{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "095c18c2-5055-4ecc-a22b-4d53335c5f24",
   "metadata": {},
   "source": [
    "# Homework 3\n",
    "\n",
    "Due 11:59PM Monday, March 11 submitted via Blackboard\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a59cd3e2-83ab-4eb7-9d1a-ae686fc09960",
   "metadata": {},
   "source": [
    "In this homework you will explore a couple of different clustering algorithms and understand the impact of various hyperparameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "4584b47c-6700-438e-898b-aa2d239e615c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Imports\n",
    "import numpy as np\n",
    "from sklearn.cluster import KMeans\n",
    "from collections import defaultdict\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import random\n",
    "from sklearn.cluster import AgglomerativeClustering"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d248872a-7b5c-459c-9f60-0c1b486af768",
   "metadata": {},
   "source": [
    "We'll be using the MNIST digits data because it is easy to understand and visualize.  The code below loads the raw data where each instance is a 28x28 grayscale image represented as a vector of 784 integers between 0 and 255.\n",
    "\n",
    "It also loads the digits (0-9) to which each image corresponds.  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6590e7e3-ebee-473e-80bf-154136bc84af",
   "metadata": {},
   "source": [
    "# Task 0: Get the data\n",
    "\n",
    "https://redirect.cs.umbc.edu/courses/undergraduate/478/spring2024/mnist_labels.txt\n",
    "https://redirect.cs.umbc.edu/courses/undergraduate/478/spring2024/mnist_data.txt\n",
    "\n",
    "Put the two files above in the same director as this notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "24a2670f-5f55-45ce-a2e8-506958795ec6",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = [np.array(list(map(int, x.split()))) for x in open('mnist_data.txt').readlines()]\n",
    "y = [int(y.strip()) for y in open('mnist_labels.txt').readlines()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "ee11a874-6c3c-4b43-bf0c-b31c5a4e621c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000\n",
      "10000\n"
     ]
    }
   ],
   "source": [
    "# Check to ensure that X and y are the same length\n",
    "print(len(X))\n",
    "print(len(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "5d892616-4a0f-4e75-b3e6-9e750171fb22",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_digit(digit):\n",
    "    \"Visualize a digit or a cluster centroid\"\n",
    "    \n",
    "    plt.imshow(digit.reshape((28,28)), cmap='gray')\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "d691ffc1-ef2d-4e74-9b8c-56febe5b577f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The digit is 3\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# You can run this cell multiple times to see different digit examples\n",
    "idx = random.randrange(len(X))\n",
    "print(\"The digit is %d\" % y[idx])\n",
    "plot_digit(X[idx])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d4fbd5e1-c220-4db6-b0c0-f7909a0330d0",
   "metadata": {},
   "source": [
    "# Task 1: Get familiar with running k-means\n",
    "\n",
    "Read through the code below because you'll be using the ideas here for the next task"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "741cfd03-18a5-40bf-b5bc-dbf607b62b23",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a KMeans object with 5 clusters and random point selection\n",
    "# for cluster initialization\n",
    "\n",
    "clst = KMeans(n_clusters = 5, init = 'random')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "97bcf7ec-90dc-486b-9d64-9876fea9ab74",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/oates/tmp/env/anaconda3/lib/python3.8/site-packages/sklearn/cluster/_kmeans.py:1416: FutureWarning: The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning\n",
