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Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/optimize/_minpack_py.py", "item_line": 590, "item_type": "function", "aliases": [ "scipy.optimize.curve_fit" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.optimize import curve_fit\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "def func(x, a, b, c):\n return a * np.exp(-b * x) + c\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nDefine the data to be fit with some noise:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "xdata = np.linspace(0, 4, 50)\ny = func(xdata, 2.5, 1.3, 0.5)\nrng = np.random.default_rng()\ny_noise = 0.2 * rng.normal(size=xdata.size)\nydata = y + y_noise\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "plt.plot(xdata, ydata, 'b-', label='data')\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nFit for the parameters a, b, c of the function `func`:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "popt, pcov = curve_fit(func, xdata, ydata)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "popt\nplt.plot(xdata, func(xdata, *popt), 'r-',\n label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nConstrain the optimization to the region of ``0 <= a <= 3``,\n``0 <= b <= 1`` and ``0 <= c <= 0.5``:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5]))\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "popt\nplt.plot(xdata, func(xdata, *popt), 'g--',\n label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.xlabel('x')\nplt.ylabel('y')\nplt.legend()\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-3690c85f223b414a.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nFor reliable results, the model `func` should not be overparametrized;\nredundant parameters can cause unreliable covariance matrices and, in some\ncases, poorer quality fits. As a quick check of whether the model may be\noverparameterized, calculate the condition number of the covariance matrix:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "np.linalg.cond(pcov)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThe value is small, so it does not raise much concern. If, however, we were\nto add a fourth parameter ``d`` to `func` with the same effect as ``a``:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def func2(x, a, b, c, d):\n return a * d * np.exp(-b * x) + c # a and d are redundant\npopt, pcov = curve_fit(func2, xdata, ydata)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "np.linalg.cond(pcov)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nSuch a large value is cause for concern. The diagonal elements of the\ncovariance matrix, which is related to uncertainty of the fit, gives more\ninformation:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "np.diag(pcov)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nNote that the first and last terms are much larger than the other elements,\nsuggesting that the optimal values of these parameters are ambiguous and\nthat only one of these parameters is needed in the model.\n\nIf the optimal parameters of `f` differ by multiple orders of magnitude, the\nresulting fit can be inaccurate. Sometimes, `curve_fit` can fail to find any\nresults:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "ydata = func(xdata, 500000, 0.01, 15)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "try:\n popt, pcov = curve_fit(func, xdata, ydata, method = 'trf')\nexcept RuntimeError as e:\n print(e)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIf parameter scale is roughly known beforehand, it can be defined in\n`x_scale` argument:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "popt, pcov = curve_fit(func, xdata, ydata, method = 'trf',\n x_scale = [1000, 1, 1])\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "popt\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "least_squares", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._lsq.least_squares:least_squares" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Minimize the sum of squares of nonlinear functions." } ] } ], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.linregress", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats._stats_py:linregress" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Calculate a linear least squares regression for two sets of measurements." } ] } ], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "f", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "xdata", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "ydata", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "p0", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "sigma", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "absolute_sigma", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "check_finite", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "bounds", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "(-inf, inf)" }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "jac", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "full_output", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "nan_policy", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "kwargs", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "VAR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "curve_fit" }, "references": [ ".. [1] K. Vugrin et al. Confidence region estimation techniques for nonlinear", " regression in groundwater flow: Three case studies. Water Resources", " Research, Vol. 43, W03423, :doi:`10.1029/2005WR004804`" ], "qa": "scipy.optimize._minpack_py:curve_fit", "arbitrary": [], "local_refs": [ "**kwargs", "absolute_sigma", "bounds", "check_finite", "f", "full_output", "ier", "infodict", "jac", "mesg", "method", "nan_policy", "p0", "pcov", "popt", "sigma", "xdata", "ydata" ] }