comparison.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.interpolate import interp1d\n",
    "import pandas as pd\n",
    "\n",
    "clight = 299792.458\n",
    "\n",
    "#MG RECONSTRUCTION PARAMETERS\n",
    "MGbins = 20\n",
    "sfac   = 10\n",
    "zmin   = 0\n",
    "zmax   = 5\n",
    "\n",
    "GRreturn = False #If set to True the last bin for mu and eta is set to 1\n",
    "                #If set to False the last bin is set to the value of the previous bin\n",
    "\n",
    "\n",
    "import camb\n",
    "\n",
    "cosmopars = {\"omch2\":0.12055785610846023,\n",
    "             \"ombh2\":0.022445,\n",
    "             \"w\":-1.0,\n",
    "             \"wa\":0.0,\n",
    "             \"H0\":67.,\n",
    "             \"ns\":0.96,\n",
    "             \"As\":2.12605e-9,\n",
    "             \"mnu\":0.06,\n",
    "             \"tau\":0.058,\n",
    "             \"num_massive_neutrinos\": 1,\n",
    "             \"nnu\": 3.046}\n",
    "\n",
    "Omegam = (cosmopars['omch2']+cosmopars['ombh2'])/(cosmopars['H0']/100)**2\n",
    "\n",
    "\n",
    "pars= camb.set_params(**cosmopars)\n",
    "\n",
    "#Computing observables\n",
    "zsamp    = np.linspace(0,5,10000)\n",
    "results  = camb.get_background(pars)\n",
    "hubble   = interp1d(zsamp,results.hubble_parameter(zsamp),kind='cubic')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "---------------ENTRIES FOR MGCAMB RECONSTRUCTION---------------\n",
      "MGbins = 20\n",
      "s_fac  = 10\n",
      "z_b    = 0.0 0.2631578947368421 0.5263157894736842 0.7894736842105263 1.0526315789473684 1.3157894736842104 1.5789473684210527 1.8421052631578947 2.1052631578947367 2.3684210526315788 2.631578947368421 2.894736842105263 3.1578947368421053 3.4210526315789473 3.6842105263157894 3.9473684210526314 4.2105263157894735 4.473684210526316 4.7368421052631575 5.0\n",
      "mu_b   = 1.297904685787941 1.2200401432968724 1.1593080364369706 1.1155108064929307 1.0848344688103075 1.0634057311469338 1.0482823879999976 1.0374381120600569 1.0295227138293717 1.0236412577917804 1.0191960971203056 1.0157827620410114 1.0131231760462882 1.0110229942853357 1.0093441716331457 1.007987115701829 1.0068789111508023 1.0059654328228929 1.0052059835881102 1.0052059835881102\n",
      "eta_b  = 1.0595809371575882 1.0440080286593745 1.031861607287394 1.023102161298586 1.0169668937620615 1.0126811462293868 1.0096564775999994 1.0074876224120113 1.0059045427658742 1.0047282515583562 1.003839219424061 1.0031565524082022 1.0026246352092578 1.002204598857067 1.001868834326629 1.001597423140366 1.0013757822301608 1.0011930865645786 1.0010411967176218 1.0010411967176218\n",
      "---------------------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "#Getting real functions\n",
    "E_11 = 0.5\n",
    "E_22 = 0.1\n",
    "mu_real  = interp1d(zsamp,1+E_11*(1-Omegam)*(cosmopars['H0']/hubble(zsamp))**2,kind='cubic')\n",
    "eta_real = interp1d(zsamp,1+E_22*(1-Omegam)*(cosmopars['H0']/hubble(zsamp))**2,kind='cubic')\n",
    "\n",
    "#Real function derivatives\n",
    "mean_z = []\n",
    "dermu_tab = []\n",
    "dereta_tab = []\n",
    "for ind in range(len(zsamp)-1):\n",