      "  super()._check_params_vs_input(X, default_n_init=10)\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<style>#sk-container-id-2 {color: black;}#sk-container-id-2 pre{padding: 0;}#sk-container-id-2 div.sk-toggleable {background-color: white;}#sk-container-id-2 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-2 label.sk-toggleable__label-arrow:before {content: \"▸\";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-2 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-2 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-2 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-2 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-2 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-2 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: \"▾\";}#sk-container-id-2 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-2 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-2 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-2 div.sk-parallel-item::after {content: \"\";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-2 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-serial::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-2 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-2 div.sk-item {position: relative;z-index: 1;}#sk-container-id-2 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-2 div.sk-item::before, #sk-container-id-2 div.sk-parallel-item::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-2 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-2 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-2 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-2 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-2 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-2 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-2 div.sk-label-container {text-align: center;}#sk-container-id-2 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-2 div.sk-text-repr-fallback {display: none;}</style><div id=\"sk-container-id-2\" class=\"sk-top-container\"><div class=\"sk-text-repr-fallback\"><pre>KMeans(init=&#x27;random&#x27;, n_clusters=5)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class=\"sk-container\" hidden><div class=\"sk-item\"><div class=\"sk-estimator sk-toggleable\"><input class=\"sk-toggleable__control sk-hidden--visually\" id=\"sk-estimator-id-2\" type=\"checkbox\" checked><label for=\"sk-estimator-id-2\" class=\"sk-toggleable__label sk-toggleable__label-arrow\">KMeans</label><div class=\"sk-toggleable__content\"><pre>KMeans(init=&#x27;random&#x27;, n_clusters=5)</pre></div></div></div></div></div>"
      ],
      "text/plain": [
       "KMeans(init='random', n_clusters=5)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Cluster the data\n",
    "clst.fit(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "26ec6d19-e08f-4459-a82e-b8d9d2d92dbe",
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_cm(clst, y):\n",
    "    \"\"\"\n",
    "    Given a cluster object that has been fit and the true labels for the data\n",
    "    return a confusion matrix where cell (i, j) is the number of times true \n",
    "    class label j occurs in cluster i.\n",
    "    \"\"\"\n",
    "    \n",
    "    cm = np.zeros((np.unique(clst.labels_).shape[0], np.unique(y).shape[0]))\n",
    "\n",
    "    for ypred, ytrue in zip(clst.labels_, y):\n",
    "        cm[ypred][ytrue] += 1\n",
    "        \n",
    "    return cm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "2172419a-9050-4e1e-b217-8d9fdfb3dcec",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_cm(cm):\n",
    "    \"Plot a confusion matrix\"\n",
    "    \n",
    "    s = sns.heatmap(cm, cmap=\"GnBu\", annot=True, fmt='g')\n",
    "    s.set(xlabel='Digit', ylabel='Cluster')\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "5b93b190-078a-48ef-b4ef-e247e1e38c51",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get the confusion matrix for the 10 cluster case\n",
    "cm = get_cm(clst, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6413133b-14fc-4d0e-a655-4185b7e6e12a",
   "metadata": {},
   "source": [
