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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Exercise 3: Least square fit with a 3rd order polynomial with iminuit"
]
},
{
"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import math"
]
},
{
"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
"outputs": [],
"source": [
"# Data x,y and dy\n",
"x = np.array([0.2 , 0.4 , 0.6 , 0.8, 1. , 1.2, 1.4, 1.6, 1.8, 2., 2.2, 2.4, 2.6, 2.8 , 3., 3.2 ,3.4, 3.6, 3.8,4.],dtype='d')\n",
"dy = np.array([0.04,0.021,0.035,0.03,0.029,0.019,0.024,0.018,0.019,0.022,0.02,0.025,0.018,0.024,0.019,0.021,0.03,0.019,0.03,0.024 ], dtype='d')\n",
"y = np.array([1.792,1.695,1.541,1.514,1.427,1.399,1.388,1.270,1.262,1.228,1.189,1.182,1.121,1.129,1.124,1.089,1.092,1.084,1.058,1.057 ], dtype='d')"
]
},
{
"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
"outputs": [],
"source": [
"# Define fit functions - a 3rd order polynomial\n",
"def pol3(a0, a1, a2, a3):\n",
" return a0 + x*a1 + a2*x**2 + a3*x**3"
]
},
{
"cell_type": "code",
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"execution_count": 4,
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"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",
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"execution_count": 5,
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"metadata": {},
"outputs": [],
"source": [
"# import minuit2 fitting library\n",
"from iminuit import Minuit"
]
},
{
"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
"outputs": [],
"source": [
"# create instance of Minuit and use LSQ function to minimize\n",
"LSQ.errordef = Minuit.LEAST_SQUARES\n",
"m = Minuit(LSQ,a0=0.01, a1=0.05 ,a2=0.01 ,a3=0.001)"
]
},
{
"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
{
"data": {
"text/html": [
"<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> 10.0e-3 </td>\n",
" <td> 0.1e-3 </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> 50.0e-3 </td>\n",
" <td> 0.5e-3 </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> 10.0e-3 </td>\n",
" <td> 0.1e-3 </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> 1.00e-3 </td>\n",
" <td> 0.01e-3 </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" </tr>\n",
"</table>"
],
"text/plain": [
"┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
"├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
"│ 0 │ a0 │ 10.0e-3 │ 0.1e-3 │ │ │ │ │ │\n",
"│ 1 │ a1 │ 50.0e-3 │ 0.5e-3 │ │ │ │ │ │\n",
"│ 2 │ a2 │ 10.0e-3 │ 0.1e-3 │ │ │ │ │ │\n",
"│ 3 │ a3 │ 1.00e-3 │ 0.01e-3 │ │ │ │ │ │\n",
"└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
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"source": [
"m.params"
]
},
{
"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Chi2/ndof = 0.8205035668806628\n"
]
}
],
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"source": [
"# run migrad for minimization\n",
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"m.migrad()\n",
"chi2 = m.fval / (len(y) - len(m.values))\n",
"print (\"Chi2/ndof =\" , chi2)"
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]
},
{
"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"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 = 13.13 </td>\n",
" <td colspan=\"3\" style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 120 </td>\n",
" </tr>\n",
" <tr>\n",
" <td colspan=\"2\" style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 7.21e-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> 1.884 </td>\n",
" <td> 0.031 </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> -0.56 </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> 2 </th>\n",
" <td> a2 </td>\n",
" <td> 0.136 </td>\n",
