uncertainty/equilibration_uncertainty_fill_in.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Day 2, Morning Session 1, MD/MC workshop\n",
    "### Dr. Michael Shirts, CU Boulder"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How do we understand the information we get out of simulations?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# load up some modules.\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy\n",
    "import scipy.stats as stats\n",
    "import pandas as pd\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's load up some typical data from a simulation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "potential = np.loadtxt('potential.dat')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-42534.128906, -42824.398438, -42978.53125 , ..., -35563.0625  ,\n",
       "       -35446.75    , -35426.964844])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "potential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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VrgUAN449HPCvGYeTjTozjcqkhZv5y2fOHFYfLHA6Ue8JMlLp40VbuPejpZSWK4My29A7SIf/g2cN4O7/LuHCYZl+H4QegzLbMCizDXef0Y/i0nISYmNYtnUvE2dt4G23Q/bso5wPifvP7M8bMzcQHyukJMXzxzG9vffj/O74HvTq2IrBGW1o2zLB7/6Fa0b1DHrNw7qn8dnSbXRzPwRCccXI7vzDHVWkQGpyHJ1Sk9hWZUaCq0/oydUnOO/tSTDnDc1gVJ8OFBSXkpwQ671p8sNrj/WeN7ZfR37M2RNy/0Sgb+epydU/Xl799TB+3JRfrdyX58PP0wT16R9+RosavgR4tGuVyDif5ruYGOG+n/ejb+dURvRox/VvLQDgu1tPJKOt/+/6vKEZfLV8B3+/6Cg6tfZPkJeM6Ob9N9jTnRqm6n1W71w5ggvdIeYA5wzJ4IJhmUz/KZdjejg1yBE90nh79ka/gSktE+M4uX8n/v2DM/ji8I4pTPzt0ewuKvHGW5NTB3Ri/OCufhOv+oqJEY7tmc5/rzuOAV1SuXhENyoqlLYhNqXWF0s0ptHILyoN+h8mkM99Oop7dWhVrR8iNkZQVc48sgvnD83wqwUE4+lHGNC1NQ+fNZD9JeX8d+EWUtzmLBFh5u0nefty/jjmcL/zT+xT2aYf777fYWktGD+4a9D3/M1xWYw5ogPd2rUMekxVbVokcNHwTN6evYnEuBhEhG9vPdF7o9+MP48KOiBCRGifkkh7t1nqifMG8c7sjQzKaON33FUn9PQms8y0ZP550ZCg98rE+/xuHxjfny+X7wjYZHZinw5+v6OA15bs9L2kuX/Pfl2C135qE2huu6pJBpw55N67+piArxHoOlol+X90tkzwf56cEEtmWgsuGVFZAx4/uCvZWWl0bZPsd+yQw9pyzlFdGdGjHUf3aAdUT2S++nVOZZn7Jeaxc/y/OA3t1pbFm/O5cezhtPUZtTc4sw0ArZNr/j8wcdYGfjn8sHpfgsESjYmowgNlrNqxj6179ldrUqqLfl1SiYkRXr4smyten8ugjNZMun5k7SfWICZGvO3qyT7fqDu3Tg52ip/O7jfjP47pXeNxIlKnJONxx2lH0K1dS+8Hd3xsDLPvGE1MjJDeKjHk12yfksjvR1ePsVViHKP6tGf6ylzuOaM/g9wPK4/7ft6P8YO7MmvdLk44vDJ5XHpMFpcek1Xn6/H4We90Hjl7IGcd1TD9B6G4+4x+DDmsjfd5alI8s+8YzfBHpgHQIdW/iS8myAd11SQDTq3mqQsG+5clBP5oHtqtLf+55lg+W7KV4tIKWrvJ5NVfDyPOrb1A5SwPdfXZkm38MsjAiENhicY0uJKyCuJjheemr+GD+TmsyXWankIZuXXawE58unibX9kRnVO9syF7ksD+epoo09NvcjBf8FomxoV1dtyUpHhvs5hHh9Tg/SIH4/cn9WLltgLvgIP0Vgns3OfUlDy1hVMGBO9wPhgiUudRfKFonRxf40CImgS6WbVDahLDu6fRr3MqHVOTWHjPWP4+bTWvfL+OQ+36aJnofLG547S+PPLpCgZ0TeXUAZ29nfdVf+e11RJDdcqATmFZUM4SjWlQ2/KLGfHoNO46/QiemLrSb9/2/OIgZ1XyndLl8mOzuOqEHrRrWfltsleHVozq075ak9bB8vyfi9bbzYZ2S2Pm7ZUj87695SQunDAz4Ei4xm7RvYEHTByK966qbG5r0yKBCvcfSrAaTajiYmO8X1LOH5pJckJs0GUnDtX5QzP4P3eaH8/Q8vpmo85MWD39xU8scu9gXrBxNy9960yB8dDk5dWOLa3h/oPj3XtFTvP5Jnffmf3p3DrZb9hoQlwMr/16uLdN+lB5E029vFrTl5wQy6TrR/KX8w7+xr7m7JieTh/LUT7NbIeqbcuEsCUZgMfPPZKHz3aGiB/ROTxfIKxGY8KmrLyCZ6at4plpq1j36Gmc/dz/ajz+myBzLv143zjvhI9xMcKHCzaTs6dhFroa3j2Nt2dvom8T/AZvGt7J/TsFnO2gMYuJEX45/DDGHNGRjvXc9OphicaEjW8/yca84InBt+oOMOuO0aQmxXPEPc50JVVXo3z7yhH1HGlwZx+VwfDu7QJ24hoTSFNKMh4iErYkA9Z0ZuogZ3cRb/usmeFRVFIWcOoW34kdg02GCfjdwf30BYPomJrkHeV1sKNn6pMlGWMOjdVoTMiueG0uK7cXcOqATt5vbapKv3umctHww3i0yrxKZ/6z8p6LK16vPlHp0xcM4siMNmS2bUH+/lJydhd5b4wEZ8LFcIyAMcY0LEs0JmSeGwB37ivxJhrP9Plvz95IZloyvTukMLZfRwqKS2ucPfalS7MZ26+j93nV2YmhaTZBGGOqs0RjQuaZZuT9eTneObTyfVYr9EwdM7JXuvdGMl/nDc3wrpY45oj6GfdvjGn8rI/GhMyzENULM9awbMtenvx8Jcc/UX1t+O9W7/Rbiz0xLob+XVK51me+L2sSMyZ6RKRGIyIPAuOBCmAHcLmqbnH3HQm8CKS6+4eparGIDAVeA5KBT4EbVFVFJBF4AxgK7AIuUNX17mtdBtzlvu1Dqvp6w1xh87Bo0x7unrSEH3Py+dsFg0nzuTHSs+Z8KP7926MZFmRVSGNM8xepGs0Tqnqkqg4GPgHuARCROODfwNWq2h8YBXjaZp4HrgR6u49T3PIrgN2q2gt4Gnjcfa004F7gaGA4cK+IhL72q+H3by/gxxxnpt0/vrvQb1nfUDxz4WBm/HmUJRljolxEEo2q+i483pLKG6/HAT+q6iL3uF2qWi4inYFUVZ2pztrTbwBnueeMBzw1lfeB0eK0y5wMfKGqeaq6G/iCyuRkavHk5yur3fuSk1dEeqvQO+jbB5jY8YwjOzP6IFYtNMY0XRHroxGRh0VkE/Ar3BoNcDigIjJVROaLyC1ueVfAd53RHLfMs28TgKqWAflAO9/yAOdUjeVKEZkrInNzcwPfnR5NVNU7PbyvXYUlnBJg9cWqC2d5pAVISv/85RBedpf2NcZEh7AlGhH5UkSWBHiMB1DVO1U1E5gIXO+eFgeMxEk+I4GzRWQ0BFx60FMLCravpnP8C1UnqGq2qma3b199/fVoU1JeEXTfkRltuPWUvn7J5a3fjQi45nr7WlZHNMZEh7ANBlDVMSEe+hYwGac/JQeYoao7AUTkU2AITr+N7/qzGcAWdzsHyARy3D6e1kCeWz6qyjnTD+JSosrmPfs5vYaO/pTEOH6Rnck1o3qSddtkwJl+/dFzBjKuX0cS42Pokd4KRWtdhtcYEx0i0nQmIr53550JrHC3pwJHikgLN2mcACxT1a1AgYiMcPtfLgUmued8BFzmbp8HfOX240wFxolIW3cQwDi3zNTg1e/WsacoeKd/SpV5x5J9ZpU9sW8Hju2ZTqfWSSEvDmaMaf4idcPmYyLSB2f48gbgagBV3S0iTwFzcJq5PlXVye4511A5vHmK+wB4GXhTRFbj1GQudF8rzx1GPcc97gFVzQv3hTV1ZVXmLPvzyX04c1AXTnpyOqXlSorPErZz7xrjt4SvMcYEEpFEo6rn1rDv3zhNZVXL5wIDApQXA+cHea1XgFcOPtLosnPfAV7733q/suxubclMa0FmWgvW5hb6rf2Sbk1jxpgQ2NdRA0BFhfKzx/3v8k9vlUg/t9P/pUuzOWtwF3q2bxWJ8IwxTZjNdWYAeGv2Rr/1Y646vge3n3aE93nP9q3424VHRSI0Y0wTZzUaA8D6nYV+z32TjDHGHAqr0US5/SXlLN+2l5aJ9k/BGBMe9ukSpXbsLSY5IZYHPl7G/83LYWSvdO++n/VOr+FMY4ypG0s0UWr4I9P8nn+3eicAH18/kr6dUyIRkjGmmbJEE2WunTiPbfnFQfcPzGjdgNEYY6KBJZoosnFXEZ8u3hbpMIwxUcZGnUWRxz5bHukQjDFRyBJNlNhRUBy0NvPiJUMbOBpjTDSxprMoUFRSxvCHp1Urv2h4JqcP7MLI3ukcmdGak/tXX2vGGGMOlSWaKLBhV+VKmbExQnmF0qtDKx4950hv+UfXj4xEaMaYKGBNZ1GgoLjMu33R8EwAstq1iFQ4xpgoY4kmCvzixZne7Y4pSYBTszHGmIZgiaYZW751L69Xmfb/rKO60j29JZeMyIpITMaY6GN9NM3UvZOW8PrMDdXKu7ZJ5uubRzV8QMaYqGU1mmYqUJJ59fJhxFiTmTGmgVmNphkqK6+oVvbdrSeS0dYGABhjGp7VaJqhrQHmMuuYmhSBSIwxxhJNs1NcWs76XYXVyuNj7U9tjIkMazprZvre/Zl3++T+HblpXB/yCksiGJExJtpZommmxhzRgRcvyY50GMYYY01nzYnvIIAh3dpGMBJjjKlkiaYZKSwp9263SU6IYCTGGFOp1kQjIteLiH09bgIKD1TOadamRXwEIzHGmEqh1Gg6AXNE5D0ROUVE7I6/Rmr0kzO823ZfpjGmsag10ajqXUBv4GXgcmCViDwiIj3DHJupg+LScvaXVjad9erQKoLRGGNMpZD6aFRVgW3uowxoC7wvIn8JY2ymDk7663Tv9sJ7xtKrQ0rkgjHGGB+1Dm8WkT8AlwE7gX8Bf1bVUhGJAVYBt4Q3RFObVdsL2OLOBvCvS7Np08IGAhhjGo9Q7qNJB85RVb9ZGlW1QkTOCE9YJlTfrdrJxS/PAuCIzqmM6dcxwhEZY4y/UBLN3wBEJM2nrEBVS1V1eViiMiHzJBmAD689NoKRGGNMYKH00cwHcoGfcJrKcoF1IjJfRIaGMzgTupTEOJLiYyMdhjHGVBNKovkMOE1V01W1HXAq8B5wLfBcOIMzNXPGaDh8R5wZY0xjEkqiyVbVqZ4nqvo5cLyq/gAkhi0yU6t1OytnaS6r0BqONMaYyAmljyZPRG4F3nGfXwDsFpFYoPoKW6bBbMgrinQIxhhTq1BqNL8EMoD/uo9MtywW+EW4AjO12+BTo7nuRLt/1hjTONVYo3FrLX9T1YuDHLK6/kMyofps6TY6t07i+1tPIsbmnDHGNFI11mhUtRxoLyJ2B2AjlFtwgCGHtbUkY4xp1ELpo1kPfC8iHwHethpVfSpcQZnaTVq4mTW5hQzLSqv9YGOMiaBQ+mi2AJ+4x6b4PA6aiDwoIj+KyEIR+VxEurjl8SLyuogsFpHlInK7zzlD3fLVIvJ3zyzSIpIoIu+65bNEJMvnnMtEZJX7uOxQYm5sbnhnIWCjzYwxjV+tNRpVvR9ARFqqamFtx4foCVW9233dPwD3AFcD5wOJqjpQRFoAy0TkbVVdDzwPXAn8AHwKnAJMAa4AdqtqLxG5EHgcuMCdyeBeIBtQYJ6IfKSqu+vpGiJm9ro87/aWPfsjGIkxxtQulIXPjhGRZcBy9/kgETmkGzVVda/P05Y4iQD3Z0sRiQOSgRJgr4h0BlJVdaY7k/QbwFnuOeOB193t94HRbm3nZOALVc1zk8sXOMmpyXt3zibv9qDMNpELxBhjQhBK09nfcD60dwGo6iLg+EN9YxF5WEQ2Ab/CqdGAkygKga3ARuCvqpoHdAVyfE7Pcctwf25yYysD8oF2vuUBzqkay5UiMldE5ubm5h7qpYVdSXnl7Us3jj08gpEYY0ztQl2PZlOVolrnOxGRL0VkSYDHePc171TVTGAicL172nD3tbsA3YGbRKQHEGhYlacWFGxfTef4F6pOUNVsVc1u3759bZcWcaVlTqKJjxXiY0P6ExpjTMSEMupsk4gcC6g7zPkPuM1oNVHVMSHG8BYwGac/5ZfAZ6paCuwQke9x+li+xblp1CMDZ5ACODWVTCDHbXJrDeS55aOqnDM9xJgatS35Tr+MDQQwxjQFoXwdvhq4jsrmq8Hu84MmIr19np4JrHC3NwIniaMlMAJYoapbgQIRGeH2v1wKTHLP+QhnYTaA84Cv3H6cqcA4EWkrIm2BcW5Zk7Y4J58fc/IBOPuogC2BxhjTqIQy6mwnTj9KfXpMRPrgzJW2ASeZATwLvAoswWn6elVVf3T3XQO8hjNIYIr7AHgZeFNEVuPUZC50484TkQeBOe5xD7j9PU3a+Ge/A+Ci4Zk8MH5AhKMxxpjahbKUc3vgd0CW7/Gq+puDfVNVPTdI+T6cIc6B9s0Fqn2yqmpxDee8ArxysHE2Rp7WspYJcdY/Y4xpEkLpo5mE00fyJSEMAjDhs3pHgXfbemeMMU1FKImmhareGvZITK0emlw5BuOaUTZbszGmaQil7eUTETkt7JGYWiXFOUs1f3jtsaS3sjXnjDFNQyiJ5gacZLNfRPaKSIGI7K31LFOv7vhwMZ8t3cbIXukcdVjbSIdjjDEhC2XU2SFNoGnqx1uzNgLQrV2LCEdijDF1E7RGIyIX+2wfV2Xf9dXPMOFS6jPlTGaaJRpjTNNSU9PZjT7b/6iy76CHNpu6Ky6tHOzXtkV8BCMxxpi6qynRSJDtQM9NGO33STStky3RGGOalpoSjQbZDvTchNEni7Z6t5MTQhmRbowxjUdNn1p9ReRHnNpLT3cb93mPsEdmvB74ZJl326qSxpimpqZEc0SDRWGC2rG32LvdISWR43qlRzAaY4ypu6CJRlU3NGQgJrB5GypXnr7rjH7ExlidxhjTtNisjI3cPR8tBeCZCwfz8yM7RzgaY4ypO+tZbsQOlJWTW3CAgV1bM36wrT1jjGmarEbTiOXsdlbS/PVxWZENxBhjDkHQGo2ILCbwMGYBVFWPDFtUBoCte5yBAF3aJEc4EmOMOXg1NZ2d0WBRmIBWuevPdLVEY4xpwmzUWSOVs7uI+z9eRtsW8WS0tURjjGm6au2jEZERIjJHRPaJSImIlNsyAeE30Z2teXdRKSI2pNkY03SFMhjgn8BFwCogGfgt1SfZNPWsU2oSAMOybO0ZY0zTFtLwZlVdLSKxqloOvCoi/wtzXFHPszTA8xcPjXAkxhhzaEJJNEUikgAsFJG/AFuBluENK7p9sWw701fmIgJpLRIiHY4xxhySUBLNJUAscD3wJyATODecQUWz8grld2/MBSAlMY4Ym3LGGNPEhbKUs2f02X7g/vCGYxbl7PFut2lpa88YY5q+mm7YfE9VfxHsxk27YTM85vtMotmltQ1rNsY0fTXVaG5wf9qNmw1oW74zG8Dtp/blxL4dIhyNMcYcuqDDm1XVs6zjtaq6wfcBXNsw4UWf7QUH6NauBVed0JPDO6ZEOhxjjDlkodxHMzZA2an1HYhxhjR/81Mu3drZoD5jTPNRUx/NNTg1lx4+yzgDpADfhzuwaPTZkm3k7y9lzBHWZGaMaT5q6qN5C5gCPArc5lNeoKp5YY0qSv3+7QUAtG+VGOFIjDGm/tQ0qWY+kA9cJCKxQEf3+FYi0kpVNzZQjFEnOSE20iEYY0y9qfU+GhG5HrgP2A5UuMUK2PDmMGmfYjUaY0zzEcrMAH8E+qjqrjDHEvX6dU4lf38p/bu0jnQoxhhTb0IZdbYJpwnNhFFFhbKr8ABDutlszcaY5iWUGs1aYLqITAYOeApV9amwRRWF/vHVarbvPcDe/aWRDsUYY+pVKIlmo/tIcB8mDKat2A7A7qKSCEdijDH1K5RJNe8HEJGWqloY/pCiU3ys04r5twsGRzYQY4ypZ6Es5XyMiCwDlrvPB4nIc2GPLMpsyy/m7KO60qN9q0iHYowx9SqUwQB/A04GdgGo6iLg+DDGFHUqKpRte4vp6C7fbIwxzUkoiQZV3VSlqLw+3lxEbhYRFZF0n7LbRWS1iKwUkZN9yoeKyGJ3399FRNzyRBF51y2fJSJZPudcJiKr3Mdl9RFzOLzwzRrKK5TEuJD+HMYY06SENLxZRI4FVEQSRORm3Ga0QyEimTgTdm70KesHXAj0B04BnnNnJQB4HrgS6O0+TnHLrwB2q2ov4Gngcfe10oB7gaOB4cC9ItIoxw5/ucwZCHBsz3YRjsQYY+pfKInmauA6oCuQAwymfpYJeBq4Bf9F1cYD76jqAVVdB6wGhotIZyBVVWeqqgJvAGf5nPO6u/0+MNqt7ZwMfKGqeaq6G/iCyuTUqCzYtIeOqYkc3cMSjTGm+QlleHMfVf2Vb4GIHMchzOAsImcCm1V1kdsC5tEV+MHneY5bVupuVy33nLMJQFXLRCQfaOdbHuCcqvFciVNb4rDDDju4izpIT3/xE6qQkmTLNhtjmqdQEs0/gCEhlPkRkS+BTgF23QncAYwLdFqAMq2h/GDP8S9UnQBMAMjOzg54TDhsyy/mmWmrALjjtL4N9bbGGNOgalqP5hjgWKC9iNzosysVqHV6YVUdE+R1BwLdAU9tJgOYLyLDcWodmT6HZwBb3PKMAOX4nJMjInFAayDPLR9V5ZzptcXdkF6Ysca7PcDmNzPGNFM19dEkAK1wklGKz2MvcN7BvqGqLlbVDqqapapZOAlhiKpuAz4CLnRHknXH6fSf7S4rXSAiI9z+l0uBSe5LfgR4RpSdB3zl9uNMBcaJSFt3EMA4t6zRSEmqzPMdbGizMaaZqmk9mhnADBF5TVU3iEiKU6z7whWMqi4VkfeAZUAZcJ2qeoZSXwO8BiTjLMg2xS1/GXhTRFbj1GQudF8rT0QeBOa4xz3Q2BZsS4p3KoZXjOwe4UiMMSZ8QumjSRGRBUAagIjsBC5T1SX1EYBbq/F9/jDwcIDj5gIDApQXA+cHee1XgFfqI85wKCguIz5WuOv0IyIdijHGhE0ow5snADeqajdV7Qbc5JaZQ7SnqITWyfFUGXlnjDHNSiiJpqWqfu15oqrTgZZhiyiKrNxeQE+b28wY08yFkmjWisjdIpLlPu4C1oU7sOZu0sLNLNi4hwFdbbSZMaZ5CyXR/AZoD3zgPtKBX4czqGhwwzsLARjQNTWygRhjTJjVdB9NEs70M72AxcBNqmrLP9azgVajMcY0czXVaF4HsnGSzKnAEw0SUZTpnm59NMaY5q2m4c39VHUggIi8DMxumJCav9U7CgD4w0m9iI2xEWfGmOatphqNt5lMVcsaIJaocd3EBQDk7Nkf4UiMMSb8aqrRDBKRve62AMnuc8GZIcB6sQ9SgrvAWY90GyVujGn+apqCptaJM83BSW+VgAhcdULPSIdijDFhZ2sHR8DW/GJG9+1IfKz9+o0xzZ990kXA1vxiurSx2ZqNMdHBEk0DKyopI39/KZ1aW6IxxkQHSzQNbGt+MQBdWidHOBJjjGkYlmga2NY9TqKxGo0xJlpYomlgS7fkA9DdhjYbY6KEJZoGtHPfAR6dsoI2LeLpaEs3G2OihCWaBvTSN2sB2FNkc5MaY6KHJZoGNOOn3EiHYIwxDc4STQPaua8EsKlnjDHRxRJNA9p3oJRhWW1596pjIh2KMcY0GEs0DeRAWTnFpRWM6tOB9imJkQ7HGGMajCWaBlJQ7Ky0kJpU04TZxhjT/FiiaSD5+52RZilJ8RGOxBhjGpYlmgayfa8zI0AHazYzxkQZSzQNZP3OIgC6trU5zowx0cUSTQP55McttEyItTnOjDFRxxJNA1m5rYDTBnYmMc4WLjXGRBdLNA1gb3EpuwpL6NWhVaRDMcaYBmeJpgFscPtnurWzGQGMMdHHEk0DWL+rEICs9BYRjsQYYxqeJZoGsMFNNN3SrEZjjIk+lmgawLqdRXRKTSI5wQYCGGOijyWaBrAoZw99OqVEOgxjjIkISzRhNnd9Hqt37OOkvh0iHYoxxkSEJZowm79xNwA/H9QlwpEYY0xkWKIJs9U79pHeKpG0lgmRDsUYYyLCEk2Ybc0vpmsbm3bGGBO9LNGE2bb8YjqmWqIxxkSviCYaEblZRFRE0t3nY0Vknogsdn+e5HPsULd8tYj8XUTELU8UkXfd8lkikuVzzmUissp9XNbQ11dRoWzes58ubWzGZmNM9IpYohGRTGAssNGneCfwc1UdCFwGvOmz73ngSqC3+zjFLb8C2K2qvYCngcfd108D7gWOBoYD94pI27BdUACb9+ynqKScwzva0GZjTPSKZI3maeAWQD0FqrpAVbe4T5cCSW6NpTOQqqozVVWBN4Cz3OPGA6+72+8Do93azsnAF6qap6q7gS+oTE4NYuW2AgC7h8YYE9UikmhE5Exgs6ouquGwc4EFqnoA6Ark+OzLcctwf24CUNUyIB9o51se4Jyq8VwpInNFZG5ubu5BXFFgK7c7iebwjjZrszEmesWF64VF5EugU4BddwJ3AONqOLc/ThOY5xgJcJjWsq+mc/wLVScAEwCys7MDHnMwftpeQNc2yaQkxdfXSxpjTJMTtkSjqmMClYvIQKA7sMjtz88A5ovIcFXdJiIZwIfApaq6xj0txz3OIwPY4rMvE8gRkTigNZDnlo+qcs70Q7+y0P20fZ/VZowxUa/Bm85UdbGqdlDVLFXNwkkIQ9wk0waYDNyuqt/7nLMVKBCREW7/y6XAJHf3RzgDBwDOA75y+3GmAuNEpK07CGCcW9YgVJWNuwptDRpjTNQLW43mIF0P9ALuFpG73bJxqroDuAZ4DUgGprgPgJeBN0VkNU5N5kIAVc0TkQeBOe5xD6hqXoNcBbC7qJTCknIy02wNGmNMdIt4onFrNZ7th4CHghw3FxgQoLwYOD/IOa8Ar9RLoHW0Kc9ZVTOzrd1DY4yJbjYzQJhs2u0mGqvRGGOinCWaMNmUtx+wRGOMMZZowmRjXhFtW8TTKjHirZPGGBNRlmjCZNnWvRxmI86MMcYSTTgUFJeyaNMeTuzTPtKhGGNMxFmiCYO8whIAMtpa/4wxxlgHQj0pKC7l+elrGH1ER2JjnNlv0lra1DPGGGOJpp6UlSvPTV9DeqtE4mKdRNOmhS3fbIwxlmjqSXJCLAD7S8tZsGYPAIfZ0GZjjLE+mvqSGBdDjEBRSRnrdu5jXL+OpLdKjHRYxhgTcZZo6omI0CIhjoLiMjbsKqJXB