From a32d2bb12f35dfe0d64b72aa23deba238863c942 Mon Sep 17 00:00:00 2001 From: Eric Charles Date: Thu, 19 Aug 2021 16:03:57 -0700 Subject: [PATCH] Dev (#16) * Add files via upload * Added project blanks * use os.path.join for audio files * Added projects * Updated docs/index.md * Add files via upload * Add files via upload * Add files via upload * Add files via upload * Add files via upload * Add files via upload * Add files via upload * Add files via upload * Add files via upload * Updates * Fix up README.md * commited clear version of nb * commited clear version of nb * Added explanations to projects * Add files via upload * Update notebooks * temp add version 2 of higgs * updated higgs search * Higgs project * Update docs/Projects.md * Updated nb * updated for final week Co-authored-by: bsafvato <86092462+bsafvato@users.noreply.github.com> --- data/Higgs.txt | 72 ++++ docs/Projects.md | 13 +- docs/Week8_after.md | 45 +++ docs/diagonal_wave_vector.png | Bin 0 -> 4083 bytes docs/diagonal_waves.png | Bin 0 -> 36444 bytes docs/index.md | 1 + nb/Project_Exoplanet.ipynb | 319 +++++++++++++----- nb/Project_Fitting_Deep_Dive.ipynb | 183 +++++++++- nb/Project_Higgs_Search.ipynb | 315 +++++++++++++---- nb/Project_Hubble_Tension.ipynb | 11 +- nb/Project_SDSS_Line_Fit_Algo.ipynb | 14 +- nb/figures/2012Higgsplot.png | Bin 0 -> 103487 bytes nb/figures/Higgs_Cut_And_Count.png | Bin 0 -> 17632 bytes nb/figures/Higgs_Events_month.png | Bin 0 -> 13595 bytes nb/figures/Higgs_Mass_fit.png | Bin 0 -> 16840 bytes .../Radial-Velocity-Method-star-orbits.png | Bin 0 -> 1130222 bytes nb/figures/higgs_cut_and_count_2.png | Bin 0 -> 17632 bytes 17 files changed, 799 insertions(+), 174 deletions(-) create mode 100644 data/Higgs.txt create mode 100644 docs/Week8_after.md create mode 100644 docs/diagonal_wave_vector.png create mode 100644 docs/diagonal_waves.png create mode 100644 nb/figures/2012Higgsplot.png create mode 100644 nb/figures/Higgs_Cut_And_Count.png create mode 100644 nb/figures/Higgs_Events_month.png create mode 100644 nb/figures/Higgs_Mass_fit.png create mode 100644 nb/figures/Radial-Velocity-Method-star-orbits.png create mode 100644 nb/figures/higgs_cut_and_count_2.png diff --git a/data/Higgs.txt b/data/Higgs.txt new file mode 100644 index 0000000..f6d68a7 --- /dev/null +++ b/data/Higgs.txt @@ -0,0 +1,72 @@ +# Mass Nevts +90.0 1169 +91.0 1191 +92.0 1182 +93.0 1128 +94.0 1096 +95.0 1098 +96.0 1060 +97.0 1112 +98.0 1092 +99.0 1130 +100.0 1138 +101.0 1102 +102.0 1092 +103.0 1099 +104.0 1073 +105.0 1068 +106.0 1094 +107.0 1093 +108.0 1052 +109.0 1053 +110.0 1016 +111.0 1083 +112.0 1054 +113.0 1044 +114.0 988 +115.0 1063 +116.0 1034 +117.0 1045 +118.0 1050 +119.0 1006 +120.0 1050 +121.0 1047 +122.0 1020 +123.0 1037 +124.0 1069 +125.0 1103 +126.0 1018 +127.0 1045 +128.0 977 +129.0 988 +130.0 934 +131.0 964 +132.0 921 +133.0 919 +134.0 977 +135.0 921 +136.0 944 +137.0 907 +138.0 900 +139.0 933 +140.0 908 +141.0 902 +142.0 880 +143.0 883 +144.0 918 +145.0 909 +146.0 905 +147.0 908 +148.0 862 +149.0 819 +150.0 842 +151.0 859 +152.0 849 +153.0 814 +154.0 876 +155.0 852 +156.0 850 +157.0 790 +158.0 870 +159.0 783 +160.0 829 diff --git a/docs/Projects.md b/docs/Projects.md index f4f5efb..d43ceef 100644 --- a/docs/Projects.md +++ b/docs/Projects.md @@ -7,6 +7,14 @@ The goals of the final projects are: 2. To give you a chance to apply some of the things we learned in this course. +The idea is for your group to work as independently or with as much guidance +as you prefer. + +If your group prefers to work independetly, great, each project notebook has some suggestions of things you might do, and +some tools to get you started. + +If your group prefers to have some guidance from the instructiors, +great, we are available to help. We have designed five different projects. The idea is that you will work in small groups, as usual, on the projects. We are expecting you to spend about three hours on the project. @@ -21,7 +29,10 @@ propose your own project, keeping in mind that you should expect to spend about Please use this sheet to express your interest in a particular project by next week. [Sign up sheet](https://docs.google.com/spreadsheets/d/12MT8zAYdLToVdNDRUBJxRjTfoQYo-oUP748JXoTYw_A/edit?usp=sharing) -We will use that to set up the breakout rooms next week. +We will use that to set up the breakout rooms for the final class sessions. + +The projects will be due on the Monday after the final class session +(i.e., Monday, August 30). ### 1. Quantifying the "Hubble Tension" diff --git a/docs/Week8_after.md b/docs/Week8_after.md new file mode 100644 index 0000000..1487ebc --- /dev/null +++ b/docs/Week8_after.md @@ -0,0 +1,45 @@ +# Follow up notes for week 8. + +### Frequency and wave number + +Last week we worked with analyzing data using Fourier analysis. + +In the first notebook we worked with audio files and looked at the frequencies that were +present is a few music clips. + +In the second notebook we worked with some data from Ben's lab studying the behavior of +electrons on a copper surface. Basically, this introduced two new concepts. + + 1. We can apply Fourier analysis to spatial data as well as time data. I.e., with music, the frequency of the + variation of the sound wave with time is what our ears hear as the pitch of the music. It turns out that the + spatial frequency of variation of the electron density with distance is released to the momentum and energy + levels of the electrons. + + 2. We can apply Fourier analysis in more that one dimension. I.e., we can study wave on surfaces, or even waves + in solid materials. + +Let's see if we can break those two concept down into pieces. + +One of the figures we had showed waves that were oriented diagonally. + +[Waves](!diagonal_waves.png) + +The bright bits might be places where the wave is higher, the dark bits where it is lower. + +The first concept we have is wave number. it is basically the spatial equivalent of frequency. I.e., it is the number +of waves that you can fit into a fixed length. Since the image above is 1m X 1m it is really easy to count how many +wave can fit in a meter. I count 10 waves along the x-axis and 5 waves along the y-axis. + +The second concept is describing the wave in more that one dimension. 