" <td> 0.030 </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> -0.012 </td>\n",
" <td> 0.005 </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> 0.000952 </td>\n",
" <td style=\"background-color:rgb(131,131,250);color:black\"> -0.0016 <strong>(-0.918)</strong> </td>\n",
" <td style=\"background-color:rgb(250,127,127);color:black\"> 0.000752 <strong>(0.822)</strong> </td>\n",
" <td style=\"background-color:rgb(153,153,250);color:black\"> -0.000105 <strong>(-0.746)</strong> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> a1 </th>\n",
" <td style=\"background-color:rgb(131,131,250);color:black\"> -0.0016 <strong>(-0.918)</strong> </td>\n",
" <td> 0.00317 </td>\n",
" <td style=\"background-color:rgb(123,123,250);color:black\"> -0.00163 <strong>(-0.975)</strong> </td>\n",
" <td style=\"background-color:rgb(250,110,110);color:black\"> 0.00024 <strong>(0.930)</strong> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> a2 </th>\n",
" <td style=\"background-color:rgb(250,127,127);color:black\"> 0.000752 <strong>(0.822)</strong> </td>\n",
" <td style=\"background-color:rgb(123,123,250);color:black\"> -0.00163 <strong>(-0.975)</strong> </td>\n",
" <td> 0.000879 </td>\n",
" <td style=\"background-color:rgb(122,122,250);color:black\"> -0.000134 <strong>(-0.988)</strong> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> a3 </th>\n",
" <td style=\"background-color:rgb(153,153,250);color:black\"> -0.000105 <strong>(-0.746)</strong> </td>\n",
" <td style=\"background-color:rgb(250,110,110);color:black\"> 0.00024 <strong>(0.930)</strong> </td>\n",
" <td style=\"background-color:rgb(122,122,250);color:black\"> -0.000134 <strong>(-0.988)</strong> </td>\n",
" <td> 2.1e-05 </td>\n",
" </tr>\n",
"</table>"
],
"text/plain": [
"┌─────────────────────────────────────────────────────────────────────────┐\n",
"│ Migrad │\n",
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
"│ FCN = 13.13 │ Nfcn = 120 │\n",
"│ EDM = 7.21e-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 │ 1.884 │ 0.031 │ │ │ │ │ │\n",
"│ 1 │ a1 │ -0.56 │ 0.06 │ │ │ │ │ │\n",
"│ 2 │ a2 │ 0.136 │ 0.030 │ │ │ │ │ │\n",
"│ 3 │ a3 │ -0.012 │ 0.005 │ │ │ │ │ │\n",
"└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
"┌────┬─────────────────────────────────────────┐\n",
"│ │ a0 a1 a2 a3 │\n",
"├────┼─────────────────────────────────────────┤\n",
"│ a0 │ 0.000952 -0.0016 0.000752 -0.000105 │\n",
"│ a1 │ -0.0016 0.00317 -0.00163 0.00024 │\n",
"│ a2 │ 0.000752 -0.00163 0.000879 -0.000134 │\n",
"│ a3 │ -0.000105 0.00024 -0.000134 2.1e-05 │\n",
"└────┴─────────────────────────────────────────┘"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
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"source": [
"# run covariance \n",
"m.hesse()"
]
},
{
"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"┌────┬─────────────────────────────────────────┐\n",
"│ │ a0 a1 a2 a3 │\n",
"├────┼─────────────────────────────────────────┤\n",
"│ a0 │ 0.000952 -0.0016 0.000752 -0.000105 │\n",
"│ a1 │ -0.0016 0.00317 -0.00163 0.00024 │\n",
"│ a2 │ 0.000752 -0.00163 0.000879 -0.000134 │\n",
"│ a3 │ -0.000105 0.00024 -0.000134 2.1e-05 │\n",
"└────┴─────────────────────────────────────────┘\n"
]
}
],
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"source": [
"#get correlation matrix\n",
"cov = m.covariance\n",
"print (cov)"
]
},
{
"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"-0.0015953258466008735\n",
"0.0007515996501623164\n"
]
}
],
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"source": [
"# access elements of the numpy arrays\n",
"print(cov[0, 1])\n",
"print(cov[0, 2])"
]
},
{
"cell_type": "code",
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"execution_count": 12,
