{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Statistical error propagation" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "from radiocalibrationtoolkit import *\n", "from IPython.display import Markdown as md" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# some global plot settings\n", "plt.rcParams[\"axes.labelweight\"] = \"bold\"\n", "plt.rcParams[\"font.weight\"] = \"bold\"\n", "plt.rcParams[\"font.size\"] = 16\n", "plt.rcParams[\"legend.fontsize\"] = 14\n", "\n", "plt.rcParams[\"xtick.major.width\"] = 2\n", "plt.rcParams[\"ytick.major.width\"] = 2\n", "\n", "plt.rcParams[\"xtick.major.size\"] = 5\n", "plt.rcParams[\"ytick.major.size\"] = 5\n", "\n", "plt.rcParams[\"xtick.labelsize\"] = 14\n", "plt.rcParams[\"ytick.labelsize\"] = 14" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# create class instances\n", "galactic_map_inst = GlobalSkyModel2016(freq_unit=\"MHz\")\n", "antenna_inst = AntennaPattern(\"./antenna_setup_files/SALLA_EW.xml\")\n", "\n", "# some constants\n", "update_antenna_conventions = {\n", " \"shift_phi\": -90,\n", " \"flip_theta\": True,\n", " \"flip_phi\": False,\n", " \"in_degrees\": True,\n", " \"add_invisible_sky\": True,\n", "}\n", "LATITUDE = -35.206667\n", "NSIDE = 64\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# first part of this examples will use this frequency and LST\n", "frequency_MHz = 45\n", "lst = 22" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# create maps\n", "rotation_parameters = create_rotation_parameters(lst, LATITUDE)\n", "rotator = Rotator(coord=[\"G\", \"C\"], rot=rotation_parameters)\n", "\n", "antenna_map_inst = antenna_inst.convert2hp(\n", " frequency=frequency_MHz, quantity=\"absolute\", **update_antenna_conventions\n", ")\n", "antenna_map = antenna_map_inst.get_map(rotator=rotator)\n", "\n", "galactic_map = hp.ma(\n", " hp.pixelfunc.ud_grade(galactic_map_inst.generate(frequency_MHz), NSIDE)\n", ").copy()\n", "galactic_map.mask = create_local_mask(NSIDE, rotation_parameters)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Propagation to effective temperature and voltage squared spectral denisty" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# set up the relative standard deviations on galactic emission and the antenna gain\n", "rstd_galactic_map = 0.5\n", "rstd_antenna_map = 0." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Error propagation from sky temperature and antenna gain to effective temperature" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# analytical calculation\n", "rstd_teff = np.sqrt(rstd_galactic_map**2 + 4*rstd_antenna_map**2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " \\begin{equation}\n", "\\rho_{T\\mathrm{eff}}(t,f,\\alpha,\\delta) = \\frac{\\sigma_{T\\mathrm{eff}}(t,f,\\alpha,\\delta)}{T_{\\mathrm{eff}}(t,f,\\alpha,\\delta)} = \\sqrt{ \\rho_{T}^2(t,\\alpha,\\delta) + 4 \\rho_H^2 (t,f,\\alpha,\\delta) \t}\n", "\\end{equation}\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# experimental test by distorting the galactic and antenna maps\n", "r = (\n", " distort_array(galactic_map, rstd_galactic_map)\n", " * distort_array(antenna_map, rstd_antenna_map) ** 2\n", ") / (galactic_map * antenna_map**2)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# compare\n", "print('Experimentally propagated: {:.2f}'.format(np.std(r)))\n", "print('Analytical calculation: ', rstd_teff)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# just a plot comparison of how the distorted