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   "source": [
    "Exercise 4: Least square fit to data"
   ]
  },
  {
   "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": "code",
   "execution_count": null,
   "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": null,
   "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": null,
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import Minuit object\n",
    "from iminuit import Minuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# run covariance  \n",
    "m.hesse()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#get covariance matrix\n",
    "m.covariance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#get correlation matrix in numpy array\n",
    "cov = m.covariance\n",
    "print (cov)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": []
  }
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