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{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Exercise 4: Least square fit to data" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from matplotlib import pyplot as plt\n", "plt.rcParams[\"font.size\"] = 20\n", "import numpy as np" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# data\n", "x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], dtype='d')\n", "dx = np.array([0.1,0.1,0.5,0.1,0.5,0.1,0.5,0.1,0.5,0.1], dtype='d')\n", "y = np.array([1.1 ,2.3 ,2.7 ,3.2 ,3.1 ,2.4 ,1.7 ,1.5 ,1.5 ,1.7 ], dtype='d')\n", "dy = np.array([0.15,0.22,0.29,0.39,0.31,0.21,0.13,0.15,0.19,0.13], dtype='d')" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# define fit function \n", "def pol3(a0, a1, a2, a3):\n", " return a0 + x*a1 + a2*x**2 + a3*x**3" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# least-squares function = sum of data residuals squared\n", "def LSQ(a0, a1, a2, a3):\n", " return np.sum((y - pol3(a0, a1, a2, a3)) ** 2 / dy ** 2)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# import Minuit object\n", "from iminuit import Minuit" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<table>\n", " <tr>\n", " <th colspan=\"5\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 3.719 </td>\n", " <td colspan=\"3\" style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 103 </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 4.33e-16 (Goal: 0.0002) </td>\n", " <td colspan=\"3\" style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n", " <td colspan=\"3\" style=\"text-align:center;background-color:#92CCA6;color:black\"> No Parameters at limit </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n", " <td colspan=\"3\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n", " </tr>\n", " <tr>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Is covariance matrix accurate?\"> Accurate </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Is covariance matrix positive definite?\"> Pos. def. </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Was positive definiteness enforced by Minuit?\"> Not forced </td>\n", " </tr>\n", "</table><table>\n", " <tr>\n", " <td></td>\n", " <th title=\"Variable name\"> Name </th>\n", " <th title=\"Value of parameter\"> Value </th>\n", " <th title=\"Hesse error\"> Hesse Error </th>\n", " <th title=\"Minos lower error\"> Minos Error- </th>\n", " <th title=\"Minos upper error\"> Minos Error+ </th>\n", " <th title=\"Lower limit of the parameter\"> Limit- </th>\n", " <th title=\"Upper limit of the parameter\"> Limit+ </th>\n", " <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n", " </tr>\n", " <tr>\n", " <th> 0 </th>\n", " <td> a0 </td>\n", " <td> -0.98 </td>\n", " <td> 0.34 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 1 </th>\n", " <td> a1 </td>\n", " <td> 2.52 </td>\n", " <td> 0.30 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 2 </th>\n", " <td> a2 </td>\n", " <td> -0.48 </td>\n", " <td> 0.06 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 3 </th>\n", " <td> a3 </td>\n", " <td> 0.0259 </td>\n", " <td> 0.0035 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", "</table><table>\n", " <tr>\n", " <td></td>\n", " <th> a0 </th>\n", " <th> a1 </th>\n", " <th> a2 </th>\n", " <th> a3 </th>\n", " </tr>\n", " <tr>\n", " <th> a0 </th>\n", " <td> 0.119 </td>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.096 <strong>(-0.931)</strong> </td>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000997 <strong>(-0.835)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a1 </th>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.096 <strong>(-0.931)</strong> </td>\n", " <td> 0.0893 </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0178 <strong>(-0.987)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000996 <strong>(0.962)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a2 </th>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0178 <strong>(-0.987)</strong> </td>\n", " <td> 0.00366 </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a3 </th>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000997 <strong>(-0.835)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000996 <strong>(0.962)</strong> </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " <td> 1.2e-05 </td>\n", " </tr>\n", "</table>" ], "text/plain": [ "┌─────────────────────────────────────────────────────────────────────────┐\n", "│ Migrad │\n", "├──────────────────────────────────┬──────────────────────────────────────┤\n", "│ FCN = 3.719 │ Nfcn = 103 │\n", "│ EDM = 4.33e-16 (Goal: 0.0002) │ │\n", "├──────────────────────────────────┼──────────────────────────────────────┤\n", "│ Valid Minimum │ No Parameters at limit │\n", "├──────────────────────────────────┼──────────────────────────────────────┤\n", "│ Below EDM threshold (goal x 10) │ Below call limit │\n", "├───────────────┬──────────────────┼───────────┬─────────────┬────────────┤\n", "│ Covariance │ Hesse ok │ Accurate │ Pos. def. │ Not forced │\n", "└───────────────┴──────────────────┴───────────┴─────────────┴────────────┘\n", "┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n", "│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n", "├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n", "│ 0 │ a0 │ -0.98 │ 0.34 │ │ │ │ │ │\n", "│ 1 │ a1 │ 2.52 │ 0.30 │ │ │ │ │ │\n", "│ 2 │ a2 │ -0.48 │ 0.06 │ │ │ │ │ │\n", "│ 3 │ a3 │ 0.0259 │ 0.0035 │ │ │ │ │ │\n", "└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n", "┌────┬─────────────────────────────────────────┐\n", "│ │ a0 a1 a2 a3 │\n", "├────┼─────────────────────────────────────────┤\n", "│ a0 │ 0.119 -0.096 0.0183 -0.000997 │\n", "│ a1 │ -0.096 0.0893 -0.0178 0.000996 │\n", "│ a2 │ 0.0183 -0.0178 0.00366 -0.000208 │\n", "│ a3 │ -0.000997 0.000996 -0.000208 1.2e-05 │\n", "└────┴─────────────────────────────────────────┘" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# create instance of Minuit and use LSQ function to minimize\n", "LSQ.errordef = Minuit.LEAST_SQUARES\n", "m = Minuit(LSQ,a0=-1.3, a1=2.6 ,a2=-0.24 ,a3=0.005)\n", "# run migrad \n", "m.migrad()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chi2/ndof = 0.6198527004952333\n" ] } ], "source": [ "# get function value at the minimum, which is per definition a chi2\n", "# obtain chi2 / degree of freedom (dof)\n", "chi2 = m.fval / (len(y) - len(m.values))\n", "print (\"Chi2/ndof =\" , chi2)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<table>\n", " <tr>\n", " <th colspan=\"5\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 3.719 </td>\n", " <td colspan=\"3\" style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 128 </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 1.06e-17 (Goal: 0.0002) </td>\n", " <td colspan=\"3\" style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n", " <td colspan=\"3\" style=\"text-align:center;background-color:#92CCA6;color:black\"> No Parameters at limit </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n", " <td colspan=\"3\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n", " </tr>\n", " <tr>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Is covariance matrix accurate?