effective temperature by 50% looks like\n", "teff = galactic_map * antenna_map**2\n", "teff_distored = distort_array(teff, rstd=rstd_teff)\n", "\n", "fontsize = {\n", " \"xlabel\": 22,\n", " \"ylabel\": 22,\n", " \"title\": 14,\n", " \"xtick_label\": 22,\n", " \"ytick_label\": 22,\n", " \"cbar_label\": 22,\n", " \"cbar_tick_label\": 22,\n", "}\n", "\n", "for m in [teff, teff_distored]:\n", " projview(\n", " m,\n", " cmap=\"jet\",\n", " min=3500,\n", " max=100000,\n", " norm=\"log\",\n", " graticule=True,\n", " graticule_labels=True,\n", " unit=\"[$m^2$K]\",\n", " xlabel=\"RA\",\n", " ylabel=\"DEC\",\n", " cb_orientation=\"vertical\",\n", " latitude_grid_spacing=30,\n", " xtick_label_color=\"white\",\n", " title='',\n", " override_plot_properties={'cbar_shrink':1},\n", " fontsize=fontsize\n", " )\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Integrated effective temperature and the voltage square spectral density\n", "\n", "- this is done in loop 100x" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def propagate_errors(rstd_teff, rstd_galactic_map, rstd_antenna_map):\n", " # calculate the effective temperature\n", " teff = galactic_map * antenna_map**2\n", "\n", " ratios_teff = np.array([])\n", " for i in range(100):\n", " # teff_distored = distort_array(teff, rstd=rstd_teff)\n", " teff_distored = (\n", " distort_array(galactic_map, rstd_galactic_map)\n", " * distort_array(antenna_map, rstd_antenna_map) ** 2\n", " )\n", " ratios_teff = np.append(\n", " ratios_teff, integrate_hpmap(teff) / integrate_hpmap(teff_distored)\n", " )\n", "\n", " # do the same with voltage square density, this should give the same results as before (more-less)\n", " # this is to check that there the extra constants have no effect\n", " ratios_v2sp = np.array([])\n", " v2sp = voltage_squared_spectral_density(antenna_map, galactic_map, frequency_MHz)\n", "\n", " for i in range(100):\n", " galactic_map_distorted = distort_array(galactic_map, rstd_galactic_map)\n", " antenna_map_distorted = distort_array(antenna_map, rstd_antenna_map)\n", "\n", " v2sp_distorted = voltage_squared_spectral_density(\n", " antenna_map_distorted, galactic_map_distorted, frequency_MHz\n", " )\n", "\n", " ratios_v2sp = np.append(ratios_v2sp, v2sp / v2sp_distorted)\n", "\n", " # figure\n", " fig, ax = plt.subplots()\n", " bins = np.histogram_bin_edges(np.concatenate([ratios_teff, ratios_v2sp]), bins=20)\n", " ax.set_title(\n", " \"Error on sky temperature: {:.2f}, error on antenna gain: {:.2f}\".format(\n", " rstd_galactic_map, rstd_antenna_map\n", " )\n", " )\n", "\n", " ax.hist(\n", " ratios_teff,\n", " bins=bins,\n", " alpha=0.5,\n", " label=r\"T$_{{\\mathrm{{eff}}}}$: Mean={:.2f}, Std={:.1e}\".format(\n", " np.mean(ratios_teff), np.std(ratios_teff)\n", " ),\n", " )\n", " ax.hist(\n", " ratios_v2sp,\n", " bins=bins,\n", " alpha=0.5,\n", " label=r\"V$^2_f$: Mean={:.2f}, Std={:.1e}\".format(\n", " np.mean(ratios_v2sp), np.std(ratios_v2sp)\n", " ),\n", " )\n", " ax.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the histograms bellow we conclude two things:\n", "\n", "- there is virtually no effect of the constants on the relative error propagation (as expected)\n", "- the error on the antenna gains is causing a bias, i.e. the whole distributions is shifted" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "propagate_errors(rstd_teff=0.5, rstd_galactic_map=0.3, rstd_antenna_map=0.2)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "propagate_errors(rstd_teff=0.5, rstd_galactic_map=0.5, rstd_antenna_map=0.