AEPSHEP2022/notebook1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# **Statistics Lecture 1 : exercises notebook**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The _Practical Statistics_ course has two components: the lectures, and practical exercises using jupyter notebooks. This is the companion notebook to lecture 1, covering the following topics:\n",
    "1. PDF means, variance and covariance\n",
    "2. Quantiles of the normal distribution\n",
    "3. The Poisson distribution and the central limit theorem\n",
    "4. Histograms and $\\chi^2$\n",
    "\n",
    "It presents some explanations followed by exercises (highlighted in $\\mathbf{\\color{red}{\\text{red}}}$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Basics\n",
    "\n",
    "The notebooks accompanying the lectures will make use of the `numpy`/`scipy`/`matplotlib` stack. It's also possible to implement these examples with other tools such as `root`, for those who are more familiar with it and have it installed. We'll go with `numpy`/`scipy`/`matplotlib` which has the advantage of being more widely available, and easy to run in tools such as `binder`, but feel free to use other implementations if you like.\n",
    "\n",
    "To start with, we need to import the relevant python packages:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # handles most of the numerical work\n",
    "import scipy.stats  # implements statistical tools (PDFs, etc)\n",
    "from matplotlib import pyplot as plt # for plotting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "numpy handles most of the underlying work. In particular operations will usually be performed on numpy arrays -- arrays of floats similar to python lists.\n",
    "For instance we can generate some events as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.09035087 2.94917786 2.74308263 1.75793111 4.0840741  3.74566598\n",
      " 3.46660394 1.32387562 2.54114835 3.86739878]\n"
     ]
    }
   ],
   "source": [
    "yvals = np.random.normal(3.0, 1.0, 10) # generate 10 events from a Gaussian(mean=3, sigma=1) distribution\n",
    "print(yvals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `scipy` package has some useful features like fully-featured PDF implementations, and `matplotlib` will be used for inline plots. As an example of how things work, we can draw a few Gaussian curves: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "xvals = np.linspace(-5,5,100) # an array of 100 points between -5 and 5 which will be used for plotting\n",
    "means = [0.0, 1.0, -2.0]   # a few mean values to plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Note that plt.plot takes 2 arrays, one array of x values and an array of y values.\n",
    "# norm.pdf is a scalar function, which takes as input a single x value and returns the value of\n",
    "# the normal PDF at x.\n",
    "# In python, the construct [ f(xval) for xval in xvals ] can be used to build the list of\n",
    "# all the f(xval) values, using only 1 line of code\n",
    "\n",
    "for mean in means:\n",
    "    yvals = [ scipy.stats.norm.pdf(xval, loc=mean, scale=1) for xval in xvals ] # compute the y-values for each x\n",
    "    plt.plot(xvals, yvals, label='mean=%g' % mean) # draw the plots\n",
    "\n",
    "plt.ylim(0,0.45)   # adjust the y range\n",
    "plt.grid(True)     # draw grid lines on the plot\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('G(x)')\n",
    "plt.title('Gaussian distributions')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now repeat with different widths, and a mean of 0 :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigmas = [1.0, 2.0, 3.0]   # a few sigma values to plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Now draw the corresponding plots by completing the code below}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "for sigma in sigmas:\n",
    "    pass # replace with real code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One can also generate random datasets and plot the result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# generate two random variables\n",
    "xvals = np.random.normal(3.0, 1.0, 10000)\n",
    "yvals = np.random.normal(50.0, 10.0, 10000)\n",
    "\n",
    "plt.subplot(121)\n",
    "plt.hist(xvals, bins=100, color='b')\n",
    "plt.subplot(122)\n",
    "plt.hist(yvals, bins=100, color='b')\n",
    "plt.figure()\n",
    "plt.scatter(xvals,yvals, color='b')\n",
    "plt.figure()\n",