\"> Accurate </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Is covariance matrix positive definite?\"> Pos. def. </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Was positive definiteness enforced by Minuit?\"> Not forced </td>\n", " </tr>\n", "</table><table>\n", " <tr>\n", " <td></td>\n", " <th title=\"Variable name\"> Name </th>\n", " <th title=\"Value of parameter\"> Value </th>\n", " <th title=\"Hesse error\"> Hesse Error </th>\n", " <th title=\"Minos lower error\"> Minos Error- </th>\n", " <th title=\"Minos upper error\"> Minos Error+ </th>\n", " <th title=\"Lower limit of the parameter\"> Limit- </th>\n", " <th title=\"Upper limit of the parameter\"> Limit+ </th>\n", " <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n", " </tr>\n", " <tr>\n", " <th> 0 </th>\n", " <td> a0 </td>\n", " <td> -0.98 </td>\n", " <td> 0.34 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 1 </th>\n", " <td> a1 </td>\n", " <td> 2.52 </td>\n", " <td> 0.30 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 2 </th>\n", " <td> a2 </td>\n", " <td> -0.48 </td>\n", " <td> 0.06 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 3 </th>\n", " <td> a3 </td>\n", " <td> 0.0259 </td>\n", " <td> 0.0035 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", "</table><table>\n", " <tr>\n", " <td></td>\n", " <th> a0 </th>\n", " <th> a1 </th>\n", " <th> a2 </th>\n", " <th> a3 </th>\n", " </tr>\n", " <tr>\n", " <th> a0 </th>\n", " <td> 0.119 </td>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.0961 <strong>(-0.931)</strong> </td>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000999 <strong>(-0.835)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a1 </th>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.0961 <strong>(-0.931)</strong> </td>\n", " <td> 0.0895 </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0179 <strong>(-0.987)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000997 <strong>(0.962)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a2 </th>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0179 <strong>(-0.987)</strong> </td>\n", " <td> 0.00366 </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a3 </th>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000999 <strong>(-0.835)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000997 <strong>(0.962)</strong> </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " <td> 1.2e-05 </td>\n", " </tr>\n", "</table>" ], "text/plain": [ "┌─────────────────────────────────────────────────────────────────────────┐\n", "│ Migrad │\n", "├──────────────────────────────────┬──────────────────────────────────────┤\n", "│ FCN = 3.719 │ Nfcn = 128 │\n", "│ EDM = 1.06e-17 (Goal: 0.0002) │ │\n", "├──────────────────────────────────┼──────────────────────────────────────┤\n", "│ Valid Minimum │ No Parameters at limit │\n", "├──────────────────────────────────┼──────────────────────────────────────┤\n", "│ Below EDM threshold (goal x 10) │ Below call limit │\n", "├───────────────┬──────────────────┼───────────┬─────────────┬────────────┤\n", "│ Covariance │ Hesse ok │ Accurate │ Pos. def. │ Not forced │\n", "└───────────────┴──────────────────┴───────────┴─────────────┴────────────┘\n", "┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n", "│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n", "├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n", "│ 0 │ a0 │ -0.98 │ 0.34 │ │ │ │ │ │\n", "│ 1 │ a1 │ 2.52 │ 0.30 │ │ │ │ │ │\n", "│ 2 │ a2 │ -0.48 │ 0.06 │ │ │ │ │ │\n", "│ 3 │ a3 │ 0.0259 │ 0.0035 │ │ │ │ │ │\n", "└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n", "┌────┬─────────────────────────────────────────┐\n", "│ │ a0 a1 a2 a3 │\n", "├────┼─────────────────────────────────────────┤\n", "│ a0 │ 0.119 -0.0961 0.0183 -0.000999 │\n", "│ a1 │ -0.0961 0.0895 -0.0179 0.000997 │\n", "│ a2 │ 0.0183 -0.0179 0.00366 -0.000208 │\n", "│ a3 │ -0.000999 0.000997 -0.000208 1.2e-05 │\n", "└────┴─────────────────────────────────────────┘" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# run covariance \n", "m.hesse()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<table>\n", " <tr>\n", " <td></td>\n", " <th> a0 </th>\n", " <th> a1 </th>\n", " <th> a2 </th>\n", " <th> a3 </th>\n", " </tr>\n", " <tr>\n", " <th> a0 </th>\n", " <td> 0.119 </td>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.0961 <strong>(-0.931)</strong> </td>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000999 <strong>(-0.835)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a1 </th>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.0961 <strong>(-0.931)</strong> </td>\n", " <td> 0.0895 </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0179 <strong>(-0.987)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000997 <strong>(0.962)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a2 </th>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0179 <strong>(-0.987)</strong> </td>\n", " <td> 0.00366 </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a3 </th>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000999 <strong>(-0.835)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000997 <strong>(0.962)</strong> </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " <td> 1.2e-05 </td>\n", " </tr>\n", "</table>" ], "text/plain": [ "┌────┬─────────────────────────────────────────┐\n", "│ │ a0 a1 a2 a3 │\n", "├────┼─────────────────────────────────────────┤\n", "│ a0 │ 0.119 -0.0961 0.0183 -0.000999 │\n", "│ a1 │ -0.0961 0.0895 -0.0179 0.000997 │\n", "│ a2 │ 0.0183 -0.0179 0.00366 -0.000208 │\n", "│ a3 │ -0.000999 0.000997 -0.000208 1.2e-05 │\n", "└────┴─────────────────────────────────────────┘" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#get covariance matrix\n", "m.covariance" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌────┬─────────────────────────────────────────┐\n", "│ │ a0 a1 a2 a3 │\n", "├────┼─────────────────────────────────────────┤\n", "│ a0 │ 0.119 -0.0961 0.0183 -0.000999 │\n", "│ a1 │ -0.0961 0.0895 -0.0179 0.000997 │\n", "│ a2 │ 0.0183 -0.0179 0.00366 -0.000208 │\n", "│ a3 │ -0.000999 0.000997 -0.000208 1.2e-05 │\n", "└────┴─────────────────────────────────────────┘\n" ] } ], "source": [ "#get correlation matrix in numpy array\n", "cov = m.covariance\n", "print (cov)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<table>\n", " <tr>\n", " <th colspan=\"5\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 3.719 </td>\n", " <td colspan=\"3\" style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 300 </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 1.06e-17 (Goal: 0.0002) </td>\n", " <td colspan=\"3\" style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n", " <td colspan=\"3\" style=\"text-align:center;background-color:#92CCA6;color:black\"> No Parameters at limit </td>\n", " </tr>\n", " <tr>\n", " <td colspan=\"2\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n", " <td colspan=\"3\" style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n", " </tr>\n", " <tr>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Is covariance matrix accurate?\"> Accurate </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Is covariance matrix positive definite?\"> Pos. def. </td>\n", " <td style=\"text-align:center;background-color:#92CCA6;color:black\" title=\"Was positive definiteness enforced by Minuit?\"> Not forced </td>\n", " </tr>\n", "</table><table>\n", " <tr>\n", " <td></td>\n", " <th title=\"Variable name\"> Name </th>\n", " <th title=\"Value of parameter\"> Value </th>\n", " <th title=\"Hesse error\"> Hesse Error </th>\n", " <th title=\"Minos lower error\"> Minos Error- </th>\n", " <th title=\"Minos upper error\"> Minos Error+ </th>\n", " <th title=\"Lower limit of the parameter\"> Limit- </th>\n", " <th title=\"Upper limit of the parameter\"> Limit+ </th>\n", " <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n", " </tr>\n", " <tr>\n", " <th> 0 </th>\n", " <td> a0 </td>\n", " <td> -0.98 </td>\n", " <td> 0.35 </td>\n", " <td> -0.34 </td>\n", " <td> 0.34 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 1 </th>\n", " <td> a1 </td>\n", " <td> 2.52 </td>\n", " <td> 0.30 </td>\n", " <td> -0.30 </td>\n", " <td> 0.30 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 2 </th>\n", " <td> a2 </td>\n", " <td> -0.48 </td>\n", " <td> 0.06 </td>\n", " <td> -0.06 </td>\n", " <td> 0.06 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", " <tr>\n", " <th> 3 </th>\n", " <td> a3 </td>\n", " <td> 0.0259 </td>\n", " <td> 0.0035 </td>\n", " <td> -0.0035 </td>\n", " <td> 0.0035 </td>\n", " <td> </td>\n", " <td> </td>\n", " <td> </td>\n", " </tr>\n", "</table><table>\n", " <tr>\n", " <td></td>\n", " <th colspan=\"2\" style=\"text-align:center\" title=\"Parameter name\"> a0 </th>\n", " <th colspan=\"2\" style=\"text-align:center\" title=\"Parameter name\"> a1 </th>\n", " <th colspan=\"2\" style=\"text-align:center\" title=\"Parameter name\"> a2 </th>\n", " <th colspan=\"2\" style=\"text-align:center\" title=\"Parameter name\"> a3 </th>\n", " </tr>\n", " <tr>\n", " <th title=\"Lower and upper minos error of the parameter\"> Error </th>\n", " <td> -0.34 </td>\n", " <td> 0.34 </td>\n", " <td> -0.3 </td>\n", " <td> 0.3 </td>\n", " <td> -0.06 </td>\n", " <td> 0.06 </td>\n", " <td> -0.0035 </td>\n", " <td> 0.0035 </td>\n", " </tr>\n", " <tr>\n", " <th title=\"Validity of lower/upper minos error\"> Valid </th>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> True </td>\n", " </tr>\n", " <tr>\n", " <th title=\"Did scan hit limit of any parameter?\"> At Limit </th>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " </tr>\n", " <tr>\n", " <th title=\"Did scan hit function call limit?