{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "german-smith",
   "metadata": {},
   "source": [
    "**NICER Tutorial**\n",
    "\n",
    "**By David A. Espinoza-Galeas**\n",
    "\n",
    "- This is a very introductory NICER tutorial with the minimum tasks you will need to start to work with NICER observations in SciServer.\n",
    "- The tutorial is intended for people that is not familiar with NICER, maybe students or researchers trying to give a try to NICER observations without getting too deep in details.\n",
    "- To get started in SciServer please go to the nice tutorial at the HEASARC: https://heasarc.gsfc.nasa.gov/docs/sciserver/\n",
    "- Any command line with a \"!\" before is a line that you can execute in the regular terminal at SciServer.\n",
    "- Feel free to share if you want.\n",
    "- If you have question or comments related with this tutorial write to 84espinozagaleas@cua.edu .\n",
    "\n",
    "**I hope you enjoy it!**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "continent-accountability",
   "metadata": {},
   "source": [
    "# Importing Modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "tested-optimum",
   "metadata": {},
   "outputs": [],
   "source": [
    "import os\n",
    "import sys\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import xspec\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "from astropy.io import fits\n",
    "import astropy.time as Time\n",
    "import astropy.units as u"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "single-penny",
   "metadata": {},
   "source": [
    "# Installing background estimators\n",
    "\n",
    "- Most of this info is in the NICER background webpage at the HEASARC: https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer_bkg_est_tools.html\n",
    "\n",
    "## Space Weather background estimator (SW)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "parallel-helicopter",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Download the nicer_bkg_estimator module\n",
    "!wget https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer_bkg_estimator_v0p6.tar.gz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "rubber-inclusion",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Untar the file\n",
    "!tar zxvf nicer_bkg_estimator_v0p6.tar.gz"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "complimentary-leone",
   "metadata": {},
   "source": [
    "- You can leave the folder in the same folder of your <code>Jupyter</code> notebook or in your ${\\tt \\$PYTHONPATH}$ (as documentation suggests)\n",
    "\n",
    "## NICER background generator (3C50)\n",
    "\n",
    "**Step 1: Get the reference model file**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "suspected-installation",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Download Model reference\n",
    "!wget https://heasarc.gsfc.nasa.gov/docs/nicer/tools/bg_models_3C50.tar.gz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "expired-constitution",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Untar files\n",
    "!tar zxfv bg_models_3C50.tar.gz"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eligible-scanner",
   "metadata": {},
   "source": [
    "- You can put the folder any place, just make sure you remember it.\n",
    "\n",
    "**Step 2: Get and install the software**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "interior-pioneer",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Where is heasoft nicer folder\n",
    "!ls $HEADAS/nicer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "smart-finder",
   "metadata": {},
   "outputs": [],
   "source": [
    "# If there is not \"tasks\" folder create one in the right place\n",
    "!mkdir $HEADAS/nicer/x86_64-pc-linux-gnu-libc2.27/tasks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "concerned-regulation",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We are going to move, but first save your working directory\n",
    "work_dir = !pwd\n",
    "\n",
    "# Move to the folder using the \"os\" module\n",
    "os.chdir('/opt/heasoft/nicer/x86_64-pc-linux-gnu-libc2.27/tasks')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "shared-burns",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Download the software\n",
    "!wget https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nibackgen3C50_v6f.tar.gz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "occupational-accident",
   "metadata": {},
   "outputs": [],
   "source": [
    "#untar files\n",
    "!tar zxvf $HEADAS/nicer/x86_64-pc-linux-gnu-libc2.27/tasks/nibackgen3C50_v6f.tar.gz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "genuine-amazon",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Move to the nibackgen3C50 folder using the \"os\" module\n",