    "plt.hist2d(xvals, yvals, (30, 30), cmap='Reds');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compute some sample means, variances and covariances:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample means: 2.999896307916171 49.91011359814221\n"
     ]
    }
   ],
   "source": [
    "mean_x = np.sum(xvals)/len(xvals)\n",
    "mean_y = np.sum(yvals)/len(xvals)\n",
    "print('sample means:', mean_x, mean_y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Now compute the variance, covariance and correlation coefficient of x and y in the same way using np.sum()}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample variances: 999 999\n",
      "sample RMSs: 31.606961258558215 31.606961258558215\n",
      "sample covariance and correlation coefficient: 999 999\n"
     ]
    }
   ],
   "source": [
    "var_x = 999 # replace with real code\n",
    "var_y = 999 # replace with real code\n",
    "print('sample variances:', var_x, var_y)\n",
    "print('sample RMSs:', np.sqrt(var_x), np.sqrt(var_y))\n",
    "cov_xy = 999 # replace with real code\n",
    "cor_xy = 999 # replace with real code\n",
    "print('sample covariance and correlation coefficient:', cov_xy, cor_xy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This can also be done more simply using numpy internal functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample means: 2.999896307916171 49.91011359814221\n",
      "sample variances: 1.004986385725867 101.59756433268522\n",
      "sample covariance and correlation coefficient: 1.004986385725867 0.006872629228453001\n"
     ]
    }
   ],
   "source": [
    "print('sample means:', np.mean(xvals), np.mean(yvals))\n",
    "print('sample variances:', np.var(xvals, ddof=1), np.var(yvals, ddof=1))\n",
    "print('sample covariance and correlation coefficient:', np.cov(xvals, yvals, ddof=1)[0,1], np.corrcoef(xvals,yvals)[0,1])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the covariance and correlations coefficient should really be zero, the non-zero value here is due to fluctuations. For a case with real correlation, one can do"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "zvals = 5*xvals + yvals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Now plot z vs. x and compute the covariance and correlation coefficient of x and z}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "pass # your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Gaussian quantiles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The *quantiles* of a PDF are the fraction of its integral within a given range of its variable. The quantiles of the normal distribution $N(x) = G(x; x_0=0,\\sigma=1)$ play an important role since they are used to define some important quantities later ($1\\sigma$ and $1\\sigma$ errors, $5\\sigma$ discovery, etc.). \n",
    "\n",
    "One defines the cumulative distribution function of normal as $\\Phi(x) = \\int\\limits_{-\\infty}^x N(t) dt$, and a two-sided version can also be defined as $\\Phi(x, y) = \\int\\limits_x^y N(t) dt$\n",
    "\n",
    "They are easy to compute using the scipy tools:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw a normal distribution as before\n",
    "xvals = np.linspace(-5,5,100)\n",
    "yvals = [ scipy.stats.norm.pdf(xval) for xval in xvals ]\n",
    "fig = plt.figure()\n",
    "plt.plot(xvals, yvals)\n",
    "plt.ylim(0,0.5)\n",
    "plt.title(\"Cumulative normal distribution\")\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('N(x)')\n",
    "\n",
    "# Now fill in part of the distribution and print the integral\n",
    "up = 1\n",
    "shaded_xvals = np.linspace(-5,up,100)\n",
    "shaded_yvals = [ scipy.stats.norm.pdf(xval) for xval in shaded_xvals ]\n",
    "plt.fill_between(shaded_xvals, shaded_yvals, alpha=0.5, color='g')\n",
    "plt.text(-4.5, 0.45, 'Phi(%g)=%g' % (up, scipy.stats.norm.cdf(up)), fontsize=20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This form is normally referred to as a *one-sided* quantile. Two-sided quantiles apply the boundary on both sides:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Now draw and compute the *two-sided* 1-sigma quantile (from -1 to +1) [Feel free to copy-and-paste the code above!]}}}$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "pass # your code here\n",
    "shaded_xvals = np.linspace(-up,up,100)\n",
    "pass # more code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's plot some quantiles for different values of the bounds. As we'll see, the tail integrals (a.k.a the *survival function*, corresponding to the unshaded areas above) are also useful, so compute all of the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "bounds = [0, 1, 2, 3, 4, 5]\n",