\"> Max FCN </th>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " </tr>\n", " <tr>\n", " <th title=\"New minimum found when doing scan?\"> New Min </th>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " <td style=\"background-color:#92CCA6;color:black\"> False </td>\n", " </tr>\n", "</table><table>\n", " <tr>\n", " <td></td>\n", " <th> a0 </th>\n", " <th> a1 </th>\n", " <th> a2 </th>\n", " <th> a3 </th>\n", " </tr>\n", " <tr>\n", " <th> a0 </th>\n", " <td> 0.119 </td>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.0961 <strong>(-0.931)</strong> </td>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000999 <strong>(-0.835)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a1 </th>\n", " <td style=\"background-color:rgb(129,129,250);color:black\"> -0.0961 <strong>(-0.931)</strong> </td>\n", " <td> 0.0895 </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0179 <strong>(-0.987)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000997 <strong>(0.962)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a2 </th>\n", " <td style=\"background-color:rgb(250,118,118);color:black\"> 0.0183 <strong>(0.878)</strong> </td>\n", " <td style=\"background-color:rgb(122,122,250);color:black\"> -0.0179 <strong>(-0.987)</strong> </td>\n", " <td> 0.00366 </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " </tr>\n", " <tr>\n", " <th> a3 </th>\n", " <td style=\"background-color:rgb(141,141,250);color:black\"> -0.000999 <strong>(-0.835)</strong> </td>\n", " <td style=\"background-color:rgb(250,106,106);color:black\"> 0.000997 <strong>(0.962)</strong> </td>\n", " <td style=\"background-color:rgb(121,121,250);color:black\"> -0.000208 <strong>(-0.993)</strong> </td>\n", " <td> 1.2e-05 </td>\n", " </tr>\n", "</table>" ], "text/plain": [ "┌─────────────────────────────────────────────────────────────────────────┐\n", "│ Migrad │\n", "├──────────────────────────────────┬──────────────────────────────────────┤\n", "│ FCN = 3.719 │ Nfcn = 300 │\n", "│ EDM = 1.06e-17 (Goal: 0.0002) │ │\n", "├──────────────────────────────────┼──────────────────────────────────────┤\n", "│ Valid Minimum │ No Parameters at limit │\n", "├──────────────────────────────────┼──────────────────────────────────────┤\n", "│ Below EDM threshold (goal x 10) │ Below call limit │\n", "├───────────────┬──────────────────┼───────────┬─────────────┬────────────┤\n", "│ Covariance │ Hesse ok │ Accurate │ Pos. def. │ Not forced │\n", "└───────────────┴──────────────────┴───────────┴─────────────┴────────────┘\n", "┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n", "│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n", "├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n", "│ 0 │ a0 │ -0.98 │ 0.35 │ -0.34 │ 0.34 │ │ │ │\n", "│ 1 │ a1 │ 2.52 │ 0.30 │ -0.30 │ 0.30 │ │ │ │\n", "│ 2 │ a2 │ -0.48 │ 0.06 │ -0.06 │ 0.06 │ │ │ │\n", "│ 3 │ a3 │ 0.0259 │ 0.0035 │ -0.0035 │ 0.0035 │ │ │ │\n", "└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n", "┌──────────┬───────────────────────┬───────────────────────┬───────────────────────┬───────────────────────┐\n", "│ │ a0 │ a1 │ a2 │ a3 │\n", "├──────────┼───────────┬───────────┼───────────┬───────────┼───────────┬───────────┼───────────┬───────────┤\n", "│ Error │ -0.34 │ 0.34 │ -0.3 │ 0.3 │ -0.06 │ 0.06 │ -0.0035 │ 0.0035 │\n", "│ Valid │ True │ True │ True │ True │ True │ True │ True │ True │\n", "│ At Limit │ False │ False │ False │ False │ False │ False │ False │ False │\n", "│ Max FCN │ False │ False │ False │ False │ False │ False │ False │ False │\n", "│ New Min │ False │ False │ False │ False │ False │ False │ False │ False │\n", "└──────────┴───────────┴───────────┴───────────┴───────────┴───────────┴───────────┴───────────┴───────────┘\n", "┌────┬─────────────────────────────────────────┐\n", "│ │ a0 a1 a2 a3 │\n", "├────┼─────────────────────────────────────────┤\n", "│ a0 │ 0.119 -0.0961 0.0183 -0.000999 │\n", "│ a1 │ -0.0961 0.0895 -0.0179 0.000997 │\n", "│ a2 │ 0.0183 -0.0179 0.00366 -0.000208 │\n", "│ a3 │ -0.000999 0.000997 -0.000208 1.2e-05 │\n", "└────┴─────────────────────────────────────────┘" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# run minos error analysis\n", "# The Minos algorithm uses the profile likelihood method to compute\n", "# (generally asymmetric) confidence intervals.\n", "m.minos()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "<matplotlib.contour.ContourSet at 0x7fa018346ee0>" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "<Figure size 640x480 with 1 Axes>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Get a 2D contour of the function around the minimum for 2 parameters\n", "# and draw a 2 D contours up to 4 sigma of a1 and a2 \n", "#m.draw_profile(\"a1\")\n", "m.draw_mncontour(\"a2\", \"a3\", cl=[1, 2, 3, 4])" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "<ValueView a0=-0.9807363849012718 a1=2.520457666696189 a2=-0.484056651556402 a3=0.02590585286811066> <ErrorView a0=0.3452141046758837 a1=0.2991369099052406 a2=0.06052732413714454 a3=0.003463469704203838>\n" ] } ], "source": [ "print(m.values,m.errors)\n", "a0_fit = m.values[\"a0\"]\n", "a1_fit = m.values[\"a1\"]\n", "a2_fit = m.values[\"a2\"]\n", "a3_fit = m.values[\"a3\"]" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "# display fitted function \n", "x_plot = np.linspace( 0.1, 10.1 , 200 )\n", "y_fit = a0_fit + a1_fit * x_plot + a2_fit * x_plot**2 + a3_fit * x_plot**3" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAloAAAHvCAYAAACMtHGxAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjYuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/P9b71AAAACXBIWXMAAA9hAAAPYQGoP6dpAABgd0lEQVR4nO3dd3gU5f7+8femk4SE3kMnFEGQElFAegcRUCxUBQtHLEex/yTY0aOox/MVBRFUEERRRECkSQcJvffeewKkJ/P7Y8hCIAkpu5ndzf26rr12dnd25pPdbPbO88w8j80wDAMRERERcTgvqwsQERER8VQKWiIiIiJOoqAlIiIi4iQKWiIiIiJOoqAlIiIi4iQKWiIiIiJOoqAlIiIi4iQKWiIiIiJOoqAlIiIi4iQKWiK3ULlyZWw2GzabjYMHD1pdToYWL15sr7FVq1ZWl5NnrVq1sv88ixcvtrqcXDl48KD9Z6hcubLV5YiIRRS0REQycH3Yy8ll48aNltY9cuTIXNWd1UVEck9BS0Tc0vWBYuTIkVaXk2vu0GLqSjyhtVMKFh+rCxARcXVNmjQhIiIiW+uWLFnSydVkLSIigqeffjrLdb7//nsuXboEQNu2balVq1Z+lCZSICloidyCO7QytGrVCsMwrC7DYVytpaJLly45bjWrXLmyJe9Jly5d6NKlS5brzJo1yx60+vXrx6BBg/KhMpGCSV2HIiIiIk6ioCUiIiLiJApaIreQnYOVMzpA99ixY7z55pvUr1+fokWLEhAQQN26dXn77be5cuXKTdvYsWMHQ4cO5bbbbiM4OJiiRYtyzz33MHHixFt2QWVneIfMhhtYu3YtQ4YMITw8nMDAQIoWLUpERATvv/9+hnVmZ5tZye3reeNjb731lv2+t956K8Oz5azsEsvqtbn+sUOHDtnvr1KlSoY/h9VdqUlJSfzwww/06dOHqlWrUrhwYYKCgqhSpQoPP/wwv/32W7a7SaOiohg2bBgNGzakaNGi+Pr6UqxYMcLDw2nevDkvvPACf/31F3Fxcemel/ZaLFmyxH5f69atM3y9Jk6c6MgfXyRPdIyWiBP8+eef9O3blwsXLqS7f9u2bURGRvLrr7/y999/U7RoUcAMCm+//Tapqanp1l+2bBnLli1j1qxZ/PTTT3h7ezusRsMwGDlyJO+++266/cbFxREVFUVUVBTffPMNCxYsoGrVqg7br7iXxYsXM2TIEPbt23fTYwcPHuTgwYNMnTqVpk2b8ssvv1C+fPkMt5OcnMzTTz/N2LFjb3rswoULXLhwgT179rBixQo+/fRT3njjDd59912H/zwi+U1BS8TBNmzYwOuvv058fDyVK1fmrrvuIjAwkG3btrF69WoANm3axIMPPsi8efN477337AdaN2zYkHr16mGz2Vi+fDl79+4FYPr06YwaNYo33njDYXWmhTuABg0aUK9ePXx9fdm4cSPr168H4MCBA9x3332sW7cOX19fh+07L3r27EndunVZs2YNUVFRQOZnBTZt2jS/y8uWkJAQ+5mB158BOGDAAAoXLnzT+pmFF2f7+eef6du3L0lJSQAEBATQtGlTKleujLe3N7t372bVqlUkJyezevVq7rrrLqKioihduvRN2xo+fHi6kFW+fHkiIiIoUaIEhmFw7tw5tm/fzq5duzKsJe31+u233zh+/DgA9913X4avTe3atfP8s4s4jCEiWapUqZIBGIBx4MCBDNdp2bKlfR0/Pz/Dz8/PGDdunJGamppuvenTpxs+Pj72dUePHm14eXkZYWFhxooVK9Ktm5KSYrz00kv2dYODg43Lly9nuP+///7bvl7Lli0zXOfAgQPparTZbEa1atWMf/7556Z1p02bZvj6+trX/+677265zUqVKmW4zo1y+nr+/fffGa4TGRlpXycyMjJb+86J62vIzfaz+9pk5/VwtOv3OWHChAzX2bp1qxEYGGhf7/nnnzfOnTt303r79u0zmjdvbl+vc+fON61z5swZ+++9t7e3MXHixJs+G2mOHz9u/Pe//zW++eabDB/Pzu+GiCtRi5aIgyUmJjJx4kQGDhx402O9evVi8ODBfP311wC88MILBAYGMn/+fGrWrJluXS8vL0aNGsWsWbPYsWMHly9fZtasWTz44IMOqbF48eIsXbqUcuXK3fT4Aw88wJo1a/j4448BmDJlCgMGDMjzft3VnDlzOHv27C3Xe+211yxrfXK0Z599ltjYWADefffdTFtTq1atyty5c2nSpAk7duzgzz//5J9//uHOO++0r7N69WqSk5MBeOihhzL8bKQpW7YszzzzjAN/EhFrKWiJONgdd9yR5RfJgw8+aA9aAE899dRNISuNl5cXDzzwgL2LLyoqyiFBC+D111/PMGSleeyxx+xBK62LrqBKO2btVoYMGeIRQWvTpk0sWrQIgPDwcF599dUs1w8KCmLEiBE8/PDDAEyePDld0IqJibEvWz2gq0h+01mHIg7Wu3fvLB+vW7durtc/cOBA7gu7wQMPPJDl47Vq1aJQoUIAnDt3jsuXLzts3+La5syZY19+4IEHsnUSRps2bezLy5cvT/dYxYoV7cvTp0/n5MmTDqhSxD2oRUvEwW677bYsH0870zBNnTp1sr3+9S0DeREaGkpYWFiW69hsNooWLWo/zT46Oprg4GCH7N/dREZGuvV8ijm1atWqdMvDhg275XOM64Z3OHLkSLrH7rzzTipVqsShQ4c4cuQIt912GwMHDqR79+40bdrUHuhFPJGCloiDhYaGZvm4j0/6j11O1k87+yuvbrXPNNefaeiofYvrSzurD2DRokX2bsTsunFYE19fXyZNmkTXrl2JiYnh/PnzfPrpp3z66af4+fnRsGFD7rnnHrp27UqLFi2w2WwO+TlEXIG6DkUcLKdfElZ8qeiLTLISHR2dp+enpKTcdF/z5s3ZtGkTjz32GEFBQfb7ExMTWb16NR999BEtW7akVq1a/Pbbb3nav4grUdASkXx146Cs4nquD0IzZszAMIwcXzJSuXJlxo8fz5kzZ1iwYAEjRoygdevW6boOd+/eTa9evRg9erTTf06R/KCgJSK5dn3XYtrp+7eS19YScb7rBxzds2ePw7dfqFAh2rZty1tvvcWiRYs4d+4c06dPp379+vZ1XnvtNY4dO+bwfYvkNwUtEcm160cxv3Dhwi3nuzt8+LDDDuj3lO5PV/w5rh+a4a+//nL6/goVKkSvXr34+++/KVOmDGB2KWa0b1d8vUSyoqAlIrkWEhJCsWLFAIiNjWX37t1Zrj9t2jSH7TsgIMC+7M4H6rviz9GtWzf78sKFC9myZUu+7Ldo0aLcfffd9tunTp26aR1XfL1EsqKgJSJ5cv0cgxMnTsx0vaNHj/LBBx84bL/Fixe3L7tzF5Mr/hwRERG0atUKMIdt6NevX7ZbIhMTE2866/DcuXPZ3vfhw4ftyxkNbuqKr5dIVhS0RCRPHnnkEfvy6NGjmT59+k3rrF69mpYtW3LhwgWHTU5dr149+/Jff/3ltsd+Xf9zOLLFL6+++OIL+7hpmzdvJiIiggULFmS6/t69e3nvvfeoUqUKK1asuGlbDRo04Msvv+TEiRMZPv/SpUu88sorrF27FgBvb286dux403rXv16//PLLLburRaymcbREJE8efvhhPvnkEzZt2kRiYiL3338/DRs2pEGDBqSkpLB582Y2bNgAwMiRI5kwYQKHDh3K836bNGlCxYoVOXz4MCdPnqRWrVp06NCBEiVK2I/jadKkicOmLHKW3r1789VXXwEwZswY1q9fT8OGDQkMDLSvM3ToUKpVq5avddWtW5cpU6bw4IMPEhsby65du2jfvj1hYWE0adKEEiVKkJiYyJkzZ9i0aRNHjx7NcnubNm3i6aefZtiwYVSrVo26detSokQJkpKSOH78OCtXruTKlSv29V999dUMB9Xt1asXr732GoZhMHv2bG6//XbuvvvudMcLPvTQQzRu3NhxL4ZIHihoiUie+Pj48Ntvv9GuXTv2798PwPr161m/fr19HZvNxuuvv86IESOYMGGCQ/br5eXFmDFj6NW "text/plain": [ "<Figure size 640x480 with 1 Axes>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot data \n", "plt.figure()\n", "plt.errorbar(x, y, dy , dx, fmt=\"o\")\n", "plt.plot(x_plot,y_fit )\n", "plt.title(\"iminuit Fit Test\")\n", "plt.xlabel('x')\n", "plt.ylabel('f(x)')\n", "plt.xlim(-0.1, 10.1)\n", "\n", "# show the plot\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.16" } }, "nbformat": 4, "nbformat_minor": 4}
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