Topic 5 Intro to data analysis.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "470f9b7d",
   "metadata": {},
   "source": [
    "# Topic 5: Intro to Data Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "324a1b2a",
   "metadata": {},
   "source": [
    "Understanding and visulizing data is a central problem in the sciences and beyond.  We will begin with the very basics and we will return to discuss more advanced methods later in the course."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8980653c",
   "metadata": {},
   "source": [
    "The starting point for most data analysis is literally reading in the data so that you can use it in python."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "7fa02074",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "955634d8",
   "metadata": {},
   "source": [
    "Here is how we open a generic text file.  We will print all the lines so you can see what the file looks line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "131885dc",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "# here is some preamble\n",
      "\n",
      "# here is some more preamble\n",
      "\n",
      "1.0,-0.6311688938249904,0.45488424301397556,1.4166838803401072,\n",
      "\n",
      "1.0909090909090908,0.1836650127781727,-1.2320204754180588,1.4640662255632122,\n",
      "\n",
      "1.1818181818181819,-1.2067884002340479,-0.9383912321268246,-1.4714065565433299,\n",
      "\n",
      "1.2727272727272727,-0.07397985995933759,1.8395768409168949,-0.0796575263423811,\n",
      "\n",
      "1.3636363636363638,-0.737654908251103,-1.2826141566048304,2.143318426612972,\n",
      "\n",
      "1.4545454545454546,-0.6352094090520055,1.196605783495026,-0.5272364016024303,\n",
      "\n",
      "1.5454545454545454,-1.3478569367857853,1.2726038119049938,-0.2387464313240688,\n",
      "\n",
      "1.6363636363636362,1.7643027277060235,-1.3358815620673254,0.42002797035543593,\n",
      "\n",
      "1.7272727272727273,1.1228898074019433,0.4373429207381663,-0.1053287323238627,\n",
      "\n",
      "1.8181818181818183,-1.9235573265400396,1.0784356569475517,-0.43585645156397873,\n",
      "\n",
      "1.9090909090909092,-2.5203822814363916,0.2925419990496477,-0.7175310482674775,\n",
      "\n",
      "2.0,-0.20414929117298058,1.8548073032642414,-1.6689147413533778,\n",
      "\n",
      "2.090909090909091,0.6621809483393832,-0.3051219239294479,1.1131763474375251,\n",
      "\n",
      "2.1818181818181817,0.3182087214669058,-0.042937245099687604,1.1752070761056175,\n",
      "\n",
      "2.2727272727272725,0.22912356066614792,0.29073506428594337,1.7818860311483344,\n",
      "\n",
      "2.3636363636363638,-0.3829663239804597,1.2152670827311336,-0.47144795060521066,\n",
      "\n",
      "2.4545454545454546,1.865932067710966,0.7675499181699784,0.12187118234859495,\n",
      "\n",
      "2.5454545454545454,-0.08836032079560019,-0.1666437043109095,0.626776483177752,\n",
      "\n",
      "2.6363636363636367,-0.1157667893960524,0.4901440432726023,0.4253528301358699,\n",
      "\n",
      "2.7272727272727275,0.24497585695544113,2.7752811189421585,0.343931167436819,\n",
      "\n",
      "2.8181818181818183,0.14888310641784397,0.9543988718029668,0.8880236302997588,\n",
      "\n",
      "2.909090909090909,-0.2114497883523676,-1.688434540970014,0.27935921952965387,\n",
      "\n",
      "3.0,1.0352247792742475,-1.0731955421188029,2.701509689543154,\n",
      "\n",
      "3.090909090909091,-0.8537914813011567,1.7509227026567182,-0.9552216026515601,\n",
      "\n",
      "3.1818181818181817,1.3722857219464086,-0.37647616008459883,-0.16025010581716082,\n",
      "\n",
      "3.272727272727273,0.053927881477298795,-1.3127359490133514,2.711507312349479,\n",
      "\n",
      "3.3636363636363638,-0.4068758845366948,0.7035038827738344,0.30365742503758647,\n",
      "\n",
      "3.4545454545454546,-0.44478474721640066,-1.4555517678140941,2.3713902873526815,\n",
      "\n",
      "3.5454545454545454,2.116426641034081,1.0109175166041757,1.7765903267105174,\n",
      "\n",
      "3.6363636363636362,0.11636504326654121,-0.39359214869578835,2.032829663570117,\n",
      "\n",
      "3.7272727272727275,0.8015329178643593,0.7273604384719508,1.227679834941189,\n",
      "\n",
      "3.8181818181818183,0.6456429398866661,-1.352494638078151,3.1188298852583616,\n",
      "\n",
      "3.909090909090909,1.613784466354094,0.04898448107888345,2.708345137790178,\n",
      "\n",
      "4.0,-1.41366205041043,1.0229177097053903,1.5199421945650475,\n",
      "\n",
      "4.090909090909091,-0.3421580136252812,-1.1187303130674546,0.7096631263875519,\n",
      "\n",
      "4.181818181818182,0.07888321542058939,0.8636770901568209,1.2864962209751158,\n",
      "\n",
      "4.272727272727273,1.8760741154369092,-1.3073459162098775,2.588484416332107,\n",
      "\n",
      "4.363636363636363,0.06585764882180982,0.8705697000713934,2.124660688963388,\n",
      "\n",
      "4.454545454545455,-0.22438214432973524,0.4027602740454191,3.066047039885218,\n",
      "\n",
      "4.545454545454545,-2.0491518156959208,0.9922030057065043,1.4647428779952012,\n",
      "\n",
      "4.636363636363637,0.5823994686118554,1.4178740616537855,1.1659646426268564,\n",
      "\n",
      "4.7272727272727275,1.1174069972630207,-0.04600732058846052,2.5796909818035645,\n",
      "\n",
      "4.818181818181818,-1.179141881111505,3.2015700262839704,0.07426732300802663,\n",
      "\n",
      "4.909090909090909,-0.5685640147773101,0.8422491785269339,1.9361054483673024,\n",
      "\n",
      "5.0,0.49560619172623754,1.378808545961349,2.772330131342585,\n",
      "\n",
      "5.090909090909091,0.24585784653847198,-0.20555502242005147,1.131185156172839,\n",
      "\n",
      "5.181818181818182,1.5643862123074455,-0.11958099935101546,1.3147100878015254,\n",
      "\n",
      "5.2727272727272725,-1.1151867423383657,0.5168985098872407,3.1046733325611573,\n",
      "\n",
      "5.363636363636363,-0.2726364668908259,-0.08699783939484457,3.2585070783888552,\n",
      "\n",
      "5.454545454545455,0.9001532285209066,1.3526696413293455,1.8728542161334232,\n",
      "\n",
      "5.545454545454546,-0.6202357596001613,2.007273180695664,1.2825433316035202,\n",
      "\n",
      "5.636363636363637,1.1819882134237294,-0.22142023759539192,2.365387520829122,\n",
      "\n",
      "5.7272727272727275,-0.5941962918244037,-0.14320348032695024,2.6040454782634064,\n",
      "\n",
      "5.818181818181818,-0.02477118308961286,0.9716562819146767,2.148810432530392,\n",
      "\n",
      "5.909090909090909,0.8327767577068002,-0.2927110377652543,3.069936739485766,\n",
      "\n",
      "6.0,-1.1556411055077778,1.4417724412287023,3.1886656103478708,\n",
      "\n",
      "6.090909090909091,-1.2168552986118735,0.4506696001559499,2.386898651226657,\n",
      "\n",
      "6.181818181818182,-0.6157921010281806,-0.497108647756321,0.05661589728718264,\n",
      "\n",
      "6.2727272727272725,-1.7671583367295978,1.135331434196452,2.817105149730652,\n",
      "\n",
      "6.363636363636364,0.13096635693994016,-0.22052459096032073,3.160826314988174,\n",
      "\n",
      "6.454545454545455,0.690294866968097,1.8963277920971895,4.487425364155156,\n",
      "\n",
      "6.545454545454546,2.371694887897769,0.27960637363250584,2.3644192911897908,\n",
      "\n",
      "6.636363636363637,1.273825402614828,0.8687499947091604,3.305655085108792,\n",
      "\n",
      "6.7272727272727275,0.37967675226481273,-1.3159423011413678,1.661604297176963,\n",
      "\n",
      "6.818181818181818,1.0948910337870426,1.3123807007747081,3.6798167626896023,\n",
      "\n",
      "6.909090909090909,-1.6205454405798412,2.9248119763275087,2.3917854722312857,\n",
      "\n",
      "7.0,0.735428855084481,1.7767204644281096,3.3694488806819884,\n",
      "\n",
      "7.090909090909091,-0.04455145140604422,-0.1371341937628388,2.642018658136439,\n",
      "\n",
      "7.181818181818182,-1.5357456259373374,1.604807734797527,4.412411518091927,\n",
      "\n",
      "7.2727272727272725,-2.009658212034761,-0.7726057755922854,4.155582143990508,\n",
      "\n",
      "7.363636363636364,2.374179182366948,1.0793284192789065,3.170149680288414,\n",
      "\n",
      "7.454545454545455,-0.6902691820562796,0.3217260254744907,2.865907753457373,\n",
      "\n",