    "os.chdir('/opt/heasoft/nicer/x86_64-pc-linux-gnu-libc2.27/tasks/nibackgen3C50/')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "conceptual-carpet",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Run the hmake\n",
    "!hmake &> hmake.log"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "original-discipline",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Install\n",
    "!hmake install &> hmake_install.log"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "severe-lightweight",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Comeback to your working directory \n",
    "os.chdir(work_dir[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "direct-update",
   "metadata": {},
   "source": [
    "# Download Observations\n",
    "\n",
    "- If you have the OBS ID and the month where the observations was allocated you can just copy the obervation folder in to your working directory."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "stuffed-costume",
   "metadata": {},
   "outputs": [],
   "source": [
    "!cp -a /home/idies/workspace/headata/FTP/nicer/data/obs/2019_05//2612011001/ ./"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "hairy-moisture",
   "metadata": {},
   "source": [
    "# Procesing NICER observation\n",
    "\n",
    "- First apply the recommended calibration process.\n",
    "\n",
    "- If you are working in a ${\\tt Jupyter}$ notebook I highly encourage you to dump the termial output into a log file.\n",
    "\n",
    "- Give it some time, can take a while."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "circular-layer",
   "metadata": {},
   "outputs": [],
   "source": [
    "!nicerl2 indir=\"./2612011001\" clobber=\"yes\" cor_range=\"1-15\" &> nicerl2.log"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "organizational-replication",
   "metadata": {},
   "source": [
    "- Create a ${\\tt mkf2}$ since you are going to need it to apply SW background estimator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "organizational-forward",
   "metadata": {},
   "outputs": [],
   "source": [
    "!niprefilter2 indir=\"./2612011001\" infile=\"./2612011001/auxil/ni2612011001.mkf\" \\\n",
    "outfile=\"./2612011001/auxil/ni2612011001.mkf2\" clobber=\"yes\" &> niprefilter2.log # &> /dev/null"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "introductory-thesis",
   "metadata": {},
   "source": [
    "NOTE:\n",
    "- The best way to run any <code>heasoft</code> tool in <code>Python</code> is using  the ${\\tt heasoftpy}$ module.\n",
    "- You can run ${\\tt heasoftpy}$ in a <code>IPython</code> terminal, but unfortunately at this moment, it is not working in <code>Jupyter</code> notebooks at SciServer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "accessible-guatemala",
   "metadata": {},
   "outputs": [],
   "source": [
    "# If you want to try the heasoftpy uncomment the next lines\n",
    "#import heasoftpy as hsp\n",
    "#res = hsp.nicerl2(indir=\"./2612011001\", clobber=\"yes\", cor_range=\"1-15\")\n",
    "#print(res.stdout)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "optional-burden",
   "metadata": {},
   "source": [
    "# Creating an Image, Lightcurve, and Spectrum\n",
    "\n",
    "- Unfortunately SciServer lack the capabilty to make you interact with a ${\\tt WX}$ server to watch LC and do cursor time filter with ${\\tt Xselect}$.\n",
    "\n",
    "- But you can write a ${\\tt Xselect}$ script to create products, take a look, and comeback to ${\\tt Xselect}$ to make a keyborad time filter. \n",
    "\n",
    "- For more about keyboard time filter see: https://heasarc.gsfc.nasa.gov/docs/asca/abc/node8.html#SECTION00852000000000000000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "desirable-ranch",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write the script\n",
    "script_xselect = open('script_xselect.xsel', 'w')\n",
    "script_xselect.write('session1\\n')\n",
    "script_xselect.write('read events ./2612011001/xti/event_cl/ni2612011001_0mpu7_cl.evt\\n')\n",
    "script_xselect.write('./\\n')\n",
    "script_xselect.write('yes\\n')\n",
    "script_xselect.write('extract all\\n')\n",
    "script_xselect.write('save all test\\n')\n",
    "script_xselect.write('exit\\n')\n",
    "script_xselect.write('no\\n')\n",
    "script_xselect.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "spoken-heater",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Run the script\n",
    "!xselect @ script_xselect.xsel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "formed-background",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Open and getting LC using astropy.fits\n",
    "fits_lc = fits.open('test.lc')\n",
    "lc_header = fits_lc[0].header\n",
    "lc = fits_lc[1].data\n",
    "date = lc_header['DATE']\n",
    "obj = lc_header['OBJECT']\n",
    "exposure = lc_header['EXPOSURE']\n",
    "tstart = float(lc_header['TSTART'])\n",