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"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 = 13.13 </td>\n",
" <td colspan=\"3\" style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 280 </td>\n",
" </tr>\n",
" <tr>\n",
" <td colspan=\"2\" style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 7.21e-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> 1.884 </td>\n",
" <td> 0.031 </td>\n",
" <td> -0.031 </td>\n",
" <td> 0.031 </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> 1 </th>\n",
" <td> a1 </td>\n",
" <td> -0.56 </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> 2 </th>\n",
" <td> a2 </td>\n",
" <td> 0.136 </td>\n",
" <td> 0.030 </td>\n",
" <td> -0.030 </td>\n",
" <td> 0.030 </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" <td> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> 3 </th>\n",
" <td> a3 </td>\n",
" <td> -0.012 </td>\n",
" <td> 0.005 </td>\n",
" <td> -0.005 </td>\n",
" <td> 0.005 </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> -0.031 </td>\n",
" <td> 0.031 </td>\n",
" <td> -0.06 </td>\n",
" <td> 0.06 </td>\n",
" <td> -0.03 </td>\n",
" <td> 0.03 </td>\n",
" <td> -0.005 </td>\n",
" <td> 0.005 </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> 0.000952 </td>\n",
" <td style=\"background-color:rgb(131,131,250);color:black\"> -0.0016 <strong>(-0.918)</strong> </td>\n",
" <td style=\"background-color:rgb(250,127,127);color:black\"> 0.000752 <strong>(0.822)</strong> </td>\n",
" <td style=\"background-color:rgb(153,153,250);color:black\"> -0.000105 <strong>(-0.746)</strong> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> a1 </th>\n",
" <td style=\"background-color:rgb(131,131,250);color:black\"> -0.0016 <strong>(-0.918)</strong> </td>\n",
" <td> 0.00317 </td>\n",
" <td style=\"background-color:rgb(123,123,250);color:black\"> -0.00163 <strong>(-0.975)</strong> </td>\n",
" <td style=\"background-color:rgb(250,110,110);color:black\"> 0.00024 <strong>(0.930)</strong> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> a2 </th>\n",
" <td style=\"background-color:rgb(250,127,127);color:black\"> 0.000752 <strong>(0.822)</strong> </td>\n",
" <td style=\"background-color:rgb(123,123,250);color:black\"> -0.00163 <strong>(-0.975)</strong> </td>\n",
" <td> 0.000879 </td>\n",
" <td style=\"background-color:rgb(122,122,250);color:black\"> -0.000134 <strong>(-0.988)</strong> </td>\n",
" </tr>\n",
" <tr>\n",
" <th> a3 </th>\n",
" <td style=\"background-color:rgb(153,153,250);color:black\"> -0.000105 <strong>(-0.746)</strong> </td>\n",
" <td style=\"background-color:rgb(250,110,110);color:black\"> 0.00024 <strong>(0.930)</strong> </td>\n",
" <td style=\"background-color:rgb(122,122,250);color:black\"> -0.000134 <strong>(-0.988)</strong> </td>\n",
" <td> 2.1e-05 </td>\n",
" </tr>\n",
"</table>"
],
"text/plain": [
"┌─────────────────────────────────────────────────────────────────────────┐\n",
"│ Migrad │\n",
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
"│ FCN = 13.13 │ Nfcn = 280 │\n",
"│ EDM = 7.21e-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 │ 1.884 │ 0.031 │ -0.031 │ 0.031 │ │ │ │\n",
"│ 1 │ a1 │ -0.56 │ 0.06 │ -0.06 │ 0.06 │ │ │ │\n",
"│ 2 │ a2 │ 0.136 │ 0.030 │ -0.030 │ 0.030 │ │ │ │\n",
"│ 3 │ a3 │ -0.012 │ 0.005 │ -0.005 │ 0.005 │ │ │ │\n",
"└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
"┌──────────┬───────────────────────┬───────────────────────┬───────────────────────┬───────────────────────┐\n",
"│ │ a0 │ a1 │ a2 │ a3 │\n",
"├──────────┼───────────┬───────────┼───────────┬───────────┼───────────┬───────────┼───────────┬───────────┤\n",
"│ Error │ -0.031 │ 0.031 │ -0.06 │ 0.06 │ -0.03 │ 0.03 │ -0.005 │ 0.005 │\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 │ 0.000952 -0.0016 0.000752 -0.000105 │\n",