      "7.545454545454546,-0.15763329565736972,-0.2569440439920063,3.368592679754569,\n",
      "\n",
      "7.636363636363637,0.9242345372372889,0.27667266297441834,3.637880473686003,\n",
      "\n",
      "7.7272727272727275,-3.1875997601455275,2.199806115530024,2.1096391833912675,\n",
      "\n",
      "7.818181818181818,-1.1718948899473165,0.10319898990813686,4.576445271388832,\n",
      "\n",
      "7.909090909090909,0.9785580968322369,0.009469202542572108,4.49391127399047,\n",
      "\n",
      "8.0,-0.8665677410547212,-0.17314343767459317,5.313216187997888,\n",
      "\n",
      "8.09090909090909,-1.6213716879341697,1.1684574860898165,3.19369658313685,\n",
      "\n",
      "8.181818181818182,-0.06367514456694123,1.4356964984967884,5.912587734389751,\n",
      "\n",
      "8.272727272727273,0.7401111090339849,0.43828138891593493,4.616231505466975,\n",
      "\n",
      "8.363636363636363,1.446697225209474,0.2686299890785137,6.166373213464694,\n",
      "\n",
      "8.454545454545455,2.0171465823853088,1.7122578969698194,4.543006941829624,\n",
      "\n",
      "8.545454545454547,0.05547192247442552,-0.6291651473329716,4.507974800623014,\n",
      "\n",
      "8.636363636363637,-0.0197020770305536,0.9233026597415765,6.920201295831565,\n",
      "\n",
      "8.727272727272727,-0.4911129241676753,1.6061208531128814,3.8010877096815032,\n",
      "\n",
      "8.818181818181818,2.069890341411209,-0.209459536509214,6.761139952443798,\n",
      "\n",
      "8.90909090909091,0.032529746059397326,-1.754907126230721,5.2962287556147905,\n",
      "\n",
      "9.0,1.4366854455763804,-0.9398701401824869,6.13375599845889,\n",
      "\n",
      "9.090909090909092,-1.0881560738374398,0.7241679159272209,7.010425652001209,\n",
      "\n",
      "9.181818181818182,0.4358544140396277,0.2484598305057409,4.543475541105149,\n",
      "\n",
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      "\n",
      "9.363636363636363,0.6720375552319127,2.6861025069373134,7.9347242752625275,\n",
      "\n",
      "9.454545454545455,0.15566217413846836,0.9488508609031605,5.960063077396186,\n",
      "\n",
      "9.545454545454545,0.5227159219087347,-0.20599329826767998,7.820105590320214,\n",
      "\n",
      "9.636363636363637,-0.23146925909689864,0.6207016561335794,5.959715623500961,\n",
      "\n",
      "9.727272727272727,-0.5167395516639268,0.7343253426310082,5.3781930296290685,\n",
      "\n",
      "9.818181818181818,-0.005145926712907867,1.8884676984294115,6.645049022367959,\n",
      "\n",
      "9.90909090909091,2.167391340402733,1.8142239879060225,6.244150828030593,\n",
      "\n",
      "10.0,0.7589063994652221,-0.4915403755397779,8.432902602792572,\n",
      "\n"
     ]
    }
   ],
   "source": [
    "f=open(\"data1.txt\",\"r\") # \"r\" means we are reading the file\n",
    "for line in f:\n",
    "    print(line)\n",
    "f.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "91373950",
   "metadata": {},
   "source": [
    "Reading the 3rd line we see that the data is organized as time,data,data,data.  Our goal is to read in this data.  We need to do two things: (1) we also want to skip over the preamble that starts with #.  (2)we want to save the data into 4 arrays, one for the time and 3 others for the time series data. Here is how we can do it.  First, we notice there is a nice command"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "13eb96cf",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True False\n"
     ]
    }
   ],
   "source": [
    "txt1='#preamble'\n",
    "txt2='data'\n",
    "print(txt1.startswith('#'),txt2.startswith('#'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31787a4f",
   "metadata": {},
   "source": [
    "Second we notice that we can split a string into pieces as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "dee2bc26",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['Welcome', 'to', 'UC', 'San', 'Diego']\n",
      "['comma,separated,words']\n",
      "['comma', 'separated', 'words']\n",
      "words separated\n"
     ]
    }
   ],
   "source": [
    "string1='Welcome to UC San Diego'\n",
    "string2='comma,separated,words'\n",
    "print(string1.split())\n",
    "print(string2.split())\n",
    "print(string2.split(','))\n",
    "print(string2.split(',')[2],string2.split(',')[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c19fbb2a",
   "metadata": {},
   "source": [
    "Using split(), we can isolate particular pieces from a given line in the file. Putting these together we can load the data as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "4ca84a23",
   "metadata": {},
   "outputs": [],
   "source": [
    "t_temp=[]\n",
    "l1_temp=[]\n",
    "l2_temp=[]\n",
    "l3_temp=[]\n",
    "f=open(\"data1.txt\",\"r\") # \"r\" means we are reading the file\n",
    "for line in f:\n",
    "    if line.startswith('#')==False:\n",
    "        spl=line.split(',')\n",
    "        t_temp.append(float(spl[0]))\n",
    "        l1_temp.append(float(spl[1]))\n",
    "        l2_temp.append(float(spl[2]))\n",
    "        l3_temp.append(float(spl[3])) \n",
    "f.close()\n",
    "t=np.array(t_temp)\n",
    "d1=np.array(l1_temp)\n",
    "d2=np.array(l2_temp)\n",
    "d3=np.array(l3_temp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "2417d85c",
   "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.scatter(t,d1)\n",
    "plt.scatter(t,d2)\n",
    "plt.scatter(t,d3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d949f6d5",
   "metadata": {},
   "source": [
    "We have successfully loaded our data and made a quick plot to see that it makes sense.  It certainly looks like some data, but what do we do with it?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a77a82e",
   "metadata": {},
   "source": [
    "### Signal and Noise"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4bdc6a13",
   "metadata": {},
   "source": [
    "The central idea in most basic analyses is that our data can be thought of as the sum of two pieces $${\\rm data} = {\\rm signal}+{\\rm noise}$$\n",
    "\n",
    "The **signal** is typically deterministic: in this case, it is some specific function of time that we are trying to measure.\n",
    "\n",
    "The **noise** is random: it is usually assumed to be uncorrelated between bins and drawn from a Gaussian with zero mean.  I.e. it is a random number drawn from a distribution that is added to our answer, but whose average is zero."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89a2a711",
   "metadata": {},
   "source": [
    "To see why this is important, let us assume that our signal is just a constant: ${\\rm signal}(t) = S$ and our noise is drawn from a Gaussian distribution with variance $\\sigma$.  Suppose our data is composed of $N$ measurements.  What we notice is that is we write\n",
    "$$\\sum ({\\rm data}- {\\rm signal}) = \\sum {\\rm noise}$$\n",
    "The right hand side is a random walk just like we did earlier in the course. We expect that average size of the RHS is set by the RMS distance of a random walk, $\\sqrt{N}\\sigma$.  On the other hand $\\sum {\\rm signal} = N S$, so as we take more data our signal will stand out, even if $S \\ll \\sigma$.  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "893f1b68",
   "metadata": {},
   "source": [
    "We can see how this works by simulating some data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7b65fc00",
   "metadata": {},
   "outputs": [],
   "source": [
    "def time_data(N,sigma=1,S=0.1):\n",
