ML-Kurs-SS2023/notebooks/02_fit_exp_fit_iMinuit.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example fit for the usage of iminuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "plt.rcParams[\"font.size\"] = 20\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Data "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "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": "markdown",
   "metadata": {},
   "source": [
    "Define fit functions -an exponential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def xp(a, b , c):\n",
    "    return a * np.exp(b*x) + c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "least-squares function: sum of data residuals squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def LS(a,b,c):\n",
    "    return np.sum((y - xp(a,b,c)) ** 2 / dy ** 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "import Minuit object"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from iminuit import Minuit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Minuit instance using LS function to minimize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "LS.errordef = Minuit.LEAST_SQUARES\n",
    "m = Minuit(LS, a=0.9, b=-0.7 , c=0.95)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run migrad , parameter c is now fixed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.fixed[\"c\"] = True\n",
    "m.migrad()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "release fix on \"c\" and minimize again"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.fixed[\"c\"] = False\n",
    "m.migrad()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Get covariance information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.hesse()\n",
    "m.params\n",
    "m.covariance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Get a 2D contour of the function around the minimum for 2 parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.minos()\n",
    "print (m.merrors['a']) # Print control information of parameter a\n",
    "m.draw_profile('b', subtract_min=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Minos algorithm uses the profile likelihood method to compute (generally asymmetric) confidence intervals. This can be plotted"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m.draw_mncontour('a', 'b', cl=[1,2,3,4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Access fit results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(m.values,m.errors)\n",
    "print (m.merrors['a'])\n",
    "print (m.merrors['a'].lower)\n",
    "print (m.merrors['a'].upper)\n",
    "a_fit = m.values[\"a\"]\n",
    "b_fit = m.values[\"b\"]\n",
    "c_fit = m.values[\"c\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Prepare data to display fitted function "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_plot = np.linspace( 0.1, 4.5 , 100 )\n",
    "y_fit =  a_fit * np.exp(b_fit*x_plot) + c_fit "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "plot data and fit results with matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.errorbar(x, y, dy , fmt=\"o\")\n",
    "plt.plot(x_plot, y_fit)\n",
    "plt.title(\"iminuit exponential Fit\")\n",
    "plt.xlim(-0.1, 4.1)\n",
    "plt.show()"
   ]
  }
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