5u12RhjwBJNvUpOiGXltgLKKpQe7S3RGGMMWKKpVy0TYlmyOR+AHu3tZk1jjAFLNPUqOSGOwpJyAHqmW43GGGPAEk29aumOPANo3cLuoTHGGLBEU68OlFUA8PDZ1ZbNMcaYqGWJph7l7y8FoIc1mxljjJclmnoU784IYAMBjDGmks0MUI9evCSbqUu30SHFbtQ0xhgPSzT1qFeHVvTq0CvSYRhjTKNiTWfGGGPCyhKNMcaYsLJEY4wxJqws0RhjjAkrSzTGGGPCyhKNMcaYsLJEY4wxJqws0RhjjAkrUdVIx9CoiEgusOEQXiId2FlP4TQV0XbN0Xa9YNccLQ7lmrupavtAOyzR1DMRmauq2ZGOoyFF2zVH2/WCXXO0CNc1W9OZMcaYsLJEY4wxJqws0dS/CZEOIAKi7Zqj7XrBrjlahOWarY/GGGNMWFmNxhhjTFhZojHGGBNWlmjqiYicIiIrRWS1iNwW6Xjqi4hkisjXIrJcRJaKyA1ueZqIfCEiq9yfbX3Oud39PawUkZMjF/3BE5FYEVkgIp+4z5v79bYRkfdFZIX7tz4mCq75T+6/6SUi8raIJDW3axaRV0Rkh4gs8Smr8zWKyFARWezu+7uISJ0CUVV7HOIDiAXWAD2ABGAR0C/ScdXTtXUGhrjbKcBPQD/gL8BtbvltwOPudj/3+hOB7u7vJTbS13EQ130j8Bbwifu8uV/v68Bv3e0EoE1zvmagK7AOSHafvwdc3tyuGTgeGAIs8Smr8zUCs4FjAAGmAKfWJQ6r0dSP4cBqVV2rqiXAO8D4CMdUL1R1q6rOd7cLgOU4/0nH43w44f48y90eD7yjqgdUdR2wGuf302SISAZwOvAvn+LmfL2pOB9ILwOoaomq7qEZX7MrDkgWkTigBbCFZnbNqvoNkFeluE7XKCKdgVRVnalO1nnD55yQWKKpH12BTT7Pc9yyZkVEsoCjgFlAR1XdCk4yAjq4hzWH38XfgFuACp+y5ny9PYBc4FW3ufBfItKSZnzNqroZ+CuwEdgK5Kvq5zTja/ZR12vs6m5XLQ+ZJZr6Eai9slmNGxeRVsB/gD+q6t6aDg1Q1mR+FyJyBrBDVeeFekqAsiZzva44nOaV51X1KKAQp0klmCZ/zW6/xHicJqIuQEsRubimUwKUNalrDkGwazzka7dEUz9ygEyf5xk41fBmQUTicZLMRFX9wC3e7lapcX/ucMub+u/iOOBMEVmP0wR6koj8m+Z7veBcQ46qznKfv4+TeJrzNY8B1qlqrqqWAh8Ax9K8r9mjrteY425XLQ+ZJZr6MQfoLSLdRSQBuBD4KMIx1Qt3dMnLwHJVfcpn10fAZe72ZcAkn/ILRSRRRLoDvXE6EpsEVb1dVTNUNQvn7/iVql5MM71eAFXdBmwSkT5u0WhgGc34mnGazEaISAv33/honP7H5nzNHnW6Rrd5rUBERri/q0t9zglNpEdFNJcHcBrOiKw1wJ2Rjqcer2skTjX5R2Ch+zgNaAdMA1a5P9N8zrnT/T2spI6jUxrTAxhF5aizZn29wGBgrvt3/i/QNgqu+X5gBbAEeBNntFWzumbgbZw+qFKcmskVB3ONQLb7e1oD/BN3VplQHzYFjTHGmLCypjNjjDFhZYnGGGNMWFmiMcYYE1aWaIwxxoSVJRpjjDFhZYnGRA0RKReRhT6PrEjH5EtEzndnTv66SnmMO2PuEncG3TnufQ6IyKci0qae3n9fLfvbiMi1Ps+7iMj79fHepnmz4c0maojIPlVtFWSf4Px/qAi0vyGIyGc4M+lWTTQXAecCv1DVCnfSz0JV3V3P7x/09+Puz8K5r2hAfb6vaf6sRmOilohkuTWI54D5QKaIPC8ic911Su73OXa9iDwiIjPd/UNEZKqIrBGRq32O+7Nb4/jR9/wq73uRWzNZIiKPu2X34Nwc+4KIPFHllM7AVk8SVNUcT5Jx40p3r2WFOyHmEhGZKCJjROR7d92R4e7x94nIzT6xLKlasxORViIyTUTmu3F6ZiJ/DOjp1gafcN9ziXtOkoi86h6/QEROdMsvF5EPROQzN46/1PHPZJqDSN+5ag97NNQDKKdydoMPgSycGZpH+ByT5v6MBaYDR7rP1wPXuNtP49xBnwK0x5mEE2AcMAFnEsIY4BPg+CoxdMGZ/qQ9zmSWXwFnufumA9kB4s5w338h8CRwlM++9UC6ey1lwED3vecBr7ixjAf+6x5/H3Czz/lLgCx3e5/7Mw5nWnjc117tvk4W/uuaeJ8DNwGvutt93WtMwlnjZS3Q2n2+AciM9L8FezTsw2o0JprsV9XB7uNst2yDqv7gc8wvRGQ+sADoj7MYlIdn/rrFwCxVLVDVXKDY7ScZ5z4W4NSQ+uLMF+VrGDBdnckcy4CJOGvBBKWqOUAf4HacxDhNREYHOHSdqi5Wp+azFJimqurGm1XTe1QhwCMi8iPwJc6U8B1rOWckzjQuqOoKnIRyuLtvmqrmq2oxzhxq3eoQi2kG4iIdgDERVujZcDvYbwaGqepuEXkN51u4xwH3Z4XPtud5HM4H9KOq+mIN71e3JXBdqnoAZ2XDKSKyHWfhqWlVDqsak2+8nv/rZfg3mften8evcGpcQ1W11J3JOtBxvmq6Lt+4yrHPnahjNRpjKqXiJJ58EekInFrH86cCvxFn7R5EpKuIdKhyzCzgBLdfJRa4CJhR04u6/UFd3O0Y4EicGsPBWI+zBAAiMgRnPZaqWuM0B5a6fS2eGkgBTnNhIN/gJChE5HDgMJyJGY2xbxbGeKjqIhFZgNPstBb4vo7nfy4iRwAznUFs7AMupnK9D1R1q4jcDnyNUwv4VFVrm3K9A/CSiCS6z2fjzKB7MP4DXCoiC3GWt/gpwDETgY9FZC5Ov9AKN/Zd7uCCJTi1q2d9znkOZyDDYpxa0+WqesD9PZgoZ8ObjTHGhJU1nRljjAkrSzTGGGPCyhKNMcaYsLJEY4wxJqws0RhjjAkrSzTGGGPCyhKNMcaYsPp/Y53HeOlsKXIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(potential)\n",
    "plt.xlabel(\"Frame of Simulation\")\n",
    "plt.ylabel(\"Potential Energy\")\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question: What should I report as the average potential energy of this simulation?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-35841.17738512487"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(potential[0:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-35179.34066243115"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(potential[500:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the uncertainty in this average?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "133.75590387085327"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "start = 50\n",
    "np.std(potential[start:])/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reporting distributions of non-normal distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1O5KIZCgVdYoMrbJ3orWbv1m3iymlhdwxL/svrReR5NPURwrlRiP8788sYW7VBL7yw63sbmr1O5KIZCAVdYpNLMjlhYduoaQwly+8sIWm811+RxKRDKOiToMppQV8/6GbudAzwEPf30KblkUVkXFQUafJNVNK+D+fvYkDzR0qaxEZFxV1Gq2YW8njaxaz/dh5PvvsJs5f6PU7kohkABV1mq2+YSpPfvYm3j/Rzqee3khLh26QKyKXp6L2wYcXTOa5z9dz+EwnDz71Dqfauv2OJCIBpqL2yYfmxvi/D93CydZuHnjqHRrPaREnERmZitpHS2dV8MN/u5Rznb088OQ7HG7p9DuSiASQitpni2snsfbhW+nuH+T+J9+h4fBZvyOJSMCoqAPgumml/PQvbmVCfpQ1z2zkR5uO+B1JRAJERR0Qc6om8tJXV7BsdiWPrdvFoy/upLd/0O9YIhIAKuoAKS3K5fnP38yXV85m7eajrHlmI6d1RohI6KmoAyYaMf7Dqmv43qcX815TG/d87022HTvvdywR8ZGKOqA+ceM0fvblZeRGIzzw5Ds8s+EgA4O6W4xIGKmoA2zBtBL+6ZEV3D4vxt++uocHnnqHA80dfscSkTRTUQfcpOI8nvncTXz7wYXsP93B6u/+VkfXIiGjos4AZsZ9i6t57eu386G5lRePrg/q6FokFFTUGaSqpIBnPld/8ej6Y9/9LU+s309334Df0UQkhVTUGWb40fXK+TH+56/2cuf/eoP/13BM0yEiWUpFnaGqSgp46s/r+YeHb6WqpIBv/uMOPv74b1m/9zTOqbBFsomKOsMtnVXBz7+yjCc+vYSuvvitvj7z7CZ2NJ73O5qIJIml4uirvr7eNTQ0JP155fJ6+wf58aYjPP76fs529nLbrAq+uGImd11TRSRifscTkcsws63OufoRt6mos097dx9rNx/lhbcO09TazczKYh5aXsf9N1VTlJfjdzwRGYGKOqT6Bgb55a6TPPfmIbYdO09JQQ5rltbyQH0Ns2MT/I4nIsOoqIWtR87x3JsH+eWukww6uH56CfcunM49C6cxpbTA73gioaeilotOtXXzyo4TvLztONsbWzGDpTPLuXfRdD66YDIVE/L9jigSSipqGdGhlk5e3tbES9uOczBxG7Drp5ewYk6M2+dWclPdJPJzoj6nFAkHFbVclnOO3U1trH//NL/d38K7R87RP+goyI2wdGYFy+dUsLC6jOunl1Kcr19GiqSCilrGpaOnn40HzvDm/hY2fNDMweb40bYZzIlN4MbqMm6sLuWG6lLmVE2gpCDX58Qime+qi9rMVgHfBaLAs865b11ufxV1dmlu72Hn8fPsaGxNfJynpaP34vbKCfnMqixmZmUxM2Pxz9WTCplcUkB5UZ7O4Rbx4KqK2syiwD7gI0AjsAVY45x7b7THqKizm3OOE63d7DreysGWTg41d3KopZODLZ20dPT8wb45EaNqYj5VJQVMLsmnckI+pYW5lBbmUjL0uSCXksIcivKi5OdEKciNUpAboTA3Sk5UF89KOFyuqL1MON4C7HfOHUw82U+Ae4FRi1qym5kxrayQaWWFf7StrbuPwy2dNJ3v4lRbD6faujnV1sPp9m4OtXSy+dBZ2rr7PS8glRMx8nIiRCNGTsSIRiKJz/EPM7BEJgMY9v2I2a/4Ty0ytklFefz0S7cl/Xm9FPV04Niw7xuBpZfuZGYPAw8D1NbWJiWcZJ6SgtzEHHbZqPs457jQO0Bbdx+tXX20dfXT1tVHV98AXX0D9PQN0N03SFffAN19A/QNDNI/6BgYdPHPA4nPg4M4wDkSnx0O4t+M9LqjbRBJklT9vsZLUY90EPJHI9459zTwNMSnPq4yl2QxM6M4P4fi/Bymlv7xUbmI/CEvE4CNQM2w76uBptTEERGRS3kp6i3AXDObaWZ5wKeAl1MbS0REhow59eGc6zezR4BfET8973nn3O6UJxMREcDbHDXOuVeBV1OcRURERqCTVEVEAk5FLSIScCpqEZGAU1GLiARcSlbPM7Nm4MhldqkEWpL+wsmhbFdG2a5MULMFNRdkb7YZzrnYSBtSUtRjMbOG0RYf8ZuyXRlluzJBzRbUXBDObJr6EBEJOBW1iEjA+VXUT/v0ul4o25VRtisT1GxBzQUhzObLHLWIiHinqQ8RkYBTUYuIBFzSi9rMVpnZXjPbb2b/cYTtZmaPJ7bvMLMlXh+bhmyfSWTaYWZvm9nCYdsOm9lOM9tmZkm/IaSHbCvNrDXx+tvM7D97fWyKc31zWKZdZjZgZuWJbal+z543s9NmtmuU7X6OtbGy+TLWPOTyZZx5zObnWKsxs/VmtsfMdpvZ10bYJ3XjzTmXtA/iy6AeAGYBecB2YMEl+6wGfkH8zjG3Apu8PjYN2ZYBkxJff2woW+L7w0BlMt+vcWZbCbxyJY9NZa5L9r8HeD0d71ni+W8HlgC7Rtnuy1jzmM2vsTZWrrSPM6/ZfB5rU4Elia8nEr/hd9q6LdlH1BdvhOuc6wWGboQ73L3AD1zcRqDMzKZ6fGxKsznn3nbOnUt8u5H43WzS4Wr+7Kl838b73GuAtUl67TE55zYAZy+zi19jbcxsfo01D+/ZaHx/zy6R7rF2wjn3buLrdmAP8fvJDpey8Zbsoh7pRriX/mFG28fLY1OdbbgvEv/bcYgDfm1mWy1+I99k8prtNjPbbma/MLPrxvnYVObCzIqAVcDPhv04le+ZF36NtfFK51jzIt3jbFz8HmtmVgcsBjZdsill483TjQPGwcuNcEfbx9NNdK+C5+c3szuJ/8+zYtiPlzvnmsysCnjNzN5PHAGkK9u7xNcC6DCz1cDPgbkeH5vKXEPuAd5yzg0/Ikrle+aFX2PNMx/G2lj8GGfj5dtYM7MJxP+C+CvnXNulm0d4SFLGW7KPqL3cCHe0fVJ9E11Pz29mNwLPAvc6584M/dw515T4fBpYR/yfM2nL5pxrc851JL5+Fcg1s0ovj01lrmE+xSX/FE3xe+aFX2PNE5/G2mX5NM7Gy5exZma5xEv6R865F0fYJXXjLckT7jnAQWAmv580v+6SfT7OH064b/b62DRkqwX2A8su+XkxMHHY128Dq9KcbQq/v0DpFuBo4j1M2fvm9bmBUuJzi8Xpes+GvU4do/9izJex5jGbL2PNQ660jzOv2fwca4n34AfAdy6zT8rGW1Lf5ESo1cR/I3oAeCzxsy8BXxr2B34isX0nUH+5x6Y527PAOWBb4qMh8fNZiTd3O7Dbp2yPJF57O/FfPi273GPTlSvx/eeBn1zyuHS8Z2uBE0Af8aOWLwZorI2VzZex5iGXL+PMSzafx9oK4tMVO4b9N1udrvGmS8hFRAJOVyaKiAScilpEJOBU1CIiAaeiFhEJOBW1iEjAqahFRAJORS0iEnD/Hzw3dj3KCQXIAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "betaf = scipy.stats.beta(0.8,4)\n",
    "x = np.linspace(0,2)\n",
    "plt.plot(x,betaf.pdf(x))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "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": [
    "num = 10000\n",
    "samples = betaf.rvs(num)\n",
    "plt.hist(samples,bins=50,density=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pause and do some calculations\n",
    "\n",
    "* What would you report as the mean and standard deviation of this distribution?\n",
    "* What might be a better way to report the behavior of this distribution?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 2.5% percentile is : 0.0025\n",
      "97.5% percentile is : 0.5675\n"
     ]
    }
   ],
   "source": [
    "xlow = np.percentile(samples,2.5)\n",
    "xhigh = np.percentile(samples,97.5)\n",
    "print(f\" 2.5% percentile is : {xlow:.4f}\")\n",
    "print(f\"97.5% percentile is : {xhigh:.4f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### What about the error in the mean of samples from this distribution?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def distribution_of_means(num,nrepeats):\n",
    "    repeats = np.zeros(nrepeats)\n",
    "    for i in range(nrepeats):\n",
    "        repeats[i] = np.mean(betaf.rvs(num))\n",
    "    return repeats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "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": [
    "plt.hist(distribution_of_means(2,1000))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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": [
    "plt.hist(distribution_of_means(5,1000))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "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": [
    "plt.hist(distribution_of_means(100,1000))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: Is the \"standard\" formula good here??"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.015634310450823783"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(samples)/np.sqrt(100-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'mean' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "Input \u001b[0;32mIn [20]\u001b[0m, in \u001b[0;36m<cell line: 1>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[43mmean\u001b[49m \u001b[38;5;241m-\u001b[39m \u001b[38;5;241m2\u001b[39m\u001b[38;5;241m*\u001b[39mstd)\n\u001b[1;32m      2\u001b[0m \u001b[38;5;28mprint\u001b[39m(mean \u001b[38;5;241m+\u001b[39m \u001b[38;5;241m2\u001b[39m\u001b[38;5;241m*\u001b[39mstd)\n",
      "\u001b[0;31mNameError\u001b[0m: name 'mean' is not defined"
     ]
    }
   ],
   "source": [
    "print(mean - 2*std)\n",
    "print(mean + 2*std)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Determining independent points: the autocorrelation time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Important: this is only valid for a _stationary_ timeseries."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pandas implements an autocorrelation function as a function of _lag_ ($\\tau$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0\n",
      "-0.11219125354023354\n"
     ]
    }
   ],
   "source": [
    "start = 500\n",
    "panda_pot = pd.Series(potential[start:]-np.mean(potential[start:]))\n",
    "print(panda_pot.autocorr(0))\n",
    "print(panda_pot.autocorr(100))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Programming Exercises\n",
    "\n",
    " * Use the `autocorr` function to calculate the entire autocorrelation function for all $\\tau$.  \n",
    " * Bonus points: Compute the autocorrelation function from scratch and compare.\n",
    " * Can you use your estimate the lag time $\\tau$ at which the system becomes uncorrelated (i.e. the autocorrelation function) to show points spaced more than $\\tau$ are independent? \n",
    " * Note: the formulas assume that the average if the timeseries is zero, so you should subtract it off first."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "start = 500\n",
    "stationary = pd.Series(potential[start:])-np.mean(potential[start:])\n",
    "n_autocorr = np.shape(stationary-1)[0]\n",
    "nlim = int(n_autocorr/2)\n",
    "acf = np.zeros(nlim)\n",
    "for i in range(nlim):\n",