    "    return np.ones(N)*S+np.random.randn(N)*sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e3213001",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11cfecdf0>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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78r54IfDBrcJ6ajK6e2h6KBPuo4cbq8Zg5FfXute/ZM7mbMxR3P5WiJMKpXj/u+UYOPoNbmfSgcvIFTc7C3y2HDZKLGIyb/8W+OL+4GPZMtzPyD1LTudWdIZz2oNfBpbpNY0/YgBZps7I+eQpYz3aWD8WfK6fuRjwSeSUUvzx9W81y9C3WvHExhEIClVp6772KUx+8iL2HY9ELiqUvQXHjjw8LcC2VjSFbbH3xCuSkQNAn8UFNbx499i6vxX1x1ah+MfY2QCs4FFxQNxNNLUvlXa/aIhqhfk9hCwGXrsaBBqDJImJwrFa4a42fOVHWJHIpK8v5q7G3FVeFcptC87ERzXXRGLkUdqjEfbknG9C1YoErmrFvp6/ejMe/nSeXiERPDtdN/g8WztftdKYUPwcLuOTbGhyNx4DZfA25uVWKzLCdPwgim3hUrGMnEVYG1XBNxApjayn9UNXR55y61Wv+OGqvwJfP4Z+xDLt2uNPHGsCFwoThWrdMn1qoAx1SaVgMhZOwsDRb+Dv738PAGhFLUuJjCYjr2vMIeWe+KIhicomzgQH5rJ1dVi4mu885anS17IB5nBzB495HL8Q55uq0+fxxBVK5Fb/3/37uy06FMBTEflbde7KjYWgd5KepcYog3p0VYmcW5pGF8jm2bqTZ+rNgpH7Ea5DpCrsTgm6OnLREuvLuauV61ymcO7k1MVrPc4RPPQhq+SFrHP2GvRUK2HQbfO7353luU5pMvKbXimcOapTd0pnBRQDe/z5fex359jQdO74lwkhP3wdUgijWmncYm0I18tXpY7wkc0zOnJfSzpWK6OWPCWvXxFOLQfd9RGOuPcTu8qgWognFOjxSTWJPFRHHgLW+a0YfnAVxsj1BjCv8d1nKnuCCqToMvKUYBBOXLiWS58WHj8a2GBt/F793EQcevdHgoRW5b+telZe3hNH28Rw3i3AyNW7UlyBJK3ZOPNXb2JUYOrIyBi5n4Y1c4HHiuQ+YcMNmuXeidCQ7CprwmPWhrDE2gJgDwKR68gHE74nrwjafZyz+e6srLl25JI7TuqgRM7/5nEZORuyuRhqlgpj5HoQ9hNKXak98jLH1VfqmXNF2qgTDdgfvrYCPW1abQ2s+Z948qyvywJfPw5s+dGbj3nnfWxbcjkBKqoVpUf2cytB1O78xze+VVJFuPUBmGLbniemWrExb9Umy9njg9uABZ8plx0FPoFcINmFmZSyezr27xwTTG3dYktKZ+BYzlnMyCuRn/B/n+GcR79Ca9Tj/Zpfh75DGKTjUfLMu9kZQz0S1nlZHblGB07OPJKPZs3IxaAFK6qYJWlJ5JtWYxo5GXulpgU6rI5bsYtP7rYCPS341IoMx4AA2J7MB167Cnj58sKDyc95GP6Y6j+H062iWnGunz0DR825xcqlylM0MWv5Rlz+9DfhCW1EDc6VRniIgsc/n4+rn52IJH0TxHVafyYuXItcnkZrP3azk3A2c5841pLSGcgk8m8WrsVHs1aiivrCYPCIe+4s4H+/cy9l3YO/2Sl2OuMd+qVi+RLmou+vx5NSRyL3qFZavEQuboCXq2/ArzMhqgK3GMrVrWnBLiCts9m5xGI+F6dfRy4nP7RBz1aYAHkvI08hj1onxswmJv7FSxcBm9X18WL4Gs5pyBmvY/iqoGPvc+MXugxVtcllbaAjdWXzVKpmEyHjMnIZHcCMZRv4HWnTKmD6S9r1smAtleoZc8x5qzZptYELVrXimMyyMXXWzA1kKejIqdAhKFgP5/l3rwJf/kOT4JAy3UfiZ0vWbQEALGX3jOzkP/nbJ7jx5WnM/bBVWDSJPJfk8ZIcVBgjF2On1FxcnnkVAEBoHhlkJYOXJmbPracjJ/b/FIt+3OKlyEdGOrsJY6puQz+yXKoicuvOeWONEzAqmbhGrDzzO02zrd++MBUH3fUhFq7ezGSV50nqmK0cw8iTVq1I9dXPng7891wr/GpEsDRkPcwgpkQOUuizrAcvxx/COY7W2rBzBk4emPQ05teejs7ghTKINq6kuThWVFw1k/173qqNWLh6M1ZusASaJb4x5+DJLxcE6/CBx0u0hAmjI9fH73L/h9m1Z0vTuDpy7lOdhtZg5IT9Ka+jx9Kx2Cc9Hb/JPCcWCNmOHVCtMCwrrtdpnuM0FVCt8DabvITPX70Zl4z5Wrl1k+rvrJMKr8iFqzczJm4FZIhjR+/7ACtmAOMf8t7jEbvW9l/IRXc6Yxk5y6y2eng79Pj4ugglMoXwJHISZORp1mqFVa3YXrMDyTIEWjZ2wDEFdR6TzmGOrGD2j7FzsN+dY7nHwemvZoJCEU/IvvHlaVxVXj7P/45JoVky8mPyH8gTUCq1WklpfOSiuegTR3ovgILghNTHmF97ur3Byah1uIyc3dhKGIFBpdZmOY89bUgVkuc6i4ywuOf73Tm2YOLGQGh++NBBnJuyOqKPXJYGlkGkGzag83djghlCNyaYyd9h2uy35PRjx9zTI5GDuh+IguhvGkaBxCuVI5C7cPXoSnWE0c2qVoJpn/xyAZZzInSyqyljR54YQ2IYOWeQBRyIJIiqWuGhrjGHG16eivV1jeC9KwHFRZk3rIsNSwCWUXNVK86Ffrux8Wu4UJDIeUinSMFqJYyRJ6paYbB0MrA5POa50PywcVOBDgFBK9bX2d8xOiilHolc5SCOy576Bh/MWC5OwE7+PImcp1rx6Mhd4uC8PV9dFU4rbzNSHiLI9z0+vReDsdDOV8jYs34eDk4V7OldklVEYY2VhIhW3v2ckcijQ6i+8JgfBh+r6EYDaZVUK3JG/vS4hRjz5ULcZ3sy2sS66VOg6EBsJlLdtkD8M6cwx9gV8qVItNNgZq/YiCufmRi4v3ZzA25+dTrqGnNKEjlv/lixdBF2evcUdMNa68aMN4Hl/HghskE9fcl6NObyOPvRrzAuxJkq6xk5BPjXfsBDB0rzAEBGY0Jft8U7kf7mhSnY1CDf0A7Dv79Y4FOthHMACoL7PpiNcXNXY/jNbwdjCXl05DbTZjfD/f148xpXtRLQkTMIrGJjSp1s3xlJvgeWTYWnj+XzwHs34fHsaOuSefTbuefikeq7AoWxgoGYPHW6Rd+Ddz/Hxg0yjFwPIvbVmMu7IVF5bZqRmZ0F6tCxI7fS7JOejm1SQecJR+KyxgtfIm8PRxosSEQAAtYGnslCUyKvawy+/+aGLK57cSoe/3w+drvtvaC0rDhwz0i/h1GpWTgz8x5a5TYAz54GPLCnFn0Obn39W3w8ayV+8cxEUEoxcPQb+PNb3wXS5XhWK2Gu7GAlcnH7OU++mOOdTBpz+djj9eNZK7UZOWB9ir9/8D021GUxZeFqKzSuE4qXZ7Uyhwn54Gfk/7en232yOerVkUv9AeK9PfuqL9XcBPxzH++EY5ffyrbMkrWN8+2V4nkJiuGtbkV18m57deQtXrWiB5Hky+5S8/RVUvvhQB3ODwVmGZLGYY6szxD7DilQtCG2SSEzkABw7cijmNyJ8Prfr8K4aZar/NGNb4N8+4rn+Zgv5+PT770u/7y3ZT0sL5z3S2md7Ov5ywaAb5dYq5A8LUhkD34cNJ9jVRx6Vivh/eDU9Af2Cs/b1p/NZhh7hIGby1MsWLPZU65qYEGKgjqk5/fPWKFxnXC+6x2VGQky7S1rgwdQbyxEmwxK5BLViv+dG9QcuCYtsurnHhk35kRh+TJfgYJhg8q355fDs3oSHnUbqloxjFwLIka2krEn5aWIpolXl8h5oJQyBwcUNNxsDs/7MAMJAJD1OmR4Njs1JXJePzt54xj8pcqy1PhT1SOB5w9/MgdnPjLOWw7zuwM24qy093T5vnUzpXSwHX7KD2sDzwuHGFDpBtKGOpkpqhgqKrbtUouwfX6mwDzNbvdP7go8C0NjLo/ZKzZ6vr8OA3AYec1GO16Oo9d/xXYOY3XkDj69m1uWw8i9Viv5kE0737M7BgVSEALrZPpbOgMbvHr9v3nUi/6MwfKlpHCsVoS0hujIvTFdRGuR4H3PZqe0hmioMEYe3gRXpl90f4sGr+cDcJLoDPqUpou+DG4sDVIoj6UlwMjZThfY7GRlh2Q2iWv9h1h76guCjQt/R9WDuLXqcYwgc5TrY78Cb8w0ZPNIIweSz7mToJtuyaTQMsOgqmJbt3GTvIW/fkyjVguOXp9EUK0AhVVdKsvsqXgSMJ6dAFZtqBMetu317CzYnq/eaPU5CiLe/B7/MPCv/YEs34YbX9wH0Jx+eAO/RC5VrThZ8vZf6tk3GU7mCct1kHIlckbPrkaaRR9zypSRyBVwTdXz7m+hGSFlZ0fOJp1GfVrMMky1YpOSIvxSPe+zebV3uSpTrUScZOan+3vpk+qK5Z2zI7Fii9ei0S4rPhqyeXxVcxneyV2Ay54quOwvX18HPLi/ez2ULEQXErQT//C1MTjwz28Jy3ftyEOQp/x20VHj+NFob46xJahYrTh1On0o5UjiNe2AN5lYKD6J/NyHPhYysYJE7lWtrN1cmNh7rJkQoASAdcLS0kligp3JI5URp+HCz8jFKf2mxvNWbfI891qpCdrA7gszlhVi5IvUOcf/X3BS8nw7s9mpBxWJnNd3tSRyosMswxh5QUfulaftQcnS9dhPgMVfFa65ErlDm+Z5ona+hekBAICNtDY0T5L6eJcOuz2erPoTjpt8SeB5fTaPLmQDOpGNOH52wTnmEF/Ux7drRuOhakttwLbrAV9fjks2eeOKsHAl8pBvm0MKSY/ONs+diKFkoU9HrlgHpa7td9ph5NVtgK8eLKQhKbD9cenK1RCpFRyrFevMTjvP0imF6gDsP3W0jwZFr1hHAElXhaZ30JCjgYHLk3L/WvVPDCULnZpcZr+pPjyGjh/OxvfjXxT21/ifg+LHzTwHukJiY36oKeEMT83j3mcbUvQx9GmJJ5FTZsOOeFx5WPND8eCgMs/OyFp/6vmbBNjNznwIXU577Juehr5rxwee12cLA/KodGFS21Cnbja4VWqp8Jm/vWcsW48D//phIB31fLFkUL3wY9xS9YSnXJUVeaHfWJi9aJn1I13jTUi8k08NbRAy35oqi03UNTLxyN8uTJxKm50CLF5tr5RSVcqOMtaEFq4j/1n6YzxQdW/BIchO8+PmhmBityB+GxyXDh73t6k+G5DK3U1Rh6AtPwIvXYpMthDz3ahWNHFjFcfzDQA8gyOqasWX78sHgLkfKlLGL80NnUK8HVPFGpxy9Ju6qpWBo9/wmB46jEzF0zWMkTmD3ZsuSNep6Q+sqI12LhnqG/XdwP3vIns3v0PQfR/MDizLASDvnj/EIv5gJT4WGWWz01UP8eLHM+W1IvVCJvaJbTFU15jz9SWNzU4OCAEWrlxnXTRs1DPuEUjk/iJakYaCNshOs2aTl5F7W1mdiAv/PQF/8J2Z6vSnqT/Y7/XpPcDkpzFs0XMRalBHIoycEPIoIWQFIWRaeOrSg1J2Ayn4PJJ0ld0C/Pu4kERhEqiz2UkY5s1udkokct8gjOqiv3ZzY2FC0ZDI36v5jfQ5tfXIrcFaDAXpur3qYbxRY4U6FS9BrQf1EQ769beh7N0C5oeCpNZBct6HqhulMimUgHpWBTkdRu7sSTpt/Nm9vsK9EnktGkPVIdYkX/hm9Rx/AxeKtLoTzX/P0Rx1lHvlt+dvhXpms9PuN1kv3Ww/1HWff3b8Qs+1w8iPvd/Rk/sU9IgeVlmGpCTyxwEckVBZRQf16Kt4EnkU1Uo4NtTLXbZZtluQYAv0yKRH6otQGMf8sFCGhVoSz9W8FeqwZ9qSXIan5ivnE3V4tx0ijAd/G8q+tV8iF4UMyIMEym2NenfykkHGN1Kgnk+nygAovBvmAICFX3iv18wFVnk9iMMYeX027+lLKY2JXgR2wmPH4lZkKbBhGT+Tf8nK5P1uqXdTuxYNgUin0hg+vABxDPzChz8wnEcdl20ovEPsIGJyJMLIKaUfAwgPXlEEzF25MTyRD2zMBbUjopLB+Y+HnafobHYShgmwErmkB+b8jByRJHI2UqBfRxzVCuOqzEuBe5T6FQdezK89HV2+/w/3mUOX/9vNqjkLZ6bfldLib8MCAw62bcFFP8yRK6gjZ1ceViL+t5OxwFGpWZ4wASoLkCqbMQYYuR/ZOuC9m9zLlAIj90vkUgFDVSJnGDmbZWzNtcBdQyU5w3XkgCWEuB7TrgmhrCQ9+A/88oyZly8BplhnJLD9o6J15ISQiwghEwghE1au5HhtRcRBd30UniiAYCCik//5Bc7+28vAnVtjSGqxIB+wpSGHgaPfwB1vy51ZeJBqFCmz2UkK0p/fs1OYP6EwtvVZ1rXcv3yNxsirEZRyCKGhK4Ues57h3p9dezYOTgXD4VaTHG7MiPZFLIgkct4kmfH5CIjGH0+10prUe+8I3bnlg/rS9GuFejhp+xGvI8391fcBsBhMV6zDHqlgyAIe0sirMXLmmzlMmN8v1Va1VQKJXIYV6+uVJwoA+MGOQ+4IcHH4qH+V5Q/85WmLaS8w+ZhRVQQleZMxckrpg5TSUZTSUd26dWuqarnwmHTav7+avwYDV3wAbFqJs30eiCzCTqaXIYwRsi76vGW5TCKfvtirGySI5qzE6g9TCfW4BvBthMNKlz2/OvMClwnWC+pyELQ1otz7AGNaGoI8ZxgFJHKRWiakii5kXSEtJ/GYKv5RfSlC8ET17fLC2fTIh3KYusa8Ryhw+he3X0aQyPXAL3/Cgh/FOdwN0WQEFCA4tFR8V/ICx6s4aNZWK0IwnYy1zS2WSsWtVtHczrvZyS5hxRLTghVrPdeEUKZTqb9ZAxOlTV/3WUjPnhrTyGGuXNWKb/AvWCPwBrTBY4K8ulgMSy3wXG9HFuLw1Fex9LzejWULbfxesAru3GzbF54X2ognsbYjwRgmlFoxxAcRsWmlHxb9YYzcy4D6kZVM3gAVSvWmAxI5xY2ZJ6V5LDUQv3yZrX3h8Akl0rjwZw0KBqIThpj7deu4aeKgRTJy3mbnMDIfI1KW+7iM4VJYH4Wn95Xigz9il9SsELqsv+xejtdqRZbZO8gImFjWGhJ5Q7YQQ8M/QMP6f5qRYD+rudL93UgVvfZ8S/vGCFYpDVB3LAGAKpLDv6rvVWLkIgbAY+StiZpEPmF+QYL8/Ss8oy+WkYeS6GKHDZ9YpneKSIEKXfRrYJWTy1NMW1Kwh66yLU76kGBAM7XNPepRrVAK9MEqXJARe9sCjpdlBG5sf8Az/7cjrkgXxq+ORO5Pm0pJVCueqgv301uSODPXi6TMD58B8AWAoYSQxYSQC5Iot1igoOgOawA5jPzNmt/hhPSnoXnzlGL/1BTU6FpyfHwnrq96Wvj4oO//iJ/M+SMAYMDqz7HvhCsYejmenT5U5YJxUGQSvAinPPglkz9Y3zUZ/gYkAFQxvYllIlkEDysA4BmL70xfhnWboqutHDSoTho+qAxlkdUK4eSvRtY76LmzAMUZD49DK9ShHTZj/PygvQCbK08pDkp9g/E1lzDP+ZSfu/B6/ksIINvsnFl7LrpgHSiAtf745gDuq74/mOnjO/HGFPmKgFJvGARVHXkGeayTOfUI6yu836+q/qudnwd/64vGKGHqJluStwtJymrlNEppL0ppFaW0L6U0GB6vGPj8ftSgAZ8y0p8K9tr4Lr6qvRw7k1kCKUc+rKPr9cQYvvxljFhlbWwNmX4vQwnj2UnEjPlI6t309dggRzA/7InVGNn4jedeChRXZl4W5hF1Jp6O/JdVL6AahcF40ZNf45fPeq16ougu6zUlcgdxVCvHpT/j5Pddc5ikk2dczeWYWnshVy3A6t9zeYobMmPQjRM3hoXrjKKBFJFvdt5XdR/ylKp/k/EPY/SLU6RJ8tR3lF34/rdFK/L4x9jZanQw8O+ptMVm3JH5F9qToJOXKvybnaKImVMXry3kaVQL6auDylatfHg7+pEV6Mtb2kmwbd1kAJa+NCx+xea6OtCbO4JOeByAJUVoDfnb+1tB8TWQzxYYXFS9vUe1oliKs4SmANfaIUzCF+kHw/TWDn5Y4zUljcLIt0n9gHE1l2nnkzNyOR2XZl5DJ7LBcy8FP9MTl9+eWHsBPKsU9k4uTwMSX5yNOhbWZqdYQNkr/S3GTvpeTw0RMlACY09xYGWQizTxWp79hXznp/+HkzMf4eL068pl+N//J/RTYFbBOEJElzN2RjdeiI1999OgWg2VzcgbNmCzQkAnPxyG88eqx0KXc3e9MREEFLm3naWqhlQCWBsby6Zq0cdGY/OaH6qrSggKVhffLFKT0O6osoIq5fOUGwdFeIalDZEpnSojD9qti9GTiJenPchapfpYyBiDczCFrKtU+VZpfoZ77X8mYYvv6Lf+ZIXnmncqO9vXhqz9GANTkvM4Y8DSkctj1PxKolbjIcy00j/2VFUrKeRRndafwCjlW+bojCt/7tvy9wJPnxRalhvvBaSy7ciLhSgSCdFwl13jxFxWbPs7Mv/SOusxCIrBTCCntmQLettMSyXmiYOL069hFLFs3ZesU9M975qy4oc35ijXpE6m2gHEcmtWsZuliZ+Ri79tN7I+0h6ACDJGPm5euE4z6GjkPertzalL8M1Cr2ncRzXXoCMKknyWw8nZMs6YO1r6PA5km52FNHnNjUE5/BJ5evEXgpRetCdbcO3kI5XpcPDBdyvw/ndBb1GdcRUGoY7c7qsUpLLtyJMAP8CVfqt441dw6vGU73uWy1nuwwKcnPkIOxPJ6SYh8OvYdkkVytLpcPump+HUzIf2ld5kl83nuRJ52Ik5ItWKKt26jDlZRi7GILIUmPWOVk/z90sCoG1NcGXixGkH+KZz0VQn+mOCIB+Iae/HZuitfgmVCzQ538TV/pljcUBqklYdgF4/ePDj4MEmqv4CSrQIyvIo2Vo6I5++NLjJE4WRszvIukFyOo+/CzdWPSVNsxk10ucyyJhl1A051VwO07AODuapVkI2eUUHEygOtCQlI13I2nZwainw9ElI58SWEn5m4teRE1BMYTa8HOyTKpgcRmPkeo5jIqioVragWmtiqaXiE6UA74HEDv5Y9Zhy+Q7GVl8jfb6DHc6aCOzPddor7P3DhJk8TQmtn+Kgohj5xvpgR4sSD4oyHYg3eA5LM9YTTHjMg+76EGThl4H0fji2zGsjmEjJpIvojFytkZzyLdVKFAYhkkYUJXKqrlqxyk0OKpOI7DBmf/42pA7tGWedFChufGV6QOplGVdSEnmUdlFh5HVUT0BpFfBu9SKXz6MjoluMOBiQWiF97uyZENBAlFDnvjrkrZsCRW+sQhts8d13VCvFOVgimtFticCfdfRbJZfLuoWFNaozwzbmKOau3ISl7Tajc0j5jgR6zP2f4hNN2mTSawp55CnRXgrqMoO3pi7l5glVrcSWyPXMOptKR15Ioz7J3lb1KP/5UydBhJxlj6cF3neKfNB0GCPXNO2spXJG3mnZ5/r+GDHBG++q/VMFaeTxUe2VmJnv67nv7C/lkTKbnTzxO/Iy0kZYo6YDJlnqktuiEBdzfl5xpzoh/Umi+jw/HKbw8qQl3M1OmUQKhJtehSEYNlY+AQ0jC6TP9RDerlW0Hk9V3cZ9phCw1vozd6wwBc/8sB30bY6jjInbqx4KhroNlKuHNiESecdV30ifW0iuvxMAc1YU1LNOyTr9KIwap+2H+gLvDbb31fJIGR25P2QkEG0ZyR431m3tFOluvV9KVKkvzgwvY3rscWY6UB3YtRnG+YTLyMPeKx4j15WwX6q5SSu9DCrfdfvNE7B3ejo/f8gESwD0JXIVAE+1cmB6sjQPr9YoY6ITCQ8HLVuR8LAXkZvdUonduoNk900oNjcE69QJZRAGEb072eE/ckhp78up1VvhiLyMtHHy5POAT+4Wpk35Oluv7BKt8uPQ1tSoZhh5FNWKqBOrDMabM4+7sTtcGhQOZkgKKv2oa1Zw0AEQ6pRGQPFC9c3SNP1oeN9SQTEOwgb0++ZmWi19TnmG8z48UHWvUl3za0/HcDJXmoYXE0cXYavEMGEkV6Qt/Ypi5H53WMCyxdVFoLFXfMtPCCDlswnvRH8UpCwgjgt/knrfOIikI/d10RHEcqNW6brnZt7ByJxXgmtKGxaVAf7TtU/EKj/MUemD6mu1y01KR66CjM82Pgyi8MUuFAJrHZ6eoFzf3qnwkybjtk1Y/rAxQkHK+qi3JkFS8llgQEkOXshQZzOGQDk0Z4jjjDRvCRk5e2xVNDtyL16p+b1SPgd+qxUrb/JxbXhIF3naKKVppQ7+2voa5AUrId1vEcrUFCRyHfD6LAt/cLMoFkFhB3WEvbORyMGXyBMqWPjIYeQU6pJ2OapWRINTmJ7TycM8O8VebWpd98qGhzzXFAS/zDyvlDcuir0SKpaSKGmJfHO6rZAhZoieZ2dobB4FHbkOeBv0nvoSUK2ckXlf+px3GhaLnLFa0XfeUYe4c06YU9igUmWysVQrMaR5EQiokps8u2HHt1rRU62o5pNhxxC9Z1Io9kqoWOqO4ElEMesiRMzINUNPhMXmGbj4Za3ywtDZF7iMB7Ztkgo4xqKayNvIuOijOIeWApCqVpwZlkBHIrfS7UzkB0nw8ybPUI5Jf4mcKCa4AHwdedj7CzY7I05OTamMKP7eRHHepgPnhKA47ClFUqACtqDbN8PatKZRHo5XF5dlXsVg8oPw+T7paagJkZhVsF2v9sJn1SGTXa5I5ocV5RAUFnI2MiSqlSp7hqVQ1xE6Hf7Fmpu1SSmWZKgkfTA9jCfVRbdaicrIOcfBFQnFWAmxaDr7m5iqlUbxhmYGOa3vUYr9njerrxM+OzH9iSeyaFTwzKAdGNWKAkohkbMf3h+qVIR4duSlNB2LtyN/QxX/BPuo70QhXuYnjZ+mPitq+cVSrSRd16K19UJdcxp5rbJLwchrQlQbW5P4Jp4pQnBw6mvMrz098CxsooivpeejoiTy6oyeekAV1GOv4auT+TBNoSMvVudXcuawJ8pu+BGXpV8JPA6j7Zg0Pw5NnHdqKon82qribqo2JSMfShZFzpuXTJ66/bpcTGlZJPEdUjSHI1Ljuc/CGPmvDh+GAb3FqpmoqChGvkv/TkAyPhMeTFy0DjsLnrGMvJKtVtSkYivNY9V3YofU/MDTqLRFHTz7pfUO5Chn1CboPRiGF2puiZxXps5Ka6pWMqRpTEebGq+sORab0/wAYmGbnbts1RVoGz06qggVpVqJFOpQATOXi92T2RlWtWMWy0U/DjIqOmBKcWzqcy4TB6LTVkrb+HKB/zSgcoVs6R9mheJHOUrkLOLI5q0JP0Rv6GaqRI0bB5XFyIsE2Qc9IV2IX6gskceQRErJ9Gqr0jggPUn4PKrKqFKcYYqJHiTcI7gckKcpiLZm0ySn9SXLcQLfLrWwqOWHWa2AFEc9bBg5gF1TYjPBfdOW269ltRItip8OSinFpAAM7Cw+BSYdMfJiuUtmTYGeCD8urhwg+8Kqm/0Oiu0tGx/Jr/BDrWJSZSyRE0KOIITMJITMJoQEDxZMCkWyWhmSEtueOmhHN+Ldmt8olRfHrby0UgxFhDNtQ2EYOdBGsBQvN8jsUmrRgEPSE5XLaqrwCuWEUEZerqoVQkgawD8A/ATAMACnEUKGxS23khFrs7PI9sxy0KKoQYxqpXImM5mbe0eFULcsylG1UmxUhx2UUcaqld0AzKaUzqWUNgB4FsBxCZRbsahJRWdcpRzwZPNqdMiuTrzcljig/aiUNkjS3LOcJy/d2EOqqA5bhaTKl5H3AcAari6273lACLmIEDKBEDJh5cqVCVRbvthvcEcM6NI6Ut5SD/j+GyclXmZT2lCXK8qZqbGQmx/qvUPH2vLegiuGj0LFqlbA3zEIjFxK6YOU0lGU0lHdunVLoNryRQq5yF2k1Iw8aRBNb8Dmikr5rjJPWl2rJd0ThZoDwhl5cSTyJByCFgPox1z3RVHcdlA0O3IRVtH26Er0A/ukkEcqIq2VIrmp4puaS7CBtio1GSVHpXxXGSPXtVrxn66VBKIcPs5DilDsm5qSAEVehEaILGOrlfEAhhBCtiKEVAM4FcCrCZRbctRBflSVCCmai2zZFEdyezR7ROS8xUInshH9U81blaaCcmbkX+QKtgky1YpuGFveQSFxwTtLNir2SotPBouK6jAfknJVrVBKswCuAPA2gO8A/IdSyj+htsJQT6si5UvRXGSJPE50ttCjtRj8M3t05HoM9FH+NtUWZHpjXdVKfWP8kLF+lHsrdkeI41cZW62AUvompXQbSulgSultSZQpqKhoRfMQVSInNHrMvjiMvFGDkT+ZPVQp3Yu5faKSY8CgtGalcrCjijU//DY/wJNOV7XSI4ypRYB/XI3Lb5t4HXHAiw+0mTKxVcrYaqUJ0bSMXIcxskghukQuCrqznrbC7Y2nSvM2UnV6VZeoz2QPUi7TQIwkVCuf5HZIgBI52BF2a/ZMz7OqkIBQfuydTn5h7t84z9HyZmGT8oOQZQ91KVfVSpOiiSXyRs1TdRykaTbyvqxY6gmPza2jWlFl5MbiJBkkwcjrEU3VFwZWnSJzCIoTnjkpBBh5BbAwD43lrFppTmAdBbIaEi6LFM1FPii6XZV4sIQdLquzgggrK0qZBmIkYX6o+s3i1VHot359eRKn68SFf1Q1RZvEhWfqMaoVoOlVK9EaPc65Nm3S/AFPEe7AUAyJfANaYf/6u7GSdlAuu6VgJVU/ICCJlU2xpE//gcTOFfV5P5aDRO43PawEidwz2RjVCppctZKNwMg30xqQfC6yuWhrISMPnx6KIZFTECygPYu2rK9k6BxonYRE3hRMi4KgdXXG/u1FOUjkflQCI/e0Y027otRR/q3gQflvdm5CDVI0Kzk8To72qTrufYuRyz+XTqdWTeusAvzSmYF3ol+Yl3srxwlt7KBYagT2y8pUK6E20iVA+atWCChLo1GtoCI2O+toDQjNSU/aluHkOv7ZkRRyrztAbwdfVfnjDGaSgDddsVGsQEgiZGmhf4St3pJwVy+aaoX5tla/sCfvotSWLMpfIqdNcoB4ubeCD+WvWmlEGtssfxNPrj4tYVoyoTpynU6trlqx0JesEqZ5LnuAcr3FRL5I+kcR2PYOa0+dON4iqHwzWtVGu9y21YV+TiUSeTmi/CXypmnH8m8FFhKJvBg2ttkIqpVBqWUAgPZUP0aLDKc3/C600+p0al3Vih8D657GR7kdAZSPiSIp0rJVBHaijzLp66JYNtMdWxf6OatCKydGPrrxQvf3XnV/x8zUYABqot2fGpMVqnThSOQfjry3aHVUFiPPiE+fTmogZZkmaaRNyxhEaKRpzKZ9Q5doOie155DCbY2nh6ZTGczlwshpwt35r8OelwoI7MTZFEt8lTrIITdFKNlnR8588pPqfx+hPC9+oF3wcW54rDLYd1+CrlhLLCsqlf75QX5k5HrX0jZez8wIyNsTME3HK0eGymLkR/xZ+CipJdYWFBpbd3K4aeRnidDgRxXJYVDXNqGMvD02KZdJkcJDuaPxWPbwkHThSCIaXRKgCatW1tf0kjIKdtJn+0pcpiWCrI/nKcHAuqeB3S92772e212pXHYi9m92jqfxXeCPrr8Nj+fk/SwMwdWIo8cPZ+RxeMNS2jm2oOLkJkVU/VUWI5cgKUa+CYXDh0WMPCfYVMslvBlbzzgknbBzn9BO24GoM3IHt2TPwdjcCOFzmZTr0BPFa7EYMTKSYORraUHHHBZmgVW9sRLjAtojNh08yCRy3mQq6i93NJ6CoXWPu9d+O/Kw/LrIIxW7LP+7U99fef3R696AaAfEOCAo8CYS1QJCAc2GkSe1tN3CLKNE5oei+/mEBVO/7XZY0KJVEZ12ZN2LUuCI7XtK80exkc4XQd+bBCP/Q+NZ7m9C5MzsudwB7u9yUa2opK9HBvWCgHBsGFud7nxUvThWHk1A+eYX1GgTSeQNEb27WRQk8uKpapsRI09mttusoFppEDjHUIlErmVRkrYGGTvYCCHSeNBnNFyHJ3NqEQ39kA0zCoL2rTJoSLcVpolyuHIxlDE0gX0S9juliDg29+CGZzA+P9S9bhr3eb06VJjcgnz3gB250x906ptOt3J/z8t7VySsSWNU+MdikzFymSOcImN26k8ZiTwcSW10fZYv6DdFduR3ZU/i3s/lqXCDVIeR01SQkQNyz7rP8sOV2+CGxvPUaQFBihCsGCbOE0W1Ugzb2nwqvvSU9zBycbqM72G2CTbGdSVycRtb94fVPYrDGu6QqFaigSc9x1WtBMv0/pXmjeFf0IAqpEXZ02phrgu+GIaRhyKJpe0x9X/EHdlT3GvR4HxCsHGTp2K1i44pI7VnepL25hGpVnQ76lzay3Mtl8gtSaJvZ7GuMJJqpQhdjyawdGUlP5kElU55mdP3tHDeeLHM9vQlcn56h77NqEU9qgObnTrSrgoIaOwVWFBH7tAWTmMc3nDYjv2QEXHytFrYCpeRF+mYN8Awcg9m0b6ewcJjyifU3yzMn6dUSIfnfp9dpHTQlLeDvJOz0oskct1B4h+gj+SOlKZNEUiD/USRyJNgEu/mdsbcfEF/n0+AkbPfnzBqBj/8apfbs8W3VdZVH4omeP8biVz046pDHDT6nNkOq/+LdhnZkElJhjh9LZWplT0MzU8Yz86UsVoJRxKMwS/x8FQr39BtxPnzYsnDG2AphFZm5h5e9zAua7wKgDiwv+67+9N/lBdbreRtduZn5G9euS+G9rSi/0WSyBPp1N74M0mo13KKqpUU8TJEndOk3s3tHIEyfWFFuV8wyainPaOBrfeGxvNQj2rPvTm0t1I5M/OFM927tfOuCF1LEAUqY60GJL4rqqoVl1bDyMORhNeb/4PrenbKrFZyhCkrRFfmqAgoCDagNbLIgBCgWqBaUR2wa6i1YakXAMuWyH11DOvdHr06WJ08ymZnXrD/8FV+KCb3P0epDAqfDTSJryNnmWU6JZbI/aoVHek1qlpJN59IR+7vL4SyOnJAx0Y7rPwltAtTrpwuP2bRvu7vNq28TNNPsR/vZ/bHRupI0zGEPCkj11StpA0jD0USqpVjR3glBe3NJWWrlZCOJZi5RVYr6bAOcvp/8WyrU1xGoLPRSEEsXbFk8nkrv5tyeW65gndcR9ti/JCr1WljVgOJq1Yk7yyzaAmb1qIySF1hRVSPX/fPvqZ3hRONTrZ/OSvRKPbprPAzY8UWz7OCDwO/tSnzUvW2QBYppn66WhwapJd4JcvCaY+OraKdAayCZsPIk1Ct7DG4u+c6SUYukkD5sF16fe8k0pGH6t62OQxPtT7bvdSRnylsxxguU7PuLacdMTuvtlx2IGO6quedWoyclciT2OxUVK3EMCWLarEj648baCvletpLGIrXszM+sty+rMjImTHjn8ScoSZk5EjhB9oVgKWjH1j3NMbkDlOkmsHWh4ifHXc/ZhzxDK5uuAyv5PYSJnMo3KZn8Q5naTaMPAlzNn+X0HbAkOhWcjpMhsPICAjGimJGKDA+v9T15AW74d5TdnLvLbY7vR8qm50EEfT0kvLSGkwyaUbuNT+USeThjE4UyyYqg5Qd7nFA/d2cevj0++8GTwhKTrXiP3zji9wwAGqWVt7okt70zrVMR35mw3WYuPs92GJ7a+c0Gn4J7QzcsBIYtL84UXVbbOq1B17O74OrGq8QJnP7VLmaHxJCTiKETCeE5Akho5IiShWH199eoCWB8vwdV1snKdORe07SVlOt+On5ML8T/rzb57wMobT5U+w7pBs6tCowhv3q7+XmcyVyXh0kuvQmc95R5eMUvljaCTByVvJLETGjSEtVK9Z9oeNYxGFXB7G+9uBR26NXB6+FRZgdeeFKJNXqjarTGq4P3PMzbJ249o15sTrGKUVkMZUnKaxEJ6zoX7DI6tRGXbVBkQIyVnphK5C0km14JYSxnQbgBAAfJ0CLGroPc3+yDZREBL6ARK6rk6TW9hsPns3OkA8rdTXnPVPYDSeEOYvRrj+bZyUxPvJIiXXkjCpJ28ZZwnRVHScoCF7M7cPQkKxnJ5Ewa0u1IqeTgOIvjacG7vvb+q72v8X2dY+417PzvbkSax0VM6I7fjYCX1x3sOee6JtQSfsm4RDEazPKSNDH7dRbqWxWzeWn2Xk3kcWUM1myq6pTd+2vUKuTX3zlgqidA+a2B41/uIgIsRg5pfQ7SunMpIhRwsWfAO16cR4k7/SdpGpFS1oUSOT2TU56ZmNHEBuClXIdSS2XL3Qs2cYdq1qpp1W4p/FEJbJEyNJUQjpy4G7GyzZp1UrQMqWAFCFKB1b9NxdcmvvLnF7lDZX7Wm5PDKp/KpBvi4aJI6+eArz3l3ffz/3N9vkkVCvuPcoKXcE0i/a5HX54Nl59E1tBIpevJthcbWp0rJoU3p0Qpb6aL3dGXhKkM4DtMOOVyOPDPzCTVK3kw1QrW+3HPA/W68kSYFiFhzvUP4r7sj/l5A8uU7MegsWDPkUIMNCSfM9r/DX+ljsxQJSOumAqHQTZ6dTqqhVLKs7aKycpI7/4E6UyPRI5xMzM0pGHE7oBKpuQxCcJ88vdIlGt8CBSrfi738ztrsDudfdj+7pHbBWgN8EJ9TcDB90oreuaQwv+FWyvct7lkgOsgyCcdaH/Hfv17gM/hvVs426iO2WOz1v1OIxdqBaKqY9eCHmgOAcq1ZQFIyeEvEcImcb5d5xORYSQiwghEwghE1auXBmdYgZeNpSEasVbRhSrFaGKgoQ4BG21P3DCwxYdDCPfuX/HYFo/o2d6UyMyaORI5awi4L4zLE/R4AqCvyQmhAC9RwI3rcXnef5BCzqtf3bD6ETc6R24QZ5kZfbaUaksdi9DutkpkdbZ+7wogxuo37lFrf10nI78dHjh01unUliOztjkTjreoFnf0G2A/X6FdaS9sK4rDx4ipYV1hqHgvC+HyfXtWND555HCPvX34pyG0fa19Q4Hb8s/9JorWGgw91/jl57SRAhTrlienTYtRTxzOJRTUUoPoZTuwPn3ik5FlNIHKaWjKKWjunWTnziuUJr9f7lJ5DKrlRCHIEIAajv8MAzp+Uv2wpw/FTZsKMA5idtb3qv5PQP32CoHdLZibgcYOYcuV7UiohteN2QAwAixu/rY476yYjxLmG7UuO5+C4loZRS+eZUwWpJjRy6Hk2JSfrB77/bGU/HnrNeaxR9USlRuPVVzQHGgutkpmrD8dGxIiRk5APzm8G3tfMHynCqcNw2k4TByK0YLcWlZTLtjM7wbummJ+WHwHdS5xAYijvbpKVGhyLLXkZcDdFx1daErkf/qsKHCQRGuvyVA3mHkTue1nHEsD0M2qY+u3X7uuZxPewE3r/WWztJl529r6wxPHtUXn40+CCKJnH2nt67aFy9e5tjMFu57Jr3WXbhvCAC5mo5WuYIN2on5rS0V1Y6ncJ974dsAS8JqhWXkmZSwX6mqVgDgpw23ur//mTs2wIz87yGydmlgPI0X5bsh328Pab0eqbQN4yNB5Izcvynup3PB7rdw63MiQvJaLGsfCv2Da+aqwsjzQlrc1CLmyPs0GhK5xwRWIlhUKXhruvr9cmXkhJDjCSGLAewJ4A1CyNvJkKUGikKsDe3jxjoN5JTnRcfWejrJkf07oV0tf0PFu9TjdKh+u7sSeUPHQVhOO+Kh2vOD5VDqZeSXfwUcdEMobd6JwLo6dFgP3HXSCNz60x3Qp2OrwoqAQR7Eo7Perld77Ny/k/tWCPxSg0i18kDuGOsdT3gwvAz7r0NeElYrrBTLG6SXNlyFaxoukXp26sIvkYuORWMZ/I9oG2rd45HIT3tWmE7EiygIHjtvV/z9tJHWDbu+La34zl9Ot2Qn9XWwGPiGDtvh0oar8Dv7EGVZf3FCSaQ8oQN8k427+ckvaUI1z9tY/XupbrjLVm0OCt+hhKoVGSilL1FK+1JKayilPSil8Q7mi4DojkCcfL6Z99TdByZTLnzSor+T/HouMHBvVyLPZdpg9/r/w4SqXfhZznq58DuV8Tw8bqfeOHevIN3ezVIniA/Bibv0RU1GzABdHXkIPBMVIZhzKt8i1S1KcOI9RUpq/cOikz3ROpN4EhL5odsXmFQ1h8O9T/bEi/n9FFUravCXIzq9Zy31LvfzR8ijCArp831P0felAA4c2j0QukIkV6Y533QWtQJf7dCnA97K747NqLV15GKJfBl1VnRiqyr33XhSbtuemFRjubXIDnuRQdUEtjrD9JGz+drmstjsLGdQEDeKnjOsHsweBVw+PmJ5vuukNuQG7C3Xt9faukfnQ4fZhffb1bXc8TPEPx0/HDcfu30gyzGewag++Y3o1xGn7NpP8FSk1yUgAkbtjg+ZjlyRkfstE5LQkY8+qrCZW5UOqlZqqmz7ZYkdub6kribdZ5HGsfUFNQ3pKd/AFfc5b13pANMqqPZ490Ueks7Kjfe4VXUa1x+5Had2GwyTc8O+enTk3lz/gX0aFmvt5aBdT9fZzbOq0lCtSC2nznoJuORTAL7JnmdtBqMjD4U1s3sZeT2qgE4DIpW3y4BOnmuSToiR05BTC50OoMDIXQHDUYP4YiJ7+upv5gG/mg0AOH23/ujieLZphNN87qK90LVtuIopwDREjNwZmBJG7r7jgIKzD+9EeH+LKkvk1y8TP2PK8EhbzmP7r0qslcsOGOTuQ8jw5xN2VHJW8Zv1hfElISP368iF3cEnBdv5RJv6MnUE8VUbmCTyOeaZnYfmA/ccTMY2wM3rgPZBs0X85A7cffII/O7IbTG8b7T4Jt4wEb7aBx8E9LROEvP2kZCJvZRWK2UJWrBa8W92xjkfcPve3o/uP+BBCdzOLLAOOepu4BffFJie05k5TDBg5uQyfW9az2Bq3Rlo282ukilBx8ZWJY6L32qFEGHsZYcJiCR2Ng3Oesm9x1M3+CnzS+QT+p4DXD3VncxcVAVtu13YdOUo4TJyh4GrbHZ2bVuDQd3acJ99cfIk93ffzq3xh+P4Zp1+sAJBqI5cMVyxULXi67Zd2lqbtIO7861X6nqMxFf5obil8RxOHb6yA5UF1ShEFufeKaCVV/jCqc8A/XdHl7Y1uGi/wX7bLXF5PqjryOUSOcBsUhtGzge7SeRxNEgogHskRs4tSGCvXdUK6FIwTXM6ZWPrHhAh0BV0zqks6DXU8yimDTA1gZjXaK/LqYSRu+aHmQLz5rmn++vs26Wd57quqgPQsb87mYnQ0JH5BvbEmEUaVRyp27mTVtCRy7DHdgO9pWrG7JDVzVVHeObZEKsVwskPoHWV1TZta/njglS1wskNN2EacxAzWwdxy+X0F8bOnw2IJZos3cl+60OAnz0KjDzLuq71TjJeay31fq+a1DPZCzL9rvECYLeLLUm+SKhwRs6aH1rI05T4K2x3TOG3k2bbo4Xl+8/MVCOKN8QUh/zwk4Dj/4XVIy4J0iLqWD6GqNQBtSa6cNr9VhcWY+Iw6su+RNYOC0AkExCvCR339DsaWbNE78t268CXfl0MPwkYFbQE2tJ3b+YQgoIzDM/80GF6XIeg3y7AtDMmeRmuoPmCDEaPkffrJD5D1YG6HXl4fVY2uSDgnG3JjSLB0kWZsne7GDj3TSu2905nelKzB1707ug12XSfEALscCJw5J3AiY+4Hsi8enUEmFPZfSGJJO05hFswrlaiE3DkHZZXepFQ4YycuOFhnQF38m79xYyKZ3e720Xi8hOUyD0MQcRtUylgxKl6Urafkat0Vh3VisJxVgQ0eOoQT+Luvl1hI1MymfA2Ox2vxpTEkiG03U58GDj6Hu4j1/09b8V8b0CGa7XiNJ0/jO2xI3oDrToizy71KZU6iXELloAtqYsskp+jyxbpbH11BTc7hQV7yvcjLZDknazO3cZswT4c+/3astgCgJp2HrrZMfP3U0bi6Z8X9kkCzVrVChj+s2C1RHQhx+UHbq2Uzru64a3giqdOYVHRjBwodJp9t7YcDfp2aiP5YJxGlUWCkyz/hVDRkTvLwAH8YPSy/hbowLLNThFUO3SngfK0zLOAjlzARBzVipKOnMFG++CEDGE3xfwSo7c7E0UmWp1O4dSGG/BO9/OB9n0wpvpnOLnh96jmSOTOwPXbkTu21v73rs+qWCqoSeTvX7u/QlkFiOLf+GsKbtwKaCHy5045vAmEMJuzeTZKKGdCL6yyCwy/XW0Gg7sVzC/FwTCSAcugldl/Ec/kDEOFMvKCK4jz0Tu1thmacmPyP0/raobBFEtHPnBfa8ed45SkDR8jV9uk0ZTAFBDYBbDpaPCde+pGXOQwcocx8qwPH88djuyuF+Oh7FHuvSfbX+iv1HcdTjcAtKpK459Xn4YDLroLIASP1pyFmbS/x9njL42nYtX541w1hMeqobrAYPwk1DXyz1mV0i1At7a1iq8kVnGwz/lXkpzO2IogkRNGIu/VoRY1tr6dV1bB7lr8BooWqr6JVWMlGgUc3tMUsciBimXkFihlPrq9LPNvdgQyOBB0xrG/OqBwkRQjB0W3dnpeosogftWKjIxwtYYWDrkZ6LMLvswPC3RYd8/C18Ucibz/lhlB8pwzRbmqlRrkDv8zE9gJ2JDu6E20+UcfDeqDaEiPdlwrlY6tLRXGZDoI+U4D3YmSEOaYOSZGvvUehV91jYoSuYSZT80PtH5Uty2sEHTCpwaq894PxLaxL/t39lv3yHXkhcmNJ5F7EeaEZuXJe+7yaAxFRNVKpMqc4nVUowmh6WtMEJb5od0h9rwC6LYtsPO5mqV4P26P9symCrs5ceH7aEy3Av42X6s8i1CKXh1aAU7Qx5ghNj2ItNmZUP3dtwN+/gE2jX4jyMjtQb0ZtahFg3s/m7MG57Quh2HEuvd9eVIY1K0NTt2N74DkZ8yBt7BXOIva7YR+GybpvYug3B16twPmFep3xyoB1qA93hl2Ow47urABm0l7VS71WUsiP6PqHmzYuFFQoZyRD79xHFAlj88SKI+qOwRl/R4+Ni1jLtyde19Eq8zfgPisVrhl2IcZt+27A7Bkhs3IC+nY+UZVtaLe1QnUZwcv/nnmLujXuRWAudaNdI2719JUqEyJnLEjdwdNKgPsebnGzrDCF2Yl8r6jkO4xTJxWCr9DkLzuVvays0+noL1zoAMrult7EyX/2T2MnFLQ2s64vfFUXJa5xZPOiYE+p1PQI4+k0vjg2gPQV8Eig4uRZwDXzMCSto5NdgJ6VIZ7WPzWek9HjTCn+2GWvb6NjM/sst6WyH+o3gpT6GDwEfLNAkxcDapvn817Vw2OJUYw1oxcIm/FqCUpCHDoH9xr9pApzwKAvRhxKnDZOGy/v7VpKdvjUFetiK+8j6ILN0fs0NP2QbHLYMxmzWanIvJ+r0gp9BrVbyLH+9bndxcHIwKAH7uMBI68y1ewvNMM6NIGD5yxM+5hDkdWjf2gBsWyNOr0L+MJsSL9LYD3NCeHkWfSxDIXY/cJQiYYZxK7ouEXOLfh1xx600B73ulRHFw91VrFcevh3bOk8YJQ6qhYvOmq0gSPZH+Cb/JbAyNOQ4O9AqmtkmycJ/pt4TqWZdkFNyOU+PvSkO7tcPzIPnj76v3w/rX7F2j1M9IQidyT9Oa1wN5XFa5ZyRrUklrtK0/53bcFulixzbf0P6DwzEdL1BgqEooTKMLuvwqWXkmjMhk5z1oir7CpFFKWH36rFR4zXU/kLsDjDnwG6LuLNA0PPxney3M4ciLQGIi6EC3j/d6FjmlhJkUsc7HBzDmTIYzckYJfz++JD/Mjg69h3+hpH0Lcu6PEg7Njf+3NZgrGjlzQhJl0CsvQBSc0/AFo09W9X8PRvxeg/j1UJDwnTZb1dLVdyoGg+WEmTXDPKTthaM92tmWIiB65RB5ClDfXOa8C+17LD3ncdWvgN/Owfvi5npVex9aF8aAskas6BCUxJpwyqlq5/Xpjp6ireD1UJiP3HPrrSA+qZl4q95xHweb59eFD8fovCk4Hgf6UqNVIEO6rb3ds5DKKo1rxXhWawfuuzknvA7twnHdCYqVk0imPtUhwM9O6HtC5tbgOBci+TliIlYwvwfl7b2WVaTfIUTv2wvA+vslf0mfW1vBDxrp5Mq2A1l25SRr9xwu6zm9+hYPmCk1C73vX7IdbjwsGbSPMcoZSAN2GAgf/XlxW685I+9RUtVVpvHbFPvz0IpIlV8KUF30oTsY9L9gpgpHIz3oRuOBd7H7F45h/+1HiPAmhwjc7WVMlFYlcbzmW4jA8v6NA2BKvEORKwSFIgkCOkx4Hco3a5QhK82L4ycDU/4SnY+C3WfZsbDE4fmQfdG9Xi7237uIkZDOF1jOoaxt8v0KyaeijKklYvMiqw40Z40vjZ+S/P2YYfn/MMPz0H58BAC7YZysmnjtbchD71N+LU3cbDr4CyMZ1i4SPstQ3MTrqIH/twmYXhJaQ9Iutu7fD1t3bBe4TJpfqV/Ga0trtrTl0POmZVYkUvUdyb/+25gb85edniPM5WgFHtdKPFxO9OKhMiZzB5KqdrB+tOkvTCSHpGTv09Q+4IMI6ZdG2OlLpyJtgoaNhr19oFyl8T85m7D5DuvJ1/pKVgmMaKCc9oupIlt4ejCtoR1Amqeug6suaUTgxJqz+Q4f1wD5bd8Vi2h31mSBT9CBdZf3jICuU00LkVFF7xFA/eA7/1nB29SdNK0SdFGLAnsC1M8WViWCfrnTjL68G2gtWSEBBsBJ8j2KiQhl54fM+2vo84MqJQAdOOMtANj2puFVN+AcJdsrk9c9AUdTaiVbq15E7OVUj8AEQhr4FgCk3HQYg5BMm1EieT3rg9fhZ6h7MoVb/KrwXnxuJToyR8y5vnofOHoXdt7IEkzh7eo2i+Oxh7TTqPOtvQDjSlakLSBFrgmpfm8E5ew1Qy5MiuDN7MjaiFdB5sF2O3jcOqI3a9RSmFOLC94Dj/yUMFuYibzPyxPxP1FGhjNwCBbFCl3YeFCG3QodQ0CWLuvRt/R7EEfW3MwORNYJtGpOkyHAHiw4jZ9JSRkeuM/Ak7c2z+ojNyLeyXd63P0GcJpXGXGLZtRMAF+xj9bUe7firIb/5oR9cCjl0e93Z2fwFr+YwZFUZub+ofa+1PI9rfAcQx5HIQdCjfS2m3Hw4tu0pP8TZQYoAH+ZH4oDMky4tugJ5InN7pwGWaWQYcra/hLFa0YMVrClq5mQYuYiA9ZnOmEH78x0XYpwUkrzZFQ/6vT+oTVUtg9WRh7e3kmrFJSqkrbrZhxM4QZv4peDCfa0NyzY1GZy+e3/Mv/0o92xW/3v6deRq4DHykHIU+q9IIg/ycU2aI/TBKAzVkb7Z6lQO9IiEJDi+M66ro22yx0FFM/JQdPSfusLrgJIPGEMilw6OCIzcsfaI7Cyjgwid2rvZyVqthAx6z2ZnzBOZIqwkVHDZAVtj/u1HeVYForcSMpqIE3CcaVukIz92RB9s00McH0aM6KqVKHAdiJh72qoV7XeLgYH7AntdCRx7X/yyNFHRjFx6gNoVXwMX8w8ADuDMF6zTevxQiWdBKc5suA6zj30pQBvAjN/9fu15qovDt++Jx87bFRfss5V2Xn3obxqK7BtCwQSc0jWLFJkfxsUgO8qe1IknIvibvGLVSpwFmEi10r9LG7zzy/31C4xBVByJnFUvdW5dJLVFEhJ5Kg0cdivQTnwwTLFQmeaHzIcVqhq6cuIJe9IyH27rQ/RpqG4LNGwEpcCn+eHY0t3n9OOXJgbsBXToD6xbGHEgEBw4tLs+nSwSjI3th2ezk1J1nrr/b4DNq4FvnlDa7ZeudEgyEuPdJ4/AhAU/Ns3qB5C2d5xwrVmqFmtF/WtHb19t9Q34qpUOrfU2EqPUW4moTEZug4IoHW4rRdSZ+LIvgBUzQN/iF5PidfazXgQmPyPZOS8yemwPzPtIgWFGYeR+5hAchFxUtQKO+Zt1zN3OZwefn/cWsGll8D54ny6ZQduutip00nQDSSpWqcv6SEAScO6rl9SoOLyVwz80uUTuVOet746f7Yie7dVMb7VUK313BYYcrk5gGSEWFySE3AngGAANAOYAOI9SujYBupTRSXZSihIiDv6O/YGO/UHf+sR7/6cPAO/fgk3pDgBWeDth1yGWN1upcMqTwNLJwQNrhdBRrfgYuU6zEgIcegv/meDwDWE51g+NypsWylvAfD7O5FexWhEMb79tvyJNBUQxP9SvhXAkcgA4eRQ/Oia3DPXKLDPDCkVcHfm7AHaglO4IYBaA6+KTpAfpkVdNAIdRu/10m8OASz8DLUFM4lDUdgC2CkYdTALlYVDZdAy86CfUOPWID/0MLSMQj9zZTI7qERxHIo9QXUowmWnVK2unocV3nW8qxOI2lNJ3mMsvAQQPzSsyjt1J4mmlgoScSJqXLi7K0CmD9y+J11QQfz9tpBvvJSqS2OwMoE036++mVdy6whFDRx7h0/jDISSOk/9tqe3u3hZl0X9jIEmrlfMBvCV6SAi5iBAygRAyYeVKvs5TF+N+dxD2GswPGBSK4SdafzuoL9N4EOlKO9mbMrKTUMoeiqNvz0FdvB6c7GZ0EUkKSFsBq5firxN4Et+xI3pjRL+O8cp19hh89zPdt7F+7HJeaBmBY8ba2tYUG5dz6wonKo6OPIJqBZGrC5TBRTpTiCW/02nRKykDhErkhJD3APB2566nlL5ip7keQBbAU6JyKKUPAngQAEaNGhVzhDkBi3Q7B1PtHpdZ8airJKFOAaDHcGCn0yUlWmX6dYCjf7ItBnZpg8OGNb0pUmxojpxHz90VuZf7AN8W7hVbvrlk/8H4bul63119s8mo0GUu3PQd+wNrF3LTi3jmM788BsA6tTr9N5xwyh2jCi9Na0fuWq3EqC+0K2RqgOsWA1VNZKFUJIQyckqp1DaPEHIOgKMBHEybxu2QrVwhTboQGbHnCPZBOBMHgEs/lT4+cngvzFr+Pbq29erqW1dncH6T2HwniD2vADpvhcJAVWOIrarTQPeBDCMvfjc4dkRv1Fal8NEsZnXXhKqV40f2weOfz8eBQ7sppXeYkYfEC98HVgqCOPnyRUFAIh98EHDxJ0CPHbz34+7AFgn+AGXRylB4uZqQwGQVgLhWK0cA+C2A/Smlm5MhSQEd+lrLQ5UNxVQayOWAX3wDdGGO2pJ94F/PAbL1SqRcedAQnL/PVmgfFlCnEnD4bdbf5dOtvzqMcd9rgdnvAYvHe24Xa24nBPjFQUNw5PBewAPuXetPrR3vu0YtpkcUjOjXMVKcac8qsm136x8vXZxJ6dA/YOLb/+Y/67Ujpy7Fcl3VFQUuHw+sXQA8VbxtMSfS4V6DOYdPGHgQ17TifgA1AN61O96XlNJLYlMVhtOes+yh2yjoxw+8HnjvJr3TYFTKtZFKkbJg4jWZFOqz0WO4xEa6Ctj+eJeRJ3s0XRCUWgN9mx6MNOXUucfl1lJZQY9croilH977Khz/2tYYQhZr1aWckuatODXdtolAnDqq0im8d