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to detect this wobble. This variation is actually very small, maybe only dozens of $m / s$ for the first exoplanets discovered, and down to $10 cm / s$ for current state of the art telescopes.\n", + "\n", + "The thing that allows us to be fully confident that we are seeing a real signal is that the wobble is periodic. Thus, by doing the analysis in the frequency domain, we can confirm the periodic signature of the wobble. \n", + "\n", + "In this project, you will consider to effect that can make it easier or harder to detect a signal.\n", + "\n", + "1. The quality of the measurements. We will smear out the measurements of the radial velocity, thus degrading the signal.\n", + "\n", + "2. The number of measurements. The longer we observe a star for, and the more measurement we take, the more chances we have to confirm the periodic signature in the radial velocity variations.\n", + "\n", + "We have provided a few simple functions, including function to degrade the date, extract the Fourier transform the of the data, and quantify both size of the signal peak and the amount of background in the Fourier transform.\n", + "\n", + "Here are some potential goals:\n", + "\n", + "1. Run through the four cases in this notebook that demonstrate idealized data, and what happens as it gets degraded.\n", + "\n", + "2. Come up with a way to define the significance of the peak in the Fourier transform and decide if constitues a discovery.\n", + "\n", + "3. Explore how the significance changes as we degrade the data, and characterize those changes.\n", + "\n", + "4. Come up with a formula to estimate how long we might have to observe to discover an exoplanet if we have a given amount of noise in our measurement of the radial velocity.\n", + "\n", + "Fun fact: the idealized data we have are using the period and radial velocity of 51 pegasi b the first exoplanet ever discovered around a \"normal\" or \"main sequence\" star.\n", + "\n", + "https://en.wikipedia.org/wiki/51_Pegasi\n" + ] + }, + { + "cell_type": "markdown", + "id": "1caef2c2", + "metadata": {}, + "source": [ + "### Function to plot the $v_{\\rm rad}$ as a function of time." ] }, { @@ -37,18 +74,39 @@ "metadata": {}, "outputs": [], "source": [ - "#_ = plt.errorbar(date, rvel, yerr=rvel_err, fmt='.')\n", - "_ = plt.scatter(date, rvel)" + "def plot_rvel(date, rvel):\n", + " _ = plt.scatter(date, rvel)\n", + " _ = plt.xlabel(r'Observation Time [days]')\n", + " _ = plt.ylabel(r'$\\Delta v_{\\rm rad} [\\frac{m}{s}]$')" + ] + }, + { + "cell_type": "markdown", + "id": "7c9182f7", + "metadata": {}, + "source": [ + "### Function to degrade data by adding noise and / or reducing number of observations" ] }, { "cell_type": "code", "execution_count": null, - "id": "e625cf02", + "id": "f067d5f5", "metadata": {}, "outputs": [], "source": [ - "from scipy.fft import fft, fftfreq" + "def degrade_data(date, rvel, noise_scale=0.01, tfrac=1.):\n", + " tmax = tfrac*np.max(date)\n", + " mask = date < tmax\n", + " return (date[mask], rvel[mask] + stats.norm(loc=0, scale=noise_scale).rvs(size=mask.sum()))" + ] + }, + { + "cell_type": "markdown", + "id": "b9d98c40", + "metadata": {}, + "source": [ + "### Function to take the Fourier transform of a time series" ] }, { @@ -58,110 +116,146 @@ "metadata": {}, "outputs": [], "source": [ - "# Number of sample points\n", - "N = len(date)\n", - "\n", - "# Arbitrary offset in data\n", - "offset = np.mean(rvel)\n", - "\n", - "# sample spacing\n", + "def do_fft(date, rvel, plot=True):\n", + " # Number of sample points\n", + " N = len(date)\n", "\n", - "T = np.mean(date[1:] - date[0:-1])\n", + " # Arbitrary offset in data\n", + " offset = np.mean(rvel)\n", "\n", - "yf = fft(rvel-offset)\n", + " # sample spacing\n", + " T = np.mean(date[1:] - date[0:-1])\n", "\n", - "xf = fftfreq(N, T)[:N//2]\n", + " yf = fft(rvel-offset)\n", + " xf = fftfreq(N, T)[:N//2]\n", "\n", - "_ = plt.plot(xf, 2.0/N * np.abs(yf[0:N//2]))\n", + " if plot:\n", + " _ = plt.plot(xf, 2.0/N * np.abs(yf[0:N//2])) \n", + " _ = plt.ylabel(\"Signal at Frequency [a.u.]\")\n", + " _ = plt.xlabel(r\"Frequency [${\\rm days}^{-1}$]\")\n", "\n", - "_ = plt.ylabel(\"Signal at Frequency [a.u.]