    "It's hard to tell what you'll see below due to randomness, but typically there is one cluster with almost all of the 0's.  Find the row for which the 0 column has a very high count.  There is also typically a column that has most of the 1's.  The 4's, 7's, and 8's tend to clump into one cluster.  Look for the row that has high counts for those columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "19e9e646-eff7-49c0-8051-ceae37437de3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Axes: xlabel='Digit', ylabel='Cluster'>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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CAG4u0ESXhC6EEEgLXQghXIYL5HNJ6EIIAdJCF0IIl5H9E0BzD0noQgiBtNCFEMJlGGSUixBCuAZpoQshhItwgXwuCV0IIUBa6EII4TJcIJ/n3jsWlf9krdM2bE+/ys4K5dQ7FhnV/e9O/rBx5ufV7MT3yuDmpPcqd/6555i3MSDH6bjI8OU2751zIxvnyvQvLXQhhECu5SKEEC7DFbpcJKELIQRyUFQIIVyGJHQhhHARLtCFLgldCCEA3AwPf0aXhC6EEEiXixBCuAwXyOeS0IUQAqSFLoQQLkMSuhBCuAgXyOeS0IUQAmSUixBCuAw3F2iiS0IXQgikD10IIVyGC+Rz3B70BgghRG6g3JTN033LUqqvUuqQUuqgUmqOUspLKRWhlNqulDqhlJqrlPKwrutpfX7CurxYduvw0LXQO1UuTNtKBdAajsUmMmTZYYY3LMOT4QHcSDUB8N6ywxyJvmF5XK8UtYvnI9mUzntL/+KwdX5ObNm0lY/Gfka6OZ02bVvxetfOOS7zlsjLUQwbPIq4K3EopWjzQis6dGyXsXzWd7P5/JP/sXrzMgIDA+wWF8BsNtP+hc6EhoUwcdIEu5admSP33/Cho9m4YQtBQYH8vGgOACtXrOGbr6Zy+tQZfvjpWypULGeXWJGXoxj23ui/36u2LenQsR2D3nmfs2fOAZCQkICfnx9zfp5pl5i3NHm2NXny+ODm5obRaGD2fPuVf6d9+NWX37B+3SaUUgTlC2TUB8MIDQ2xe5zxn37JxvWbcXd3J7xwIUaOeZ+8ef1yXCdb2KvLRSlVCHgbKK+1TlZKzQNeApoCE7TWPymlvgFeByZZ/7+qtS6plHoJ+Ahod5fi7+mhaqGH+nrwyhPhvDBrF62+24FBQdOyoQB8uuEEz83cyXMzd2Yk89oR+Sga6EPjadsYvuIIwxuUyfE2mM1mPhzzMV9P/oJffpvL8qUrOHniVI7LvcVgNNB3wNss+G0O382Zyvw5P3PqxGnAkkC2bdlB/gL57RYvsx9n/UTxEsUcUvYtjt5/LVs35+vJn/9jXsmSxRn/xUc8UeVxu8UB63vVvxcLFs/mu9lTmP/TQk6dPM24z0Yz5+eZzPl5JvUa1KHus8/YNe4tU7/7mnm//GDXZA533oedX3uF+b/8yLyFP1D7maeZMmm6Q+JUq16VBb/OZv4vP1K0aBFmTLVv3e7FzU3ZPNnACHgrpYyAD3AZqAcssC6fCbS2Pm5lfY51eX2VzW+XhyqhAxjcFF5GNwxK4eVuIDox9a7r1isVzKJDkQDsv3wdPy8jwXk8chT/4IFDFC4STnjhQrh7uNO4SUPWr92YozIzCwkJplx5yxdPnjx5iChejOjoGADGf/QFvd/p6ZC+vqjIKDZt2EKb51vZv/BMHL3/Kld5nLz+ef8xr3iJCIpFFLVbjFtuf6+KEh0Vk7Fca83q5Wtp3LSB3WM70p32oa+vb8bj5ORku7Rm7xSnRs1qGI2WjoNKj1YkKio6x3FslZUuF6VUN6XUrkxTt1vlaK0vAp8C57Ak8mvAbiBea22yrnYBKGR9XAg4b32tybp+vuzU4aHqcom+kcq3O8+x5s0a3DSl88eZOP44E0ezcmH0rlWcHjUi2HY2jvEbT5Jm1oT6ehKZcDPj9VEJKYT5ehJ7jy+B+25DVAz584dlPA/NH8qB/YdyVK+7uXTxMkcOH6NipQqsX7uRkLAQSpct5ZBYH4+bQN93e5GYmOSQ8m9x5v5zJst7dZyKlSpkzNu7ex9B+YIoUrSw3eMpBT3eeBul4PkX29D2xTZ2j/Fv//tiEksWL8XX15ep337t8Hi/LvyNRk2edXicW7LyHaW1ngJMuXM5KhBLqzsCiAfmA41zvIE2cHoLXSn1anZfm9fTSL2SITSYspU6k7bg7W6gRfkwJmw8SbPp23lx1k78vdx5o6r9W2POlpSYRP8+g3l3UB8MBgMzpsyk+1tdHRJrw/pNBAUFUr6CffqW/2uSkpLo3/c93h3YG1/fPBnzly9dTaOmjklI3/4whZ9+/p6vJn/OvDkL2L1rr0PiZNardw9WrPmNps0b8dPs+Q6NNXXytxiMBpo2d0oeBCx96LZO9/EscFprHaO1TgMWAjWBAGsXDEA4cNH6+CJQ2LoNRsAfuJKdOjyILpeRd1uQ+WfM1W1LbltevWggF68lczU5DVO6ZtXxGB4r6J/R4k4za345eJlHClh+xkXfSCG/n1fG68P8PIm6kZKjjQ8NCyEyMirjeXRkNGE5PDj0b2lpJvr3eY8mzRpRr0EdLpy/wKWLl2n/XEeaN2hDdFQML7ftQmxMtt7z2+zbs5/16zbR5NlWDHxnCDu372LwgGF2KfvfnLH/nOnv96oh9RrUyZhvMplYt3o9DRs7JqGHhVmOHQXlC6Ju/TocdOKvnKbNGrNm1TqHlb/olyVs2rCZDz8a5dSx4W5K2TzdxzmgmlLKx9oXXh/4C1gHtLWu0xlYZH282Poc6/K1WmudrTpk50X3o5Taf5fpABB2t9dpradoratorasEVmt+2/LLCSk8WjAvXkbLZlcrEsipK0n/6BevXzKE47GJAKw9EUurCpYDiJUK5CUhxZyj7haAChXLc+7seS5cuEhaahrLl63kmbq1clRmZlprRg/7gIjiRXmlS3sASpUuyepNS1my6heWrPqF0LAQflzwHcEh2epmu03vfj1ZtW4Jy1Yv4qPPPuDJp6ow9uNRdin73xy9/5zJ8l59SETxYrzSuf0/lu3YtotixYsSlj/U7nGTk5JJTEzMeLz1j+2ULFXC7nEyO3v2XMbj9es2EuGAYxJgGQE1c8YsPp/4Kd7eXvd/gR25GZTN071orbdjObi5BziAJc9OAQYC/ZRSJ7D0kd86sjwdyGed3w8YlN06OKoPPQxoBFz913wF/JHdQvdfvs7KYzEs6PQk5nTN4egbzNt/kcnPP0aQjzsKOBJzg5ErjwKw8dQVahfPx/Ku1bmZZmbIssPZDZ3BaDQyeEh/enR9m/T0dFq3aWHXP6Z9e/bz++LllCxdgvbPdQKgZ5/uPF27ht1iPEiO3n+D3h3Krp17iI+Pp2G95vTo2Q1//7yM+/BTrsbF0+v/+lKmTGkmTf0yx7H27d3P778tp2SpErR/3tLA6tn7TZ6uXYMVy1bTqIljDoZeuRJHv7cHAGAymWnSrBE1a1W3W/l32oebN27hzJlzuLm5UaBAfoYMH+iQODOmziQ1LZXub/QCLAdGhw7Pdn7LEnv+GtBaDweG/2v2KaDqHda9Cbxgj7gqmy37exeq1HTgW6315jssm6217nC/Msp/stb+G3YXe/pVdlYoTNrstFhGZXBaLGdxxOf1bsxOfK8Mbk56r5y3+5zK2xiQ42z8zPc7bN47GzpVzZXnlTqkha61fv0ey+6bzIUQwtnkWi5CCOEibDmlP7eThC6EELjGxbkkoQshBODm9tCdOH8bSehCCAG4QI+LJHQhhADpQxdCCJcho1yEEMJFuEA+l4QuhBAgLXQhhHAZ97tGy8NAEroQQiAtdCGEcBmS0IUQwkW4wKhFSehCCAGg1MN/KUpJ6EIIARjcJKELIYRLcIEeF0noQggB4CZdLo6z6W3n3YFeO/E2LjHJsU6L1WjiufuvZAf7BzzplDgAU4/ucVqsLqUecVoso5P+FAfvPOKUOAAfPFnaabHswQUGueTehC6EEM4kCV0IIVyEdLkIIYSLMEhCF0II1yBdLkII4SLkxCIhhHARD/8dRW2og1LKoJTq64yNEUKIB0UpbfOUW903oWutzUB7J2yLEEI8MAY3bfOUW9na5bJFKTURmAsk3pqptXbeWR5CCOFA/6WrLT5m/X9UpnkaqGfXrRFCiAdEOfGMcUexKaFrres6ekOEEOJBcoVhizYd2FVKhSmlpiulllmfl1dKve7YTRNCCOdxU9rm6X6UUgFKqQVKqSNKqcNKqepKqSCl1Cql1HHr/4HWdZVS6kul1Aml1H6l1BPZroON630HrAAKWp8fA/pkN6gQQuQ2Stk+2eALYLnWuizwKHAYGASs0VqXAtZYnwM0AUpZp27ApOzWwdaEHqy1ngekA2itTYA5u0GFECK3MSht83QvSil/oDYwHUBrnaq1jgdaATOtq80EWlsftwK+1xbbgAClVIHs1MHWhJ6olMqH5UAoSqlqwLXsBBRCiNwoKy10pVQ3pdSuTFO3TEVFADHAt0qpvUqpaUqpPECY1vqydZ1IIMz6uBBwPtPrL1jnZZmto1z6AYuBEkqpLUAI8EJ2AgohRG6Ulastaq2nAFPustgIPAH00lpvV0p9wd/dK7der5UDzlCyNaEfAp4BymC5U9NRXONMWSGEAOw6yuUCcEFrvd36fAGWhB6llCqgtb5s7VKJti6/CBTO9Ppw67wsszWhb9VaP4ElsQOglNqD5VvogUm4nsDYEZ9w6sRplFK8N2ognp4efDJ6PKmpqRgMBt4d0pfyj9j37kdNnm1Nnjw+uLm5YTQamD1/5v1fdA/jR37B9s07CQj0Z/K8rwCYNXk2y39dgX+gPwBd/q8TVZ+ukvGa6Mhour3Qk1e6tadtx+dsjtXlySK88GhBNHAs5gaDlvxFqjmdvrVL0LhcKOnpMHvvBWbtOk9eLyNjm5ancKA3qaZ0Bv/+F8djE+8b4362bNrKR2M/I92cTpu2rXi9a+ccl5luTmf+gLHkCQqg+ZCeXDhwhC3fLSTdZCKkRBHq9eyIm8EAwMWDx9g0Yz7pZjPefr60GdMv23HNZjOdXnqD0NAQJnz1MRcvXGLIgOFci79O2fJlGDX2fdzd3XNcv3+z9z5c904/DN5eKOWGMrjx9IhRXD93joMzv8WUkoJPvmAe7d4Dd29vAK6fP8fB777FlHwT3BQ1h43A4OGRpZgjho5h44YtBAUFsmDRbAAGvjOEM6ctd9pKSEjAz8+PuQtn5ahutnKz0zh0rXWkUuq8UqqM1vooUB/4yzp1BsZZ/19kfcli4C2l1E/AU8C1TF0zWXLPhK6Uyo+lL8dbKfU4f99HNS/gk52A9vT5RxOpVrMqH44fRVpaGjeTbzK0/0he696F6rWe4o9N2/hqwjd8NeMLu8ee+t3XBAYG2KWsBi3q06JdMz4dNuEf89t0aHXXZD1l/HSq1KicpThhvp50rFKYplO3kmJK5/PWj9CsfBgKKJDXi8aTt6KBIB9LAupevRiHoxPouXA/xYN8GN6oLJ3n5OzkYLPZzIdjPmbytImEhYXSoV1n6tStRYmSxXNU7v7f1xIYnp/UpJvo9HTWfPk9rUb2JqBgGNvn/MaRddso/2xNUhKT2DBlDi3e74VfSBBJ8ddzFPenH+YTEVGUxMQkACZOmESHju1o2ORZxo76hEULl9C2XZscxfg3R+3DagMH4+Hnl/H8wLfTKduuPfnKluX8xg2cXvo7pZ9vS7rZzJ+TJ/NotzfJW6QIqTcScDNm/Tp/LVo3o12Htrw/+O/zFT/67IOMx599/AW+vr45qlNW2Hkcei/gR6WUB3AKeBVLr8Y865Dvs8CL1nWXAk2BE0CSdd1suV+3SSPgUyw/AT7LNPUD3rvXC5VSZZVS9ZVSvv+a3zi7G5vZjYQb7Nv9Jy2eawaAu7s7fnn9UEqRmJhoXSeR4JBge4RzqEeeqIhfXr/7r2j1x/qthBUKo2jxIlmOZXRTeBndMCiFt7sb0TdS6PBEOBO3nMpon8QlpQFQMtiXbWeuAnAqLolC/l7k88laK+zfDh44ROEi4YQXLoS7hzuNmzRk/dqNOSrzRuxVzuw+SPlnawJwMyERN6OBgIKWY06FHy3LyW17ATi2cSfFqz2GX0gQAD4BebMdNyoyms2bttLq+RYAaK3ZuWMP9RrUAaBZyyZsWLsp2+XfjSP24Z0kRkYSVKYMAMEVKhK5excAsQcP4le4MHmLWD5/Hr5+KLes98BWrvI4/v533v9aa1atWEPjZg2yufVZZ89ruWit92mtq2itK2mtW2utr2qtr2it62utS2mtn9Vax1nX1VrrnlrrElrrR7TWu7Jbh3t+rWqtZwIzlVLPa61/trVQpdTbQE8sYy+nK6V6a61v/bz4EFie3Q2+5dLFywQEBfDB++M4fuwkZcuVps/AXvQZ8BZ9u/dn4meTSNeayd9PzGmo2ygFPd54G6Xg+Rfb0PZF+7bAblk873dW/76O0uVK0rXv6/jl9SU5KZl5M39m7FejWTDrlyyVF3Ujhenbz7K+59OkmNLZfPoKW07HMb5VRZqWC6NB6VCuJqUyetVRzl5N5kh0Ag3KhLLrQjyVCuSloL8X+fN6ciUpNdt1io6KIX/+sIznoflDObD/0D1ecX+bZ8ynRqc2pCWnAOCV15d0czrRJ84SWrIoJ7fu5Uas5Ysp/lIU6WYzv7w/nrTkm1RqVo+ydatlK+74j7/k7b49SEqytM6vxV/Dz88Xo7W1Gpo/hOjomBzV7U4csQ9RsOPTjwFFkbp1KVKnLr6FChG1Zw/5K1fm8s4dJMfFAZAYaekN2PHpx6QmJFDgqWqUaNosZ/H/Zc/ufQTlC6Jo0aw3WrIrN19F0Va2fq2GK6XyWs9omqaU2qOUaniP9bsClbXWrYE6wPtKqd7WZXf9YZN5KNDMaT/cc4PMZjPHDh+jzYutmDlvGl7e3syaMZuF8xbxdv+e/LpqPr3792Ts8I9trKLtvv1hCj/9/D1fTf6ceXMWsHvXXrvHaN62Cd/+OoWvZ39BUHAgUydMB+CHKbN5rkMrvH28s1xmXi8j9UuFUO/rLTz9v034uBtoWSE/HgY3Uk3pPP/dDubtu8jYZuUBmLz1DHm9jCx67Sk6VinM4agE0tNz14f+zK4DePv7EVqiaMY8pRSN3nmdzd/OZ/6Acbh7e2a0IHV6OjEnz9F8SE9aDHubXQuWEn8pKstxN23YQmBQAOUqlLVbXR6k6kOG8vTI0Tz5zrucXbOauKNHqPTaG5xbu4bNw4dhvnkz4xiETk/n6vFjPPZmD6q/N5So3buI/SuHXyj/snzpSho3dV7rHCzJ0NYpt7K14+s1rfUXSqlGQD6gIzALWHmX9d201jcAtNZnlFJ1gAVKqaLcI6FnHgp0JeXyPTNHaFgIIWEhVKhkST51GzzDrBmz2b/3AH0H9gKgXsM6jB3xiY1VtF1YWCgAQfmCqFu/Dgf3H6JylcftGiMwX2DG48ZtGjG8j6Wf8cjBY2xa8wfTvvyOxIRElJvCw8ODlu2a37fMGsWCuHAtmavJli6VlUdjeDzcn6iEFFYetRxwX3kshrHNKgCQmGpm8O9/Zbx+bY+anItPzlG9QsNCiIz8O4FGR0YTFhqS7fIuHznJ6Z37ObvnIKY0E2lJyaz6/Fsa9HmV5z54F4Bz+/4i/pKlfnnyBVLYLw/uXp64e3lSsHwpYs9cyOiesdWfew+wad0W/ti0jZSUVBITE/l03BckJNzAZDJhNBqJjowhNAd1uxt770MAr0BLF5Rn3ryEPVGZ+FOnKN6kKVX7DwDgRuRlov/8M2PdoDJlMvrbQyo9yvUzZwguXyFH23CLyWRi7er1zJ6Xs8EGWfVfaqHfSsJNsZzRdIh7JGYsw3Meu/XEmtybA8HAI9nYztvkC85HWFgoZ61HxHdt301E8aIEh+Rj7659AOzevofCRcLtES5DclJyRh99clIyW//YTslSJewaA+BKbFzG4z/WbaWYtQX62bSP+P636Xz/23Rat2/JS6++YFMyB7h0/SaPFfTHy2h526sXC+RUbBKrj8XwVFHLH3TVIoGcuWqpn5+nEXfrNUVffLQgu87Hk5iasxOEK1Qsz7mz57lw4SJpqWksX7aSZ+rWynZ51V9pTZdpY+k0+QMa9XudQo+UoUGfVzMOdprT0tjzy0oqNrLEiKhaicuHT5JuNpOWkkrUsdMEFsqf5bhv9enO72t+YfGKBXz4yQierFqZMR8Np8qTj7N21XoAfl+8jNp1n8523e7G3vvQlJKCKTk543HsoYP4FQon5bplH+r0dE4uXkyRupZr9IU88ggJFy5gTkkh3Wwm7ugRfAtm6zyYO9q+dSfFIooRlj/UbmXa4r/UQt+tlFqJ5QyowUopP6yXAbiLToAp8wzr5QI6KaUmZ2tL76Dv4LcZOXgMaWkmCoYXYMjoQdSqW5PPP5qI2WzGw8ODgcPfsVc4AK5ciaPf25ZWi8lkpkmzRtSsVT1HZY597xP27z7A9fjrvNK0C69068D+3Qc4dew0KEVYgVDeHtIzx9u+/9J1VhyN5tfXnsKUrjkclcBP+y7gZTTwWcuKdHmyCElpJoYsPQxAieA8fNS8PFrDidhE3lv6130i3J/RaGTwkP706Po26enptG7TwiFfiHsXreLsroNonU7FRrUJf8TSNRIUXoAij5fnp75jUEpR/tma5Ctqv2T0Vt8eDBkwgkn/m0qZsqVo9ZxtX7ZZYe99mHrtGrv/ZxkJps3pFKxWnZBKlTi9cgVn16wGIH/lKoTXqg2Ae548RDRqzJaRI0BBaKVHCX3ssSzHHfTu++zeuYf4+Hga1WtB955dafN8S1YsW+X07hbI2olFuZXS2qYrh7lhuSb6Ka11vPUyAIW01vsdtWH363KxJx9j1vujsysyKfr+K9lJo4nnnBJn/4AnnRIHYMqR3U6L1aWUXX5M2sTDkLPRQ7YavOOIU+IAfPBkaafF8jEG5njQ4Zh9G23OOUMfq50rL7Zrawv91u/GSsoVLhoshBD/4gqZzdaE3j/TYy+gKrAbuWOREMJFuEKXi613LGqR+blSqjDwuSM2SAghHgRX6HzI+vm6FhcA+14gRQghHiAXyOe2JXSl1P8g48zwWwdIc3ZRDyGEyEXud+OKh4GtLfTM1xYwAXO01lscsD1CCPFA/Jf60J17ypYQQjiZy3e5KKUOwN0vEqy1rmT3LRJCiAfAFU79v18L/Tks9707/6/5hbHcE08IIVxCbj6l31b3q8MELHfPOJt5wnKD6An3ea0QQjw0lNI2T7nV/VroYVrrA/+eqbU+oJQq5phNEkII5/svjHIJuMcy510ARQghHOy/0OWySynV9d8zlVJvYDn1XwghXMJ/oculD/CLUupl/k7gVQAPwDH3XRNCiAfA5Yctaq2jgBpKqbpARevs37XWax2+ZUII4UT/pROL1gHrHLwtQgjxwPwXDooKIcR/wn/5aosOZ1QGp8VSTuw9G7MrwWmx9vR/wilxrqc5r06vlnbeycn1px91WqwNb9jnBsv3M6qK/W/3dzfpdz/JPFdygXyeexO6EEI403+mD10IIVydtNCFEMJFSAtdCCFchCR0IYRwEf+FU/+FEOI/QSll82RjeQal1F6l1BLr8wil1Hal1Aml1FyllId1vqf1+Qnr8mLZrYMkdCGEwHJQ1NbJRr2Bw5mefwRM0FqXBK4Cr1vnvw5ctc6fYF0vWyShCyEE9m2hK6XCgWbANOtzBdQDFlhXmQm0tj5uZX2OdXl9ZevPgH+RhC6EENi9hf45MABItz7PB8RrrU3W5xeAQtbHhbDeFc66/Jp1/SyThC6EEICbUjZPSqluSqldmaZut8pRSjUHorXWTr/EuIxyEUIIwC0LveNa6ynAlLssrgm0VEo1BbyAvMAXQIBSymhthYcDF63rX8Ryn+YLSikj4A9cyV4dhBBCoJTt071orQdrrcO11sWAl4C1WuuXsVyxtq11tc7AIuvjxdbnWJev1Vpna1C8JHQhhMBykT5b/2XTQKCfUuoElj7y6db504F81vn9gEHZDSBdLkIIgWMun6u1Xg+stz4+BVS9wzo3gRfsEU8SuhBCYDko+rCThC6EEDj3vgiO8tAndLPZTOeXuhISGsyErz5Ga82k/01lzcp1GNwMPN+uNe1ebnv/grIRt/0LnQkNC2HipAk5KsuUlMjxWVNJunQelKJUp25cPbiPK3/uRik33P3yUqpzdzwDAjEl3uDY91O4GRuFm9GdUp3eJE+hwlmOGXk5imGDRxF3JQ6lFG1eaEWHju0yls/6bjaff/I/Vm9eRmBgQI7qBzB31gKWLFyKUoripSIYPGoA/d7sT1JSMgBX4+IpV7EMYz8fnaM4KSkpdO3ck7TUNMxmE/Ub1OXNt95g7uwFzJk1jwvnL7J60+8EZLNO7SsVpFXZMDSaE1eSGL3+OME+Hox5tgz+XkaOxCQyfO0xTOmaDpUK0rJsGGatiU9OY/T6E0TeSMlW3OFDR7NxwxaCggL5edEcAMZ/+iUb12/G3d2d8MKFGDnmffLm9ctW+bfcbf+NGDKGPbv24eubx7I9HwyhTNnSOYrl7M/g/bjCAcWHPqH/9MN8ikUUJTExEYAlvy4lKjKa+Yt/xM3NjbgrVx0S98dZP1G8RDFu3EjMcVmn5n1PYIVHKfdmH9JNJtJTU/ApEE7Rli8CcGntcs7/vpCSL7/O+eWL8C1clPI9+pEUeZGTc77jkb5DshzTYDTQd8DblCtfhsTERF554VWqVa9K8ZIRRF6OYtuWHeQvkD/HdQOIiYrh59m/MOuXGXh6eTKs/yjWLF/LV999kbHO0H4jeLpujRzH8vDw4JsZX+Lj44MpzcTrnXpQo1Y1Hn28ErWeqcmbr76V7bJD8njQrmIB2s3dS4o5nQ8blKFByRBqFglkzv5LrDoZy6BaJWhVNoyf/4rkaGwinRf+SYopnefL56dXtWIMWZ29uyC1bN2clzq8wNDBIzPmVatelbf7/B9Go5HPP5vIjKkz6fNO9usHd99/AG+/05NnG9bNUfmZOfMzaItsnpyZqzzUX0pRkdFs2bSVVs83z5j387xFvNG9C25ulqoF5Qt0QNwoNm3YQpvnW+W4LFNyEteOHyGsZh0A3IxGjD55MHr7ZKxjTk3JOD0t6fJF/MtYblfmk78QKVdiSL1+LctxQ0KCKVe+DAB58uQhongxoqNjABj/0Rf0fqenXQ8Smc1mUlJSMJnM3Ey+SXBIcMayxBuJ7N6xl1p1a+Y4jlIKHx/LvjOZTJhMJpRSlC1XmoKFCuS4fIObwtPohkGBl9GNK4mpVCnoz9pTsQD8fiyaZyKCANh96RopJsuJggeiEgj19ch23MpVHievf95/zKtRsxpGo6VNVunRikRFRWe7/Fvutv8cwdmfwftxwLVcnM5hLXSlVFVAa613KqXKA42BI1rrpfaKMeHjL+nV9/9ISkrKmHfh/EVWLV/L+jUbCQwM4J3BvSlSNOtdEvfy8bgJ9H23F4mJSfdf+T5uxkbj7uvH8ZmTSbx4Ft8iERR/sRMGTy/O/DqX6O2bMHr78EjfoQDkCS/Clb078S9VloTTJ7gZF0vq1St45PXP9jZcuniZI4ePUbFSBdav3UhIWAily5bKcd1uCQkL4aXOL9C2UXs8vDypWr0KVWtUyVi+ad0WKj/1OHmsP+dzymw20/HF1zh/7iIvtH+OipXsc7/OmMRUfvjzIotfqUKKKZ3tF+I5HHuDhFQTZuuo4agbKYTkuT1xtywXxtZzjvm1CPDrwt9o1ORZu5R1p/23YO4vfP3lZKZN+pYnq1WmV98eeHhk/wvq3xz9GbSFtNDvQik1HPgSmKSUGgtMBPIAg5RSWe8fuINNG7YQGBRIuQpl/jE/LTUNT08Pvp87jdZtWzB62Dh7hMuwYf0mgoICKV+hnF3K0+np3Dh/hgLPPMvjQ8bi5uHJhRWLASjWuh1Vx04kpGpNLq1fCUB4o5aYkhPZO2Ywl9avxLdwMXDL/tuYlJhE/z6DeXdQHwwGAzOmzKT7W13tUbUMCdcT2LzuD+Yu/ZFfV80jOTmZFUtWZSxfvWwtzzapZ7d4BoOB2T/PZOmaXzh04C9OHD9ll3L9PAw8UyyI1j/uoumsnXgb3ahe+P6/ABuXCqFciC+z9l2877rZMXXytxiMBpo2b2