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "propagate_errors(rstd_teff=0.5, rstd_galactic_map=0., rstd_antenna_map=0.25)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Analytical calculations of the error propagation to the integrated sky temperature" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We attempt to calculate the following equation other assumption of isotropic effective temperature distorted by 50%\n", "\n", " \\begin{equation}\n", "\\rho_{G\\mathrm{sky}} (t,f) = \\sqrt{ \\frac{\\sum_{\\delta} \\sin^2{\\delta} }{N_{\\alpha} \\left( \\sum_{\\delta} \\sin{\\delta} \\right)^2} } \\rho_{T\\mathrm{eff}}(t,f,\\alpha,\\delta)\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# check what is the size of the theta and phi when integrating (should be 1000x500)\n", "# so, for half sphere it is 1000x250\n", "# check the value of the propagated error at 250 then\n", "PHI, THETA, grid_map = hpmap2grid(galactic_map)\n", "phi_delta = abs(np.diff(PHI[0, :])[0])\n", "phi_theta = abs(np.diff(THETA[:, 0])[0])\n", "print(PHI[0, :].size, THETA[:, 0].size)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "propagated_err_analytical = np.array([])\n", "\n", "Ns = [10, 100, 250, 500, 1000, 10000, 50000, 100000]\n", "for N in Ns:\n", " sum_1 = 0\n", " sum_2 = 0 \n", " for x in np.linspace(0, np.pi/2, N):\n", " sum_1 += np.sin(x)**2\n", " sum_2 += np.sin(x)\n", "\n", " propagated_err_analytical = np.append(propagated_err_analytical, np.sqrt(sum_1/sum_2**2 /(N*4))*rstd_teff)\n", " \n", "print(propagated_err_analytical)\n", " " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "fig, ax = plt.subplots()\n", "ax.plot(Ns, np.log10(propagated_err_analytical))\n", "ax.scatter(Ns[2], np.log10(propagated_err_analytical)[2], c='r', label='our used resolution')\n", "ax.set_xlabel('N$_{\\\\theta}$')\n", "ax.set_ylabel('propagated log$_{10}$($\\\\rho$)')\n", "ax.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Propagation to the power" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# check the propagation to the finally calculated power at a range of LST and frequency values\n", "lst_range = np.asarray(list(range(24))) + 0.5\n", "freq_Mhz_range = range(30, 81, 1)\n", "\n", "hw_file_path = \"./antenna_setup_files/HardwareProfileList_realistic.xml\"\n", "hw_dict = read_hw_file(hw_file_path)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def propagated_errors_2_power(rstd_antenna_map=0.2, rstd_galactic_map=0.3):\n", "\n", " lst_range = np.asarray(list(range(24))) + 0.5\n", " freq_Mhz_range = range(30, 81, 1)\n", " impedance_func = hw_dict[\"IImpedance\"][\n", " \"antenna_EW\"\n", " ]\n", "\n", " power_density_DF = calculate_power_spectral_density(\n", " antenna_inst=antenna_inst,\n", " galactic_map_inst=galactic_map_inst,\n", " lst_range=lst_range,\n", " freq_Mhz_range=freq_Mhz_range,\n", " latitude=LATITUDE,\n", " update_antenna_conventions=update_antenna_conventions,\n", " impedance_func=impedance_func,\n", " distort_antenna_map=rstd_antenna_map,\n", " distort_galactic_map=rstd_galactic_map\n", " )\n", "\n", " power_DF = integrate_spectral_density(power_density_DF, integrated_MHz_bands=power_density_DF.columns.values)\n", "\n", " hw_reponse_1 = dB2PowerAmp(\n", " hw_dict[\"RResponse\"][\"LNA\"](power_DF.columns)\n", " )\n", " hw_reponse_2 = dB2PowerAmp(\n", " hw_dict[\"RResponse\"][\"digitizer\"](power_DF.columns)\n", " )\n", " hw_reponse_3 = dB2PowerAmp(\n", " hw_dict[\"RResponse\"][\"cable_fromLNA2digitizer\"](power_DF.columns)\n", " )\n", " hw_reponse_4 = dB2PowerAmp(\n", " hw_dict[\"RResponse\"][\n", " \"impedance_matching_EW\"\n", " ](power_DF.columns)\n", " )\n", "\n", "\n", " power_in_HW_DF = power_DF.multiply(\n", " hw_reponse_1 * hw_reponse_2 * hw_reponse_3 * hw_reponse_4\n", " )\n", "\n", " # to piko and round\n", " power_in_HW_DF_distorted = (power_in_HW_DF*1e+12).round(3)\n", "\n", "\n", " power_DF_normal = pd.read_csv('./simulated_power_datasets/Salla_EW_GSM16.csv', index_col=0)\n", " power_DF_normal.columns = power_DF_normal.columns.astype(float)\n", "\n", " ratio_DF = power_DF_normal/power_in_HW_DF_distorted.values\n", "\n", " # figures\n", "\n", " fig = px.imshow(\n", " ratio_DF.T - 1, width=600, aspect=\"cube\", color_continuous_scale=\"jet\"\n", " )\n", " fig.update_layout(\n", " # title=\"\",\n", " xaxis=dict(title=\"LST\", tickprefix=\"\", ticksuffix=\"\", dtick=2),\n", " yaxis=dict(\n", " title=\"frequency [MHz]\",\n", " tickprefix=\"\",\n", " ticksuffix=\"\",\n", " range=(30, 80),\n", " tick0=0,\n", " dtick=10,\n", " autorange=False,\n", " ),\n", " coloraxis=dict(\n", " colorbar=dict(\n", " title=dict(\n", " text=\"power [pW]\",\n", " side=\"right\",\n", " ),\n", " tickprefix=\"\",\n", " ticksuffix=\"\",\n", " ),\n", " ),\n", " font=dict(\n", " # family=font,\n", " size=20,\n", " color=\"black\",\n", " ),\n", " )\n", " fig.update_layout(\n", " coloraxis=dict(\n", " colorbar=dict(title=dict(text=\"zundistorted/distorted power - 1\", side=\"right\")),\n", " # cmin=0,\n", " # cmax=40,\n", " )\n", " )\n", " fig.show()\n", "\n", " # fig, ax = plt.subplots()\n", " # ax.plot(ratio_DF.mean(axis=0).values)\n", " # ax.set_xlabel('frequency [MHz]')\n", "\n", " # fig, ax = plt.subplots()\n", " # ax.plot(ratio_DF.mean(axis=1).values)\n", " # ax.set_xlabel('LST')\n", "\n", " # ratios_power = ratio_DF.values.flatten()\n", "\n", " # fig, ax = plt.subplots()\n", "\n", " # ax.hist(\n", " # ratios_power,\n", " # bins=20,\n", " # alpha=0.5,\n", " # label=r\"Mean={:.2f}, Std={:.1e}\".format(\n", " # np.mean(ratios_power), np.std(ratios_power)\n", " # ),\n", " # )\n", "\n", " # ax.legend()\n", " # ax.set_xlabel(r'$\\frac{\\mathrm{undistorted}}{\\mathrm{distorted}}$ power')\n", " # ax.set_ylabel('entries')\n", " return ratio_DF" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ratio_DF_1 = propagated_errors_2_power(rstd_antenna_map=0.2, rstd_galactic_map=0.3)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ratio_DF_2 = propagated_errors_2_power(rstd_antenna_map=0.25, rstd_galactic_map=0.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ratio_DF_3 = propagated_errors_2_power(rstd_antenna_map=0., rstd_galactic_map=0.5)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ratio_DF_2aa = propagated_errors_2_power(rstd_antenna_map=0.20, rstd_galactic_map=0.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ratio_DF_2a = propagated_errors_2_power(rstd_antenna_map=0.15, rstd_galactic_map=0.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ratio_DF_2b = propagated_errors_2_power(rstd_antenna_map=0.10, rstd_galactic_map=0.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ratio_DF_2c = propagated_errors_2_power(rstd_antenna_map=0.05, rstd_galactic_map=0.