\")\n", - "_ = plt.xlabel(r\"Frequency [${\\rm days}^{-1}$]\")\n", + " freq_max = xf[np.argmax(np.abs(yf[0:N//2]))]\n", + " period = 1./freq_max\n", "\n", - "freq_max = xf[np.argmax(np.abs(yf[0:N//2]))]\n", - "period = 1./freq_max\n", + " _ = plt.annotate(r\"$f \\sim %0.2f {\\rm days}^{-1}$\" % freq_max, (2.0, 35))\n", + " _ = plt.annotate(r\"$P \\sim %0.2f {\\rm days}$\" % period, (2.0, 32))\n", + " \n", + " return xf, 2.0/N * np.abs(yf[0:N//2])" + ] + }, + { + "cell_type": "markdown", + "id": "f6079b09", + "metadata": {}, + "source": [ + "### Function to esimate the peak in the Fourier transform, and also to esimate the noise level" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3c1b8761", + "metadata": {}, + "outputs": [], + "source": [ + "def fft_noise_stats(xf, yf):\n", + " min_bin = np.searchsorted(xf, 0.5)\n", + " other_data = yf[min_bin:]\n", + " mean = np.mean(other_data)\n", + " std = np.std(other_data)\n", + " peak = np.max(yf)\n", + " print(\"The mean and standard deviation of the FFT away from the peak is %0.2f %0.2f\" % (mean, std))\n", + " print(\"The value at the mean is %.2f\" % peak)" + ] + }, + { + "cell_type": "markdown", + "id": "e189a6a1", + "metadata": {}, + "source": [ + "# Case 1: using idealized data\n", "\n", - "_ = plt.annotate(r\"$f \\sim %0.2f {\\rm days}^{-1}$\" % freq_max, (2.0, 35))\n", - "_ = plt.annotate(r\"$P \\sim %0.2f {\\rm days}$\" % period, (2.0, 32))" + "This case represents some very idealized measurements. There is no instrumental error, and we observe the star a few times a day for 40 days. This results in a very clear and convincing signal." ] }, { "cell_type": "code", "execution_count": null, - "id": "62ab0a65", + "id": "46028e55", "metadata": {}, "outputs": [], "source": [ - "data = np.loadtxt(open(\"../data/Vela_Flux.txt\", 'rb'), usecols=range(7))\n", + "data = np.loadtxt(open(\"../data/51peg_model_rvs.txt\", 'rb'), usecols=range(2))\n", "\n", "# This is how we pull out the data from columns in the array.\n", + "date = data[:,0] - np.min(data[:,0])\n", + "rvel = data[:,1]\n", "\n", - "# This is the date in \"Mission Elapsesd Time\"\n", - "# For the Fermi mission, this is defined to be the number of seconds since the start of 2001.\n", - "date_MET = data[:,0]\n", - "# This is the offset in seconds between the Fermi \"MET\" and the UNIX \"epoch\" used by matplotlib\n", - "MET_To_Unix = 978336000\n", - "\n", - "# These are the numbers of photons observed from Vela each week in the \"low\" Energy Band (100 MeV - 800 MeV)\n", - "nObs_LE = data[:,1]\n", - "\n", - "# These are the number of photons expected from Vela each week, under the assumption that it is \n", - "# not varying at all, and the only differences depend on how long we spent looking at Vela\n", - "# that particular weeek\n", - "nExp_LE = data[:,2]\n", - "\n", - "# These are the band bounds, in MeV\n", - "LE_bounds = (100., 800.)\n", - "\n", - "# This is the \"significance\" of the variation for each week. We will discuss this more later\n", - "signif_LE = data[:,3]\n", - "\n", - "#nObs_HE = data[:,4]\n", - "#nExp_HE = data[:,5]\n", - "#signif_HE = data[:6]\n", - "#HE_bounds = (800., 10000.)\n", - "\n", - "dates = [datetime.datetime.fromtimestamp(date + MET_To_Unix) for date in date_MET]\n" + "plot_rvel(date, rvel)" ] }, { "cell_type": "code", "execution_count": null, - "id": "12dd89e0", + "id": "7f61a024", "metadata": {}, "outputs": [], "source": [ - "# Number of sample points\n", - "N = len(date_MET)\n", - "\n", - "# Arbitrary offset in data\n", - "offset = 0.\n", - "\n", - "# sample spacing\n", - "\n", - "T = np.mean(date_MET[1:] - date_MET[0:-1])/(24*3600)\n", - "\n", - "yf = fft(signif_LE-offset)\n", - "\n", - "xf = fftfreq(N, T)[:N//2]\n", - "\n", - "_ = plt.plot(xf, 2.0/N * np.abs(yf[0:N//2]))\n", - "\n", - "_ = plt.ylabel(\"Signal at Frequency [a.u.]\")\n", - "_ = plt.xlabel(r\"Frequency [${\\rm days}^{-1}$]\")\n", - "\n", - "max_idx = np.argmax(np.abs(yf[0:N//2]))\n", - "freq_max = xf[max_idx]\n", - "period = 1./freq_max\n", - "\n", - "_ = plt.ylim(0, 0.30)\n", - "#_ = plt.annotate(r\"$f \\sim %0.2E {\\rm days}^{-1}$\" % freq_max, (0.02, 0.27))\n", - "_ = plt.annotate(r\"$P \\sim %0.0f {\\rm days}$\" % period, (0.02, 0.25))\n", + "xf, yf = do_fft(date, rvel)\n", + "fft_noise_stats(xf, yf)" + ] + }, + { + "cell_type": "markdown", + "id": "4748692f", + "metadata": {}, + "source": [ + "# Case 2: shorter observation, but still idealized\n", "\n", - "freq_max = xf[np.argmax(np.abs(yf[0:max_idx-1]))]\n", - "period = 1./freq_max\n", + "In this case we still have no measurment error, but we only observed for 20 days. We still get a very nice clear signal." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "bcc38876", + "metadata": {}, + "outputs": [], + "source": [ + "less_data = degrade_data(date, rvel, tfrac=0.5)\n", + "plot_rvel(less_data[0], less_data[1])" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "bc75e5f1", + "metadata": {}, + "outputs": [], + "source": [ + "xf, yf = do_fft(less_data[0], less_data[1])\n", + "fft_noise_stats(xf, yf)" + ] + }, + { + "cell_type": "markdown", + "id": "dde73862", + "metadata": {}, + "source": [ + "# Case 3, full observation time, but noisy data\n", "\n", - "#_ = plt.annotate(r\"$f \\sim %0.2E {\\rm days}^{-1}$\" % freq_max, (0.00, 0.22))\n", - "_ = plt.annotate(r\"$P \\sim %0.0f {\\rm days}$\" % period, (0.00, 0.20))\n" + "In this case we have about $100 \\frac{m}{s}$ of noise in the measurements of $v_{\\rad}$. Even thought we can't really see a signal in the time series, we can see a pretty clear signal in the Fourier transform." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "b5a79eb2", + "metadata": {}, + "outputs": [], + "source": [ + "noisy_data = degrade_data(date, rvel, noise_scale=100.)\n", + "plot_rvel(noisy_data[0], noisy_data[1])" ] }, { @@ -170,12 +264,55 @@ "id": "8728575d", "metadata": {}, "outputs": [], + "source": [ + "xf, yf = do_fft(noisy_data[0], noisy_data[1])\n", + "fft_noise_stats(xf, yf)" + ] + }, + { + "cell_type": "markdown", + "id": "b45eef6c", + "metadata": {}, + "source": [ + "# Case 4, less observation time, and noisy data\n", + "\n", + "In this case we have both noise measurement and a shorter observation, and the signal is getting really marginal." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ac3b8f9a", + "metadata": {}, + "outputs": [], + "source": [ + "noisy_short_data = degrade_data(date, rvel, noise_scale=100., tfrac=0.5)\n", + "plot_rvel(noisy_short_data[0], noisy_short_data[1])" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c9240e6d", + "metadata": {}, + "outputs": [], + "source": [ + "xf, yf = do_fft(noisy_short_data[0], noisy_short_data[1])\n", + "fft_noise_stats(xf, yf)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "b6f7a40a", + "metadata": {}, + "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "id": "fea3456f", + "id": "2dd83d91", "metadata": {}, "outputs": [], "source": [] diff --git a/nb/Project_Fitting_Deep_Dive.ipynb b/nb/Project_Fitting_Deep_Dive.ipynb index ae6ed27..672ea81 100644 --- a/nb/Project_Fitting_Deep_Dive.ipynb +++ b/nb/Project_Fitting_Deep_Dive.ipynb @@ -15,6 +15,49 @@ "import datetime" ] }, + { + "cell_type": "markdown", + "id": "dc92e659", + "metadata": {}, + "source": [ + "# Project, a deeper dive into the math behind fitting model parameters.\n", + "\n", + "Recall that in weeks 5 and 6 we explored fitting model parameters, and in particular, fitting data on the excess counts from Vela using a model with two parameters, a slope and an offset.\n", + "\n", + "It this project we are going to explore the math behind the fitting a bit more. In particular we are going to simplify the model a bit by removing the slope, so that we are only fitting for the offset. \n", + "\n", + "This model is simple enough that we can take the derivatives by hand, and we can solve for the minimum \n", + "\n", + "Some things you might do:\n", + "\n", + "Recall that the $\\chi^2$ is given by:\n", + "\n", + "$\\chi^2 = \\sum_i (\\frac{n_i - m_i(x)}{\\sigma_i})^2$\n", + "\n", + "Where $n_i$ is the number of counts from week $i$ (i.e., 'excess_counts' in the next cell) and $\\sigma_i$ is uncertainty on that number (i.e., 'sigma_counts' in the next cell).\n", + "\n", + "And in our case our model is trivial:\n", + "\n", + "$m(x) = p_0$, leaving use with\n", + "\n", + "$\\chi^2 = \\sum_i (\\frac{n_i - p_0}{\\sigma_i})^2$\n", + "\n", + "\n", + "Some goals for this project could include:\n", + "\n", + "1. Take the first and second partial derivativs of the $\\chi^2$ with respect to $p_0$. If you aren't sure how to take a derivative of an expression inside a sum you can try just writing out the first three terms of the sum and taking the derivatives of those:\n", + "\n", + "$\\chi^2 = (\\frac{n_0 - p_0}{\\sigma_i})^2 + (\\frac{n_i - p_0}{\\sigma_i})^2 + (\\frac{n_2 - p_0}{\\sigma_i})^2 + ...$\n", + "\n", + "2. Write out your own expression of the first and second derivatives of $\\chi^2$ with respect to $p_0$. Implement them as functions, plot them and verify your results by comparing them to the functions that we wrote that date the derivatives numerically.\n", + "\n", + "3. Then you can explicitly solve for best fit value of $p_0$ by solving for the place where the first derivative is zero. This will give you an expressing for $p_0$ in terms of the $n_i$ and $\\sigma_i$.\n", + "\n", + "4. You can explicitly solve for uncertainty on $p_0$ ($\\sigma_{p0}$) by noting that they correspond to the points where the $\\chi^2$ function changes by 1 with respect to the minimum, and also by noting that the $\\chi^2$ function is a parabola, you can also write as $\\Delta \\chi^2 = a (\\delta p_0)^2$. (For this step you can either demonstrate that for our case $a = \\frac{1}{2} \\frac{\\partial^2\\chi^2}{\\partial p_0^2}$ or you can just take our word on that.)\n", + "\n", + "6. Finally, you can compare the formulas you got for $p_0$ and $\\sigma_{p0}$ to the formulas for taking the weighted average by inverse variance weighting. \n" + ] + }, { "cell_type": "code", "execution_count": null, @@ -22,7 +65,7 @@ "metadata": {}, "outputs": [], "source": [ - "data = np.loadtxt(open(\"../data/Vela_Pulsar.txt\", 'rb'), usecols=range(7))\n", + "data = np.loadtxt(open(\"../data/Vela_Flux.txt\", 'rb'), usecols=range(7))\n", "\n", "# This is how we pull out the data from columns in the array.\n", "\n", @@ -55,9 +98,19 @@ "sigma_counts = np.sqrt(nObs_LE)" ] }, + { + "cell_type": "markdown", + "id": "26c5a297", + "metadata": {}, + "source": [ + "# Defining the model and the cost function\n", + "\n", + "This next cell defines the model and the cost function" + ] + }, { "cell_type": "code", - "execution_count": 1, + "execution_count": null, "id": "055923f6", "metadata": {}, "outputs": [], @@ -76,7 +129,23 @@ " return np.sum(chi2)\n", "\n", "def cost_function(params):\n", - " return 0.5*chi2_function(years_since_mid_2014, excess_counts, sigma_counts, linear_function, params)" + " return 0.5*chi2_function(years_since_mid_2014, excess_counts, sigma_counts, constant_function, params)\n", + "\n", + "def plot_data(dates, excess_counts, sigma_counts, p0, p0err):\n", + " _ = plt.errorbar(dates, excess_counts, sigma_counts)\n", + " _ = plt.errorbar(dates, np.ones(len(dates))*(p0-p0err), np.ones(len(dates))*(p0+p0err))\n", + " _ = plt.xlabel(r'Date [year]')\n", + " _ = plt.ylabel(r'Counts')" + ] + }, + { + "cell_type": "markdown", + "id": "911da631", + "metadata": {}, + "source": [ + "# Fitting the data for p_0, the offset number of counts\n", + "\n", + "This next cell fits the model, prints out the result, and plots both the data and the best fit." ] }, { @@ -85,7 +154,113 @@ "id": "2e541137", "metadata": {}, "outputs": [], - "source": [] + "source": [ + "result = optimize.minimize(cost_function, x0=[0.])