    "    acf[i] = stationary.autocorr(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "gh = pd.plotting.autocorrelation_plot(panda_pot)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### When does the ACF become essentially zero?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What's your guess? Post in the notes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fitting ACF to an exponential to estimate correlation time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import curve_fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "def exponential(x, r): #function f(x, r) = e^(r*x)\n",
    "    #return np.e ** (r * x)\n",
    "    return np.exp(r*x) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "#This function calculates tau by using an exponential fit.\n",
    "#To do this it uses scipy.optimize curve_fit\n",
    "#curve_fit will optimize any function to fit given data\n",
    "\n",
    "def tau_calc(ac_data, function = exponential): #takes in data and a python function\n",
    "    x_data = np.arange(len(ac_data))\n",
    "    pars, cov = curve_fit(f=function, xdata=x_data, ydata=ac_data, p0=[0], bounds=(-np.inf, np.inf))\n",
    "    #curve fit returns an np.array of optimally fit paramters(pars) and their coverience(cov)\n",
    "    #pars in this case will return an optimized value of k to fit the dataset\n",
    "    return -1/pars #tau = -1/k\n",
    "# scipy curve_fit documentation: https://towardsdatascience.com/basic-curve-fitting-of-scientific-data-with-python-9592244a2509"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[6.26820455]\n"
     ]
    }
   ],
   "source": [
    "tau = tau_calc(acf)\n",
    "print(tau)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This $\\tau$ is the time it takes to go from 1 to $1/e$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "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": [
    "x_data = np.arange(100)\n",
    "plt.plot(x_data,acf[0:100], label='autocorrelation') \n",
    "plt.plot(exponential(x_data, -1/tau), label = 'tau_calc estimation')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Key point*: If the data fits an exponential well, we can treat it as uncorrelated samples if they samples are $2\\tau$ apart. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-35194.81663986712\n"
     ]
    }
   ],
   "source": [
    "start = 700\n",
    "mean = np.mean(potential[start:])\n",
    "print(mean)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "nsamples = (len(potential)-start)/(2*6.3)\n",
    "#nsamples = (len(potential)-start)\n",
    "stderr_mean = np.std(potential[start:])/np.sqrt(nsamples-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What should we use instead of `nsamples` above?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean = -35194.81663986712, sderr_mean = 26.726525287464483\n"
     ]
    }
   ],
   "source": [
    "print(f\"Mean = {mean}, sderr_mean = {stderr_mean}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-35179.34066243115"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-35179.34066243115"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What we are probably interested in is the average potential energy per molecule, not the total potential energy. There were 900 water molecules in the simulation above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can estimate the heat of vaporization $H_{vap}$ by:\n",
    "    \n",
    "\\begin{eqnarray}\n",
    "H_{vap} &=& H_{gas}-H_{liquid} \\\\\n",
    "        &=& U_{gas} + PV_{gas} - (U_{liquid} + PV_{liquid}) \\\\ \n",
    "        &=& \\left(U_{gas} - U_{liquid}\\right) - P \\left(V_{gas}-V_{liquid}\\right) \\\\ \n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the kinetic energy of liquid and vapor is the same, then this is"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{eqnarray}\n",
    "U_{pot,gas} - U_{pot,liquid} + P(V_{gas}- V_{liquid})\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we assume ideality of gas ($PV=nRT$), and zero internal energy (which is valid for rigid water, like TIP3P or SPC/E), then we get:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{eqnarray}\n",
    "       &=& - U_{pot,liquid} + P\\left( \\frac{nRT}{P} - V_{liquid}\\right) \\\\\n",
    "        &=& - U_{pot,liquid} + nRT - PV_{liquid} \\\\\n",
    "\\mathrm{Molar} &=& -\\frac{U_{pot,liquid}}{N} + RT - \\frac{PV_{liquid}}{N}\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "What is the $H_{vap}$ of water predicted by this simulation? What is the uncertainty in the estimate? How does it compare to the experimental $H_{vap}$ of water is 40.7 kJ/mol?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hvap = 41.598 +/- 0.030\n"
     ]
    }
   ],
   "source": [
    "print(f\"Hvap = {-mean/900 + (300)*(0.00831)-(0.0821)*(0.01802)/900:.3f} +/- {stderr_mean/900:.3f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Automated tool for equilibration detection and correlation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use scipy statistical tests to see if two parts of the distributions are within the uncertainties of each other.  NOTE: have to use uncorrelated samples, or it will erroneously say that they are NOT within uncertainties of each other."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A higher T-statistic means more difference between the data sets\n",
    "\n",
    "A lower P-value indicates a low probability that the difference in means was by chance\n",
    "\n",
    "P-value < 0.05 suggest they are two different data sets! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise:  Use this code to estimate better than \"eyeballing\" what fraction is equibrated  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "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": [
    "plt.plot(potential[600:])\n",
    "plt.xlabel(\"Frame of Simulation\")\n",
    "plt.ylabel(\"Potential Energy\")\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ttest_indResult(statistic=1.333787248936898, pvalue=0.19890601164672067)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "start = 600\n",
    "cut = 800\n",
    "stop = 1000\n",
    "per_ind_sample = 20  # the number of indices between independent samples\n",
    "# compare \n",