    "one_sided = [ scipy.stats.norm.cdf(up) for up in bounds ]\n",
    "one_sided_tail = [ scipy.stats.norm.sf(up) for up in bounds ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Compute the two-sided integrals for each value of \"bounds\"}}}$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Bound</th>\n",
       "      <th>1-sided</th>\n",
       "      <th>1-tail</th>\n",
       "      <th>2-sided</th>\n",
       "      <th>2-tail</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.500000</td>\n",
       "      <td>5.000000e-01</td>\n",
       "      <td>-999.0</td>\n",
       "      <td>-999.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0.841345</td>\n",
       "      <td>1.586553e-01</td>\n",
       "      <td>-999.0</td>\n",
       "      <td>-999.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2.0</td>\n",
       "      <td>0.977250</td>\n",
       "      <td>2.275013e-02</td>\n",
       "      <td>-999.0</td>\n",
       "      <td>-999.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>3.0</td>\n",
       "      <td>0.998650</td>\n",
       "      <td>1.349898e-03</td>\n",
       "      <td>-999.0</td>\n",
       "      <td>-999.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4.0</td>\n",
       "      <td>0.999968</td>\n",
       "      <td>3.167124e-05</td>\n",
       "      <td>-999.0</td>\n",
       "      <td>-999.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>5.0</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>2.866516e-07</td>\n",
       "      <td>-999.0</td>\n",
       "      <td>-999.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Bound   1-sided        1-tail  2-sided  2-tail\n",
       "0    0.0  0.500000  5.000000e-01   -999.0  -999.0\n",
       "1    1.0  0.841345  1.586553e-01   -999.0  -999.0\n",
       "2    2.0  0.977250  2.275013e-02   -999.0  -999.0\n",
       "3    3.0  0.998650  1.349898e-03   -999.0  -999.0\n",
       "4    4.0  0.999968  3.167124e-05   -999.0  -999.0\n",
       "5    5.0  1.000000  2.866516e-07   -999.0  -999.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "two_sided = [ -999 for up in bounds] # replace with real code\n",
    "two_sided_tail = [ -999 for up in bounds ] # replace with real code\n",
    "\n",
    "# Pretty-print the result\n",
    "import pandas as pd\n",
    "import jinja2\n",
    "labels = [ 'Bound', '1-sided', '1-tail', '2-sided', '2-tail' ]\n",
    "data = np.array([ bounds, one_sided, one_sided_tail, two_sided, two_sided_tail ]).T\n",
    "pd.DataFrame(data, columns=labels)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Poisson distributions and the Central-limit theorem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First let's get acquainted with Poisson distributions. As explained in the lecture, they are critical to describe counting processes, and become progressively more Gaussian with increasing event rates, so we can check a few rate hypotheses:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Rate</th>\n",
       "      <th>Mean</th>\n",
       "      <th>Variance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.1</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>10.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>10.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Rate  Mean  Variance\n",
       "0   0.1   0.1       0.1\n",
       "1   1.0   1.0       1.0\n",
       "2   3.0   3.0       3.0\n",
       "3  10.0  10.0      10.0"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xvals = np.linspace(0, 20, 21)\n",
    "rates = [ 0.1, 1, 3, 10 ]\n",
    "\n",
    "for i, rate in enumerate(rates) :\n",
    "  plt.subplot(221 + i)\n",
    "  plt.vlines(xvals, 0, scipy.stats.poisson.pmf(xvals, rate), colors='b', lw=3, alpha=0.5, label='rate=%g' % rate)\n",
    "  plt.legend()\n",
    "\n",
    "means     = [ scipy.stats.poisson.stats(mu=rate, moments='m') for rate in rates ]\n",
    "variances = [  scipy.stats.poisson.stats(mu=rate, moments='v')  for rate in rates ]\n",
    "pd.DataFrame(np.array([ rates, means, variances]).T, columns=[ 'Rate', 'Mean', 'Variance' ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "nexp = 10000\n",
    "nterms = [ 1, 2, 3, 5, 10, 100 ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Exercise: check the central limit theorem by drawing Poisson events:}}}$\n",
    "- For $nterms=1$, draw $nexp=10000$ $Poisson(1)$ events (`n = np.random.poisson(1)`) and histogram them\n",
    "- For $nterms=2$, draw pairs of $Poisson(1)$ events and histogram their averages\n",