\n", + "pars = result['x']\n", + "fmin = result['fun']\n", + "p0_best = pars[0]\n", + "cov = result['hess_inv']\n", + "p0_err = np.sqrt(cov[0,0])\n", + "\n", + "print(\"Fitter result:\")\n", + "print(result)\n", + "\n", + "print(\"\")\n", + "print(\"Human readable version ---------------\")\n", + "print(\" p0 best fit value: %.1f +- %.1f counts\" % (p0_best, p0_err))\n", + "print(\" Minimum value of cost function: %.1f\" % fmin)\n", + "print(\" Minimum value of chi**2: %.1f\" % (2*fmin))\n", + "\n", + "plot_data(dates, excess_counts, sigma_counts, p0_best, p0_err)" + ] + }, + { + "cell_type": "markdown", + "id": "ee3f5dce", + "metadata": {}, + "source": [ + "# Numerical derivatives and plots of them\n", + "\n", + "These functions will compute the derivates of the $\\chi^2$ with respect to $p_0$ and also make plots of them." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "eb488c2a", + "metadata": {}, + "outputs": [], + "source": [ + "def chi2_scan(cost_func, xvals):\n", + " chi2_vals = np.zeros(xvals.shape)\n", + " for i, xv in enumerate(xvals):\n", + " chi2_vals[i] = 2*cost_func([xv])\n", + " return chi2_vals\n", + "\n", + "def numerical_deriv(cost_func, xvals):\n", + " dx = np.mean(xvals[1:] - xvals[:-1]) / 50.\n", + " chi2_0 = chi2_scan(cost_func, xvals)\n", + " chi2_1 = chi2_scan(cost_func, xvals + dx)\n", + " return (chi2_1 - chi2_0)/dx\n", + "\n", + "def numerical_2nd_deriv(cost_func, xvals):\n", + " dx = np.mean(xvals[1:] - xvals[:-1]) / 50.\n", + " deriv_0 = numerical_deriv(cost_func, xvals)\n", + " deriv_1 = numerical_deriv(cost_func, xvals + dx)\n", + " return (deriv_1 - deriv_0)/dx\n", + "\n", + "def plot_chi2(cost_func, xvals):\n", + " yvals = chi2_scan(cost_function, xvals)\n", + " _ = plt.plot(xvals, yvals)\n", + " _ = plt.xlabel(r'$p_0$ [counts]')\n", + " _ = plt.ylabel(r'$\\chi^2$')\n", + " \n", + "def plot_dchi2_dp0(cost_func, xvals):\n", + " yvals = numerical_deriv(cost_function, xvals)\n", + " _ = plt.plot(xvals, yvals)\n", + " _ = plt.xlabel(r'$p_0$ [counts]')\n", + " _ = plt.ylabel(r'First Derivative: $\\frac{\\partial\\chi^2}{\\partial p_0}$')\n", + " \n", + "def plot_d2chi2_dp02(cost_func, xvals):\n", + " yvals = numerical_2nd_deriv(cost_function, xvals)\n", + " _ = plt.plot(xvals, yvals)\n", + " _ = plt.xlabel(r'$p_0$ [counts]')\n", + " _ = plt.ylabel(r'Second Derivative: $\\frac{\\partial^2\\chi^2}{\\partial p_0^2}$')\n", + " _ = plt.ylim(-1, 1)\n", + "\n", + "# This is the grid we will use for plotting\n", + "xvals = np.linspace(-10, 10, 101)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "05aa9837", + "metadata": {}, + "outputs": [], + "source": [ + "plot_chi2(cost_function, xvals)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9abfbc7f", + "metadata": {}, + "outputs": [], + "source": [ + "plot_dchi2_dp0(cost_function, xvals)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1ea24d1e", + "metadata": {}, + "outputs": [], + "source": [ + "plot_d2chi2_dp02(cost_function, xvals)" + ] } ], "metadata": { diff --git a/nb/Project_Higgs_Search.ipynb b/nb/Project_Higgs_Search.ipynb index 8d83bf9..9f3eec9 100644 --- a/nb/Project_Higgs_Search.ipynb +++ b/nb/Project_Higgs_Search.ipynb @@ -7,19 +7,77 @@ "metadata": {}, "outputs": [], "source": [ + "import os\n", "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", - "import scipy.stats as sps" + "import scipy.stats as sps\n", + "import scipy.optimize as optimize" ] }, { - "cell_type": "code", - "execution_count": null, - "id": "66614ebc", + "cell_type": "markdown", + "id": "61de816d", "metadata": {}, - "outputs": [], - "source": [] + "source": [ + "# Searching for the Higgs boson\n", + "\n", + "Recall that in weeks 7 of this course we searched for dark matter in a simplified data set.\n", + "\n", + "This week we are going to seach for the Higgs Boson. In spirit this is similar to the dark matter search,\n", + "but there are going to be a few differences.\n", + "\n", + "1. The detectors are quite different. The detectors at CERN that were used to discover the Higgs particle are massive, basically the size of a 6 story building (about 150 long and 75 feet tall). The detectors capture information from many particles that are produced when two protons collide at very nearly the speed of light. Complicated data analysis algorithms then filter that data to find events that might have contained a Higgs particle. The last step of the analysis is to combined together the particles that were flagged as possibly resulting from the Higgs particle's decay, as seeing if they are consistent with the mass of the Higgs particle.\n", + "\n", + "2. In the dark matter search in week 7, we used the log(s2) (i.e., the log of the electron charge signal) to distinguish signal from background. For the Higgs search we as going to be using using the mass computed from combined different sets of particles collected in the detector. For background events that mass will be randomly distributed, and we will model it as a linear function. For signal events that mass will be centered of the mass of the Higgs particle ( $m_{H} = 125.1 \\frac{\\rm GeV}{c^2}$ ). This means that in the Higgs search, we will be cutting on both sides of the signal. I.e., the signal events are $ | m - m_{H} | < w $, where $w$ is the width of the cut window.\n", + "\n", + "3. In this search, we have much more data, even at the last step of the data analysis. Both the rate of signal events and background events is much higher that it was for the dark matter search.\n", + "\n", + "\n", + "You can try two different ways of doing the search.