    "scipy.stats.ttest_ind(potential[start:cut:per_ind_sample],potential[cut:stop:per_ind_sample],equal_var=True)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pymbar.timeseries tools "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pymbar import timeseries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "[t0, g, Neff_max] = timeseries.detectEquilibration(potential)\n",
    "# t0 is the initial point detected as starting the stationary point.\n",
    "# g is the estimate of the correlation time (approximately 2tau)\n",
    "# Neff_max is an estimate of the _effective_ number of samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "745\n",
      "5.662475\n",
      "45.386513\n"
     ]
    }
   ],
   "source": [
    "print(t0)\n",
    "print(g)\n",
    "print(Neff_max)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can subsample the data, picking every $n$th particle so that all the points are statistically independent."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "89"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newdata = timeseries.subsampleCorrelatedData(potential[500:],g=5.66)\n",
    "len(potential[500:])\n",
    "len(newdata)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0,\n",
       " 6,\n",
       " 11,\n",
       " 17,\n",
       " 23,\n",
       " 28,\n",
       " 34,\n",
       " 40,\n",
       " 45,\n",
       " 51,\n",
       " 57,\n",
       " 62,\n",
       " 68,\n",
       " 74,\n",
       " 79,\n",
       " 85,\n",
       " 91,\n",
       " 96,\n",
       " 102,\n",
       " 108,\n",
       " 113,\n",
       " 119,\n",
       " 125,\n",
       " 130,\n",
       " 136,\n",
       " 142,\n",
       " 147,\n",
       " 153,\n",
       " 158,\n",
       " 164,\n",
       " 170,\n",
       " 175,\n",
       " 181,\n",
       " 187,\n",
       " 192,\n",
       " 198,\n",
       " 204,\n",
       " 209,\n",
       " 215,\n",
       " 221,\n",
       " 226,\n",
       " 232,\n",
       " 238,\n",
       " 243,\n",
       " 249,\n",
       " 255,\n",
       " 260,\n",
       " 266,\n",
       " 272,\n",
       " 277,\n",
       " 283,\n",
       " 289,\n",
       " 294,\n",
       " 300,\n",
       " 306,\n",
       " 311,\n",
       " 317,\n",
       " 323,\n",
       " 328,\n",
       " 334,\n",
       " 340,\n",
       " 345,\n",
       " 351,\n",
       " 357,\n",
       " 362,\n",
       " 368,\n",
       " 374,\n",
       " 379,\n",
       " 385,\n",
       " 391,\n",
       " 396,\n",
       " 402,\n",
       " 408,\n",
       " 413,\n",
       " 419,\n",
       " 424,\n",
       " 430,\n",
       " 436,\n",
       " 441,\n",
       " 447,\n",
       " 453,\n",
       " 458,\n",
       " 464,\n",
       " 470,\n",
       " 475,\n",
       " 481,\n",
       " 487,\n",
       " 492,\n",
       " 498]"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newdata"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bootstrapping for complicated uncertainties.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a difference in statistics between a **population** and a **sample**. The population is all the possible observations out there. For instance if I were an epidemiologist, this might be all the people in the U.S. or all the children in Massachusetts.\n",
    "\n",
    "\n",
    "<img width=\"304\" height=\"232\" alt=\"Image result for population versus sample\" src=\"http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_BiostatisticsBasics/Sampling3.jpg\">\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But this can also be applied to measurements in the lab as well such as possible voltage values from this battery. If you ran an infinite number of experients, you would get an infinite number of measurements, but either in this case of the whole US population case: accessing the actual population is virtually impossible.\n",
    "\n",
    "<img width=\"304\" height=\"223\" alt=\"Image result for thermometer gif\" src=\"https://media.giphy.com/media/26FL3uMhARSAvIZZS/giphy.gif\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a scientist, you only have access to a sample. Part of designing an experiment is choosing how big your sample should be.\n",
    "\n",
    "But a key problem is: if you change your sample, it could change your sample mean and sample variance significantly. So the question is, how can you understand the variation in the **population** by only looking at your **sample**.\n",
    "\n",
    "Let me repeat: the question, which the bootstrap is trying to answer, is to understand the variation in the  **population** by understanding the  variation at your **sample**.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's say we were in the lab, and we were making a calibration curve, say absorbance against concentration using UV-vis measurements. You generate the following plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def generate_data(xlim=[0,5],ylim=[0,100],npoints=20,noise=1.0,seed=None,noise_model=\"even\"):\n",
    "    '''\n",
    "    This function just generates some random data that might look like at UV-vis output.\n",
    "    Inputs: range of data in the x and y direction, the number of points, and the level of noise.\n",
    "    Add random number seed to allow repetability\n",
    "    \n",
    "    We have two types of noise; even noise, and the other exponential noise model.\n",
    "    '''\n",
    "    np.random.seed(seed) # so we can control the random noise\n",
    "    x = np.linspace(xlim[0], xlim[1], npoints)\n",
    "    y_raw = np.linspace(ylim[0], ylim[1], npoints)\n",
    "    if noise_model == \"even\":\n",
    "        y = y_raw + noise*np.random.rand(npoints)\n",
    "    else:\n",
    "        y = y_raw + np.exp(x*np.random.rand(npoints)) - 1\n",
    "    return x, y "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xmin = 0\n",
    "xmax = 5\n",
    "\n",
    "ymin = 0\n",
    "ymax = 100\n",
    "\n",
    "npoints = 50\n",