    "fits_lc.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "seeing-butter",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Take a look at using matplotlib\n",
    "fig, ax = plt.subplots(figsize=(8,8))\n",
    "ax.plot(lc['TIME'], lc['RATE'])\n",
    "ax.set_ylabel('Rates (cnts/s)', fontsize=16)\n",
    "ax.set_xlabel('Time (seconds)', fontsize=16)\n",
    "ax.set_title(obj+' Date='+date+' Exposure='+str(exposure))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "communist-anthropology",
   "metadata": {},
   "outputs": [],
   "source": [
    "# If you want to make a keyboard time filter here is an example\n",
    "to = tstart+0.0\n",
    "tf = tstart+1000.0\n",
    "script_xselect = open('script_xselect.xsel', 'w')\n",
    "script_xselect.write('session1\\n')\n",
    "script_xselect.write('read events ./2612011001/xti/event_cl/ni2612011001_0mpu7_cl.evt\\n')\n",
    "script_xselect.write('./\\n')\n",
    "script_xselect.write('yes\\n')\n",
    "script_xselect.write('filter time scc\\n')\n",
    "script_xselect.write(str(to)+','+str(tf)+'\\n')\n",
    "script_xselect.write('x\\n')\n",
    "script_xselect.write('extract all\\n')\n",
    "script_xselect.write('save all test2\\n')\n",
    "script_xselect.write('exit\\n')\n",
    "script_xselect.write('no\\n')\n",
    "script_xselect.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "alike-herald",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Run the script to apply filter\n",
    "!xselect @ script_xselect.xsel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "derived-happiness",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Open a getting LC (again)\n",
    "fits_lc2 = fits.open('test2.lc')\n",
    "lc_header2 = fits_lc2[0].header\n",
    "lc2 = fits_lc2[1].data\n",
    "date2 = lc_header2['DATE']\n",
    "obj2 = lc_header2['OBJECT']\n",
    "exposure2 = lc_header2['EXPOSURE']\n",
    "fits_lc2.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plain-video",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Taking a new look at\n",
    "fig, ax = plt.subplots(figsize=(8,8))\n",
    "ax.plot(lc2['TIME'], lc2['RATE'])\n",
    "ax.set_ylabel('Rates (cnts/s)', fontsize=16)\n",
    "ax.set_xlabel('Time (seconds)', fontsize=16)\n",
    "ax.set_title(obj2+' Date='+date2+' Exposure='+str(exposure2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cross-rotation",
   "metadata": {},
   "source": [
    "# Estimating background"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "juvenile-guidance",
   "metadata": {},
   "source": [
    "## Space Weather (SW) estimator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "tropical-tuesday",
   "metadata": {},
   "outputs": [],
   "source": [
    "#reading space weather python module\n",
    "from nicergof.bkg import bkg_estimator as bk"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "royal-heating",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating mkf3\n",
    "mkf2 = '2612011001/auxil/ni2612011001.mkf2'\n",
    "bk.add_kp(mkf2, clobber=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "composite-banks",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Estimating background\n",
    "pha = 'test.pha'\n",
    "mkf3 = '2612011001/auxil/ni2612011001.mkf3'\n",
    "bevt = '/home/idies/workspace/headata/FTP/caldb/data/nicer/xti/pcf/30nov18targskc_enhanced.evt'\n",
    "bkg_chan, bkgspectot, btotexpo = bk.mk_bkg_spec_evt(pha, mkf3, bevt=bevt)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "informed-cocktail",
   "metadata": {},
   "source": [
    "## NICER Background Generator (3C50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fundamental-membership",
   "metadata": {},
   "outputs": [],
   "source": [
    "!nibackgen3C50 rootdir=\"./\" obsid=\"2612011001\"  bkgidxdir=\"./bg_models_3C50\" bkglibdir=\"./bg_models_3C50\" gainepoch=\"2020\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "asian-haiti",
   "metadata": {},
   "source": [
    "# Reading spectrum with Xspec"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "demographic-management",
   "metadata": {},
   "outputs": [],
   "source": [
    "xspec.AllData.clear()\n",
    "xspec.AllModels.clear()\n",
    "\n",
    "s1 = xspec.Spectrum('./test.pha')\n",
    "s1.response = '/home/idies/workspace/headata/FTP/caldb/data/nicer/xti/cpf/rmf/nixtiref20170601v002.rmf'\n",
    "s1.response.arf = '/home/idies/workspace/headata/FTP/caldb/data/nicer/xti/cpf/arf/nixtiaveonaxis20170601v004.arf'\n",
    "#s1.background = './test_bkg.pha'\n",
    "s1.background = './nibackgen3C50_bkg.pi'\n",
    "date = s1.fileinfo('DATE-OBS')\n",
    "exposure = s1.fileinfo('EXPOSURE')\n",
    "xspec.Plot.device = '/null'\n",
    "#Plot.xAxis = 'angstrom'\n",
    "xspec.Plot.xAxis = 'keV'\n",
    "xspec.Plot.add = 'True'\n",
    "xspec.Plot.setRebin(5.0, 10)\n",
    "xspec.Plot('data')\n",
    "x1 = np.asarray(xspec.Plot.x(1))\n",