yX8u60/97q052ff5vD93Oncf3adWZO/8EuscA5n0FbZOXU/9zKUV0ubbGc/lob6Am01lqPBhoB7e72oszXR/hu2vf3DLB/7wE2rdtCq0YvMKT/CHbt2MOwQaMIDQuhTv3aANSpX5sTx07aq04A7Nuzn/XrNtHk2VYMfGcIO7fvYvCAYdkuzzMgCM+AIPwiSgIQ/MRT3Dh35h/rhFStyZW9OwAwevtQunN3Hh86ltJdepCWcB2v4NBsxU5LM9G/z3s0adaIeg3qcOH8BS5dvEz75zrSvEEboqNieLltF2JjsnUWcoZd2/ZQoFB+AoMCMLobeaZ+LQ7++RcA8VevcfjgEapb+2ntyS+vH1WqPsHWzdvsUl7V8AAuXU8h/qYJc7pm3ekrPJrfDz8PIwbr33iYrycxiakZr3mykD+vPhHOu8sOk5Zu/zviLPplCZs2bObDj0bZvYWZef8FhwSjlMLDw4MWrZtx6MDh+xdgA2d9Bm0hXS53Z9Jam4EkpdRJrfV1AK11slIq/W4vynx9hGup0ff89Pfs052efboDsHvnXn74bg6jxg1j4oRv2L1zL4XCC7Jn1z67d7f07teT3v16ArBzx25mfvsDYz8ele3yPPwD8AzKR1LkJXzyFyT+yEF8ChQiOeoy3mGWPt+4P3fjHVYQsIyIcfPwxM1oJGrzOvKWKvuP/nZbaa0ZPewDIooX5ZUu7QEoVbokqzf93SPWvEEbZs37NscjDELzh3Jo/2FuJt/E08uT3dv3UMbad7p+1UZq1K6Gp6d9fr5fjbuK0WjEL68fN2+msH3rTjq/9opdyo68kULFMD88jW6kmNJ5slAAh2NusPvSNeoVD2bVyVialQ5lw5k4AErny8Pg2iXovfQvrt5Ms8s2ZLZl01ZmzpjFtJnf4O3tZZcy77b/YmNiCQ4JRmvNhrUbKVGqeI5jOfMzaAsZtnh3qUopH611ElD51kyllD9/X07SITq//jLDBo1izvfz8PbxZsjIgY4MZxfF23Xm2IyvSDeb8AoOpXSnNzn+w1SSoy6DUngGBVOyg+Va+EmRFzn23TeWg1cFwinVMXs/Tfft2c/vi5dTsnQJ2j/XCbB8ST5dO+cjTf6tQqVy1GlQm9df6o7BYKBU2ZK0bNsMgDUr1vHKay/ZLVZszBWGDxlDujmddJ1Og0b1qFWnJj/9MJ/vv/2RK7FxvPRcJ2rWqs77owZnqexD0TdYcyqWWc8/illrjsYm8stfkWw+G8cHDcrQvWoRjsUmsvhwFABvVy+Gt7uBsQ0sX16RN1J5d3n2WraD3h3Krp17iI+Pp2G95vTo2Y0ZU2eSmpZK9zd6AZYDo0OHZ/usceDu+6/7a724ejUerTVlypRi8PD+OYoDzv0M2sLt4c/nqGxeA+behSrlqbW+bcCtUioYKKC1PnC/Mu7XQrcnT4Ons0LRc+MJp8X6olaEU+IkmXJ+cNhW3gb7tERtUX969oYYZseGN+xz4PZ+TBmX43Y8pZw3iM7XGJTjdLw5cqPNOefp/LVzZfp3SAv9TsncOj8WiHVETCGEyIncfLDTVg/9iUVCCGEPLpDPJaELIQTIQVEhhHAZ0kIXQggXIS10IYRwEQ/1ha2sJKELIQTOHWbpKJLQhRCC3H1Kv60koQshBK5xtUVJ6EIIAbhCG10SuhBC4ArpXBK6EEIAMmxRCCFchvShCyGEy5CELoQQLkG6XIQQwkW4QI+LJHQhhLB4+DN6rk3oHm72ucekLQKf/cJpsfb/+rzTYjnzTkLOcvbGOafF+r1LMafFikqOcUqclPSbTokDEOwV7LRYvnbIZG6S0IUQwkW4QJ/Lw381GiGEsAOVhX/3LEepwkqpdUqpv5RSh5RSva3zg5RSq5RSx63/B1rnK6XUl0qpE0qp/UqpJ7JbB0noQgiB/RI6YALe0VqXB6oBPZVS5YFBwBqtdSlgjfU5QBOglHXqBkzKbh0koQshhB1prS9rrfdYHycAh4FCQCtgpnW1mUBr6+NWwPfaYhsQoJQqkJ3YktCFEALLmaJZmLoppXZlmrrdpcxiwOPAdiBMa33ZuigSCLM+LgScz/SyC9Z5WSYHRYUQgqydWKS1ngJMuWd5SvkCPwN9tNbXM19aQGutlVI6m5t6V9JCF0II7NqHjlLKHUsy/1FrvdA6O+pWV4r1/2jr/ItA4UwvD7fOyzJJ6EIIAZZhi7ZO9yxGKWA6cFhrPT7TosVAZ+vjzsCiTPM7WUe7VAOuZeqayRLpchFCCOx6nmhNoCNwQCm1zzrvPWAcME8p9TpwFnjRumwp0BQ4ASQBr2Y3sCR0IYTAfhfn0lpv5u7fD/XvsL4GetojtiR0IYQA5FouQgjhIuQGF0II4SLkeuhCCOEiJKELIYSrePjzuSR0IYQAaaELIYTLkIOiuciPs35i4YJFaK15rm0rXunUPstlfDOgBU2qlSYmPpEqr30DwHPPlGNIl2coWySEWj2mseeY5QSuepWLM7pbPTyMBlJNZt77ZjUb9p4BwN3oxoTeTaj9aDHStWbE9LX8uvHIXeN+PnoiOzfvwj/Qn69/stw96dSx03w17htSU9IwGAz0GNiNMhVKceP6DT4fPZHIi1G4e7jT+/2eFCtRNMt1BZg7awFLFi5FKUXxUhEMHjUADw93pk6cwbqVGzAYDLR+oQVtX34uW+XfL5anp+WuVJ+Pm8jSX5exctvv2So7NuoKX478mmtx10BBg9b1ad6uCXMmz2PHxl24ubnhH5iXt97vTlBIEBfOXOSrMZM5dfQ0Hbq3o9XLzXNQr/n8lqle740ayLgRn3Dk0FGMRiPlKpZlwPv9MLpn/U/ts5Gfs33zTgIC/Zky72sAZk3+kWW/rsA/0B+AV/+vE1WffhKAn76dx/JFqzC4udGjfzeqVK9sU5yYqFg+H/E/4uOuAdCoTQNavtSMhGsJfDxkAtGXowktEMrAD/vhm9eX9cs38vP3v4IGbx8vegzsRkTpYlmu39nT5xg2YGTG84sXLtP1/16lSctGvN9/JJcvRVKgYH5GfzqCvHn9slx+VrlCC11ZxrTnPsmmeJs37MTxkwx8dyg//PQt7u5Ger7ZhyHDBlKkaOH7vxgIavAlADUrFSExOZVpg1tnJPQyRYJJ15qJ/ZoxeNKqjIT+aMn8RF+9weUrNyhfLITfPn6ZEi9+DsDQLs9gcHNj5Ix1KAVBft5cuZ4M3PkWdAf3HMLLx4vxI77MSOjv9xpJq/YtqFLjCXZu2c3Ps35l3DejmfHlTLy8vejQtR3nz1xg0sdT+fDrkbeVCeDv6X/XOsdExdCzSx9m/TIDTy9PhvUfRbWnq6I17N25j/dGD8DNzY2rV64SmC/Qpv2Y1VhNWzXmyKGjzP9xIZvWbrYpoUcnR98272rsVa7GxlO8bATJicn07/IeAz9+h3yhQfjk8QHg97nLuXDmAm8OfINrcdeIiYxl+4Zd+ObNc9eEHuYddsf5mev1f11688Mv3+Lp5cn7/UdS/emnCAwKoNrTTwEwYtAYHqtciTYvtrpnWYlpt98u8MCeg3j5ePHJsPH/SOhePt680PGfX7JnT51j7JCP+XLmBOJirjDo/4YyfeFkDAbDP9a70y3o4mKvcjX2KiXKFicpMZl+nQbw3icDWLNkPX7+vrTt3IYFM3/hxvUbdOnVkcP7j1C4WDi+eX3Z/cce5kydx6ffjrut3Kzcgs5sNtPq2bZM/XESC3/6FT9/Pzq9/jLfT/+RhOs36Nn3zXu+Pp9ngRxn4wuJp23OOeF5InJl9nfatVyUUt87quxTp87wSKUKeHt7YTQaqVzlcdasXp/lcrbsP0ecNfHecvRcLMfPX7lt3T9PRHL5yg0A/joTg5enOx7ulj+ezk0e45PZmwHQmoxkfjcVn6iA320tEEVSouWPPOlGEvmCgwA4d/o8lao8AkDhYuFEX47m6pX4LNXzFrPZTEpKCiaTmZvJNwkOCWbRvMV0ebMjbm6Wj0ZOk/m9YpnNZr4eP5kefe945VGbBQYHUrxsBADeebwJL1aIuOi4jGQOkHLzJreOevkH+VOyfAmMRsOdisuSzPVKSU4hOCQf1WtVy7jMavmKZYmOis1W2Y88UfEOn4s727phG3Ua