    "    mean_z.append(0.5*(zsamp[ind+1]+zsamp[ind]))\n",
    "    dermu_tab.append((mu_real(zsamp[ind+1])-mu_real(zsamp[ind]))/(zsamp[ind+1]-zsamp[ind]))\n",
    "    dereta_tab.append((eta_real(zsamp[ind+1])-eta_real(zsamp[ind]))/(zsamp[ind+1]-zsamp[ind]))\n",
    "\n",
    "\n",
    "der_mu_real = interp1d(mean_z,dermu_tab,kind='cubic')\n",
    "der_eta_real = interp1d(mean_z,dereta_tab,kind='cubic')\n",
    "\n",
    "tauder_mu_real = interp1d(mean_z,[-hubble(zi)/clight*dermu_tab[ind] for ind,zi in enumerate(mean_z)],\n",
    "                          kind='cubic')\n",
    "tauder_eta_real = interp1d(mean_z,[-hubble(zi)/clight*dereta_tab[ind] for ind,zi in enumerate(mean_z)],\n",
    "                          kind='cubic')\n",
    "\n",
    "#Binned function\n",
    "z_lim  = np.linspace(zmin,zmax,MGbins)\n",
    "z_b = []\n",
    "for ind in range(len(z_lim)-1):\n",
    "    z_b.append(0.5*(z_lim[ind+1]+z_lim[ind]))\n",
    "    \n",
    "mu_b  = mu_real(z_b)\n",
    "eta_b = eta_real(z_b)\n",
    "print('')\n",
    "print('---------------ENTRIES FOR MGCAMB RECONSTRUCTION---------------')\n",
    "print('MGbins = {}'.format(MGbins))\n",
    "print('s_fac  = {}'.format(sfac))\n",
    "print('z_b    =',*z_lim, sep=\" \")\n",
    "if GRreturn:\n",
    "    mu_b  = np.append(mu_b, 1)\n",
    "    eta_b = np.append(eta_b, 1)\n",
    "else:\n",
    "    mu_b  = np.append(mu_b, mu_b[-1])\n",
    "    eta_b = np.append(eta_b, eta_b[-1])\n",
    "print('mu_b   =',*mu_b, sep=\" \")\n",
    "print('eta_b  =',*eta_b, sep=\" \")\n",
    "print('---------------------------------------------------------------')\n",
    "\n",
    "mu  = np.zeros((len(zsamp)))\n",
    "eta = np.zeros((len(zsamp)))\n",
    "for ind,z in enumerate(zsamp):\n",
    "    if z<z_lim[0]:\n",
    "        mu[ind]  = mu_b[0]\n",
    "        eta[ind] = eta_b[0]\n",
    "    elif z>z_lim[-1]:\n",
    "        mu[ind]  = mu_b[-1]\n",
    "        eta[ind] = eta_b[-1]\n",
    "    else:\n",
    "        mu[ind]  =  mu_b[0]\n",
    "        eta[ind]  =  eta_b[0]\n",
    "        for ind2 in range(len(z_lim)-2):\n",
    "            mu[ind] += (0.5*(mu_b[ind2+1]-mu_b[ind2])*\n",
    "                           (1+np.tanh(sfac*(z-z_lim[ind2+1])/(z_lim[ind2+1]-z_lim[ind2]))))\n",
    "            eta[ind] += (0.5*(eta_b[ind2+1]-eta_b[ind2])*\n",
    "                           (1+np.tanh(sfac*(z-z_lim[ind2+1])/(z_lim[ind2+1]-z_lim[ind2]))))\n",
    "            \n",
    "            \n",
    "#Binned derivative\n",
    "\n",
    "\n",
    "der_z_lim = z_b\n",
    "der_z_b   = []\n",
    "for ind in range(len(der_z_lim)-1):\n",
    "    der_z_b.append(0.5*(der_z_lim[ind+1]+der_z_lim[ind]))\n",
    "der_mu_b  = [(mu_b[ind+1]-mu_b[ind])/(z_lim[ind+1]-z_lim[ind]) for ind in range(len(z_b)-1)]\n",
    "der_eta_b = [(eta_b[ind+1]-eta_b[ind])/(z_lim[ind+1]-z_lim[ind]) for ind in range(len(z_b)-1)]\n",
    "\n",
    "dermu  = np.zeros((len(zsamp)))\n",
    "dereta = np.zeros((len(zsamp)))\n",
    "for ind,z in enumerate(zsamp):\n",
    "    if z<der_z_lim[0]:\n",
    "        dermu[ind]  = der_mu_b[0]\n",
    "        dereta[ind] = der_eta_b[0]\n",
    "    elif z>der_z_lim[-1]:\n",