    "\n",
    "# in the absence of noise, the slope is 25 and the intercept is 0, \n",
    "# but because of the heteroscedasticity, the linear fit will be somewhat biased.\n",
    "\n",
    "x, y = generate_data(xlim=[xmin,xmax],ylim=[ymin,ymax],noise=1.0,npoints=npoints,noise_model=\"exponenential\", seed=1)\n",
    "\n",
    "\n",
    "results = stats.linregress(x,y)\n",
    "slope = results.slope\n",
    "intercept = results.intercept\n",
    "y_fit = intercept + np.array([xmin,xmax])*slope\n",
    "original_data = pd.DataFrame({'x': x, 'y':y})\n",
    "original_data.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we plot the data and the fit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,6))\n",
    "ax.scatter(original_data['x'], original_data['y'], alpha=0.5, label='raw data')\n",
    "ax.plot(np.array([0,5]), y_fit, linewidth=4, label='linear fit')\n",
    "ax.set_xlim([0, 5])\n",
    "ax.set_ylim([0, 200])\n",
    "ax.legend(loc=\"lower right\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But uh oh. You can see that some are those points have error getting a little large. You want to get some indication of how your calibration curve would change if you repeated the experiment again. \n",
    "\n",
    "Instead of actually repeating the experiment, you decide to use bootstrapping, and *estimate the variation within your sample as a replacement for the variation within your population.*\n",
    "\n",
    "To do this, you will generate \"new\" samples, picking from the data **with replacement**, pretending the data you collected is **entirely representative** of the actual population) **with replacement** and perform a regression on this resampling of your sample.\n",
    "**With replacement**, means you can draw the same point multiple times, is an important statement. We are assuming our population is much, much, bigger than our sample, but has the exact same distribution as our sample.  So when we withdraw the value at has x=1.456, there are still infinte more points with x=1.456 remaining. So the next point you pick is just as likely to be x=1.456 as it was the first time you drew it.\n",
    "\n",
    "You will generate some code to do this. First, though, we will see what happens if we resample from the original population, then we will see what bootstrapping can do. \n",
    "\n",
    "So first, we are drawing from the _population_, because we are regenerating the entire dataset, including with new amounts of noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def draw_from_population(data,npoints=20,ndraws=10):\n",
    "    '''\n",
    "    Draw from the population using the generate_data function,\n",
    "    and determining the slope and intercept for each process. \n",
    "    '''\n",
    "    fig, ax = plt.subplots(figsize=(8,6))\n",
    "    ax.scatter(data['x'], data['y'], alpha=0.5, label='raw data')\n",
    "\n",
    "    xmin = data['x'].min()\n",
    "    xmax = data['x'].max()\n",
    "    \n",
    "    # initializing outputs\n",
    "    slope = np.zeros(ndraws)\n",
    "    intercept = np.zeros(ndraws)\n",
    "\n",
    "    for i in range(ndraws):\n",
    "        # vvvv - these are the lines to change for bootstrap\n",
    "        #x, y = generate_data(xlim=[xmin,xmax],ylim=[ymin,ymax],npoints=npoints,noise=10.0) # don't set the seed, so we get new random data.\n",
    "        x, y = generate_data(xlim=[xmin,xmax],ylim=[ymin,ymax],npoints=npoints,noise=1.0,noise_model=\"exponenential\") # don't set the seed, so we get new random data.\n",
    "        sample = pd.DataFrame({'x': x, 'y':y})\n",
    "        # ^^^^ \n",
    "        results = stats.linregress(sample['x'].values,sample['y'])\n",
    "        slope[i] = results.slope\n",
    "        intercept[i] = results.intercept\n",
    "        y_fit = intercept[i] + np.array([0,5])*slope[i]\n",
    "        ax.plot(np.array([xmin,xmax]), y_fit, linewidth=2, color='b', alpha=0.2)\n",
    "    return slope, intercept"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "draw_from_population(original_data,npoints=20,ndraws=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can repeat this, like, 5000 times, generate a distribution of slopes and intercepts, and use the distributions to calculate confidence intervals on the slope and intercept:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "slopes, intercepts = draw_from_population(original_data,npoints=50,ndraws=5000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.hist(slopes,bins=30)\n",
    "plt.title('slope values')\n",
    "plt.figure()\n",
    "plt.hist(intercepts,bins=30)\n",
    "plt.title('intercept values')\n",
    "ci = 95\n",
    "\n",
    "print('mean slope: {:.2f} +/- {:.2f} ({:2d}% CI: {:.2f} - {:.2f})'.format(np.mean(slopes), \n",
    "                                                       np.std(slopes,ddof=1), ci,\n",
    "                                                       np.percentile(slopes, ((100-ci)/2)),\n",
    "                                                       np.percentile(slopes, ((100+ci)/2))))\n",
    "\n",
    "print('mean intercept: {:.2f} +/- {:.2f} ({:2d}% CI: {:.2f} - {:.2f})'.format(np.mean(intercepts), \n",
    "                                                       np.std(intercepts,ddof=1), ci,\n",
    "                                                       np.percentile(intercepts, ((100-ci)/2)),\n",
    "                                                       np.percentile(intercepts, ((100+ci)/2))))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how these values compare to the ones estimated by the standard linear estimator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "results = stats.linregress(original_data['x'],original_data['y'])\n",
    "print(f\"slope + standard error of slope = {results.slope:.2f} +/- {results.stderr:.2f}\")\n",