\n", + "\n", + "1. \"Cut and count\". Basically you define a signal region and background region and show that there are more events in the signal region that in the background region. In our case, since the background is not constant, it make sense to pick two background regions, one on either side of the signal, and to average them.\n", + "\n", + "\n", + "\n", + "\n", + "2. \"Fitting\". In this case you make signal and background models and use a fit optimizer to extract the size of the signal.\n", + "\n", + "\n", + "\n", + "\n", + "For comparison, here is a figure from the Higgs discover paper:\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "id": "568c46d3", + "metadata": {}, + "source": [ + "Some goals for this project could include:\n", + "\n", + "1. Optimize the width of the cut window\n", + "\n", + "2. Estimate the significance you might expect as a function of time using the \"cut and count\" analysis\n", + "\n", + "3. Apply the \"cut and count\" analysis to the real data.\n", + "\n", + "4. Apply the \"fitting\" analysis to the real data, and compare the results with the \"cut and count\" analysis." + ] + }, + { + "cell_type": "markdown", + "id": "a35a1e1e", + "metadata": {}, + "source": [ + "# Some useful stuff\n", + "\n", + "First make our lives easy by using the true values for the Higgs particle mass and the width of the mass peak.\n", + "\n", + "Then we will suppose that we are going to be histograming Higgs mass data in units of $1 {\\rm GeV}{c^2}$, as was done in the plots above. And we will look at all the data between 90 and 160 ${\\rm GeV}{c^2}$" + ] }, { "cell_type": "code", @@ -28,124 +86,259 @@ "metadata": {}, "outputs": [], "source": [ - "xgrid = np.linspace(0., 10., 51)" + "Higgs_Mass = 125.1 # In Units of MeV / c**2\n", + "Higgs_Width = 4.2 # In Units of MeV / c**2\n", + "\n", + "mass_grid = np.linspace(90., 160., 71)" ] }, { - "cell_type": "code", - "execution_count": null, - "id": "86f5de87", + "cell_type": "markdown", + "id": "e7349a3e", "metadata": {}, - "outputs": [], "source": [ - "nbkg = 40.\n", - "nsig = 5*np.sqrt(nbkg)" + "# Some useful functions for cut and counting\n", + "\n", + "You can use these three functions to:\n", + "\n", + "1. `passed_cuts`: tells how many signal and background events from the idealized data pass your cuts if you use a cut window of a particular width, using models for how much signal and background we expect. \n", + "\n", + "2. `extract_sig_from_data`: tells you many events are in your signal region on the \"real\" data. Since it is real data, you don't know if they are signal or background.\n", + "\n", + "3. `extract_bkg_from_data`: tells you many events are in your background region on the \"real\" data. " ] }, { "cell_type": "code", "execution_count": null, - "id": "94034cf6", + "id": "7fd03b26", "metadata": {}, "outputs": [], "source": [ - "def plot_trial(nsig, nbkg):\n", - " x_cent = 0.5*(xgrid[1:] + xgrid[0:-1])\n", - " mask = np.bitwise_and(x_cent > 4.5, x_cent < 5.5)\n", - " bkg_model = nbkg*(10./50.)*np.ones((50))\n", - " sig_model = nsig*(10./50.)*sps.norm(loc=5.0, scale=0.10).pdf(x_cent)\n", - " tot_model = bkg_model + sig_model\n", - " fake_data = np.random.poisson(tot_model)\n", - " _ = plt.bar(x_cent, fake_data, width=0.2)\n", - " plt.xlabel(\"Variable [a.u.]\")\n", - " plt.ylabel(\"Counts [per 0.2 a.u.]\")\n", - " bkg_exp = bkg_model[mask].sum()\n", - " sig_exp = sig_model[mask].sum()\n", - " roi_evt = fake_data[mask].sum()\n", - " pvalue = sps.poisson(mu=bkg_exp).sf(roi_evt)\n", - " signif = sps.norm().isf(pvalue)\n", - " plt.title(r\"$n_{\\rm ex} = %0.3f$, $n_{obs} = %i$, $p=%0.2e$, $s=%0.1f\\sigma$\" % (bkg_exp, roi_evt, pvalue, signif))\n", - " print(\"Expected background: %0.3f\" % bkg_exp)\n", - " print(\"Expected signal: %0.3f\" % sig_exp)\n", - " print(\"Observed events: %i\" % fake_data[mask].sum())\n", - " print(\"P-value: %0.2e\" % pvalue)\n", - " " + "def passed_cuts(cut_width, masses, model_sig, model_bkg):\n", + " mask = np.abs(masses - 125) < cut_width\n", + " n_sig = np.sum(model_sig[mask])\n", + " n_bkg = np.sum(model_bkg[mask])\n", + " return n_sig, n_bkg\n", + "\n", + "def extract_sig_from_data(cut_width, masses, nevts):\n", + " mask = np.abs(masses - 125) < cut_width\n", + " return np.sum(nevts[mask])\n", + "\n", + "def estimate_bkg_from_data(cut_width, masses, nevts):\n", + " mask_bkg_lo = np.abs(masses-105) < cut_width\n", + " mask_bkg_hi = np.abs(masses-145) < cut_width\n", + " mask_bkg = np.bitwise_or(mask_bkg_lo, mask_bkg_hi)\n", + " bkg_estimate = 0.5 * np.sum(nevts[mask_bkg])\n", + " return (bkg_estimate, np.sqrt(bkg_estimate))" ] }, { - "cell_type": "code", - "execution_count": null, - "id": "dac5b639", + "cell_type": "markdown", + "id": "5e1b508d", "metadata": {}, - "outputs": [], "source": [ - "def plot_5_sigma(nbkg):\n", - " nsig = 5*np.sqrt(nbkg)\n", - " plot_trial(nsig, nbkg)" + "# Some useful functions optimize data and setting expectations\n", + "\n", + "You can use these three functions to:\n", + "\n", + "1. `plot_nexp_passed_cuts`: makes a plot of how many signal and background events from the idealized data pass your cuts if you as a function of the cut width.\n", + "\n", + "2. `find_sig2noise`: makes of plot of the signal to noise, i.e., the number of signal events divided by the uncertainty in the background, as this is the key quantity for understanding the significance of the signal.\n", + "\n", + "3. `sig2noise_v_time`: shows you how the signal to noise will increase with time, assuming you keep taking data." ] }, { "cell_type": "code", "execution_count": null, - "id": "86eb1c24", + "id": "92f80e7b", "metadata": {}, "outputs": [], "source": [ - "plot_5_sigma(10)" + "def plot_nexp_passed_cuts(masses, model_sig, model_bkg):\n", + " sig_cts = np.zeros(26)\n", + " bkg_cts = np.zeros(26)\n", + " widths = np.linspace(0, 25, 26)\n", + " for i, width in enumerate(widths):\n", + " sig_cts[i], bkg_cts[i] = passed_cuts(width, masses, model_sig, model_bkg)\n", + " _ = plt.plot(widths, sig_cts, label=\"Signal\")\n", + " _ = plt.plot(widths, bkg_cts, label=\"Background\")\n", + " _ = plt.yscale('log')\n", + " _ = plt.xlabel(r\"Cut Width [GeV / $c^2$]\")\n", + " _ = plt.ylabel(r\"Events [per GeV / $c^2$ / month]\")\n", + " \n", + "def find_sig2noise(mass_grid, model_sig, model_bkg, plot=True):\n", + " sig_cts = np.zeros(26)\n", + " bkg_cts = np.zeros(26)\n", + " widths = np.linspace(0, 25, 26)\n", + " for i, width in enumerate(widths):\n", + " if i == 0:\n", + " continue\n", + " sig_cts[i], bkg_cts[i] = passed_cuts(width, mass_grid, model_sig, model_bkg)\n", + " sig2noise = np.zeros(26)\n", + " sig2noise[1:] = sig_cts[1:]/np.sqrt(bkg_cts[1:])\n", + " if plot:\n", + " _ = plt.plot(widths, sig_cts/np.sqrt(bkg_cts))\n", + " _ = plt.xlabel(r\"Cut Width [GeV / $c^2$]\")\n", + " _ = plt.ylabel(r\"$\\frac{n_{\\rm sig}}{\\sqrt{n_{\\rm bkg}}}$ for one month\")\n", + " return sig2noise\n", + "\n", + "def sig2noise_v_time(mass_grid, model_sig, model_bkg, plot=True):\n", + " max_s2n = np.zeros(24)\n", + " best_cut = np.zeros(24)\n", + " n_months_array = np.arange(24)\n", + " for n_months in n_months_array:\n", + " if n_months == 0:\n", + " continue\n", + " s2n = find_sig2noise(mass_grid, n_months*model_sig, n_months*model_bkg, plot=False)\n", + " max_s2n[n_months] = np.max(s2n)\n", + " best_cut[n_months] = np.argmax(s2n)\n", + " if plot:\n", + " _ = plt.scatter(n_months_array, max_s2n)\n", + " _ = plt.xlabel(r\"Time [months]\")\n", + " _ = plt.ylabel(r\"$\\frac{n_{\\rm sig}}{\\sqrt{n_{\\rm bkg}}}$ for N months\")\n", + "\n", + " return max_s2n" ] }, { - "cell_type": "code", - "execution_count": null, - "id": "2a5acd60", + "cell_type": "markdown", + "id": "e5c79d33", "metadata": {}, - "outputs": [], "source": [ - "plot_5_sigma(100)" + "# Some useful functions for fitting the data\n", + "\n", + "The important one here is `fitAndPlotResult`, which will find you data an plot the results.\n", + "\n", + "You need to provide it with:\n", + "\n", + "1. masses: the array of mass points\n", + "\n", + "2. nevs: the number of events observed a each mass\n", + "\n", + "3. ref_mass: the reference mass for the background model (use $125 \\frac{\\rm GeV}{c^2}$)\n", + "\n", + "4. init_pars: guesses for the initial parameters.\n", + "\n", + "The three parameters are: \n", + "\n", + "1. The total number of signal events.\n", + "\n", + "2. The number of background events in the bin at the reference mass.\n", + "\n", + "3. The slope of the background model, in events per bin." ] }, { "cell_type": "code", "execution_count": null, - "id": "2218e8ba", + "id": "b1b79b47", "metadata": {}, "outputs": [], "source": [ - "plot_5_sigma(1000)" + "from functools import partial\n", + "\n", + "def Gauss(x, nsig, mu, sigma):\n", + " return nsig*sps.norm(loc=mu, scale=sigma).pdf(x)\n", + "\n", + "def poly1(x, ref_mass, offset, slope):\n", + " return offset + (x-ref_mass)*slope\n", + "\n", + "def model_func(x, ref_mass, nsig, offset, slope):\n", + " return Gauss(x, nsig, Higgs_Mass, Higgs_Width) + poly1(x, ref_mass, offset, slope)\n", + "\n", + "def generic_chi2(params, data_vals, model, x, ref_mass):\n", + " model_vals = model(x, ref_mass, *params)\n", + " return np.sum(((data_vals - model_vals)**2)/data_vals)\n", + "\n", + "def cost_func(data_vals, model, x, ref_mass):\n", + " return partial(generic_chi2, data_vals=data_vals, model=model, x=x, ref_mass=ref_mass)\n", + "\n", + "def fitAndPlotResult(masses, nevts, ref_mass, init_pars):\n", + " our_cost_func = cost_func(nevts, model_func, masses, ref_mass=ref_mass)\n", + " result = optimize.minimize(our_cost_func, x0=np.array(init_pars))\n", + " fit_pars = result['x']\n", + " cov = result['hess_inv']\n", + " model_fit = model_func(masses, ref_mass, *fit_pars)\n", + " background_fit = poly1(masses, ref_mass, fit_pars[1], fit_pars[2])\n", + " print(\"Best Fit ---------\")\n", + " print(\"N Signal: %.1f [Events]\" % fit_pars[0])\n", + " print(r\"Higgs Peak: %.4f $[\\frac{\\rm GeV}{c^2}]$\" % Higgs_Mass)\n", + " print(r\"Higgs Width: %.4f $[\\frac{\\rm GeV}{c^2}]$\" % Higgs_Width)\n", + " print(r\"Background at 125 GeV: %.2f $[{\\rm Events} / \\frac{\\rm GeV}{c^2}]\" % fit_pars[1])\n", + " print(r\"Background slope: %.2f $[{\\rm Events} / \\frac{\\rm GeV}{c^2} / \\frac{\\rm GeV}{c^2}]$\" % fit_pars[2])\n", + " _ = plt.errorbar(masses, nevts, yerr=np.sqrt(nevts), fmt='.', label=\"data\")\n", + " _ = plt.plot(masses, background_fit, label=\"background model\")\n", + " _ = plt.plot(masses, model_fit, label=\"full model\")\n", + " _ = plt.xlabel(r\"mass $[\\frac{\\rm GeV}{c^2}]$\")\n", + " _ = plt.ylabel(r\"Events $[{\\rm per }\\frac{\\rm GeV}{c^2}]$\")\n", + " _ = plt.legend()\n", + " return (fit_pars[0], np.sqrt(cov[0,0]))" ] }, { - "cell_type": "code", - "execution_count": null, - "id": "16440af6", + "cell_type": "markdown", + "id": "0485e2c2", "metadata": {}, - "outputs": [], "source": [ - "plot_5_sigma(10000)" + "# Here are your models for the signal and background.\n", + "\n", + "Note that they are expressed in terms of events per month." ] }, { "cell_type": "code", "execution_count": null, - "id": "300e22b9", + "id": "7e1d8738", "metadata": {}, "outputs": [], "source": [ - "plot_5_sigma(100000)" + "ref_mass = 130.\n", + "nsig_per_month = 20.