    "        dermu[ind]  = der_mu_b[-1]\n",
    "        dereta[ind] = der_eta_b[-1]\n",
    "    else:\n",
    "        dermu[ind]  =  der_mu_b[0]\n",
    "        dereta[ind]  =  der_eta_b[0]\n",
    "        for ind2 in range(len(der_z_lim)-2):\n",
    "            dermu[ind] += (0.5*(der_mu_b[ind2+1]-der_mu_b[ind2])*\n",
    "                           (1+np.tanh(sfac*(z-der_z_lim[ind2+1])/(der_z_lim[ind2+1]-der_z_lim[ind2]))))\n",
    "            dereta[ind] += (0.5*(der_eta_b[ind2+1]-der_eta_b[ind2])*\n",
    "                           (1+np.tanh(sfac*(z-der_z_lim[ind2+1])/(der_z_lim[ind2+1]-der_z_lim[ind2]))))  \n",
    "            \n",
    "taudermu = [-hubble(zi)/clight * dermu[ind] for ind,zi in enumerate(zsamp)]\n",
    "taudereta = [-hubble(zi)/clight * dereta[ind] for ind,zi in enumerate(zsamp)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#LOAD MGCAMB RESULTS\n",
    "\n",
    "MGfuncs= pd.read_csv(\"./MGCAMB/recon_MGCAMB_debug_MG_fncs.dat\",delimiter=\"   \",header=None, skiprows=1, \n",
    "                     names=['k','a','mu','gamma','mudot','gammadot','Q','R','Phi','Psi','dPhi','dPsi'])\n",
    "singlek = MGfuncs[(MGfuncs.k == MGfuncs['k'].iloc[0])]\n",
    "\n",
    "\n",
    "\n",
    "MGz_recon   = -1+1/singlek['a'].values[:]\n",
    "MGmu_recon  = singlek['mu'].values[:]\n",
    "MGeta_recon = singlek['gamma'].values[:]\n",
    "MGdermu_recon  = singlek['mudot'].values[:]\n",
    "MGdereta_recon = singlek['gammadot'].values[:]\n",
    "MGPhi_recon = singlek['Phi'].values[:]\n",
    "MGPsi_recon = singlek['Psi'].values[:]\n",
    "MGderPhi_recon = singlek['dPhi'].values[:]\n",
    "MGderPsi_recon = singlek['dPsi'].values[:]\n",
    "\n",
    "\n",
    "MGfuncs= pd.read_csv(\"./MGCAMB/pkpar_MGCAMB_debug_MG_fncs.dat\",delimiter=\"   \",header=None, skiprows=1, \n",
    "                     names=['k','a','mu','gamma','mudot','gammadot','Q','R','Phi','Psi','dPhi','dPsi'])\n",
    "singlek = MGfuncs[(MGfuncs.k == MGfuncs['k'].iloc[0])]\n",
    "\n",
    "MGpk = 0. #Here read pk\n",
    "\n",
    "\n",
    "MGz_real   = -1+1/singlek['a'].values[:]\n",
    "MGmu_real  = singlek['mu'].values[:]\n",
    "MGeta_real = singlek['gamma'].values[:]\n",
    "MGdermu_real  = singlek['mudot'].values[:]\n",
    "MGdereta_real = singlek['gammadot'].values[:]\n",
    "MGPhi_real = singlek['Phi'].values[:]\n",
    "MGPsi_real = singlek['Psi'].values[:]\n",
    "MGderPhi_real = singlek['dPhi'].values[:]\n",
    "MGderPsi_real = singlek['dPsi'].values[:]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "#Check mu eta functions\n",
    "\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Comparison $\\mu(z)$',fontsize=20)\n",
    "plt.plot(MGz_recon,MGmu_recon,label=r'MGCAMB recon',color='black',lw=3,ls=':',zorder=10)\n",
    "plt.plot(MGz_real,MGmu_real,label=r'MGCAMB $\\mu(z)=1+E_{11}\\Omega_{\\rm DE}$',color='purple',lw=3,ls=':',zorder=10)\n",
    "plt.plot(zsamp,mu,label=r'tanh reconstruction',color='#8E001C',lw=3,zorder=1)\n",
    "plt.plot(zsamp,mu_real(zsamp),label=r'$\\mu(z)=1+E_{11}\\Omega_{\\rm DE}$',color='#FFB300',lw=3,ls='-',zorder=5)\n",
    "plt.errorbar(z_b,mu_b,label=r'bins',color='green',ls='',marker='*',markersize=10,zorder=20)\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$\\mu(z)$',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Comparison $\\eta(z)$',fontsize=20)\n",
    "plt.plot(MGz_recon,MGeta_recon,label=r'MGCAMB recon',color='black',lw=3,ls=':',zorder=10)\n",
    "plt.plot(MGz_real,MGeta_real,label=r'MGCAMB $\\eta(z)=1+E_{22}\\Omega_{\\rm DE}$',color='purple',lw=3,ls=':',zorder=10)\n",
    "plt.plot(zsamp,eta,label=r'tanh reconstruction',color='#8E001C',lw=3,zorder=1)\n",
    "plt.plot(zsamp,eta_real(zsamp),label=r'$\\eta(z)=1+E_{22}\\Omega_{\\rm DE}$',color='#FFB300',lw=3,ls='-',zorder=5)\n",
    "plt.errorbar(z_b,eta_b,label=r'bins',color='green',ls='',marker='*',markersize=10,zorder=20)\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$\\eta(z)$',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Comparison code derivative $\\mu(z)$',fontsize=20)\n",
    "plt.plot(MGz_recon,MGdermu_recon,label=r'MGCAMB recon',color='black',lw=2,ls=':',zorder=10)\n",
    "plt.plot(MGz_real,MGdermu_real,label=r'MGCAMB $\\eta(z)=1+E_{22}\\Omega_{\\rm DE}$',\n",
    "         color='purple',lw=3,ls=':',zorder=10)\n",
    "plt.plot(zsamp,taudermu,label=r'tanh reconstruction',color='#8E001C',lw=3,zorder=1)\n",
    "plt.plot(mean_z,tauder_mu_real(mean_z),label=r'$\\mu(z)=1+E_{11}\\Omega_{\\rm DE}$',color='#FFB300',\n",
    "         lw=3,ls='-',zorder=5)\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$d\\mu(z)/d\\tau$',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Comparison code derivative $\\eta(z)$',fontsize=20)\n",
    "plt.plot(MGz_recon,MGdereta_recon,label=r'MGCAMB recon',color='black',lw=3,ls=':',zorder=10)\n",
    "plt.plot(MGz_real,MGdereta_real,label=r'MGCAMB $\\eta(z)=1+E_{22}\\Omega_{\\rm DE}$',\n",
    "         color='purple',lw=3,ls=':',zorder=10)\n",
    "plt.plot(zsamp,taudereta,label=r'tanh reconstruction',color='#8E001C',lw=3,zorder=1)\n",
    "plt.plot(mean_z,tauder_eta_real(mean_z),label=r'$\\mu(z)=1+E_{11}\\Omega_{\\rm DE}$',color='#FFB300',\n",
    "         lw=3,ls='-',zorder=5)\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$d\\eta(z)/d\\tau$',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Comparison derivative $\\mu(z)$',fontsize=20)\n",
    "plt.plot(zsamp,dermu,label=r'tanh reconstruction',color='#8E001C',lw=3,zorder=1)\n",
    "plt.plot(mean_z,der_mu_real(mean_z),label=r'$\\mu(z)=1+E_{11}\\Omega_{\\rm DE}$',color='#FFB300',lw=3,ls='-',zorder=5)\n",
    "plt.errorbar(der_z_b,der_mu_b,label=r'bins',color='green',ls='',marker='*',markersize=10,zorder=20)\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$d\\mu(z)$/dz',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "#plt.ylim([-2,2])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Comparison derivative $\\eta(z)$',fontsize=20)\n",
    "plt.plot(zsamp,dereta,label=r'tanh reconstruction',color='#8E001C',lw=3,zorder=1)\n",
    "plt.plot(mean_z,der_eta_real(mean_z),label=r'$\\eta(z)=1+E_{22}\\Omega_{\\rm DE}$',color='#FFB300',lw=3,ls='-',zorder=5)\n",
