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This function exhibits strong\nnon-linearity and non-monotonicity.\n\nRemember, Sobol' indices assumes that samples are independently\ndistributed. In this case we use a uniform distribution on each marginals.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.stats import sobol_indices, uniform\nrng = np.random.default_rng()\ndef f_ishigami(x):\n f_eval = (\n np.sin(x[0])\n + 7 * np.sin(x[1])**2\n + 0.1 * (x[2]**4) * np.sin(x[0])\n )\n return f_eval\nindices = sobol_indices(\n func=f_ishigami, n=1024,\n dists=[\n uniform(loc=-np.pi, scale=2*np.pi),\n uniform(loc=-np.pi, scale=2*np.pi),\n uniform(loc=-np.pi, scale=2*np.pi)\n ],\n rng=rng\n)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "indices.first_order\nindices.total_order\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nConfidence interval can be obtained using bootstrapping.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "boot = indices.bootstrap()\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThen, this information can be easily visualized.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import matplotlib.pyplot as plt\nfig, axs = plt.subplots(1, 2, figsize=(9, 4))\n_ = axs[0].errorbar(\n [1, 2, 3], indices.first_order, fmt='o',\n yerr=[\n indices.first_order - boot.first_order.confidence_interval.low,\n boot.first_order.confidence_interval.high - indices.first_order\n ],\n)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "axs[0].set_ylabel(\"First order Sobol' indices\")\naxs[0].set_xlabel('Input parameters')\naxs[0].set_xticks([1, 2, 3])\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "_ = axs[1].errorbar(\n [1, 2, 3], indices.total_order, fmt='o',\n yerr=[\n indices.total_order - boot.total_order.confidence_interval.low,\n boot.total_order.confidence_interval.high - indices.total_order\n ],\n)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "axs[1].set_ylabel(\"Total order Sobol' indices\")\naxs[1].set_xlabel('Input parameters')\naxs[1].set_xticks([1, 2, 3])\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.tight_layout()\nplt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-896728bb561cb560.png" } }, { "__type": "Text", "__tag": 4046, "value": "\n.. note::\n\n By default, `scipy.stats.uniform` has support ``[0, 1]``.\n Using the parameters ``loc`` and ``scale``, one obtains the uniform\n distribution on ``[loc, loc + scale]``.\n\nThis result is particularly interesting because the first order index\n:math:`S_{x_3} = 0` whereas its total order is :math:`S_{T_{x_3}} = 0.244`.\nThis means that higher order interactions with :math:`x_3` are responsible\nfor the difference. Almost 25% of the observed variance\non the QoI is due to the correlations between :math:`x_3` and :math:`x_1`,\nalthough :math:`x_3` by itself has no impact on the QoI.\n\nThe following gives a visual explanation of Sobol' indices on this\nfunction. Let's generate 1024 samples in :math:`[-\\pi, \\pi]^3` and\ncalculate the value of the output.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.stats import qmc\nn_dim = 3\np_labels = ['$x_1$', '$x_2$', '$x_3$']\nsample = qmc.Sobol(d=n_dim, seed=rng).random(1024)\nsample = qmc.scale(\n sample=sample,\n l_bounds=[-np.pi, -np.pi, -np.pi],\n u_bounds=[np.pi, np.pi, np.pi]\n)\noutput = f_ishigami(sample.T)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nNow we can do scatter plots of the output with respect to each parameter.\nThis gives a visual way to understand how each parameter impacts the\noutput of the function.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "fig, ax = plt.subplots(1, n_dim, figsize=(12, 4))\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "for i in range(n_dim):\n xi = sample[:, i]\n ax[i].scatter(xi, output, marker='+')\n ax[i].set_xlabel(p_labels[i])\nax[0].set_ylabel('Y')\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.tight_layout()\nplt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-36fa33ffb0075d71.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nNow Sobol' goes a step further:\nby conditioning the output value by given values of the parameter\n(black lines), the conditional output mean is computed. It corresponds to\nthe term :math:`\\mathbb{E}(Y|x_i)`. Taking the variance of this term gives\nthe numerator of the Sobol' indices.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "mini = np.min(output)\nmaxi = np.max(output)\nn_bins = 10\nbins = np.linspace(-np.pi, np.pi, num=n_bins, endpoint=False)\ndx = bins[1] - bins[0]\nfig, ax = plt.subplots(1, n_dim, figsize=(12, 4))\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "for i in range(n_dim):\n xi = sample[:, i]\n ax[i].scatter(xi, output, marker='+')\n ax[i].set_xlabel(p_labels[i])\n for bin_ in bins:\n idx = np.where((bin_ <= xi) & (xi <= bin_ + dx))\n xi_ = xi[idx]\n y_ = output[idx]\n ave_y_ = np.mean(y_)\n ax[i].plot([bin_ + dx/2] * 2, [mini, maxi], c='k')\n ax[i].scatter(bin_ + dx/2, ave_y_, c='r')\nax[0].set_ylabel('Y')\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.tight_layout()\nplt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-592e9204c2451526.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nLooking at :math:`x_3`, the variance\nof the mean is zero leading to :math:`S_{x_3} = 0`. But we can further\nobserve that the variance of the output is not constant along the parameter\nvalues of :math:`x_3`. This heteroscedasticity is explained by higher order\ninteractions. Moreover, an heteroscedasticity is also noticeable on\n:math:`x_1` leading to an interaction between :math:`x_3` and :math:`x_1`.\nOn :math:`x_2`, the variance seems to be constant and thus null interaction\nwith this parameter can be supposed.\n\nThis case is fairly simple to analyse visually---although it is only a\nqualitative analysis. Nevertheless, when the number of input parameters\nincreases such analysis becomes unrealistic as it would be difficult to\nconclude on high-order terms. Hence the benefit of using Sobol' indices." } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "func", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "n", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "dists", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "saltelli_2010" }, { "__type": "SigParam", "__tag": 4030, "name": "rng", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "random_state", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "sobol_indices" }, "references": [ ".. [1] Sobol, I. M.. \"Sensitivity analysis for nonlinear mathematical", " models.\" Mathematical Modeling and Computational Experiment, 1:407-414,", " 1993.", ".. [2] Sobol, I. M. (2001). \"Global sensitivity indices for nonlinear", " mathematical models and their Monte Carlo estimates.\" Mathematics", " and Computers in Simulation, 55(1-3):271-280,", " :doi:`10.1016/S0378-4754(00)00270-6`, 2001.", ".. [3] Saltelli, A. \"Making best use of model evaluations to", " compute sensitivity indices.\" Computer Physics Communications,", " 145(2):280-297, :doi:`10.1016/S0010-4655(02)00280-1`, 2002.", ".. [4] Saltelli, A., M. Ratto, T. Andres, F. Campolongo, J. Cariboni,", " D. Gatelli, M. Saisana, and S. Tarantola. \"Global Sensitivity Analysis.", " The Primer.\" 2007.", ".. [5] Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and", " S. Tarantola. \"Variance based sensitivity analysis of model", " output. Design and estimator for the total sensitivity index.\"", " Computer Physics Communications, 181(2):259-270,", " :doi:`10.1016/j.cpc.2009.09.018`, 2010.", ".. [6] Ishigami, T. and T. Homma. \"An importance quantification technique", " in uncertainty analysis for computer models.\" IEEE,", " :doi:`10.1109/ISUMA.1990.151285`, 1990." ], "qa": "scipy.stats._sensitivity_analysis:sobol_indices", "arbitrary": [], "local_refs": [ "dists", "func", "method", "n", "res", "rng" ] }