1sbDw538hfJTsHABjh46ZtNrg4IDKVG2OAA+ebwJjyjElZg4dmzcSb1mdQCo16wO2zfsBKBcpbL45vUFoEzF0sRGx2WxZrfbtX0PhQoXokDB/Gxat4WmLRsD0LRlYzat3Zzj8m2hsjDlVg7pclFKLf73LKCuUioAQGvd0p7xSpYszsQvJhEffw1PT082b/qD8hXK2TPEPbWpXY59xy+TmmbGP48nAMNfq0utR4ty+tJV+n65jOiriVkqs1u/1xj29ihmfDGTdK35dNqHAESUKsbWdduo+Hh5jh46TnRkDFeirxCYLyBL5YeEhfBS5xdo26g9Hl6eVK1ehao1qjBy0BjWrljPxrWbCQj0p/fAtyhcNDxLZdsaa/6PP1OzTg2CQ/LlqPzMoi/FcPrYGUpVLAnAj5PmsmHZRnx8fRj51ft2iwO36vUizzd6CU8vT56sXoWqNZ7MWG5KM7FiySp6D3zLrnF/m7eENb+vpVS5knTr+wZ+eX2Jjb5CuUfKZqwTHBrMlejbGyL3E3UpmlNHz1CmQini4+IJCrZ8oQfmCyA+Lv629VctXkPl6o9nuy63rF6+lgZN6gEQFxeX8ZnIFxxEXFzOvzBs4gJ96I5qoYcD14HxwGfWKSHT4zvKfNH46VO/szlY8RIRvPp6J3p07UXPN3tTpmzpjBamo5UrFsKYbvV5a7ylu8BocCM81J9tB89T482pbP/rAmO7N8hyuUt/Xs4bfV/luyVT6drnVb4YY/nJ/UKn50i8kUivl/uxZN5SSpSOwM2Q9bomXE9g87o/mLv0R35dNY/k5GRWLFlFWmoaHh7uTJsziRbPNWPc8E+yXLYtsZb/tpJ1KzfyfPs2OS7/luSkm3wyeAKv9umU0Tp/uUc7piz+itqNarJswQq7xQK4fj2Bzeu2MG/pbH5dNZ+byTdZsWRVxvLPPvycRytX4tEnKtktZvO2Tfn216l8PftLgoKDmDJhmt3KTk5KZtygT3mjXxd8fH3+sUzd4SqD+3cdZNXitXR+65UcxU1LS2Pz+i3Ua1jntmVK2Xa5Wnuw5+VzHxRHZb0qwG5gCJZLQa4HkrXWG7TWG+72Iq31FK11Fa11lde7dslSwDbPt2TO/O+Z8f1k/PL6UbRYkRxsvm0KBfsxd9SLvDFuEacvXQUs3SuJyan8uukwAAvX/8VjpfNnuew1v6+nRt1qADz9bA2O/XUcAB9fH/oM68X/fhxPvxFvcy3+OvkL3ruv9052bdtDgUL5CQwKwOhu5Jn6tTj451+EhIVQu34tAGrXf5qTx09nuWxbYs34eiYXz1+kfYuOvNCkAzdvpvBS847ZjmEymfhk8ARqNapJtbpVb1teq9HTbFu3IyfVuM2ubbspUKhARr1q16/FgT8PATDjm5nEX71Gr3f/z64xA/MFYjAYcHNzo0mbRhndKsGh+YiJislYLzY6lnyhtv/yMZlMjBv4Kc80qpXxuQsICiAu1vK5jou9SkDg38dkTh8/w8QPJjHkk4HkDcjZAcutm7dTulxpgvJZuhWDgoKIjbH8uoiNuUJgkH26/e4nK3csyq0cktC11ula6wlYLgM5RCk1EQePqIm7YvlZdvlSJGtXr6dJs0aODId/Hk8WjmvP+1PXsPXg+X8sW7r1GLUfKwZAnSciOHIm632oQSGBHNhjSQ5/7jxAwcKWWwzeSEgkLS0NgBWLVlPhsfK3taZsEZo/lEP7D3Mz+SZaa3Zv30PRiCLUqluTvTv3AbBv15857m65W6wXO7Zl0doFzF82m/nLZuPl5clPS2Zlq3ytNV9/MIXwYgVp2aFZxvxL5/6+pPTOjbsoVLRgjuuSWVj+MA7t/+sf9SoWUYTfFv7Ojj92MmLcULv/UrwS+3f3wx/rtmaMcKpW+ynWr9xIamoakRcjuXj+EmUqlLapTK01/xv9NeER4bR+uUXG/Kq1q7D29/UArP19PVVrW7qTYiJjGDvwU/qO7GWXfbpq2RoaNPn7IoRP16nB0sXLAVi6eDm16tbMcQxbuEIL3SmjXJRSzYCaWuv3bH1NVka5ALzasRvX4q9hNBp5Z2Afnqr25P1fZHVrlMvMoc9R67GiBPv7EH01kdHfrefq9WTGv92EYH8f4m/cZP/JKFoO+JGBr9Sif4eanLj49x9Yi/4/EBOfRJEwf6YPbo2/rxex15J486NFnI++Dtx5lMvHQ8dzYPdBrscnEJDPn5e7vkShooWYMn46ZpMZD08P/m9AN0qWK8Hh/UeZMPJLlFIUKV6Y3kN7Zhyg+rd7jXIBmP71d6xdsR6DwUCpsiUZOOIdUm6mMuq9D4m+HI23jxfvDu1LyTIlbN6XWYnl4eGRsbxhtWbZHuVyeN8RhnYfSZEShTMSaIce7VizeD2Xzl1CKUVI/hDeHPg6+UKDuHolngFdhpCcmIxyU3h5e/HFT5/84yAq3H+Uy616rVmxDoPBQOmyJRk44l0aVGtKWIGwjPKeqVeLV7t3umc5dxrlMva9j9m/+wDX4q8TmC+Ajt1eZv/uA5w8dgqlFGEFQnl7yFsZB8xnT5/LysWrMBgMdH+nK0/WrHJbmXca5fLXvsMM6vY+RUsWwU1Z9l/H/+tA6Qql+Pi9z4iJiiU0fwgDPuyHn78f/xsziT/WbSM0fwgABoMb47//+LZybRnlkpyUTJtG7ViwdDa+fpbP8bX4awx9dyRRkVHkLxDGmE9HkNc/7z3Lsccol6jk8zbnnDDvwrkyq7vEsMWcupXQneFOCd1R7pfQH0Z3SuiOYktCt5c7JXRHuFNCd5SsDFvMKXsk9OjkCzbnnFDv8FyZ0F3mxCIhhMiRXJmis0YSuhBC4BpnikpCF0IIJKELIYTLyM3DEW0lCV0IIZAWuhBCuIyHP51LQhdCCAvpchFCCNfg5gJtdEnoQgiB9KELIYTrkC4XIYRwDQ9/OpeELoQQgHS5CCGE65AuFyGEcA2uMMrFaTeJFkKI3MyeN7hQSjVWSh1VSp1QSg1ywuYDktCFEMJCZWG6VzFKGYCvgCZAeaC9Uqq8ozY7M0noQgiBXVvoVYETWutTWutU4CeglcMrAJb7CbrSBHRzpTgS6+GK5Yp1cuVYOdlGYFemqVumZW2BaZmedwQmOmO7XLGF3s3F4kishyuWK9bJlWNli9Z6ita6SqZpyoPeJpAuFyGEsLeLQOFMz8Ot8xxOEroQQtjXTqCUUipCKeUBvAQsdkZgVxyH7qyfPs78iSWxHp5YrlgnV45ld1prk1LqLWAFYABmaK0POSO2snbaCyGEeMhJl4sQQrgISehCCOEiXCahO+tUW6XUDKVUtFLqoKNiZIpVWCm1Tin1l1LqkFKqtwNjeSmldiil/rTGGumoWNZ4BqXUXqXUEgfHOaOUOqCU2qeU2uXgWAFKqQVKqSNKqcNKqeoOilPGWp9b03WlVB8Hxepr/TwcVErNUUp5OSKONVZva5xDjqqPy3vQA/TtNMjfAJwEigMewJ9AeQfFqg08ARx0Qr0KAE9YH/sBxxxYLwX4Wh+7A9uBag6sWz9gNrDEwfvwDBDs6PfKGmsm8Ib1sQcQ4ISYBiASKOqAsgsBpwFv6/N5QBcH1aMicBDwwTJYYzVQ0hnvmytNrtJCd9qptlrrjUCcI8q+Q6zLWus91scJwGEsf2SOiKW11jesT92tk0OOmCulwoFmwDRHlP8gKKX8sXzZTwfQWqdqreOdELo+cFJrfdZB5RsBb6WUEUuyveSgOOWA7VrrJK21CdgAPOegWC7LVRJ6IeB8pucXcFDie1CUUsWAx7G0nB0Vw6CU2gdEA6u01o6K9TkwAEh3UPmZaWClUmq3UsqRZyBGADHAt9aupGlKqTwOjHfLS8AcRxSstb4IfAqcAy4D17TWKx0RC0vrvJZSKp9Sygdoyj9PzhE2cJWE7tKUUr7Az0AfrfV1R8XRWpu11o9hObOtqlKqor1jKKWaA9Fa6932LvsuntZaP4Hlync9lVK1HRTHiKUrbpLW+nEgEXDoZVOtJ620BOY7qPxALL90I4CCQB6l1CuOiKW1Pgx8BKwElgP7ALMjYrkyV0noD+xUW0dTSrljSeY/aq0XOiOmtatgHdDYAcXXBFoqpc5g6Rqrp5T6wQFxgIxWJlrraOAXLN1zjnABuJDpV80CLAnekZoAe7TWUQ4q/1ngtNY6RmudBiwEajgoFlrr6Vrrylrr2sBVLMeMRBa4SkJ/YKfaOpJSSmHpkz2stR7v4FghSqkA62NvoAFwxN5xtNaDtdbhWutiWN6ntVprh7T6lFJ5lFJ+tx4DDbH8tLc7rXUkcF4pVcY6qz7wlyNiZdIeB3W3WJ0DqimlfKyfxfpYjuM4hFIq1Pp/ESz957MdFctVucSp/9qJp9oqpeYAdYBgpdQFYLjWerojYmFpzXYEDlj7tgHe01ovdUCsAsBM68X53YB5WmuHDil0gjDgF0suwgjM1lovd2C8XsCP1kbFKeBVRwWyfkE1AN50VAyt9Xal1AJgD2AC9uLY0/J/VkrlA9KAnk46qOxS5NR/IYRwEa7S5SKEEP95ktCFEMJFSEIXQggXIQldCCFchCR0IYRwEZLQxQOnlDJbrxp4yHq1x3eUUm7WZVWUUl/aUMYf1v+LKaU6OHqbhciNZNiieOCUUje01r7Wx6FYTijZorUeno2y6gDvaq2b23UjhXgISAtd5CrWU/S7AW8pizq3rpluPZt1lbUlP00pdVYpFWxddutKkeOwXORpn1Kq74OphRAPhiR0ketorU9hOeM39F+LhmO5VEAFLNdKKXKHlw8CNmmtH9NaT3DslgqRu7jEqf/iP+NpoA2A1nq5UurqA94eIXIVaaGLXEcpVRzLpVOjH/S2CPEwkYQuchWlVAjwDTBR337EfgvwonW9hkDgHYpIwHK7PiH+cyShi9zA+9awRSz3klwJ3Okm1SOBhtYbdL+A5V6aCf9aZz9gtg5/lIOi4j9Fhi2Kh4ZSyhMwWy+XXB3L3YEee8CbJUSuIQdFxcOkCDDPetJRKtD1AW+PELmKtNCFEMJFSB+6EEK4CEnoQgjhIiShCyGEi5CELoQQLkISuhBCuIj/B5aVMF0lCurxAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_cm(cm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "a45c43f5-0497-4a9f-8930-dccc0aee0dd5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster ID 0 centroid\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster ID 1 centroid\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster ID 2 centroid\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster ID 3 centroid\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster ID 4 centroid\n"
     ]
    },
    {
     "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": [
    "# Look at the centroids for all of the clusters\n",
    "for i in range(len(clst.cluster_centers_)):\n",
    "    print(\"Cluster ID %d centroid\" % i)\n",
    "    plot_digit(clst.cluster_centers_[i])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85e51e69-791a-42f1-a1c9-1124b9b3b078",
   "metadata": {},
   "source": [
    "# Task 2: Varying k in k-Means\n",
    "\n",
    "For values of k in {2, 10, 15} \n",
    "* cluster the digits\n",
    "* print the confusion matrices\n",
    "* plot the cluster centroids\n",
    "\n",
    "Note that for k = 2, 2-means must combine distinct digits.  For k = 10 you would hope that it has a cluster for each digit in which that digit is by far the most common.  For k = 15 the expectation is that there may be clusters, for example, for the same digit but different ways of writing it (e.g., 9 with a straight bottom vs. 9 with a curled tail at the bottom).\n",
    "\n",
    "For each of the three cases, write a couple of paragraphs on what you observe about the way k-means has clustered the data, why it has done it that way (e.g., certain digits are visually similar, be sure to say which ones and why), and why the cluster centroids look the way they do.  You don't have to talk about all of the clusters, especially in the k = 15 case, but you do need to mention several of them in each case."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06b67c07-4405-4221-8932-d2462cbf5879",
   "metadata": {},
   "source": [
    "# Task 3: Agglomerative Clustering\n",
    "\n",
    "Read the scikit documentation on Agglomerative Clustering.  There are two important hyperparameters:\n",
    "* linkage, which can take on values ‘ward’, ‘complete’, ‘average’, ‘single’\n",
    "* metric, which can take on values “euclidean”, “l1”, “l2”, “manhattan”, “cosine”\n",
    "\n",
    "Explore combinations of those hyperparameters with n_clusters = 10 and see how close you can get to the ideal of one majority digit per cluster.  \n",
    "* Write a couple of paragraphs explaining what you tried, what worked and what didn't with example confusion matrices.  Include an explanation of **why** you think the best results were a result of the hyperparameters chosen in that case.  Note that you may need to dig through the scikit documentation or use the internet to get more information on their effects."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9f83b067-193a-48bb-82dd-af315f63eaff",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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