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "fig, ax = plt.subplots()\n", "\n", "labels = [\n", " r\"$\\rho_{ant}$=0.20, $\\rho_{sky}$=0.30\",\n", " r\"$\\rho_{ant}$=0.00, $\\rho_{sky}$=0.50\",\n", " r\"$\\rho_{ant}$=0.25, $\\rho_{sky}$=0.00\",\n", " r\"$\\rho_{ant}$=0.20, $\\rho_{sky}$=0.00\",\n", " r\"$\\rho_{ant}$=0.15, $\\rho_{sky}$=0.00\",\n", " r\"$\\rho_{ant}$=0.10, $\\rho_{sky}$=0.00\",\n", " r\"$\\rho_{ant}$=0.05, $\\rho_{sky}$=0.00\",\n", "]\n", "\n", "bins = np.histogram_bin_edges(\n", " np.concatenate(\n", " [\n", " ratio_DF_1.values.flatten(),\n", " ratio_DF_3.values.flatten(),\n", " ratio_DF_2.values.flatten(),\n", " ratio_DF_2aa.values.flatten(),\n", " ratio_DF_2a.values.flatten(),\n", " ratio_DF_2b.values.flatten(),\n", " ratio_DF_2c.values.flatten(),\n", " ]\n", " ),\n", " bins=100,\n", ")\n", "bins -= 1\n", "results = []\n", "\n", "for i, df in enumerate(\n", " [\n", " ratio_DF_1,\n", " ratio_DF_3,\n", " ratio_DF_2,\n", " ratio_DF_2aa,\n", " ratio_DF_2a,\n", " ratio_DF_2b,\n", " ratio_DF_2c,\n", " ]\n", "):\n", " ratios_power = df.values.flatten() - 1\n", " ax.hist(ratios_power, bins=bins, alpha=0.5, label=labels[i])\n", " stats = r\"$\\sigma$={:.3f}, $\\mu$={:.3f}\".format(\n", " np.abs(np.std(ratios_power)), round(np.mean(ratios_power), 3)+0\n", " )\n", " # print(labels[i] +', '+ stats)\n", " temp = labels[i] + \", \" + stats\n", " numbers = re.findall(r\"[-+]?\\d*\\.?\\d+(?:[eE][-+]?\\d+)?\", temp)\n", " # print(numbers)\n", " results.append([float(i) for i in numbers])\n", "\n", "ax.legend()\n", "ax.set_xlabel(r\"$\\frac{\\mathrm{(undistorted \\: power)}}{\\mathrm{(distorted \\: power)}}$ - 1\")\n", "ax.set_ylabel(\"entries\")\n", "\n", "s = (\n", " r\"\"\"| $\\rho_{{ant}}$ | $\\rho_{{sky}}$ | $\\rho$ | $b$ |\n", "|----------------|----------------|---------|---------|\n", "| {:.2f} | {:.2f} | {:.1e} | {:.2f} |\n", "| {:.2f} | {:.2f} | {:.1e} | {:.2f} |\n", "| {:.2f} | {:.2f} | {:.1e} | {:.2f} |\n", "| {:.2f} | {:.2f} | {:.1e} | {:.2f} |\n", "| {:.2f} | {:.2f} | {:.1e} | {:.2f} |\n", "| {:.2f} | {:.2f} | {:.1e} | {:.2f} |\n", "| {:.2f} | {:.2f} | {:.1e} | {:.2f} |\"\"\"\n", ").format(*np.asarray(results).flatten())\n", "\n", "md(print(\"RESULTS:\"))\n", "md(s)" ] }, { "cell_type": "code", "execution_count": 106, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\\begin{tabular}{rrrr}\n", "$\\rho_{ant}$ & $\\rho_{sky}$ & $\\rho$ & $b$ \\\\\n", "20.0 & 30.0 & 0.3 & -3.8 \\\\\n", "0.0 & 50.0 & 0.3 & 0.0 \\\\\n", "25.0 & 0.0 & 0.3 & -5.9 \\\\\n", "20.0 & 0.0 & 0.2 & -3.9 \\\\\n", "15.0 & 0.0 & 0.2 & -2.2 \\\\\n", "10.0 & 0.0 & 0.1 & -1.0 \\\\\n", "5.0 & 0.0 & 0.1 & -0.2 \\\\\n", "\\end{tabular}\n", "\n" ] }, { "data": { "text/html": [ "\n", 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$\\rho_{ant}$$\\rho_{sky}$$\\rho$$b$
20.030.00.3-3.8
0.050.00.30.0
25.00.00.3-5.9
20.00.00.2-3.9
15.00.00.2-2.2
10.00.00.1-1.0
5.00.00.1-0.2
\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "results_df = pd.DataFrame(results, columns=['$\\\\rho_{ant}$', '$\\\\rho_{sky}$', '$\\\\rho$', '$b$'])*100\n", "s = results_df.style.format('{:.1f}').hide(axis='index')\n", "latex_table = s.to_latex()\n", "print(latex_table)\n", "display(s)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the distributions above we see that:\n", " \n", "- the error on the antenna gain is indeed causing a bias\n", "- the propagated error is of order of $10^{-3}$ just like was analytically calculated\n", "\n", "And from the color maps we learned that there is no LST or frequency correlation after propagating the error to the power." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.12" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }