lecture_11_trees_and_ensembles.ipynb

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   "source": [
    "# Tree-Based Methods & Ensembling Techniques (Notebook Development In Progress...)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Intro"
   ]
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    "<br /><br />\n",
    "\n",
    "### Motivation\n",
    "\n",
    "Most of the models we've learned so far (linear/logistic regression, svm) and models we will learn in the upcoming classes (neural networks) involve <b>gradient-based weight optimization (using SGD).</b>\n",
    "\n",
    "In this class we will go through tree-based models, known as decision trees, that are different in a sense that they <b>do not require</b> weight optimization via SGD. Besides, tree-based models are <b>nonlinear</b>.\n",
    "\n",
    "Besides, we will learn about model ensembling - powerful technique in ML of combining multiple models in different ways in order to get much powerful models. Particularly, we will examine two most popular and widely used ways of decision tree ensembling - <b>Random Forests</b> and <b>Gradient Boosting</b>.\n",
    "\n",
    "As usual, there will be accompanying examples in Scikit-Learn."
   ]
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    "\n",
    "### Learning Objectives\n",
    "\n",
    "At the end of this class, listeners will be able to:\n",
    "\n",
    "<ul>\n",
    "    <li>Understand key concepts of decision trees and their ensembles.</li>\n",
    "    <li>Understand intuition behind different types of model ensembling.</li>\n",
    "    <li>Implement simple models using Scikit-Learn and Python.</li>\n",
    "</ul>"
   ]
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   "source": [
    "### Reading Material (ordered by difficulty)\n",
    "\n",
    "Decision Trees (for Classification and Regression)\n",
    "\n",
    "<ul>\n",
    "    <li>https://www.youtube.com/watch?v=GZuweldJWrM&list=PLnZuxOufsXnvftwTB1HL6mel1V32w0ThI&index=20 - Bloomberg's ML lecture explaining decision trees for classification and regression. (<b>recommended prerequisite for this class</b>)</li>\n",
    "    <li>https://scikit-learn.org/stable/modules/tree.html - Good explanation of Tree methods with Cons and Pros and practical tips</li>\n",
    "    <li>T. Hastie, R. Tibshirani and J. Friedman -  Elements of Statistical Learning, SECTION 9.2 - In-depth explanation of theory behind classification/regression trees.</li>\n",
    "</ul>\n",
    "\n",
    "Ensembling\n",
    "\n",
    "<ul>\n",
    "    <li>\n",
    "        https://www.youtube.com/watch?v=f2S4hVs-ESw&list=PLqIj-nyfGu55TQpzjv_kwZcz3pJSW9bnH&index=22 - Bloomberg's ML lecture explaining tree ensembles. (<b>recommended prerequisite for this class</b>)\n",
    "    </li>\n",
    "    <li>http://blog.kaggle.com/2017/01/23/a-kaggle-master-explains-gradient-boosting/</li>\n",
    "    <li>https://bloomberg.github.io/foml/#lectures Bloomberg's ML course page (gradient boosting section)</li>\n",
    "    <li>https://statweb.stanford.edu/~jhf/ftp/trebst.pdf - In-depth theory of Gradient Boosting (Advanced)</li>\n",
    "</ul>\n",
    "\n",
    "<br /><br />"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "from sklearn.datasets import load_boston\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the first part of this class we gonna use toy dataset that comes with Scikit-Learn library. The dataset is about predicting house prices in Boston using 13 continuous features: https://scikit-learn.org/stable/datasets/index.html\n",
    "\n",
    "These are the features of the data:\n",
    "\n",
    "CRIM - per capita crime rate by town\n",
    "\n",
    "ZN - proportion of residential land zoned for lots over 25,000 sq.ft.\n",
    "\n",
    "INDUS - proportion of non-retail business acres per town\n",
    "\n",
    "CHAS - Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n",
    "\n",
    "NOX - nitric oxides concentration (parts per 10 million)\n",
    "\n",
    "RM - average number of rooms per dwelling\n",
    "\n",
    "AGE - proportion of owner-occupied units built prior to 1940\n",
    "\n",
    "DIS - weighted distances to five Boston employment centres\n",
    "\n",
    "RAD - index of accessibility to radial highways\n",
    "\n",
    "TAX - full-value property-tax rate per $10,000\n",
    "\n",
    "PTRATIO - pupil-teacher ratio by town\n",
    "\n",
    "B - 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town\n",
    "\n",
    "LSTAT - % lower status of the population\n",
    "\n",
    "TARGET (value we need to estimate) - Median value of owner-occupied homes in $1000’s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now load the data and do some EDA (Exploratory Data Analysis)"
   ]
  },
  {
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   "execution_count": 2,
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    {
     "data": {
      "text/plain": [
       "(506, 14)"
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     "execution_count": 2,
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   "source": [
    "# loading the data\n",
    "X, y = load_boston(return_X_y=True)\n",
    "\n",
    "# feature names\n",
    "cols = ['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD', 'TAX', 'PTRATIO', 'B', 'LSTAT']\n",
    "\n",
    "# creating the dataframe for better visualizations and data processing\n",
    "df = pd.DataFrame(np.c_[X, y], columns=cols + ['TARGET']); df.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CRIM</th>\n",
       "      <th>ZN</th>\n",
       "      <th>INDUS</th>\n",
       "      <th>CHAS</th>\n",
       "      <th>NOX</th>\n",
       "      <th>RM</th>\n",
       "      <th>AGE</th>\n",
       "      <th>DIS</th>\n",
       "      <th>RAD</th>\n",
       "      <th>TAX</th>\n",
       "      <th>PTRATIO</th>\n",
       "      <th>B</th>\n",
       "      <th>LSTAT</th>\n",
       "      <th>TARGET</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00632</td>\n",
       "      <td>18.0</td>\n",
       "      <td>2.31</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.538</td>\n",
       "      <td>6.575</td>\n",
       "      <td>65.2</td>\n",
       "      <td>4.0900</td>\n",
       "      <td>1.0</td>\n",
       "      <td>296.0</td>\n",
       "      <td>15.3</td>\n",
       "      <td>396.90</td>\n",
       "      <td>4.98</td>\n",
       "      <td>24.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.02731</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>6.421</td>\n",
       "      <td>78.9</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>396.90</td>\n",
       "      <td>9.14</td>\n",
       "      <td>21.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.02729</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>7.185</td>\n",
       "      <td>61.1</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>392.83</td>\n",
       "      <td>4.03</td>\n",
       "      <td>34.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.03237</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>6.998</td>\n",
       "      <td>45.8</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>394.63</td>\n",
       "      <td>2.94</td>\n",
       "      <td>33.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.06905</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>7.147</td>\n",
       "      <td>54.2</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>396.90</td>\n",
       "      <td>5.33</td>\n",
       "      <td>36.2</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      "text/plain": [
       "      CRIM    ZN  INDUS  CHAS    NOX     RM   AGE     DIS  RAD    TAX  \\\n",
       "0  0.00632  18.0   2.31   0.0  0.538  6.575  65.2  4.0900  1.0  296.0   \n",
       "1  0.02731   0.0   7.07   0.0  0.469  6.421  78.9  4.9671  2.0  242.0   \n",
       "2  0.02729   0.0   7.07   0.0  0.469  7.185  61.1  4.9671  2.0  242.0   \n",
       "3  0.03237   0.0   2.18   0.0  0.458  6.998  45.8  6.0622  3.0  222.0   \n",
       "4  0.06905   0.0   2.18   0.0  0.458  7.147  54.2  6.0622  3.0  222.0   \n",
       "\n",
       "   PTRATIO       B  LSTAT  TARGET  \n",
       "0     15.3  396.90   4.98    24.0  \n",
       "1     17.8  396.90   9.14    21.6  \n",
       "2     17.8  392.83   4.03    34.7  \n",
       "3     18.7  394.63   2.94    33.4  \n",
       "4     18.7  396.90   5.33    36.2  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's visualize distribution of each feature using boxplot\n",
    "\n",
    "We can see that feature scale highly differs, but, fortunately, tree-based models <b>doesn't require</b> feature re-scaling (subtract mean and divide by standard deviation or something similar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1080x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "df.plot(kind='box', rot=90, figsize=(15, 5))\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's see correlation (Pearson's correlation) matrix of features and target. \n",
    "We can observe that some of the features (e.g LSTAT, RM) are in <b>high correlation</b> with the target, that usually means that they are <b>good predictors</b> for estimating target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<seaborn.matrix.ClusterGrid at 0x1a1ce1c668>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ZSXaN7mMl3b1ndA8AALAVVdXuJLsXDS1098Lk2IOTPC/Jcyf/rCQ/nOQvquqXu/t1q9UXyDbeLoEHAAAOTZPwtXCQw2cmuW93f3bR2Ieq6u1JXjf5rMiSRQAAgNkcsSSMJUkmY0dMU0AgAwAAmM3VVfU9Swer6pZJ9k5TwJJFAACA2fx6krdW1W8meX+STnKXJGckecY0BQQyAACAGXT32VX1mSS/lOQpOfBQj/OTPLK7PzRNDYEMAABgRpPg9dhZrxfIAAAAZlBVr8+BZYrL6u4HrVZDIAMAAJjN89ZbQCADAACYzc909+PXU8Bj7wEAAGZz0noLmCEDAACYzbWr6uQceLriv9Pd/7haAYEMAABgNscleX6WD2Sd5JTVCghkAAAAs7mgu1cNXStxDxkAAMAgAhkAAMBsnrHcYFXdoqpOn6aAQAYAADCD7v7ra7ar6kZV9fNV9bdJ3pnk2GlquIcMAABgBlV1TJKHJjktyW2SvDbJrbv7+GlrCGQAAACz+UqS9yX51STv7u6uqoeupYAliwAAALN5VpJdSV6a5JlV9b1rLSCQAQAAzKC7X9Ddd0vyoBx4F9nZSW5eVc+oqttMU0MgAwAAWIfu/nR3P6e775jkLkmun+RN01zrHjIAAIA56e6PVNWvJTl/mvPNkAEAAMygqq5XVc+sqhdX1f3qgKckuSDJI6apYYYMAABgNn+c5NIk70nyxCSnJzkyyUO6+4PTFBDI+A6PfOW7R7cwF3/+2HuObmFufuWf3ju6hbm5/MrRHczPJdtogcGV+0Z3MB91xOgO5ueI2xw1uoW52fvxI0e3MDf7P/X3o1uYmyNvfLvRLczNdfdun79cjrjZnUa3MDdXXfyR0S1slltP7htLVf1hkq8m+Z7u/ua0BbbP/1EAAABsrquv2ejufUk+s5YwlpghAwAAmNWdquqyyXYlOXqyX0m6u6+3WgGBDAAAYDa7uvvq1U87OEsWAQAAZvMP6y0gkAEAAMym1lvAkkUAAIDZ3Liqnn6wg93926sVEMgAAABmsyPJdbOOmTKBDAAAYDZf6u4z11PAPWQAAACzcQ8ZAADAID9RVb+Y5PuSfCTJH3X33rUUMEMGAAAwmxckuXMOhLEHJHn+WguYIQMAAJjN7br7jklSVX+U5H1rLWCGDAAAYDZXX7Ox1qWK1zBDBgAAMJs7VdVlk+1KcvRkv5J0d19vtQICGQAAwAy6e8d6a1iyCAAAMIhABgAAMIhABgAAMIhABgAAMIhABgAAMIhABgAAMIhABgAAMIhABgAAMIgXQwMAAMygqh620vHu/svVaghkW0BVnZFk16iv3917Rn1tAADYyqpqd5Ldi4YWunthsv2aJB+cfJKkFp3XSQSyQ8QuoQgAALaeSfhaOMjhhyd5VJKTkrwuyZ919wVrqe8eMgAAgBl092u7+9QkP5bkn5M8v6reXVU/Nm0NgQwAAGB9rkjyjSSXJblO1nA7kiWLAAAAM6iqeyd5dJK7Jnlrkhd293lrqSGQAQAAzOZtST6c5N1Jjkry2Kp67DUHu/u/rFZAIAMAAJjNf8qBpynOTCADAACYQXe/fL01Zgpko9+bdYg5YXQDAADA/FXV67PCDFl3P2i1GrPOkHlv1pSqas/oHgAAgA3xvPUWsGQRAABgBt39roMdq6p7TFNDIAMAAJhBVe1I8sgkxyV5c3d/tKoemORZSY5OcvJqNQQyAACA2fxRklskeV+SF1XV55L8hyRndPfZ0xQQyAAAAGZz5yQndff+qtqV5KtJvq+7L562wLU2rDUAAIDt7aru3p8k3X1Fkk+uJYwlZsgAAABmdduq+vBku5J876L9dPdJqxUQyAAAAGZzpyTHJvnCkvFbJrlomgKWLAIAAMzmBUku6+7PLf4kuXxybFUCGQAAwGxO6O4PLx3s7vOSnDBNAYEMAABgNrtWOHb0NAUEMgAAgNmcW1VPWjpYVU9I8v5pCnioBwAAwGx+Mclrq+ox+bcAduckRyZ56DQFBDIAAIAZdPeXk/xIVd07yR0mw2/o7rdPW0MgAwAAWIfufkeSd8xyrUDGdzj6+O3xS+JX/um9o1uYm+f8wN1HtzA3F1/+1dEtzM2zzrtgdAvzc4PtcTvxTW48uoP52bFjdAfz8/ld2+PXV5LsvfIbo1uYmyNr+/x3yTb6XvrrXxzdwtxc/s2lr+XiYLbPr2AAAIBDjEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwiEAGAAAwyM7RDbCxquqMJLtWOqe792xONwAAcGipqt1Jdi8aWujuhUXH753kKUm+fzL0T0le3N3vnKa+QLb97RK4AABgNpPwtbDcsar6iSQvTnJmkt9IUkl+KMnLqurJ3f3G1eoLZAAAALM5PclDuvtDi8Y+WFXnJfndJKsGMveQAQAAzOamS8JYkqS7P5zk2GkKCGQAAACz+daMx/6VJYsAAACz+d6qOmeZ8Upy62kKCGQAAACzefAKx543TQGBDAAAYAbd/a7lxqvqFklOTbLs8cXcQwYAALBOVXWjqvr5qvrbJO/MlA/1MEMGAAAwg6o6JslDk5yW5DZJXpvk1t19/LQ1BDIAAIDZfCXJ+5L8apJ3d3dX1UPXUsCSRQAAgNk8K8muJC9N8syq+t61FhDIAAAAZtDdL+juuyV5UA486v7sJDevqmdU1W2mqSGQAQAArEN3f7q7n9Pdd0xylyTXT/Kmaa51DxkAAMCcdPdHqurXkpw/zflmyAAAAGZQVderqmdW1Yur6n51wFOSXJDkEdPUMEMGAAAwmz9OcmmS9yR5YpLTkxyZ5CHd/cFpCghkAAAAs7n15L6xVNUfJvlqku/p7m9OW0Ag23hXVNWeVc45YRP6AAAA5uvqaza6e19VfWYtYSwRyDZcd5+12jlTBDYAAGDruVNVXTbZriRHT/YrSXf39VYrIJABAADMoLt3rLeGpywCAAAMIpABAAAMIpABAAAMIpABAAAM4qEeW1BVnZFk15zKnbCWk7vn9FUHu/zK0R3Mz8WXf3V0C3Nz02vfaHQLc7Pr6AtGtzA3V22TH83d8frb5BtJ8uWr9o9uYW6+sHeb/MWSpKpGtzA3ffklo1uYmysv/8roFuZm567rj25hbq6++vLRLRwyBLKtaVd375lHIY/UBwCArWv7/DgRAADgECOQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAArFFVXXeFY987bR2BDAAAYO0+VFWPXDxQVbuq6r8nefO0RXbOvS22miuqas9KJ3T3iscBAOBwVVW7k+xeNLTQ3QtJ7pfkxVX1pCQ/n+T2SZ6X5OwkJ09bXyDb5rr7rNE9AADAoWoSvhaWGf/nJA+oqtOTfDzJxUn+Y3efv5b6liwCAACsUVXtrKpnJvnZJP85yXlJXlRV37+WOgIZAADA2n0gyXFJfri7F7r7IUlekOR1VfXcaYsIZAAAAGv3+O5+cnd/45qB7v6rJD+YZN+0RQQyAACANeru9x/k0A8nucG0dTzUAwAAYB2q6geTnJbkkUk+k+Qvpr1WIAMAAFijqrpNklOTPDrJJUlenaS6+95rqSOQAQAArN3Hk/xdkp/s7guSpKqettYi7iEDAABYu4fnwLvH3lFVf1BV90lSay0ikAEAAKzd67v7UUlum+SdSZ6W5NiqemlV3W/aIgIZAADA2r0vSbr7W939P7v7gUmOT/LBJGdMW0QgAwAAWLt/tzyxu7/W3b/f3adMW8RDPQAAANbuxlX19IMd7O7fnqaIQAYAALB2O5JcNzM8yGMxgQwAAGDtvtTdZ663iHvIAAAA1m5dM2PXEMgAAADW7j7zKCKQAQAArFF3f20edQQyAACAQQQyAACAQQQyAACAQQQyAACAQQQyAACAQbwYmu9w1WX7R7cwF5dso581POu8C0a3MDe7jt4+38v/uMvdR7cwN6d/9L2jW5iLj26TP7+SZN++0R3MT+3t0S3MzZFH33B0C3Pz+U+9bnQLc/O1y744uoW5ue2Rx4xuYW6OOOLao1s4ZGyf/2sFAAA4xAhkAAAAgwhkAAAAgwhkAAAAgwhkAAAAgwhkAAAAgwhkAAAAgwhkAAAAgwhkAAAAg+wc3QBJkiuqas+i/RMG9QEAAGwigWwL6O6zFu8vCWcAAMA2ZckiAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADAIAIZAADADKrqN9dbQyADAACYzf3XW2DnPLpg7u5ZVXs264t196Z9LQAAOJRU1e4kuxcNLXT3wmR7R1XdIEktd213f221+gLZ1rRTSAIAgPEm4WvhIIdvm+T9WT6QdZJbr1ZfIAMAAJjNx7r75PUUcA8ZAADAIAIZAADAbF643GBV7aqqR0xTQCADAACYQXe//JrtqtpRVQ+oqlcm+VySR01Twz1kAAAAM6qq/yPJaUl+Isn7ktwjya26+/JprhfIAAAAZlBVFyb5fJKXJjm9u79ZVZ+ZNowlliwCAADM6i+SHJcDyxN/sqqukwOPu5+aQAYAADCD7n5qkhOS/HaSeyf5ZJIbV9Ujq+q609QQyAAAAGbUB7y9u5+UA+HstCQPSfLZaa4XyAAAAGZQVS9fvN/dV3f367v7tCS3mKaGQAYAADCbkw52oLu/PU0BT1kEAACYzbWr6uQktdzB7v7H1QoIZAAAALM5Lsnzs3wg6ySnrFZAIAMAAJjNBd29auhaiXvIAAAABhHIAAAAZvOMxTtVdURVnVxVN5m2gEAGAAAwm4dV1e2TpKq+K8mHkrwyyQeq6tHTFBDIAAAAZvOj3X3+ZPtnknyyu++Y5IeT/NdpCghkAAAAs7lq0fZ9k5ydJN198bQFBDIAAIDZfL2qHjh5F9k9krw5SapqZ5KjpyngsfcAAACz+dkkL0py0yS/uGhm7D5J3jBNAYEMAABgBt39yST3X2b8LVX1A9PUEMj4Dvu/tm90C3Nx5fb4Ng64wfZZWXzV9vlWcvpH3zu6hbn5rTvcfXQLc3HmJ94zuoW5OfH6R4xuYW7OPnb7fC9HXeemo1uYm1ve/jGjW5ibm136z6NbmJtL/uVDo1uYmy9d8qnRLWwFT0/yO6udtI3+9wgAAGDLqGlOEsgAAADmr6c5yZJFAACAGVTVN7N88Kp4yiIAAMDG6e5j1lvDkkUAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBDIAAIBBBLKt6bOjGwAAADaeQLY1fXZ0AwAAwMYTyAAAAAYRyAAAAAYRyAAAAAYRyAAAANaoqm67aPuoJcfuPm2dnfNsis1TVWck2TWPWt29Zx51AABgu6mq3Ul2Lxpa6O6FJH+a5IcmY+9ZtJ0k/2PJ/kEJZIeuXYIUAABsrEn4WljmUB1ke7n9g7JkEQAAYO36INvL7R+UGTIAAIC1O76qXpQDs2HXbGeyf9y0RQQyAACAtTt90fZ5S44t3T8ogQwAAGCNuvsV86jjHjIAAIA1qqp7VtVjF+2/pqrePvmcMm0dM2QAAABr9xtJnrJo//uTPD7JdZI8K8nbpylihgwAAGDtrtfdH1u0/6nufn93/22SY6YtIpABAACs3fUX73T3wxbtHjttEYEMAABg7T5eVT+xdLCqHpjkE9MWcQ8ZAADA2j0tyRuq6qeS/ONk7IeT/EiSB05bxAwZAADAGnX3BUlOSvJ3SU5Icssk70ryn5I8ddo6ZsgAAABm0N1XJnlZVZ2c5NFJfj3JZ5L8xbQ1BDIAAIA1qiDQ+j8AACAASURBVKrbJDk1B4LYJUlenaS6+95rqSOQAQAArN3Hc2C54k9Oli+mqp621iLuIQMAAFi7hye5OMk7quoPquo+SWqtRQQyAACANeru13b3o5LcNsk7c+Cpi8dW1Uur6n7T1hHIAAAAZtTd3+ru/9ndD0xyfJIPJjlj2usFMgAAgDno7q919+939ynTXiOQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADLJzdANsLTtuuD1+SdQRozuYn5vceHQH83PH62+fnwF99LL9o1uYmzM/8Z7RLczFs7//P4xuYW5ef9EHRrcwP/tGNzA/ey//yugW5mb/1d8e3cLcXLGN/rscsfPao1uYmyN2HDW6hUPG9vm/IwAAgEOMQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADDIztENsKwrqmrPKuecsAl9AAAAG0gg24K6+6zVzpkisAEAAFucJYsAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAABzUFVHVNXJVXWTaa8RyAAAAGZQVb9XVbefbH9Xkg8leWWSD1TVo6epsXMD+2MTVdUZSXbNcm1375lvNwAAsD1U1e4kuxcNLXT3wmT7R7v75ybbP5Pkk939kKq6aZI3Jfmz1eoLZNvHLsEKAADmaxK+Fg5y+KpF2/dN8v9Nrrm4qqaqb8kiAADAbL5eVQ+sqpOT3CPJm5OkqnYmOXqaAmbIAAAAZvOzSV6U5KZJfrG7L56M3yfJG6YpIJABAADMoLs/meT+y4y/JclbpqkhkAEAAMygqn43SS8a6iRfTfKO7n73NDUEMgAAgNmct8zYdyf5rap6dXf/zmoFBDIAAIAZdPcrlhuvqt9L8r+SrBrIPGURAABgjrr729Oea4YMAABgTiaPvP/pJBdOc75ABgAAMIOq+mYOPMhj8VugL0/yrhx4JP6qBDIAAIAZdPcx660hkAEAAMxoskTxAUluOxn6WJK3dPfeaa73UA8AAIAZVNXNk5yf5JeS3DzJcUn+a5LzJ8dWZYYMAABgNr+Z5KVL3zdWVf8lyXOTPG61AgIZAADAbO7e3Y9fOtjdL6qqT0xTwJJFAACA2az0vrHLpylghgwAAGA231VVD1tmvJJcb5oCAhkAAMBs3pXkJw9y7G+nKSCQAQAAzKC7f2a9NQQyAACAGVTVY1c43N39x6vVEMgAAABmc5dlxioHljEel0QgAwAA2Ajd/ZRrtquqkjwmyTOSvDfJc6apIZABAADMqKp2Jnl8kl9K8g9Jfqq7p3oHWSKQscS+z105uoW5OOI2R41uYW527Bjdwfx8+ar9o1uYm337RncwPyde/4jRLczF6y/6wOgW5uYnb37y6Bbm5k92vm10C3PT+64e3cLcHHGdY0e3MDf79l4xuoW52buNvpdvX/2t0S1siqr6hSRPTfK2JPfv7s+ttYZAdui6oqr2LNo/YVAfAABwuPrdJF9Jcs8krz+wajHJgfvIurtPWq2AQHaI6u6zFu8vCWcAAMDGu9V6CwhkAAAAM5hlieJSAhkAAMAMquqbSXq5QzmwZPF6q9UQyAAAAGbQ3cest8a15tEIAAAAayeQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAADCKQAQAAzKCqHn6Q8SOr6temqbFzvi0x0BVVtWeWC7t7pusAAGC7q6rdSXYvGlro7oXJ9u6qemKS/9zdn5mc/4AkL0jy5mnqC2TbRHefNboHAADYbibha+Egx/5jVT06yVur6k+T3CHJjZM8qrs/NE19gQwAAGB2f57k9kmeluTrSU7p7k9Oe7F7yAAAAGZQVfdM8oEkN0xyiyRPTvL6qjqzqo6apoZABgAAMJvfSfLE7v757r60u89O8kNJjkpiySIAAMAGumt371880N3fSvKMqnr5NAUEMgAAgBl09/6qukmSX8iB+8g6yceSvKS7/2maGpYsAgAAzKCq7pHk3MnuK5P8yWT7fZNjqzJDBgAAMJvnJ3lId39g0djrquq1SX4/yd1WK2CGDAAAYDbXWxLGkiTd/cEkx0xTQCADAACYTVXVDZYZ/O5MmbUEMgAAgNm8IMlfV9WPVdUxk8+9krxpcmxV7iEDAACYQXcvVNVFSf5bDjxlMUnOT/Lfu/v109QQyAAAAGbU3X+V5K9mvV4gAwAAmEFVPXuFw93d/221GgIZAADAbL61zNh1kjwhyQ1zYCnjigQyAACAGXT386/Zrqpjkjw1yc8keVUOvKNsVQIZAADAjCaPuH96ksckeUWSH+ruS6e9XiADAACYQVX9VpKHJVlIcsfu/t9rreE9ZAAAALP5pSQ3T/KrSS6qqssmn29W1WXTFDBDBgAAMIPuXvcElxkyAACAQQQyAACAQQQyAACAQQQyAACAQTzUg+9w9HnfGN3CXOz9+JGjW5ibz+/aPj83+cLeHt3C3NQ2+l7OPvaI0S3Mx77RDczPn+x82+gW5ubVv36f0S3MzalX7R/dwtxcdeT2+bulr3eX0S3MzbX/+YrRLczND37rp0e3cMjYPr8bAQAADjECGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwCACGQAAwAyq6klVdeJku6rq/62qy6rqw1X1Q9PU2LmxLW5dVXVGkl2j+9gKunvP6B4AAGArqqrdSXYvGlro7oXJ9lOTvHyy/egkJyW5VZKTk7wwyY+uVv+wDWRJdgkiAADASibha+Egh/d299WT7QcmeWV3X5LkrVX1/0xT35JFAACA2eyvqptV1a4k90ny1kXHjp6mwOE8QwYAALAez05yXpIdSc7p7vOTpKp+LMmnpykgkAEAAMygu/+qqm6Z5JjuvnTRoXOTnDpNDUsWAQAAZtTde68JY5MnLZ6S5EVJLpjmeoEMAABgHarqblX1wiSfS3JOkr9LcttprhXIAAAAZlBVz6mqTyX5zSQfyYHH3f9Ld79iyRLGg3IPGQAAwGx2J/lEkpcm+avuvqKqei0FzJABAADM5qZJnpPkQUkuqKo/TnJ0VU098WWGDAAAYAbdvS/Jm5K8afIusgcmuXaSL1bV27r7tNVqCGQAAADr1N1XJHlNktdU1TFJHjbNdQIZAADADKrq6eut4R4yAACA2Ryz6PPLS/avO00BM2QAAAAz6O7fuGa7qh6yeH9aZsgAAADWb02Pu7+GQAYAADCIJYsAAAAzqKqP5N9mxr6vqj68+Hh3n7RaDYEMAABgNg9LcmySLywZv2WSi6YpYMkiAADAbF6Q5LLu/tziT5LLJ8dWJZABAADM5oTu/vDSwe4+L8kJ0xQQyAAAAGaza4VjR09TQCADAACYzblV9aSlg1X1hCTvn6aAh3oAAADM5heTvLaqHpN/C2B3TnJkkodOU0Ag4zvs+PYVo1uYi/2f+vvRLczN3iu/MbqFuamq0S3MzZFH33B0C3Nz1HVuOrqFudh7+VdGtzA3ve/q0S3MzalX7R/dwty86jn3Hd3C3DzquW8f3cLc1M2PGN3C3Owd3cAc7b3hdUe3sCm6+8tJfqSq7p3kDpPhN3T31L/JBDIAAIB16O53JHnHLNe6hwwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGAQgQwAAGBOqupGVVXTni+QAQAAzKCq7l5V76yqv6yqk6vqo0k+muTLVXX/aWrs3NgWSZKqOiPJrtF9HEx37xndAwAAbEVVtTvJ7kVDC929MNl+cZJnJfmuJG9P8oDufm9V3TbJnyV582r1BbLNsUvoAQCAQ88kfC0c5PDO7v7rJKmqM7v7vZNrPj7tqkVLFgEAAGazf9H2t5cc62kKmCEDAACYzZ2q6rIkleToyXYm+1PdsiSQAQAAzKC7d6y3hiWLAAAAg8w6Q3ZFVe2ZZyMDnDC6AQAA4PA2UyDr7rPm3chm2waBEgAAOMRZsggAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADCIQAYAADDIztENsLXsvfijo1uYiyNvfLvRLczNkbV9fm7Sl18yuoW5+fynXje6hbm55e0fM7qFudh/9bdHtzA3R1zn2NEtzM1VR26fP8Me9dy3j25hbl79zFNGtzA3L/nMe0e3MDfv+l+jO5ijS/eO7uCQsX3+lAQAADjECGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAACDCGQAAPz/7d171Fxleffx72USCAhoBVSEahAPWCsgh5IKukBexbrigXoIKR7SVRupiL4gYNBXG+pbpbaU2kK1ARXpqoIulQoq1ipQKiAVCBAUEUQ8IHJUQIkQuPrH3gM7k+ewZ7KfuScP389az8rsw9xz3bNn7z2/2YdIKsRAJkmSJEmFGMgkSZIkqRADmSRJkiQVYiCTJEmSpEIMZJIkSZJUyNzSBaidiFgOzJ+JtjNzxUy0K0mSJG3sImIZsKwxamVmrqyn/eNUz83Md0zXvoFs4zHf4CRJkiSNVh2+Vk4y+VBgNfBZ4GYgBm3fQCZJkiRJw9kOeB2wGFgLnAl8PjPvatuA15BJkiRJ0hAy847M/Fhm7g8sBR4PXBMRb2zbhkfIJEmSJGkDRMTuwBLgJcBXgcvaPtdAJkmSJElDiIjjgEXA94AzgGMzc+0gbRjIJEmSJGk47wN+COxa/30wIqC6uUdm5i7TNWAgkyRJkqTh7LihDRjIJEmSJGkImXnTROMjYg5wMDDh9CbvsihJkiRJQ4iIrSLi2Ig4KSJeGpXDqU5jfH2bNjxCJkmSJEnD+VfgLuBi4C3A0cAmwKsyc1WbBgxkkiRJkjScp2fm8wAi4lTgduCpmXlP2wY8ZVGSJEmShvNA70FmPgjcOEgYg0f3EbI1EbFiRK+1YESvI0mSJGl0do2Iu+vHAWxWD/due7/VdA08agNZZh4/qtcaYfCTJEmSNCKZOWdD2/CURUmSJEkqxEAmSZIkSYUYyCRJkiSpEAOZJEmSJBViIJMkSZKkQgxkkiRJklSIgUySJEmSCjGQSZIkSVIhBjJJkiRJKsRAJkmSJEmFzC1dwGwTEcuB+X2j94uIFRvY9IINfH4r99z5g1G8zIzbYu1vS5fQnZg9v5v89je3li6hM3fe/bPSJXRmu7tuKF1CJ9bMos/Xg2vXlC6hM7nVXqVL6Ew8ZV7pEjpz8o2XlC6hM4ftuLB0CZ258PKLSpfQmYc2nT3fX2aagax78zNzRXNERKzoHzeoDgKdJEmSpDFjdJUkSZKkQgxkkiRJklSIgUySJEmSCjGQSZIkSVIhBjJJkiRJKsRAJkmSJEmFGMgkSZIkqRADmSRJkiQVYiCTJEmSpEIMZJIkSZJUiIFMkiRJkgoxkEmSJElSIQYySZIkSSrEQCZJkiRJhRjIJEmSJKkQA5kkSZIkFWIgkyRJkqRCDGSSJEmSVIiBTJIkSZIKMZBJkiRJUiEGMkmSJEkaUESc1kU7BjJJkiRJGtwuXTQyt4tGJEmSJOlRZvOIeD4QE03MzMvbNGIg23isiYgVM9FwZs5Iu5IkSdLGLiKWAcsao1Zm5kpge+AEJg5kCby4TfsGso1EZh5fugZJkiTp0aYOXysnmHR9ZrYKXVPxGjJJkiRJKsRAJkmSJEmD+9BkEyLiqW0bMZBJkiRJ0uCW9x5ExDf6pp3VthEDmSRJkiQNrnkzjydMMW1KBjJJkiRJGlxO8nii4Ul5l0VJkiRJGtwTI+JIqqNhvcfUw9u2bcRAJkmSJEmDOwXYcoLHAKe2bcRAJkmSJEkDyszjumjHQCZJkiRJA4qI5wI7ZeaX6uETgcfVk0/KzMvbtONNPSRJkiRpcMcDtzeGDwS+DJwHvL9tIx4hkyRJkqTBbZeZFzWG787MzwNExFvbNuIRMkmSJEkaXPMmHmTmwsbgE9s2YiCTJEmSpMHdHBF794+MiIXAzW0b8ZRFSZIkSRrcu4EzI+I0oHcDjz2ANwOL2zbiETJJkiRJGlBmXgosBOYAS+u/xwAL62mteIRMkiRJkoaQmb9ggjsqRsQ+mfmtNm0YyCRJkiRpQBExB3g9sD3w1cy8JiIWAe8BNgOe36YdA5kkSZIkDe7jwO8ClwL/FBE3AX8ILM/Ms9o2YiCTJEmSpMHtCeySmQ9FxHyq/yT6GZl5yyCNeFMPSZIkSRrc/Zn5EEBmrgGuGzSMgUfIJEmSJGkYO0fEVfXjAHaqhwPIzNylTSMGMkmSJEka3HO6aMRApnU8dssdSpfQiXnb7Vq6hM7kL39WuoTOzJ3/+NIldGbnTbYsXUJn7rjtytIldGLe3M1Ll9CZtWvXlC6hM5vfMHv6srZ0AR264KLSFXTnwstnT2fOeM0LSpfQmQ9ff0npEmZcZt400fiI2Af4E+CwNu0YyLq3ICJW9I2bPXsjSZIkSeuIiN2oQtjrgRuBL7R9roGsez/KzBWli5AkSZI0cyLiWcDBwBLgDuBMIDJz/0HaMZBJkiRJ0uCuBS4EXpGZ1wNExBGDNuJt7yVJkiRpcK8BbgHOi4hTIuIAqjssDsRAJkmSJEmDOzszFwM7A+cDRwBPioiPRsRL2zZiIJMkSZKkwV0KkJm/zsx/y8xFwA7AKmB520YMZJIkSZI0uPVOT8zMOzPzXzLzxW0b8aYekiRJkjS4bSPiyMkmZubft2nEQCZJkiRJg5sDbMEQN/JoMpBJkiRJ0uB+npl/taGNeA2ZJEmSJA1ug46M9RjIJEmSJGlwB3TRiIFMkiRJkgaUmXd20Y6BTJIkSZIKMZBJkiRJUiEGMkmSJEkqxEAmSZIkSYUYyCRJkiSpEAOZJEmSJBViIJMkSZKkQgxkkiRJklSIgUySJEmSCjGQSZIkSVIhBjJJkiRJKmRu6QI0uYhYDsyf6dfJzBUz/RqSJEnSxigilgHLGqNWZubKetqbpnpuZp4+XfsGsvE237AkSZIklVOHr5WTTN5rgnEBvALYHjCQSZIkSdJMyMzDe48jIoBDgHcDlwB/3aYNA5kkSZIkDSki5gJLgXcB3wZem5nfb/t8A5kkSZIkDSEiDgPeCXwDeFlm3jRoGwYySZIkSRrOPwG3AvsCZ1dnLQLVdWSZmbtM14CBTJIkSZKG8zbgXCAnmLa4TQP+P2SSJEmSNJyTgU8AazPzpuYfcHCbBgxkkiRJkjScq4DPAJdExOv6psUE86/HQCZJkiRJw8nMPAU4ADgmIj4ZEZv3prVpwEAmSZIkSRsgM68D/hD4BXBFROzd9rne1EOSJEmShvPwaYmZuRZYHhHnUp3GuG2bBgxkkiRJkjSc4/pHZOb5EbEH8NY2DRjIJEmSJGkImXnWJOPvAo5v04bXkEmSJElSIQYySZIkSSrEQCZJkiRJhRjIJEmSJKkQA5kkSZIkFWIgkyRJkqRCDGSSJEmSVIiBTJIkSZIKMZBJkiRJUiEGMkmSJEkqxEAmSZIkSYXMLV3ALLQmIlZ01NaCjtppbdNtnjPql5wR999ydekSOvObe35SuoTOPPDAb0qX0Jl58zYvXUJnfn7HD0qX0Il5czYtXUJn7nvg16VL6Mxuv35j6RI6s3brLUqX0J271pauoDMPbTp7ji98+PpLSpfQmWOesbB0CRsNA1nHMvP4rtrqMNhJkiRJGkOz5ycFSZIkSdrIGMgkSZIkqRADmSRJkiQVYiCTJEmSpEIMZJIkSZJUiIFMkiRJkgoxkEmSJElSIQYySZIkSSrEQCZJkiRJhRjIJEmSJKkQA5kkSZIkFWIgkyRJkqRCDGSSJEmSVIiBTJIkSZIKMZBJkiRJUiEGMkmSJEkqxEAmSZIkSYUYyCRJkiSpEAOZJEmSJBViIJMkSZKkQgxkkiRJklSIgUySJEmSCjGQSZIkSVIHImLriDgoIvZo+5y5M1mQNtiaiFgx0y+SmTP+GpIkSdLGKCKWAcsao1Zm5sp62jnA8sxcHRHbAZcD3wF2ioiVmfkP07VvIBtjmXl86RokSZKkR7M6fK2cZPKOmbm6fvynwNcz800RsSXwLWDaQOYpi5IkSZI0nAcajw8AvgKQmfcAD7VpwCNkkiRJkjScn0TE4cBPgd2BcwEiYjNgXpsGPEImSZIkScP5M+C5wFJgcWb+sh6/EPhkmwY8QiZJkiRJQ8jMW4FDJxh/HnBemzYMZJIkSZI0hIg4G8jJpmfmK6drw0AmSZIkScP5uw1twEAmSZIkSUPIzAt6jyNi23rcbYO04U09JEmSJGlIEfGXEXE7cC1wXUTcFhHvb/t8A5kkSZIkDSEijgD2BfbKzK0z83eAvYF96mnTMpBJkiRJ0nDeBCzJzBt7IzLzh8Ab6mnTMpBJkiRJ0nDmZebt/SPr68j8j6ElSZIkaQbdP+S0h3mXRUmSJEkazq4RcfcE4wOY36YBA5kkSZIkDSEz52xoG56yKEmSJEmFGMgkSZIkqRADmSRJkiQVYiCTJEmSpEIMZJIkSZJUiIFMkiRJkgoxkEmSJElSIQYySZIkSSrEQCZJkiRJhURmlq5BjzIRsSwzV5auowv2ZTzZl/E0W/oyW/oB9mVc2ZfxZF/G02zoi0fIVMKy0gV0yL6MJ/synmZLX2ZLP8C+jCv7Mp7sy3ja6PtiIJMkSZKkQgxkkiRJklSIgUwlbNTn+faxL+PJvoyn2dKX2dIPsC/jyr6MJ/synjb6vnhTD0mSJEkqxCNkkiRJklSIgUySJEmSCjGQaVoR8WBErIqI1RHxuYjYvh5eFRG3RMTPGsOb9M1/dkQ8vq+9IyJiTUQ8rh4+sPH8eyPi+/Xj0yNiv4g4p/HcV0fEVRFxbURcHRGvnoH+PjkizoiIGyLiuxHxlYh4VkTcV9f13bq2efX8D9cYEUsjIiPigEZ7B9XjXtt1rS37M9DyqMftFxG/iogr6uXxXxGxaPTVrysitp7ms9d7r3duPGfPuu+b1MM7RcQPI2KrEdZ97wTjnh0R59e1fy8iVk61LjSe95G634+ph/+08Zz76/ViVUQcP6r+1XVM9N4/MyLOqdelyyLivIh4UT1taUTc1qh9VUT83ohrvrf+d0Fd++GNaSdFxNL68WkRcWNEXBkR19Xr//b97TSGl0bESfXj9ZbziPqWEXFCY/ioiFjRGF5Wb0evjYhLI2Lfevycelm9qDHvf0TE60ZRd9MU2+LVffOtiIijGsNzI+L2iPhQ33yL6m3alXV7bx1VX9pqbK+vjIjLI+IFpWtqY7L9TGPd+kBj3m0i4oHeOjLDdQ2zz1gQk+zv6+l/UK/TP6iX0Zcj4nn1tBV9r7EqIhZPt12fwf4f1FfLqoh4KCL+Yqpt3rhpfL6uqdeNI+ORfWDzO9iTotrn9Nbxr5StfACZ6Z9/U/4B9zYe/xtwZGN4BXDUFPN/Cnhv3/RLgQuBpRO81vnAno3h/YBz6se7AtcDO9bDO9bDu3TY1wAuBg5tjNsNeCGwuh6eA3wTOGSCGpcCVwGnNp5/JrAKeO0YLL9Wy6PZp8Z78CPggNKfx2k+e5+t+7Kib/w/A++pH58LLCm1DBrjvga8qjH8vL7p66wL9bjHAD8GLgH2m6DNHwHbFFoe67z3wHzgOuCVjXl+v/c5q9eVkwp/hu6t/10A/KLenmxSjzupUetpvfW33kYcUfdtk2Y7jXYf7tt0y3kG+7YGuLH3eQCOaiybRcBljWm715+rJ9fDewNXA/OAJcDXCiybabfFjfHrbAuAlwPfAm7gkWvl5wE3AzvUw5sCzy75+ZvqM1k/PhC4oHRNQ9T98H6mXrduAK5oTP8Lqn3iSNf//s9JPW69fUZd82T7+ydRbWdf0Jh/X+DVk71G3+udT992fcTvwTLgAuDpU23zxu2v7/P1ROA/gePq4f145DvYvwDvbMzb2ffDmf7zCJkGdSHwjAHmvxho/pK8E7AF8P+odvSDOAr4YGbeCFD/+yHg6AHbmcr+wAOZ+bHeiMxcBfykMfwgVYjZfv2nA9V79AcRMS8itqB6v1Z1WOOGGGp51O/BXwFvn+kCh1W/1/sAfwYc3Df5PcBbIuIYYF5mfmbU9U1gO+CnvYHMvLrFc/YHVgMfZfD1Z8ZM8t4fAlycmV/qzZeZqzPztNFX2MptwDeAN081U1ZOBG4B/qhFu8Ms5y6spbrz2BETTHs3cHRm3l7XdDnVl+jD6uFvAxdRfbn8YG/8iE27LZ7CEuAjVCFzYT1uS2AucEfd1m8z8/udVty9rYC7ShcxhHX2M8B9wPciYs96eDFVECpqmn0GMOH+/u3ApzLzosY8/52ZZ81wuRssIp4FvB94I/AQLbd54yYzb6UKlm+PiOib3L+9vWqUtW0IA5lai4i5VF9AWn2hiIg5wAHAlxqjlwCfoQotz46IJw5QwnOpftVt+k49viu/P8FrrCMi5lP9gnzuJLMkSiljUgAABrNJREFU1a83BwKvYt3+F9PB8rgc2HmK6aW9Gjg3M68D7oyI3XsTMvOXwN9QBfi3Faqv34nANyPiq1GdNvr4aZ/xyPL6IrCoeRpNYRO998+l+sxMpXkqz6qI2GzGK53a8cC76nVlOm3Xh2GWc1dOBg6JxunItTbb0mOB/wt8OjOvn7kSJzXVtnin5ucGOLQ3of4MHQCcQ7WuLAHIzDuptn03RcRnIuKQ3ilPY2azul/XAqcCH5juCeNkkv0MwBnAwRGxA/Ag1dHK0ibdZ/RMsL9vs107ovH5PK/TiodU7ys+TXX07seNSYNs88ZGZv6QKsP0f2c5Gfh4VKfHvzcinjL66oYzjhsjjZ/N6p3ed6h+cfx4y/nvAJ4AfL0x7WDgjMx8CPgCMMh1CUEVdqYbN1N2avTrx9P88nIGVV8PpvpSUFJXy6P/l6hxs4Tqfaf+t/8I0h9RnaIx0uuUJpOZnwSeA3yO6pSLSyJi08nmj+oauJcDZ2Xm3cC3gZeOoNQ2pnvviYgvRnV9yRcao8/MzN0af/eNotjJ1EfdLwX+pMXs060PWbc50HLuUv05OR14R4vZ+7elLwJ+RRWMxs0Nzc8N8LHGtEXAeZn5G+DzwEG9L5uZ+RaqsHAp1RkXnxhx3W3cV/drZ+BlwOkTHAUYR1PtZ6AKNC+h2jacOeLaJjPVdqvV/j4ivh3VtaEfaYw+sfH53H9GKh/cB4BrMvOM5sgBt3njZr31IjO/RnU65ilUP5hdERHbjrqwYRjI1MZ9jY3L4Zl5f5v5gacBm1Cf7hIRuwDPBL4eET+iCgODnHZ1DbBn37jdge8O0Eab19hjkmk31P16BrAwIl45WSOZeSnVF5lt6l/fSupqeTwf+N7MljqciNgaeDFwat2Xo6mOvkQ9fRHwOKqjln8bEZuXqrUpM2/OzE9k5quoTjGb6svvy6j6cHXdx30Zg9MWJ3vvqdal5lHKg6iurXrC6KscyAepTumbbv/YXB/uqwNzzxOA23sDAy7nrv0D1SlZj22M+y7rb+ce3pZGxGOBD1Mt120j4uUjqLPfVNviqSwB/k/9WbwM2Jrq9EegOmW0PuX0JcBrOqhzxmTmxcA2wMbwhXLC/UxP/b3hMuBdVEG5qOn2GUy+v+/fru0NvI9q2zyWImI/qs/6ZJcctN3mjY2IeDrVkdZb+6dl5p2Z+enMfCPwP1Q/Lo29jebN18YnM39F9cvsUfXh8iVUF84uqP+eAmwfEU9r2eTfAcdGxAKo7oREdW3QCZM+Y3DfBDaNiD/vjYiIvah2MgBk5s+B5VSn9Ezl2Lq+sbAhy6MOb++jOh1gHL0WOD0zn1b35Xepbmiwb30K0wnAYfX1O/8OvLdgrQBExMvikTt1Ppnqi+PPpnjKEuAtveVFdVObl45BuJzsvb8O2Kfvh4vStU4rM6+lCiYT3lU0Ku+gulahdxrTBcAb6umbAa8HzquHB13OnapP1fssVSjr+TDwN/WXUiJiN6qw/M/19PcDn63fi7cBJ9anbo3StNviflHdOXVf4KmN9eQwYElEbFF/Me3ZDbhpJgrvSlR3/ptDfd3bxmCC/UzTCcC7M3Mc+jPpPqM50wT7+5OBpbHu3S/HdrsWEb8DfBJ4U2beM9E8023zxk19xOtjVDeFyb5pL+7tEyNiS2AnqjO7xp6BTDMqM68AruSR0/e+2DfLF5nkYtoJ2lpF9SvO2fX59WcDx9Tju6o3gYOAl0R1q+VrqC5s7z/f/Sxg84h44RRtfTUzx+L88Z4Bl8cLo77tPdVO6B2Z+Y2RFTuYJazfl89TnYbxPqrT/HpHUldQXcvwzNGVx+YR8dPG35FUpxuujogrqe7Ed3Rm3jLRk+sdzIHAl3vjMvPXwH8Dr5j58qc01Xu/CDg0qv9m4GKqm8f8/8Z8/deQjcstvv8a2KFv3N/Wy+o6YC9g/8bZAu8E/rg+xekS4HOZ+V/1tNbLeQadQHWkBYCsbrTyCeCielt6CvCGzPx5VP/1wEFU70Fvu/s1qm3vyAywLW76Y+Cbmfnbxrh/B15JFWyOifqW48BxVCF03PSuIVtFdWrfm+sbS2w0+vYzzfHXZOanylS1nqm2W/0e3t/X6+5i4EMRcX1EXEQV7pq38D+ib7u2oPvyWzuU6jqrj8a6110u7ptvom3eOOmtF9dQXaP/H1TrcL89gO9ExFVUN5c5NTP/Z4R1Di36wqUkSZIkaUQ8QiZJkiRJhRjIJEmSJKkQA5kkSZIkFWIgkyRJkqRCDGSSJEmSVIiBTJIkSZIKMZBJkiRJUiH/C+OPxdU5XDwMAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 1080x1080 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "corr = df.corr()\n",
    "sns.clustermap(corr, figsize=(15, 15), cmap='mako', method='centroid')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TARGET     1.000000\n",
       "LSTAT      0.737663\n",
       "RM         0.695360\n",
       "PTRATIO    0.507787\n",
       "INDUS      0.483725\n",
       "TAX        0.468536\n",
       "NOX        0.427321\n",
       "CRIM       0.388305\n",
       "RAD        0.381626\n",
       "AGE        0.376955\n",
       "ZN         0.360445\n",
       "B          0.333461\n",
       "DIS        0.249929\n",
       "CHAS       0.175260\n",
       "Name: TARGET, dtype: float64"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# here we sort features in decreasing order of high correlation with target variable.\n",
    "\n",
    "np.abs(corr['TARGET']).sort_values(ascending=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now lets move on prediction.\n",
    "\n",
    "Let's imagine our goal is to optimize MSE (Mean Squred Error).\n",
    "\n",
    "In that case, trivial solution using constant predictor (always predicting same value irrespective of the input) would be the <b>sample mean</b> of targets in training dataset. One can proof mathematically, that for MSE the optimal constant predictor is indeed a mean. You can google it up :)\n",
    "\n",
    "Let's find out what the MSE is in this case (for simplicity we don't split the data and do everything on training set for now)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "84.4195561561656"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def mse(x, y):\n",
    "    return np.mean((y - x) ** 2)\n",
    "\n",
    "constant_predictor = df['TARGET'].mean()\n",
    "\n",
    "mse(constant_predictor, df['TARGET'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see empirically that deviations from optimal constant predictor causes MSE to grow, i.e MSE is optimal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(85.41955615616551, 85.41955615616557)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eps = 1\n",
    "\n",
    "mse(constant_predictor - eps, df['TARGET']), mse(constant_predictor + eps, df['TARGET'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ok, now let's try to do slightly better. \n",
    "\n",
    "Let's try to divide our training set in <b>two groups</b> and for each group, calculate <b>their own mean</b>. If we make this split in a smart way, intuitively the two means would be better than using just <b>single global mean</b> as we did in previous example.\n",
    "\n",
    "One way to do a split is to select on of the features of the data, let's say X, and some threshold, T, and make:<br /><br />\n",
    "group 1 - {all points having $X<T$}<br />\n",
    "group 2 - {all points having $X>=T$}<br />\n",
    "\n",
    "Then we will have 2 constant predictors - one for each group - and, for every input we first gonna determine its group, and then use group's mean to estimate the target.\n",
    "\n",
    "This intuition is captured by Decision Tree with depth 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.tree import DecisionTreeRegressor, plot_tree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Text(167.4, 163.07999999999998, 'RM <= 6.941\\nmse = 84.42\\nsamples = 100.0%\\nvalue = 22.533'),\n",
       " Text(83.7, 54.360000000000014, 'mse = 40.273\\nsamples = 85.0%\\nvalue = 19.934'),\n",
       " Text(251.10000000000002, 54.360000000000014, 'mse = 79.729\\nsamples = 15.0%\\nvalue = 37.238')]"
      ]
     },
     "execution_count": 10,
     "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": [
    "# defining decision tree with depth 1.\n",
    "dt = DecisionTreeRegressor(max_depth=1)\n",
    "\n",
    "# fitting the decision tree on our data/\n",
    "model = dt.fit(X, y)\n",
    "\n",
    "# visualizing the tree\n",
    "plot_tree(model, feature_names=cols, proportion=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, decision tree chose a feature \"RM\" and a threshold T=6.941 to do the split. You can also see the proportion of samples in two groups and value of the mean used for prediction.\n",
    "\n",
    "Let's verify this picture using python by our own:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(19.93372093023257, 37.23815789473684, 40.272839643050304, 79.7292018698061)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "left = df[df['RM'] < 6.941]['TARGET']\n",
    "right = df[df['RM'] > 6.941]['TARGET']\n",
    "\n",
    "left.mean(), right.mean(), mse(left.mean(), left), mse(right.mean(), right)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Intuitively, we can go on with splitting and apply same splitting logic recursively on groups (leaf nodes)...\n",
    "\n",
    "For example, the tree with depth=2 can be seen below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Text(167.4, 181.2, 'RM <= 6.941\\nmse = 84.42\\nsamples = 100.0%\\nvalue = 22.533'),\n",
       " Text(83.7, 108.72, 'LSTAT <= 14.4\\nmse = 40.273\\nsamples = 85.0%\\nvalue = 19.934'),\n",
       " Text(41.85, 36.23999999999998, 'mse = 26.009\\nsamples = 50.4%\\nvalue = 23.35'),\n",
       " Text(125.55000000000001, 36.23999999999998, 'mse = 19.276\\nsamples = 34.6%\\nvalue = 14.956'),\n",
       " Text(251.10000000000002, 108.72, 'RM <= 7.437\\nmse = 79.729\\nsamples = 15.0%\\nvalue = 37.238'),\n",
       " Text(209.25, 36.23999999999998, 'mse = 41.296\\nsamples = 9.1%\\nvalue = 32.113'),\n",
       " Text(292.95, 36.23999999999998, 'mse = 36.628\\nsamples = 5.9%\\nvalue = 45.097')]"
      ]
     },
     "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": [
    "dt = DecisionTreeRegressor(max_depth=2)\n",
    "model = dt.fit(X, y)\n",
    "plot_tree(model, feature_names=cols, proportion=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b>General notes on Decision Trees for Regression</b>\n",
    "\n",
    "<ul>\n",
    "    <li>Generally, there is no need to use binary trees (i.e we can have three groups instead of two) and no need to select single feature for splitting. However, the approach we used is most widely used and popular.</li>\n",
