{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "43f086df",
   "metadata": {},
   "source": [
    "# PHYS 310 Class 4 (Data Analysis class number 3)\n",
    "Skeleton notebook for in-class exercises\n",
    "\n",
    "Tom Solomon, Feb 2023"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5972accc",
   "metadata": {},
   "outputs": [],
   "source": [
    "# First, start with the standard stuff at the top\n",
    "import numpy as np\n",
    "from scipy import optimize\n",
    "import urllib\n",
    "\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "00f416ac",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Following is an Ipython magic command that puts figures in notebook.\n",
    "%matplotlib notebook \n",
    "        \n",
    "# M.L. modifications of matplotlib defaults\n",
    "# Changes can also be put in matplotlibrc file, \n",
    "# or effected using mpl.rcParams[]\n",
    "mpl.style.use('classic') \n",
    "plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures\n",
    "plt.rc('axes', labelsize=16, titlesize=14)\n",
    "plt.rc('figure', autolayout = True) # Adjusts supblot params for new size"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "782bf598",
   "metadata": {},
   "source": [
    "# Recall from last week:  \n",
    "### Average $\\bar{x} = \\frac{1}{N}\\Sigma_ix_i$\n",
    "### Weighted average $\\bar{x}_{weighted} = \\frac{\\Sigma_iw_ix_i}{\\Sigma_iw_i}$\n",
    "### where weights $w_i = \\frac{1}{\\alpha_i^2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "383d7763",
   "metadata": {},
   "source": [
    "<hr>\n",
    "Question:  when is it appropriate to average different results (with or without using weights)?\n",
    "\n",
    "Let's say, e.g., two different groups measure Plank's constant _h_ from the cut-off wavelength of x-rays emanating from x-ray anodes. Let's say that one group uses a Molybdenum anode and finds results $6.8 \\pm 0.2$, $6.9 \\pm 0.2$, $6.7 \\pm 0.3$ and $7.0 \\pm 0.2$  (all $\\times 10^{-34}$ Js), and another group uses a copper anode and finds results $4.8 \\pm 0.3$, $5.1 \\pm 0.2$, $4.7 \\pm 0.4$ and $4.9 \\pm 0.2$ (also all $\\times 10^{-34}$ Js)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d8a1ead9",
   "metadata": {},
   "outputs": [],
   "source": [
    "xvalues = np.array([1,2,3,4,5,6,7,8])\n",
    "hvalues = np.array([6.8,6.9,6.7,7.0,4.8,5.1,4.7,4.9])\n",
    "hmean = np.mean(hvalues)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8e7a10c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.scatter(xvalues,hvalues)\n",
    "plt.axhline(hmean, c='r', label='mean')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de9c900d",
   "metadata": {},
   "source": [
    "### Question:  Does taking the \"mean\" of the data above produce a better estimate of the true value of _h_?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c8cc42c",
   "metadata": {},
   "source": [
    "## What if one group measured _h_, all with the same electrode and got the following values?\n",
    "$6.82 \\pm 0.16$, $6.9 \\pm 0.2$, $4.8 \\pm 0.2$, $5.13 \\pm 0.16$, $6.72 \\pm 0.13$, $4.7 \\pm 0.2$, $4.9 \\pm 0.2$ and $7.0 \\pm 0.4$  (all $\\times 10^{-34}$ Js)?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1290981c",
   "metadata": {},
   "outputs": [],
   "source": [
    "xvalues = np.array([1,2,3,4,5,6,7,8])\n",
    "hvalues = np.array([6.8,6.9,4.8,5.1,6.72,4.7,4.9,7.0])\n",
    "hmean = np.mean(hvalues)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3059a4b0",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.scatter(xvalues,hvalues)\n",
    "plt.axhline(hmean, c='r', label='mean')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51032c10",
   "metadata": {},
   "source": [
    "### Question:  Does taking the \"mean\" of the data above produce a better estimate of the true value of _h_?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "308de63d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0a83653e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e8984489",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7c96dc6c",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "76fccab6",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "200aede5",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "96024a51",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "21501a7d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "27113853",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c203db87",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e5773c95",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "6185998a",
   "metadata": {},
   "source": [
    "#### What if we add the error bars to the plot?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ceb15fa8",
   "metadata": {},
   "outputs": [],
   "source": [
    "xvalues = np.array([1,2,3,4,5,6,7,8])\n",
    "hvalues = np.array([6.82,6.9,4.8,5.13,6.72,4.7,4.9,7.0])\n",
    "alphavalues = np.array([0.16, 0.2, 0.2, 0.16, 0.13, 0.2, 0.2, 0.4])\n",
    "# In class, add a calculation of weighted mean, call it wmean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4d45042f",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.errorbar(xvalues,hvalues,alphavalues, fmt = 'ko')\n",
    "#plt.axhline(wmean, c='r', label='weighted mean')  #Uncomment this when you have determined weighted mean\n",
    "plt.xlabel('experiment number')\n",
    "plt.xlim(0,9)\n",
    "plt.ylabel('h ');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "993536c6",
   "metadata": {},
   "source": [
    "Go back two cells, add a line or two to calculate the weighted mean. (Hint:  a dot product makes the numerator easy.  you can take the dot product of two matrices a and b either with numpy.dot -- i.e., np.dot(a,b) -- or by using the @ sign -- i.e., a @ b.)  Then plot the weighted mean on the plot above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5b9ca08c",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "4aa486a5",
   "metadata": {},
   "source": [
    "Next, plot the residuals $x_i - \\bar{x}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2985beee",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "a38933a2",
   "metadata": {},
   "source": [
    "And then plot the normalized residuals ($(x_i - \\bar{x})/\\alpha_i$)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa8495d1",
   "metadata": {},
   "source": [
    "Plot residuals, then normalized residuals.  Then square the residuals and add them up -- chi-squared."
   ]
  }
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