\n", + "nbkg_per_mev_per_month = 40.\n", + "bkg_slope_per_mev_per_month = -0.2\n", + "model_bkg = poly1(mass_grid, ref_mass, nbkg_per_mev_per_month, bkg_slope_per_mev_per_month)\n", + "model_sig = Gauss(mass_grid, nsig_per_month, Higgs_Mass, Higgs_Width)\n", + "\n", + "_ = plt.scatter(mass_grid, model_sig, label=\"Signal\", marker='.')\n", + "_ = plt.scatter(mass_grid, model_bkg, label=\"Background\", marker='.')\n", + "_ = plt.scatter(mass_grid, model_sig+model_bkg, label=\"Combined\", marker='.')\n", + "_ = plt.xlabel(r\"Mass [GeV/$c^2$]\")\n", + "_ = plt.ylabel(r\"Counts [per GeV/$c^2$ / month]\")\n", + "_ = plt.legend()" + ] + }, + { + "cell_type": "markdown", + "id": "855cb5c5", + "metadata": {}, + "source": [ + "# Here is your \"real\" data.\n", + "\n", + "Note that that is simulated for 24 months of data." ] }, { "cell_type": "code", "execution_count": null, - "id": "550e92d9", + "id": "fbc66963", "metadata": {}, "outputs": [], - "source": [] + "source": [ + "data = np.loadtxt('../data/Higgs.txt')\n", + "masses = data[:,0]\n", + "nevts = data[:,1]\n", + "errors = np.sqrt(nevts)\n", + "_ = plt.errorbar(masses, nevts, yerr=errors, fmt='.')\n", + "_ = plt.xlabel(r\"Mass [GeV/$c^2$]\")\n", + "_ = plt.ylabel(r\"Counts [per GeV/$c^2$]\")" + ] }, { "cell_type": "code", "execution_count": null, - "id": "86efca62", + "id": "0c75268c", "metadata": {}, "outputs": [], "source": [] diff --git a/nb/Project_Hubble_Tension.ipynb b/nb/Project_Hubble_Tension.ipynb index 7ce7f80..4acf8ab 100644 --- a/nb/Project_Hubble_Tension.ipynb +++ b/nb/Project_Hubble_Tension.ipynb @@ -20,9 +20,9 @@ "source": [ "# Project, exploring the Hubble \"Tension\"\n", "\n", - "Recall that in weeks 1 and 2 of this course we explored some measurements of the Hubble constant and used them to study some simple statistics, and to try using inverse variance weighting to make a weighted average of measurements that took into account the fact that the some of the measurements had smaller uncertainties than others.\n", + "Recall that in weeks 1 and 2 of this course we explored some measurements of the Hubble constant and used them to study some simple statistics, and to uise inverse variance weighting to make a weighted average of measurements that took into account the fact that the some of the measurements had smaller uncertainties than others.\n", "\n", - "It turns out that this is actually one of the biggest open questions in cosmology. Roughly speaking, measurements of the Hubble constant can be divided into two types. \n", + "It turns out that the consistency (or potential inconsistency) is the measurements of the Hubble constant is actually one of the biggest open questions in cosmology. Roughly speaking, measurements of the Hubble constant can be divided into two types. \n", "\n", "1. \"Early time\" measurements that use our knowledge of cosmology, and look at structures in the very early universe, e.g., using the primoridal background of light left over from the Big Bang that is now visible to us as a bath of faint microwave radiation, to determine the Hubble parameter. These measurements are more sensitive to our understanding of cosmology.\n", "\n", @@ -30,8 +30,7 @@ "\n", "We've provided you with a new data file, which is a sub-set of the more recent Hubble data measurements, including an extra column telling you if a particular measurement is type 1 or type 2.\n", "\n", - "\n", - "Here are some things that you might try and do:\n", + "Some goals for this project might include:\n", "\n", "1. Make a nice plot that summarize the difference between \"Early Time\" results and \"Late Time\" results\n", "\n", @@ -48,7 +47,7 @@ "metadata": {}, "outputs": [], "source": [ - "data = np.loadtxt(open(\"../data/Hubble_Extra.txt\", 'rb'), usecols=[1,2,3,4])\n" + "data = np.loadtxt(open(\"../data/Hubble_Extra.txt\", 'rb'), usecols=[1,2,3,4])" ] }, { @@ -92,7 +91,7 @@ " err = std / np.sqrt(nvals)\n", " wts = 1/(errors*errors)\n", " wmean = np.sum(values*wts)/np.sum(wts)\n", - " werr = np.sqrt(1/np.sum(H0_weights))\n", + " werr = np.sqrt(1/np.sum(wts))\n", " return (nvals, mean, std, err, wmean, werr)" ] }, diff --git a/nb/Project_SDSS_Line_Fit_Algo.ipynb b/nb/Project_SDSS_Line_Fit_Algo.ipynb index 39498a3..fcfa412 100644 --- a/nb/Project_SDSS_Line_Fit_Algo.ipynb +++ b/nb/Project_SDSS_Line_Fit_Algo.ipynb @@ -30,7 +30,7 @@ "\n", "This notebook has the pieces you will need to do that.\n", "\n", - "Here are some things that you might try and do:\n", + "Some goals for this project might include:\n", "\n", "1. Write a function to extract initial guesses for the fit parameters.\n", "\n", @@ -166,7 +166,7 @@ "def fitAndPlotResult(cutout_wl, cutout_flux, ref_lambda, init_pars):\n", " our_cost_func = cost_func(cutout_flux, model_func, cutout_wl, ref_lambda=ref_lambda)\n", " result = optimize.minimize(our_cost_func, x0=np.array(init_pars))\n", - " fit_parspars = result['x']\n", + " fit_pars = result['x']\n", " model_fit = model_func(cutout_wl, ref_lambda, *fit_pars)\n", " background_fit = poly1(cutout_wl, ref_lambda, fit_pars[3], fit_pars[4])\n", " print(\"Best Fit ---------\")\n", @@ -198,7 +198,7 @@ "metadata": {}, "outputs": [], "source": [ - "_ = plt.plot(wavelength, best_fit)\n", + "_ = plt.plot(wavelength, flux)\n", "_ = plt.xlabel(r\"$\\lambda [nm]$\")\n", "_ = plt.ylabel(\"Flux [arcane units]\")\n", "_ = plt.title(\"A spectrum from SDSS\")" @@ -327,14 +327,6 @@ "print(\"The flux value in the largest bin is: \", cutout_flux[np.argmax(cutout_flux)])" ] }, - { - "cell_type": "code", - 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