"│ a1 │ -0.0016 0.00317 -0.00163 0.00024 │\n",
"│ a2 │ 0.000752 -0.00163 0.000879 -0.000134 │\n",
"│ a3 │ -0.000105 0.00024 -0.000134 2.1e-05 │\n",
"└────┴─────────────────────────────────────────┘"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
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"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",
examples are automatically written updated 03_ml_intro.odp
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"execution_count": 13,
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"metadata": {},
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"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.contour.ContourSet at 0x7f783a26f280>"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
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"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_mncontour(\"a1\", \"a2\", cl=[1, 2, 3, 4])\n"
]
},
{
"cell_type": "code",
examples are automatically written updated 03_ml_intro.odp
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"execution_count": 14,
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"metadata": {},
examples are automatically written updated 03_ml_intro.odp
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"outputs": [
{
"data": {
"text/plain": [
"(array([0.07628965, 0.07748762, 0.07868559, 0.07988356, 0.08108153,\n",
" 0.0822795 , 0.08347748, 0.08467545, 0.08587342, 0.08707139,\n",
" 0.08826936, 0.08946733, 0.0906653 , 0.09186328, 0.09306125,\n",
" 0.09425922, 0.09545719, 0.09665516, 0.09785313, 0.0990511 ,\n",
" 0.10024908, 0.10144705, 0.10264502, 0.10384299, 0.10504096,\n",
" 0.10623893, 0.1074369 , 0.10863488, 0.10983285, 0.11103082,\n",
" 0.11222879, 0.11342676, 0.11462473, 0.1158227 , 0.11702068,\n",
" 0.11821865, 0.11941662, 0.12061459, 0.12181256, 0.12301053,\n",
" 0.12420851, 0.12540648, 0.12660445, 0.12780242, 0.12900039,\n",
" 0.13019836, 0.13139633, 0.13259431, 0.13379228, 0.13499025,\n",
" 0.13618822, 0.13738619, 0.13858416, 0.13978213, 0.14098011,\n",
" 0.14217808, 0.14337605, 0.14457402, 0.14577199, 0.14696996,\n",
" 0.14816793, 0.14936591, 0.15056388, 0.15176185, 0.15295982,\n",
" 0.15415779, 0.15535576, 0.15655373, 0.15775171, 0.15894968,\n",
" 0.16014765, 0.16134562, 0.16254359, 0.16374156, 0.16493953,\n",
" 0.16613751, 0.16733548, 0.16853345, 0.16973142, 0.17092939,\n",
" 0.17212736, 0.17332533, 0.17452331, 0.17572128, 0.17691925,\n",
" 0.17811722, 0.17931519, 0.18051316, 0.18171113, 0.18290911,\n",
" 0.18410708, 0.18530505, 0.18650302, 0.18770099, 0.18889896,\n",
" 0.19009693, 0.19129491, 0.19249288, 0.19369085, 0.19488882]),\n",
" array([7.76355388e+03, 7.45301172e+03, 7.14880716e+03, 6.85094020e+03,\n",
" 6.55941083e+03, 6.27421905e+03, 5.99536487e+03, 5.72284829e+03,\n",
" 5.45666930e+03, 5.19682790e+03, 4.94332410e+03, 4.69615790e+03,\n",
" 4.45532929e+03, 4.22083827e+03, 3.99268485e+03, 3.77086903e+03,\n",
" 3.55539080e+03, 3.34625016e+03, 3.14344712e+03, 2.94698168e+03,\n",
" 2.75685383e+03, 2.57306357e+03, 2.39561091e+03, 2.22449585e+03,\n",
" 2.05971838e+03, 1.90127850e+03, 1.74917622e+03, 1.60341154e+03,\n",
" 1.46398445e+03, 1.33089495e+03, 1.20414305e+03, 1.08372875e+03,\n",
" 9.69652035e+02, 8.61912920e+02, 7.60511400e+02, 6.65447475e+02,\n",
" 5.76721145e+02, 4.94332410e+02, 4.18281270e+02, 3.48567725e+02,\n",
" 2.85191775e+02, 2.28153420e+02, 1.77452660e+02, 1.33089495e+02,\n",
" 9.50639250e+01, 6.33759500e+01, 3.80255700e+01, 1.90127850e+01,\n",
" 6.33759500e+00, 8.32667268e-11, 0.00000000e+00, 6.33759500e+00,\n",
" 1.90127850e+01, 3.80255700e+01, 6.33759500e+01, 9.50639250e+01,\n",