    "- Repeat for the $nterms$ values provided above, to get increasingly Gaussian distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "for nterm in nterms :\n",
    "    pass # replace with real code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, the distribution becomes more Gaussian as we increase the number of terms, with a variance that grows as $\\sqrt n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Histograms and the $\\chi^2$ test\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We consider a 20-bin histogram of data over $[0,1]$, which we assume is coming from a linear distribution, $f(x) \\propto (1 - x/2)$. We also assume that we have a histogram of data, and would like to decide whether the data is compatible with the assumed linear distribution. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7fe766863be0>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "nbins = 20\n",
    "nevts = 1000\n",
    "np.random.seed(0)\n",
    "xvals = np.linspace(0.5, nbins - 0.5, nbins)\n",
    "expected_yields = np.array([ (1 - i/2/nbins) for i in range(0, nbins) ])\n",
    "expected_yields *= nevts/np.sum(expected_yields)\n",
    "dataset = [ np.random.poisson(y) for y in expected_yields ]\n",
    "plt.figure()\n",
    "plt.bar(xvals, expected_yields, yerr=np.sqrt(expected_yields), color='b')\n",
    "plt.scatter(xvals, dataset, color='orange', zorder=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Compute chi2 values}}}$\n",
    "- Compute the chi2 values between the dataset and the expected yields as the sum of `(d - y)/np.sqrt(y)`\n",
    "- Compute the chi2 p-value using `scipy.stats.chi2.sf(chi2, nbins)`\n",
    "- Compute the significance for the pvalue using `scipy.stats.norm.isf(pvalue)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "chi2: 19980\n",
      "p(chi2): 999\n",
      "signif(chi2): 999\n"
     ]
    }
   ],
   "source": [
    "chi2 = sum( [ 999 for d,y in zip(dataset, expected_yields) ]) # replace 999 by real code\n",
    "pvalue = 999   # replace 999 by real code\n",
    "signif = 999   # replace 999 by real code\n",
    "print('chi2:', chi2)\n",
    "print('p(chi2):', pvalue)\n",
    "print('signif(chi2):', signif)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The compatibility between the generated dataset the true distribution may be larger or smaller depending on the random fluctuations that were generated, but generally the agreement should be quite good.\n",
    "\n",
    "One can check that the distribution is doing what it is expected by generating datasets:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fe76650f910>]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ndata = 10000\n",
    "all_data = [ [ np.random.poisson(y) for y in expected_yields ] for i in range(0, ndata) ]\n",
    "all_chi2 = [ sum( [ ((d-y)/np.sqrt(y))**2 for d,y in zip(data, expected_yields) ]) for data in all_data ]\n",
    "bins = np.linspace(0,50,100)\n",
    "plt.hist(all_chi2, bins=bins, color='b');\n",
    "plt.plot(bins, scipy.stats.chi2.pdf(bins, 20)*ndata/2, color='orange')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The p-value we have computed above is the tail integral in the distribution above, and we see that it agrees quite well with the generated datasets.\n",
    "\n",
    "If we now consider a prediction with the opposite slope, this will be generally much worse:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7fe7663c9a00>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "new_dataset = np.flip(expected_yields)\n",
    "plt.figure()\n",
    "plt.bar(xvals, new_dataset, yerr=np.sqrt(new_dataset), color='b')\n",
    "plt.scatter(xvals, dataset, color='orange', zorder=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\color{red}{\\rightarrow\\text{Recompute the chi2, chi2 pvalue, and chi2 signifiance for the new dataset}}}$\n",
    "\n",
    "You should find a very large $\\chi^2$, a small $p$-value and a large significance, all indicative of a significant disagreement between the dataset and the prediction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "pass # replace with real code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, one can repeat the whole study with a lower number of events per dataset, e.g. 10 instead of 1000. In this regime, the Poisson distributions in each bin are not Gaussian enough, and the computed $\\chi^2$ does not in fact follow a $\\chi^2$ distribution. In this case, one needs to move away from the $\\chi^2$ formula and use a statistical model than accounts for Poisson effects."
   ]
  }
 ],
 "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}