83+6N0xYrjmFoS4Visc98kmQNtu1jFhKtjnautfAJW9S+3HR78+ECs21JWajBLD/qaZamD3i0tLig/Hj+yLaT98i16KTKmgI4+2otl3SFect0074P1I2cOISrBQObbu3jY8kUFle3bGQpmYqiWFnh1q3bMqYyPuQC2i1YoUZfxNz997IM7ecwDnZHo++tvmi4MjMrInL9jd0r8rMHL11VP5tm9LR8tl5KZTKkC3jXT8DuOBuwlYxoycECI8dIKHg7frgRcu3ZNzLFzyUKaq2rbskBwAYlAatFxGXsaDvvJRHn6elY5dBkQ8vtCFWh9XHgpH3Q10GeKeEG9QPqjoMLYGxUJERsxwBOdnTcZ0sZIhaWGldWfgoOuBkFOQDJoeLfiLGIk8FDEYQcfW1bj20G3w9M/3SJCg4lvDNCsoHjnWvMJLtEwY1YpB8rA3O39x8JASE9LCkUk61GsEElIE2TgePQZKaLmM3EghYkQ/CNUpIClKhCj386vLApmaUlOA967ZHzOWbSg1Gc0eLZeRG4lcAj0XfReONYP/TNJWnYHhJ8WmSkjRwH2B+Z/wnrRsqISgKDIGdm2DgV2b/jBiKY66G9iwrNRUJIqWy8gNwqFtfWhvueSz3vu/nZcIOUKc/h/ruDgDL1R15C1Nptn1glJTkDha7mZni+u9Gug61Aqs9JM79PI5sW/yueRpsnHTMcOwTY+2Xo+/6tYxIvo1Yyj2cbPZWfkwErlBEFW1wKWf6ecTqVYSxO6DukQ7Ad5ACCPTVD5arkRukDyco8SKKJEbGBgEYRi5QXJwJHK/jtygrNESBPITRvYpNQlFhVGtGCQHV7ViJPKywX6/AbrK7fmbu5PVnD8d6R7k3FxhGLlBciBGIi87HHR9qSkoOdLNnYujJapWOg8qNQXNF65qpYQHXBhoo/mzueaPlieRn/8OsGpWqaloniBGtVKJaOaalRaBlsfI23az/hkkj5SxWqlExNKRX/al2tm5BkWF+QIGycFsdrY8dN+u1BQYoCXqyA2KB7PZaWBQEhiJ3CA5mM3OeLj0c6BV8Y92M2h+MIzcIDmYzc546LF9qSkwqFDEUq0QQm4lhEwhhEwihLxDCOmdFGEGFYgmCJplYGAQRFwd+Z2U0h0ppTsBeB3A7+OTZFCxMC76BgYlQSxGTildz1y2gTk+vWXDiUduVCsGBk2K2DpyQshtAM4GsA7AgZJ0FwG4CAD69+8ft1qDcoTZ7DQwKAlCJXJCyHuEkGmcf8cBAKX0ekppPwBPAbhCVA6l9EFK6ShK6ahu3YxDTrOEoyM3ErmBQZMiVCKnlB6iWNbTAN4AcFMsigwqF12GAH13BQ69tdSUGBi0KMRSrRBChlBKv7cvjwUwIz5JBhWLTDVw4XulpsLAoMUhro78dkLIUAB5AAsAXBKfJAMDg6bAm1fui3HzzKHVzQGxGDml9MSkCDEwMGhaDOvdHsN6ty81GQYJwMRaMTAwMKhwGEZuYGBgUOEwjNzAwMCgwmEYuYGBgUGFwzByAwMDgwqHYeQGBgYGFQ7DyA0MDAwqHIaRGxgYGFQ4CKVNH3mWELISlidoFHQFsCpBcioB5p1bBsw7twzEeecBlNJA1MGSMPI4IIRMoJSOKjUdTQnzzi0D5p1bBorxzka1YmBgYFDhMIzcwMDAoMJRiYz8wVITUAKYd24ZMO/cMpD4O1ecjtzAwMDAwItKlMgNDAwMDBgYRm5gYGBQ4agoRk4IOYIQMpMQMpsQMrrU9CQBQkg/QshYQsh3hJDphJCr7PudCSHvEkK+t/92YvJcZ7fBTELI4aWjPh4IIWlCyERCyOv2dbN+Z0JIR0LI84SQGfb33rMFvPMv7X49jRDyDCGktrm9MyHkUULICkLINOae9jsSQnYhhEy1n/2dEEKUiaCUVsQ/AGkAcwAMAlANYDKAYaWmK4H36gVgZ/t3OwCzAAwDcAeA0fb90QD+Yv8eZr97DYCt7DZJl/o9Ir77NbAO7X7dvm7W7wzgCQAX2r+rAXRszu8MoA+AeQBa2df/AXBuc3tnAPsB2BnANOae9jsC+ArAngAIgLcA/ESVhkqSyHcDMJtSOpdS2gDgWQDHlZim2KCULqWUfmP/3gDgO1gD4DhYAx/235/av48D8CyltJ5SOg/AbFhtU1EghPQFcBSAh5nbzfadCSHtYQ34RwCAUtpAKV2LZvzONjIAWhFCMgBaA1iCZvbOlNKPAazx3dZ6R0JILwDtKaVfUIur/5vJE4pKYuR9ACxirhfb95oNCCEDAYwEMA5AD0rpUsBi9gC628maSzvcC+A3sA7udtCc33kQgJUAHrPVSQ8TQtqgGb8zpfQHAH8FsBDAUgDrKKXvoBm/MwPdd+xj//bfV0IlMXKevqjZ2E4SQtoCeAHA1ZTS9bKknHsV1Q6EkKMBrKCUfq2ahXOvot4ZlmS6M4AHKKUjAWyCteQWoeLf2dYLHwdLhdAbQBtCyJmyLJx7FfXOChC9Y6x3ryRGvhhAP+a6L6xlWsWDEFIFi4k/RSl90b693F5uwf67wr7fHNphbwDHEkLmw1KRHUQIGYPm/c6LASymlI6zr5+Hxdib8zsfAmAepXQlpbQRwIsA9kLzfmcHuu+42P7tv6+ESmLk4wEMIYRsRQipBnAqgFdLTFNs2DvTjwD4jlJ6N/PoVQDn2L/PAfAKc/9UQkgNIWQrAENgbZJUDCil11FK+1JKB8L6jh9QSs9E837nZQAWEUKG2rcOBvAtmvE7w1Kp7EEIaW3384Nh7QE153d2oPWOtvplAyFkD7utzmbyhKPUO76au8NHwrLqmAPg+lLTk9A77QNrCTUFwCT735EAugB4H8D39t/OTJ7r7TaYCY2d7XL8B+AAFKxWmvU7A9gJwAT7W78MoFMLeOdbAMwAMA3Ak7CsNZrVOwN4BtYeQCMsyfqCKO8IYJTdTnMA3A/b817ln3HRNzAwMKhwVJJqxcDAwMCAA8PIDQwMDCochpEbGBgYVDgMIzcwMDCocBhGbmBgYFDhMIzcwMDAoMJhGLmBgYFBheP/AaaU5RzxKbw2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Sdata=time_data(1000)\n",
    "noSdata=time_data(1000,S=0)\n",
    "plt.plot(Sdata)\n",
    "plt.plot(noSdata)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "d27a6881",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(103.3687315787275, -45.98886932180953)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sdata.sum(),noSdata.sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75047f7e",
   "metadata": {},
   "source": [
    "We get a bigger answer for the case with $S=0.1$ even though they look identical on the time series.  Let's check this isn't an accident"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "070050f1",
   "metadata": {},
   "outputs": [],
   "source": [
    "S_list=[]\n",
    "noS_list=[]\n",
    "for i in range(1000):\n",
    "    S_list.append(time_data(1000).sum())\n",
    "    noS_list.append(time_data(1000,S=0).sum())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "99c5afdf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_h,ax_h=plt.subplots(figsize=(12,6))\n",
    "ax_h.hist(S_list,bins=20)\n",
    "ax_h.hist(noS_list,bins=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5c932a8",
   "metadata": {},
   "source": [
    "We see that there is a clear difference between the two distributions.  We can repeat it with more points to see that this difference becomes more pronounced.  If we make it 10000 data points, we should expect the $S=0.1$ distribution to move to around 1000, while the $S=0$ will be rought between -100 and 100 at 1$\\sigma$.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "617041a9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11d4836d0>]"
      ]
     },
     "execution_count": 12,
     "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": [
    "Sdata=time_data(10000)\n",
    "noSdata=time_data(10000,S=0)\n",
    "plt.plot(Sdata)\n",
    "plt.plot(noSdata)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "afd4b099",
   "metadata": {},
   "outputs": [],
   "source": [
    "S_Llist=[]\n",
    "noS_Llist=[]\n",
    "for i in range(1000):\n",
    "    S_Llist.append(time_data(10000).sum())\n",
    "    noS_Llist.append(time_data(10000,S=0).sum())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "afb3e990",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_h,ax_h=plt.subplots(figsize=(12,6))\n",
    "ax_h.hist(S_Llist,bins=20)\n",
    "ax_h.hist(noS_Llist,bins=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd6fd603",
   "metadata": {},
   "source": [
    "Now suppose we want to measure the value of $S$ in our data.  Suppose the true value of $S$ is $S_0$.  Let's just try to guess a value of $S$ and calculate \n",
    "$$ \\sum ({\\rm data}-S)^2 = \\sum (S_0+{\\rm noise} - S)^2$$\n",
    "Now notice that the sum over $N$ is just $N$ times the average, so as we take $N$ to be large, this is just $N$ times the statistical average.  This is useful because we know the statistical properties of the noise:\n",
    "$$\\sum (S_0+{\\rm noise} - S)^2 = N(S_0-S)^2 + N \\sigma^2$$\n",
    "Notice that this is the sum of two positive terms and the answer is minimized when $S=S_0$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12013389",
   "metadata": {},
   "source": [
    "So let's see how accurate a measure of $S$ we can make by just minimizing this quantity.  I happen to know that the noise has $\\sigma=1$ so this quantity is equivalent to $\\chi^2$ (otherwise it would be $\\sigma^2 \\times \\chi^2$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "4e398e49",
   "metadata": {},
   "outputs": [],
   "source": [
    "data_ref=time_data(10000)\n",
    "def chisqr(S):\n",
    "    return ((data_ref-S)**2).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "e76092db",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11cfbbe80>]"
      ]
     },
     "execution_count": 16,
     "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": [
    "Srange=np.linspace(-1,1,100)\n",
    "chi_out=np.zeros(len(Srange))\n",
    "for i in range(len(Srange)):\n",
    "    chi_out[i]=chisqr(Srange[i])\n",
    "plt.plot(Srange,chi_out)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce5ede35",
   "metadata": {},
   "source": [
    "Given what we know about equation solving, we could find this minimum ourselves.  However, finding minimima or maxima rapidly becomes a hard problem, so we are going to make our life easier by using a built in minimization tool:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "a601e0a0",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import minimize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "e02cd5ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      fun: 9748.101315983986\n",
       " hess_inv: array([[5.00000022e-05]])\n",
       "      jac: array([0.00024414])\n",
       "  message: 'Desired error not necessarily achieved due to precision loss.'\n",
       "     nfev: 48\n",
       "      nit: 1\n",
       "     njev: 18\n",
       "   status: 2\n",
       "  success: False\n",
       "        x: array([0.0916426])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimize(chisqr,0.4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "766a1954",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      fun: 9880.134110111303\n",
       " hess_inv: array([[5.00000002e-05]])\n",
       "      jac: array([0.00024414])\n",
       "  message: 'Desired error not necessarily achieved due to precision loss.'\n",
       "     nfev: 88\n",
       "      nit: 1\n",
       "     njev: 38\n",
       "   status: 2\n",
       "  success: False\n",
       "        x: array([0.02035976])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_ref2=time_data(10000,S=0)\n",
    "minimize(lambda x:((data_ref2-x)**2).sum(),0.4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3cf093cd",
   "metadata": {},
   "source": [
    "Minimize works more or less like our Newton solver (at this level): it takes a function and an initial point and finds a nearby point where the function is minimized (presumably where the derivative vanishes).  \n",
    "\n",
    "Using this insight, we see that minimizing this function does a pretty good job of finding the correct value of $S$.  When we put in $S_0=0.1$ we measured $S=0.0916426$ and when we put $S_0=0$ we measured $S=0.02035976$.  In a future class, we will talk about how to give this measurements error-bars."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab47359f",
   "metadata": {},
   "source": [
    "## Linear Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2bb2f0e4",
   "metadata": {},
   "source": [
    "The natural next step is to fit a line to the data that we uploaded from a file.  This is a thing one typically does as a first guess given data that we don't totally understand: a line has enough freedom to capture some overall behavior, but is sufficiently simple that we aren't going to overfit (i.e. the function is just finding patterns in the noise). "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2cd81a0e",
   "metadata": {},
   "source": [
    "There are lots of build in functions that will give you a best fit line.  For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "2c0204f7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.044053980142284774 -0.17318253969528752 0.10331235714633374 0.30636705556359395 0.04284396708900212\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "LinregressResult(slope=0.044053980142284774, intercept=-0.17318253969528752, rvalue=0.10331235714633374, pvalue=0.30636705556359395, stderr=0.04284396708900212, intercept_stderr=0.26108947495494694)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "slope1, intercept1, r_value1, p_value1, std_err1 = stats.linregress(t,d1)\n",
    "print(slope1, intercept1, r_value1, p_value1, std_err1)\n",
    "stats.linregress(t,d1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "36a1e601",
   "metadata": {},
   "outputs": [],
   "source": [
    "slope2, intercept2, r_value2, p_value2, std_err2 = stats.linregress(t,d2)\n",
    "slope3, intercept3, r_value3, p_value3, std_err3 = stats.linregress(t,d3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aba6515a",
   "metadata": {},
   "source": [
    "What most such algorithms are doing is the same as what we did above, but now where we assume that $${\\rm data} = a t + b + {\\rm noise}$$ where $t$ is time, $a$ and $b$ are constants.  Following what we did before, you guess that if you calculate  $$\\sum ({\\rm data}-a t - b)^2$$\n",
    "you can minimize with respect to $a$ and $b$ to find your \"best fit\" line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "5eb31447",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      fun: 9880.134110111303\n",
       " hess_inv: array([[5.00000002e-05]])\n",
       "      jac: array([0.00024414])\n",
       "  message: 'Desired error not necessarily achieved due to precision loss.'\n",
       "     nfev: 88\n",
       "      nit: 1\n",
       "     njev: 38\n",
       "   status: 2\n",
       "  success: False\n",
       "        x: array([0.02035976])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimize(lambda x:((data_ref2-x)**2).sum(),0.4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "067a5dde",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.04405392, -0.17318235])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimize(lambda X:((d1-X[0]*t-X[1]*np.ones(len(t)))**2).sum(),np.array([0.1,0.1]),method='Nelder-Mead', tol=1e-6)['x']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "b1db639e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.044053980142284774, -0.17318253969528752)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "slope1, intercept1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "817ed0ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.06501928, 0.11842078])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimize(lambda X:((d2-X[0]*t-X[1]*np.ones(len(t)))**2).sum(),np.array([0.1,0.1]),method='Nelder-Mead', tol=1e-6)['x']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "ee0a7e19",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0650193686177917, 0.11842051937262404)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "slope2, intercept2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "8d9454ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.73878142, -1.35719998])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimize(lambda X:((d3-X[0]*t-X[1]*np.ones(len(t)))**2).sum(),np.array([0.1,0.1]),method='Nelder-Mead', tol=1e-6)['x']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "f67f3d9b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.7387814055506847, -1.3571997755399319)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "slope3, intercept3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aef9e8a1",
   "metadata": {},
   "source": [
    "We see that our minimization procedure is reproducing the linear regression results, pretty close to exactly.  However, this does not guarantee that we have a good fit to the data. To see how well we are doing we are going to plot the data vs the best fit line, and also a histogram of our error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "111dbf5d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 9 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig_h,ax_h=plt.subplots(figsize=(12,12),ncols=3,nrows=3)\n",
    "ax_h[0,0].scatter(t,d1)\n",
    "ax_h[0,0].scatter(t,slope1*t+intercept1)\n",
    "ax_h[0,1].scatter(t,d2)\n",
    "ax_h[0,1].scatter(t,slope2*t+intercept2)\n",
    "ax_h[0,2].scatter(t,d3)\n",
    "ax_h[0,2].scatter(t,slope3*t+intercept3)\n",
    "ax_h[1,0].hist(d1-slope1*t-intercept1,bins=20)\n",
    "ax_h[1,1].hist(d2-slope2*t-intercept2,bins=20)\n",
    "ax_h[1,2].hist(d3-slope3*t-intercept3,bins=20)\n",
    "ax_h[2,0].scatter(t,d1-slope1*t-intercept1)\n",
    "ax_h[2,1].scatter(t,d2-slope2*t-intercept2)\n",
    "ax_h[2,2].scatter(t,d3-slope3*t-intercept3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "672cd0fb",
   "metadata": {},
   "source": [
    "These plots look pretty good.  What about the $\\chi^2$ after removing the best fit line: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "5f02789b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(123.878341340528, 114.2267169116914, 124.2774840597678)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((d1-slope1*t-intercept1)**2).sum(),((d2-slope2*t-intercept2)**2).sum(),((d3-slope3*t-intercept3)**2).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "b6b2036d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(125.69249389635905, 139.79811466358296, 1232.430424286052)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(d1**2).sum(),(d2**2).sum(),(d3**2).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0252da0b",
   "metadata": {},
   "source": [
    "At first sight, everything looks fine.  Interestingly, not all the data was made with a straight line.  The first two, d1 and d2 were, but d3 was actually made with a parabola:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "3d214c28",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x11de5ef10>"
      ]
     },
     "execution_count": 32,
     "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": [
    "plt.plot(t,slope3*t+intercept3)\n",
    "plt.plot(t,0.07*t**2)\n",
    "plt.scatter(t,d3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "6b400c83",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "107.98045329875646"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((d3-0.07*t**2)**2).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66da80db",
   "metadata": {},
   "source": [
    "## More general $\\chi^2$ minimization"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65dfd15a",
   "metadata": {},
   "source": [
    "We can now see that pattern for finding a set of best fit parameters for any model given the data. Suppose with think our signal is described by a model $S(t; \\vec X)$ where $\\vec X$ is a list of parameters decribing the model.  Now we caculate $$ \\sum (data-S(t;\\vec X))^2$$ and we find the minimum in terms of the components of $X$.  \n",
    "\n",
    "For example, we can add allow our data to be modelled by a parabola instead of just a line as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "1b19b8a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.06909539, -0.02126793,  0.25711986])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimize(lambda X:((d3-X[0]*t**2-X[1]*t-X[2]*np.ones(len(t)))**2).sum(),np.array([0.1,0.1,0.1]),method='Nelder-Mead', tol=1e-6)['x']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7deca122",
   "metadata": {},
   "source": [
    "For simple models like this one, this is a relatively straightforward minimization problem: after adding up all the data, we have quadratic polynomial in a small number of variables.  However, for more complicated models (and data) finding the minimum can be extremely difficult."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "7f6ba2e7",
   "metadata": {},
   "outputs": [],
   "source": [
    "def model(X,n=8):\n",
    "    model=np.zeros(len(t))\n",
    "    for i in range(n):\n",
    "        model+=X[i]*t**i\n",
    "    return model\n",
    "def g(X,n=8):\n",
    "    return ((d3-model(X,n))**2).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "a506adbf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9831016539991.152"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(0.1*np.ones(8))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "23fce34c",
   "metadata": {},
   "outputs": [],
   "source": [
    "Xsol=minimize(g,0.1*np.ones(8),method='Nelder-Mead', tol=1e-6)['x']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "d076ef6a",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x1211c9bb0>"
      ]
     },
     "execution_count": 38,
     "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": [
    "plt.plot(t,model(Xsol))\n",
    "plt.scatter(t,d3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "0d4844dc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "223.83679406449997"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(Xsol)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "854ebe56",
   "metadata": {},
   "source": [
    "We will see more advanced strategies for dealing with these kinds of higher dimensional problems.  We are really just scratching the surface of an enormous problem - many AI/machine learning techniques boil down to minimizing some nonlinear function of a large number of parameters.  In some cases, it is more or less obvious what it is you want to minimize and all the new ideas correspond to better or faster ways to find the minima."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5e020bc",
   "metadata": {},
   "source": [
    "### Saving Data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81bbec24",
   "metadata": {},
   "source": [
    "The final thing you might like to do is to write your own data to a file.  Sometimes you do a long calculation and you don't want to have to recaclulate it every time you want to make a figure, for example.\n",
    "\n",
    "The most basic form of writing to a file is the opposite of what we did above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "ecaff71a",
   "metadata": {},
   "outputs": [],
   "source": [
    "f=open(\"save_data_test.txt\",\"w\") # 'w' means writes (over the existing file), \n",
    "f.write('# here is some preamble\\n')\n",
    "f.write('# here is some more preamble\\n')\n",
    "t=np.linspace(1,10,100)\n",
    "for i in range(100):\n",
    "    f.write(str(t[i])+',')\n",
    "    for j in range(3):\n",
    "        f.write(str(np.random.randn())+',')\n",
    "    f.write('\\n')\n",
    "f.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36650053",
   "metadata": {},
   "source": [
    "Often you may want to save something that is already in the form of an array or a python friendly object.  There are LOTS of ways to do this (pandas is a popular library with lots of options, pickle is also popular).  Hoever, the key idea is always the same: some library has a way of saving and reading a type of object that is pretty fast.  E.g. numpy has a save option:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "4b019163",
   "metadata": {},
   "outputs": [],
   "source": [
    "outdata=np.random.randn(100)\n",
    "with open('save_random.npy', 'wb') as f: # wb means write binary\n",
    "    np.save(f,outdata)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "ffc5ab0e",
   "metadata": {},
   "outputs": [],
   "source": [
    "with open('save_random.npy', 'rb') as f: # rb means read binary\n",
    "    a = np.load(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "25371d2c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-7.23372587e-01,  2.59738992e+00,  1.05298953e+00,  1.53972070e-01,\n",
       "       -2.34497900e-03,  7.07501742e-01, -6.92818374e-01,  2.40949381e-01,\n",
       "       -1.87995340e+00,  8.63528676e-01, -5.35938921e-01, -1.05898457e+00,\n",
       "        1.32122008e+00,  2.80440635e-01,  1.86641189e+00,  1.00204161e+00,\n",
       "        1.29875524e+00, -2.15975895e-01, -1.98798934e+00, -3.78778308e-01,\n",
       "        6.08542178e-01,  1.93885775e-01,  1.08457808e+00,  2.05273181e+00,\n",
       "        1.80960530e-01,  1.69398744e+00,  7.51389444e-01,  1.92691378e+00,\n",
       "       -3.09312143e-01, -8.62031008e-01, -9.12359852e-02,  1.79892384e+00,\n",
       "       -1.02401419e-01, -9.93243788e-01,  1.09336854e+00, -5.44485716e-01,\n",
       "       -1.90137420e+00,  3.86576073e-01, -1.63614745e+00, -7.28121415e-01,\n",
       "       -3.03125045e-01, -7.50005819e-01, -1.08057423e+00,  1.05903041e-01,\n",
       "       -3.91944476e-01,  1.38349343e-02, -2.67588561e-01,  7.29266710e-01,\n",
       "       -1.41866676e+00, -2.11140981e-01,  1.43183556e+00,  5.46536576e-01,\n",
       "       -1.61083880e-01,  1.59726183e-02,  2.12993025e-01, -1.34513135e+00,\n",
       "        4.02868998e-01,  8.22481140e-01,  1.32723787e-01, -6.98032214e-01,\n",
       "       -5.43537635e-02, -6.51973007e-02, -5.85898854e-01, -2.51363227e-02,\n",
       "       -6.42139524e-01,  1.53036613e+00,  2.08035946e+00,  4.37940705e-01,\n",
       "        5.20049471e-01, -1.20962804e+00, -5.97779993e-01,  4.49505259e-02,\n",
       "        3.27302819e-01,  4.52625232e-01,  1.03646701e-01,  6.00298584e-01,\n",
       "        6.49145813e-01,  7.34009379e-01,  4.60632400e-01, -7.54182519e-01,\n",
       "        4.38878830e-01,  1.81687083e-01, -5.43440213e-01,  1.05668977e+00,\n",
       "       -8.30545303e-01,  2.02550729e-01,  1.25763505e+00,  4.41801376e-01,\n",
       "        8.99973157e-01,  4.03005054e-01,  6.39935838e-01,  1.20840509e+00,\n",
       "        5.75388738e-01,  5.08574422e-02,  5.09325511e-01,  4.75262729e-01,\n",
       "        9.15958367e-01,  7.75227021e-01, -2.74923399e-02, -1.26020339e+00])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "665c2342",
   "metadata": {},
   "source": [
    "## Summary"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34f0cea5",
   "metadata": {},
   "source": [
    "We have just started to scratch the surface of how do deal with data.  However, we were able to see the most basic problems in data analysis in some relatively simple form: (1) data cleaning - getting your data stored in a useful way, (2) data visulization - understanding what your data represents, (3) data analysis - separating the signal from the noise, (4) store your work for easy use in the future."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8fbfd572",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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