" 1.33089495e+02, 1.77452660e+02, 2.28153420e+02, 2.85191775e+02,\n",
" 3.48567725e+02, 4.18281270e+02, 4.94332410e+02, 5.76721145e+02,\n",
" 6.65447475e+02, 7.60511400e+02, 8.61912920e+02, 9.69652035e+02,\n",
" 1.08372875e+03, 1.20414305e+03, 1.33089495e+03, 1.46398445e+03,\n",
" 1.60341154e+03, 1.74917622e+03, 1.90127850e+03, 2.05971838e+03,\n",
" 2.22449585e+03, 2.39561091e+03, 2.57306357e+03, 2.75685383e+03,\n",
" 2.94698168e+03, 3.14344712e+03, 3.34625016e+03, 3.55539080e+03,\n",
" 3.77086903e+03, 3.99268485e+03, 4.22083827e+03, 4.45532929e+03,\n",
" 4.69615790e+03, 4.94332410e+03, 5.19682790e+03, 5.45666930e+03,\n",
" 5.72284829e+03, 5.99536487e+03, 6.27421905e+03, 6.55941083e+03,\n",
" 6.85094020e+03, 7.14880716e+03, 7.45301172e+03, 7.76355388e+03]))"
]
},
"execution_count": 14,
"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"
}
],
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"source": [
"m.draw_profile(\"a2\",subtract_min=True)"
]
},
{
"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"-0.029647391105505568\n",
"0.029647391215961143\n"
]
}
],
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"source": [
"# access fit results by parameter name and get minos asymetric errors\n",
"print (m.merrors['a2'].lower)\n",
"print (m.merrors['a2'].upper)"
]
},
{
"cell_type": "code",
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"execution_count": 16,
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"metadata": {},
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"<ValueView a0=1.884312126113227 a1=-0.555095569066645 a2=0.135589233702581 a3=-0.012129739293476468> <ErrorView a0=0.030846480016973075 a1=0.056345427583826234 a2=0.029649793193055433 a3=0.004578762777736633>\n",
"<ErrorView a0=0.030846480016973075 a1=0.056345427583826234 a2=0.029649793193055433 a3=0.004578762777736633>\n"
]
}
],
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"source": [
"# more print out\n",
"print (m.values,m.errors)\n",
"print (m.errors)"
]
},
{
"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
"outputs": [],
"source": [
"# Access fit results\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",
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"execution_count": 18,
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"metadata": {},
"outputs": [],
"source": [
"# display fitted function \n",
"x_plot = np.linspace( 0.1, 4.1 , 100 )\n",
"y_fit = a0_fit + a1_fit * x_plot + a2_fit * x_plot**2 + a3_fit * x_plot**3\n"
]
},
{
"cell_type": "code",
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"execution_count": 19,
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"metadata": {},
examples are automatically written updated 03_ml_intro.odp
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"outputs": [
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
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"source": [
"plt.figure()\n",
"\n",
"plt.errorbar(x, y, dy , fmt=\"o\")\n",
"plt.plot(x_plot,y_fit ) \n",
"plt.xlabel('x')\n",
"plt.ylabel('f(x)')\n",
"plt.title('iminuit exponential Fit')\n",
"#plt.axis([0,30,-1.2,1.2])\n",
"\n",
"# show the plot\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
examples are automatically written updated 03_ml_intro.odp
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"name": "python3"
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},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
examples are automatically written updated 03_ml_intro.odp
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"version": "3.8.16"
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}
},
"nbformat": 4,
"nbformat_minor": 4
}
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