    "y1 = np.asarray(xspec.Plot.y(1))\n",
    "x_err1 = np.asarray(xspec.Plot.xErr(1))\n",
    "y_err1 = np.asarray(xspec.Plot.yErr(1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "pressed-holder",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(15,5))\n",
    "ax.plot(x1, y1, label=\"2612011001\")\n",
    "ax.set_xlim(0.2, 15)\n",
    "ax.set_ylim(1e-2,15.0)\n",
    "ax.set_yscale('log')\n",
    "ax.set_xscale('log')\n",
    "ax.set_xlabel('Energy (keV)',fontsize=16)\n",
    "ax.set_ylabel('Normalized Counts s$^{-1}$ keV$^{-1}$', fontsize=16)\n",
    "#title('NICER Cycle 2 Eta Car Spectra',fontsize = 20)\n",
    "plt.minorticks_on()\n",
    "plt.tick_params(axis='both', which='both',right = True, top=True,direction='in')\n",
    "plt.legend(fontsize=12, ncol=2, numpoints=1, loc=3)\n",
    "#savefig('./Eta_Car_spectrum.png', dpi=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "lesser-suggestion",
   "metadata": {},
   "source": [
    "# Fitting a Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bridal-cathedral",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Reading a model with Xspec\n",
    "model = xspec.Model('tbabs * bvapec')\n",
    "model.TBabs.nH.values          = [1.0]\n",
    "model.bvapec.kT.values         = [1.0]\n",
    "model.bvapec.He.values         = [1.0]\n",
    "model.bvapec.C.values          = [1.0]\n",
    "model.bvapec.N.values          = [1.0]\n",
    "model.bvapec.O.values          = [1.0]\n",
    "model.bvapec.Ne.values         = [1.0]\n",
    "model.bvapec.Mg.values         = [1.0]\n",
    "model.bvapec.Al.values         = [1.0]\n",
    "model.bvapec.Si.values         = [1.0]\n",
    "model.bvapec.S.values          = [1.0]\n",
    "model.bvapec.Ar.values         = [1.0]\n",
    "model.bvapec.Ca.values         = [1.0]\n",
    "model.bvapec.Fe.values         = [1.0]\n",
    "model.bvapec.Ni.values         = [1.0]\n",
    "model.bvapec.Redshift.values   = [0.0]\n",
    "model.bvapec.Velocity.values   = [200]\n",
    "model.bvapec.norm.values       = [1.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "senior-baseline",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating a pandas DataFrame to save parameters\n",
    "df_nicer = pd.DataFrame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "severe-pride",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Performing the fit with some other commented options that you may be interested \n",
    "\n",
    "#xspec.XspecSettings.abund = 'wilm'\n",
    "#xspec.Fit.statMethod = 'chi'\n",
    "xspec.Fit.query = \"yes\"\n",
    "xspec.Fit.perform()\n",
    "xspec.Plot.device = \"/null\"\n",
    "xspec.Plot.xAxis = \"keV\"\n",
    "#xspec.Plot.add = 'True'\n",
    "#Plot.xAxis = \"angstrom\"\n",
    "#s.ignore('**-1.7 10.0-**')\n",
    "#xspec.Plot.setRebin(3.0, 10)\n",
    "xspec.Plot('data')\n",
    "x = np.asarray(xspec.Plot.x()) \n",
    "y = np.asarray(xspec.Plot.y())\n",
    "x_err = np.asarray(xspec.Plot.xErr())\n",
    "y_err = np.asarray(xspec.Plot.yErr())\n",
    "mod = xspec.Plot.model()\n",
    "xspec.AllModels.calcFlux(\"2.0 10.0\")\n",
    "df_nicer.at['2612011001', 'FLUX_2.0-10.0'] = s1.flux[0]\n",
    "df_nicer.at['2612011001', 'Fit_Statistic'] = xspec.Fit.statistic\n",
    "df_nicer.at['2612011001', 'dof'] = xspec.Fit.dof\n",
    "red_chi = xspec.Fit.statistic/xspec.Fit.dof\n",
    "for c in model.componentNames:\n",
    "    for p in model.__getattribute__(c).parameterNames:\n",
    "        df_nicer.at['2612011001', c+'_'+p] = model.__getattribute__(c).__getattribute__(p).values[0]\n",
    "        df_nicer.at['2612011001', c+'_'+p+'_err'] = model.__getattribute__(c).__getattribute__(p).sigma\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "essential-hypothetical",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Take a look to the dataframe with parameters for OBS ID 2612011001\n",
    "df_nicer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "following-nurse",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Taking a look at the spectrum and the fitting \n",
    "\n",
    "fig, ax = plt.subplots(2, sharex=True, figsize=(15, 8), gridspec_kw={'height_ratios': [3, 1]})\n",
    "fig.subplots_adjust(hspace=0)\n",
    "ax[0].errorbar(x, y, xerr=x_err, yerr=y_err, fmt='.', alpha=0.7)\n",
    "ax[0].plot(x,mod, label='$\\chi^{2}_{red} = $'+str(red_chi))\n",
    "ax[0].legend()\n",
    "ax[0].minorticks_on()\n",
    "ax[0].tick_params(axis='both', which='both',right = True, top=True,direction='in')\n",
    "ax[0].set_ylabel('normalized $counts$ $s^{-1}$ $chan^{-1}$')\n",
    "ax[1].set_xlabel('Energy (keV)')\n",
    "ax[0].set_xlim(0.4, 12)\n",
    "ax[0].set_ylim(1e-2,17.0)\n",
    "ax[0].set_yscale('log')\n",
    "ax[0].set_xscale('log')\n",
    "ax[1].plot(x, y-mod)\n",
    "ax[1].set_ylabel('Residual')\n",
    "plt.show()"
   ]
  }
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