    "plt.errorbar(der_z_b,der_eta_b,label=r'bins',color='green',ls='',marker='*',markersize=10,zorder=20)\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$d\\eta(z)/dz$',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "#plt.ylim([-2,2])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Bardeen potentials',fontsize=20)\n",
    "plt.plot(MGz_real,MGPhi_real,color='#8E001C',lw=3,ls='-',zorder=0,label=r'$\\Phi$ model')\n",
    "plt.plot(MGz_recon,MGPhi_recon,color='black',lw=3,ls='--',zorder=10,label=r'$\\Phi$ recon')\n",
    "plt.plot(MGz_real,MGPsi_real,color='#FFB300',lw=3,ls='-',zorder=0,label=r'$\\Psi$ model')\n",
    "plt.plot(MGz_recon,MGPsi_recon,color='green',lw=3,ls=':',zorder=10,label=r'$\\Psi$ recon')\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$f(z)$',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "plt.ylim([-0.009,-0.006])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title(r'Bardeen potentials derivatives',fontsize=20)\n",
    "plt.plot(MGz_real,MGderPhi_real,color='#8E001C',lw=3,ls='-',zorder=0,label=r'$\\dot{Phi}$ model')\n",
    "plt.plot(MGz_recon,MGderPhi_recon,color='black',lw=3,ls='--',zorder=10,label=r'$\\dot{\\Phi}$ recon')\n",
    "plt.plot(MGz_real,MGderPsi_real,color='#FFB300',lw=3,ls='-',zorder=0,label=r'$\\dot{\\Psi}$ model')\n",
    "plt.plot(MGz_recon,MGderPsi_recon,color='green',lw=3,ls=':',zorder=10,label=r'$\\dot{\\Psi}$ recon')\n",
    "plt.xlabel(r'$z$',fontsize=20)\n",
    "plt.ylabel(r'$f(z)$',fontsize=20)\n",
    "plt.xlim([0,z_lim[-1]])\n",
    "plt.ylim([-0.5e-6,0.5e-6])\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "recon_CMB = np.loadtxt('./MGCAMB/recon_lensedCls.dat')\n",
    "recon_Pk  = np.loadtxt('./MGCAMB/recon_matterpower.dat')\n",
    "\n",
    "\n",
    "pkpar_CMB = np.loadtxt('./MGCAMB/pkpar_lensedCls.dat')\n",
    "pkpar_Pk  = np.loadtxt('./MGCAMB/pkpar_matterpower.dat')\n",
    "\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title('CMB spectra',fontsize=20)\n",
    "plt.plot(pkpar_CMB[:,0],100*(recon_CMB[:,1]-pkpar_CMB[:,1])/pkpar_CMB[:,1],color='#DA0914',\n",
    "         ls='-',zorder=0,label=r'TT',lw=3)\n",
    "plt.plot(pkpar_CMB[:,0],100*(recon_CMB[:,2]-pkpar_CMB[:,2])/pkpar_CMB[:,2],color='#FCDE03',\n",
    "         ls='-',zorder=0,label=r'EE',lw=3)\n",
    "plt.plot(pkpar_CMB[:,0],100*(recon_CMB[:,3]-pkpar_CMB[:,3])/pkpar_CMB[:,3],color='#630356',\n",
    "         ls='-',zorder=0,label=r'BB',lw=3)\n",
    "plt.xlabel(r'$\\ell$',fontsize=20)\n",
    "plt.ylabel(r'$[C_\\ell^{\\rm recon}-C_\\ell^{\\rm latepar}]/C_\\ell^{\\rm latepar}$ [%]',fontsize=20)\n",
    "#plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)\n",
    "\n",
    "\n",
    "\n",
    "plt.figure(figsize=(10, 8))\n",
    "plt.title('Matter power spectrum',fontsize=20)\n",
    "plt.plot(pkpar_Pk[:,0],100*(recon_Pk[:,1]-pkpar_Pk[:,1])/pkpar_Pk[:,1],color='#8E001C',ls='-',\n",
    "         zorder=0,label=r'Planck par',lw=3)\n",
    "plt.xlabel(r'$k$',fontsize=20)\n",
    "plt.ylabel(r'$[P^{\\rm recon}(k)-P^{\\rm latepar}(k)]/P^{\\rm latepar}(k)$ [%]',fontsize=20)\n",
    "#plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.legend(bbox_to_anchor=(1.04,0.5), loc=\"center left\",fontsize=20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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