    "print(f\"intercept + standard error of intercept = {results.intercept:.2f} +/- {results.intercept_stderr:.2f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They seem statistically consistent!  Both the constant term and the x term are within the error bars; or equivalently, the values are well within the distributions. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Now, to the bootstrap\n",
    "\n",
    "To pull a sample with replacement, we will take advantage of `pandas` sampling capabilities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "original_data.head()\n",
    "subsample = original_data.sample(n=original_data.x.count(), replace=True)\n",
    "print(subsample)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that some samples occur more than once, and some don't appear at all! That is what with replacement means!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Hacking Time**: That code draws *one* bootstrap sample. Copy the `draw_from_population` function from above to create a new function `bootstrap_from_data` to plot the lines from _multiple bootstrap samples_ of the original dataset instead of multiple draws from the population."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def draw_from_bootstrap(data,ndraws=500):\n",
    "    '''\n",
    "    Draw from the boostrap and determining the slope and intercept for each process. \n",
    "    '''\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that bootstrap means will not be exactly the same as the population means, as each draw from the population will be different, and if we determine the bootstrap means more precisely, they will converge to the value for that sample, which should be well WITHIN the distribution of populations means (it will be have the distribution of the population distribution of means).  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once you have the code working, then run the line below - you should hopefully get something similar to the draw from population!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "slopes_boot, intercepts_boot = draw_from_bootstrap(original_data,ndraws=5000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Hacking time, Part 2**: plot the histograms of the distribution of slopes and intercepts of the population and the bootstrap samples and compare. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.hist(slopes,bins=30,alpha=0.4,density=True,label='population')\n",
    "plt.hist(slopes_boot,bins=30,alpha=0.4,color='orange',density=True,label='bootstrap')\n",
    "plt.legend()\n",
    "plt.title('slope values')\n",
    "ci = 95\n",
    "print('mean slope (bootstrap): {:.2f} +/- {:.2f} ({:2d}% CI: {:.2f} - {:.2f})'.format(np.mean(slopes_boot), \n",
    "                                                       np.std(slopes_boot,ddof=1),ci,\n",
    "                                                       np.percentile(slopes_boot, ((100-ci)/2)),\n",
    "                                                       np.percentile(slopes_boot, ((100+ci)/2))))\n",
    "plt.show()\n",
    "\n",
    "plt.figure()\n",
    "plt.hist(intercepts,bins=30,alpha=0.4,density=True,label='population')\n",
    "plt.hist(intercepts_boot,bins=30,alpha=0.4,color='orange',density=True,label='bootstrap')\n",
    "plt.title('intercept values')\n",
    "plt.legend()\n",
    "ci = 95\n",
    "print('mean intercept (bootstrap): {:.2f} +/- {:.2f} ({:2d}% CI: {:.2f} - {:.2f})'.format(np.mean(intercepts_boot), \n",
    "                                                       np.std(intercepts_boot,ddof=1),ci,\n",
    "                                                       np.percentile(intercepts_boot, ((100-ci)/2)),\n",
    "                                                       np.percentile(intercepts_boot, ((100+ci)/2))))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bootstrapping for complicated function distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bootstrapping is generalizable to any statistical quantities you may be interested in. One useful example is error propagation. If you know the error in raw datasets, and you want to know how that error will impact downstream calculations, you can use bootstrapping to do it **without having to do the calculus** involved in standard error propagation, or if the error propagation is really impossible to do. \n",
    "\n",
    "For example, try obtaining the distribution of the log of the absolute value of the (intercept/slope) in the linear fit above. What is the average and the distribution of this quantity? (Why this quantity? No reason, really, except that it would be REALLY HARD to get the distribution of errors using any standard error propagation). Compare this to what one would get using bootstrapping.\n",
    "\n",
    "Similarly, bootstrapping can be used to estimate error in ANY fitting procedure extending beyond linear regression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "vals = np.sin(np.log(np.fabs(slopes/intercepts)))\n",
    "vals_boot = np.sin(np.log(np.fabs(slopes_boot/intercepts_boot)))\n",
    "plt.hist(vals,bins=30,alpha=0.4, density=True,label='population')\n",
    "plt.hist(vals_boot,bins=30,alpha=0.4,density=True,color='orange',label='bootstrap')\n",
    "plt.title('f(slope/intercepts) values')\n",
    "plt.legend()\n",
    "ci = 95\n",
    "\n",
    "print('mean sin(log(abs(slope/intercept))): {:.2f} ({:2d}% CI: {:.2f} - {:.2f})'.format(np.mean(vals), ci,\n",
    "                                                       np.percentile(vals, ((100-ci)/2)),\n",
    "                                                       np.percentile(vals, ((100+ci)/2))))\n",
    "print('mean sin(log(abs(slope/intercept))): (Bootstrap): {:.2f} ({:2d}% CI: {:.2f} - {:.2f})'.format(np.mean(vals_boot), ci,\n",
    "                                                       np.percentile(vals_boot, ((100-ci)/2)),\n",
    "                                                       np.percentile(vals_boot, ((100+ci)/2))))"
   ]
  }
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