auto_data_analysis/manage_data_and_samples/images/thumb/sphx_glr_plot_sample_correlation_thumb.png :alt: :ref:sphx_glr_auto_data_analysis_manage_data_and_samples_plot_sample_correlation.py .. raw:: html
.. toctree:: :hidden: /auto_data_analysis/manage_data_and_samples/plot_sample_correlation .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/polynomial_chaos_metamodel/images/thumb/sphx_glr_plot_chaos_draw_validation_thumb.png :alt: :ref:sphx_glr_auto_meta_modeling_polynomial_chaos_metamodel_plot_chaos_draw_validation.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/polynomial_chaos_metamodel/plot_chaos_draw_validation .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/polynomial_chaos_metamodel/images/thumb/sphx_glr_plot_chaos_ishigami_grouped_indices_thumb.png :alt: :ref:sphx_glr_auto_meta_modeling_polynomial_chaos_metamodel_plot_chaos_ishigami_grouped_indices.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/polynomial_chaos_metamodel/plot_chaos_ishigami_grouped_indices .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/sensitivity_analysis/images/thumb/sphx_glr_plot_sensitivity_sobol_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_sensitivity_analysis_plot_sensitivity_sobol.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/sensitivity_analysis/plot_sensitivity_sobol