    "    <li>In practice, trees experience <b>low bias and very high variance</b> - i.e they tend to overfit. One of the ways to control tree complexity is to limit it's depth or to set the restriction on the minimal number of datapoints required for node to be split further. Usually, ensembling approaches (follow up in this class) are used to lower the variance.</li>\n",
    "    <li>For <b>nominal categorical variables</b> (i.e ones that can't be ordered) splitting is done by partitioning the set of values for the variable. For example, if we have X={A, B, C} than we can have {A} {B, C} as a split.</li>\n",
    "</ul>\n",
    "\n",
    "Refer here, for more notes: https://scikit-learn.org/stable/modules/tree.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To be continued..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.model_selection import train_test_split, cross_val_score\n",
    "from sklearn.utils import shuffle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Var: 0.20231209692307692\n"
     ]
    }
   ],
   "source": [
    "X_train, X_test, Y_train, Y_test = train_test_split(X, y, train_size=0.95)\n",
    "\n",
    "preds = []\n",
    "for i in range(100):\n",
    "    \n",
    "    X_tr, Y_tr = shuffle(X_train, Y_train)[:100]\n",
    "\n",
    "    Y_pred = DecisionTreeRegressor(max_depth=10).fit(X_tr, Y_tr).predict(X_test)\n",
    "\n",
    "    preds.append(Y_pred)\n",
    "    \n",
    "    \n",
    "print('Var:', np.mean((np.array(preds) - np.mean(preds, axis=0))**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ensembling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Model ensembling is a technique in Machine Learning to <b>stabilize (reduce variance)</b> of a model's prediction on a unseen samples. It is especially useful with inherently <b>high variance</b> models, such as decision trees.\n",
    "\n",
    "The idea is to take multiple models trained for the same task and combine their predictions, usually, by averaging or majority voting in case of regression and classification, respectively.\n",
    "\n",
    "To see why it works (i.e reduces model variance), let's take a look at following fact:\n",
    "\n",
    "Let's $X_1, X_2, ..., X_n \\sim^{iid} X$ and $\\bar{X_n}=\\frac{1}{n}\\sum_{i=1}^n{X_i}$ be a sample average. If $Var(X_i)=\\sigma^2$, then $Var(\\bar{X_n})=Var(\\frac{1}{n}\\sum_{i=1}^n{X_i})=\\frac{\\sigma}n$\n",
    "\n",
    "This fact can be translated in ML terminology: If we take $N$ \"independent\" models with some variance and average their predictions, we should get linear decay of variance. By independence we mean models trained on independent samples drawn from underlying data distribution, however, in practice, it's impossible, so we use <b>Cross-Validation</b> to simulate independent samples, or so called <b>Bootstrap Sampling (pseudo-independent sample can be generating by sampling from sample itself, with replacement)</b> that usually works good.</b>\n",
    "\n",
    "<br />\n",
    "\n",
    "Alternative intuitive interpretation of <b>single model VS ensemble</b> is to think of it as a decision by single person VS decision by group of people - the latter is more reliable and stable.\n",
    "\n",
    "<br /><br />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b><big>Random Forests</big></b>\n",
    "\n",
    "In practice, we almost always use decision tree ensemble (a.k.a <b>Random Forest</b>) instead of a single tree. Random forest consists of multiple decision trees, usually each trained on bootstrap sample of the train set. Besides, it's common practice select a <b>small random subsample</b> of features for every individual tree, and build those trees on those features. Number of trees in a forest highly varies from task to task, can be 10, 100 or even 1000."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.ensemble import RandomForestRegressor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Var:   0.20069091959478835\n"
     ]
    }
   ],
   "source": [
    "X_train, X_test, Y_train, Y_test = train_test_split(X, y, train_size=0.95)\n",
    "\n",
    "preds = []\n",
    "for i in range(100):\n",
    "    \n",
    "    X_tr, Y_tr = shuffle(X_train, Y_train)[:100]\n",
    "\n",
    "    Y_pred = RandomForestRegressor(max_depth=10, n_estimators=50).fit(X_tr, Y_tr).predict(X_test)\n",
    "\n",
    "    preds.append(Y_pred)\n",
    "    \n",
    "    \n",
    "print('Var:  ', np.mean((np.array(preds) - np.mean(preds, axis=0))**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b><big>Boosting</big></b>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In previous section we've seen how to combine multiple models (trained on same data) by averaging or majority voting. That method is usually called <b>Bagging (Bootstrap Aggregating)</b>. As opposed to that, there is a second way of combining models called <b>Boosting</b>. \n",
    "\n",
    "As opposed to bagging, where all models do prediction in parallel and then we average the predictions, here we have a prediction pipeline: First model ingests the input data and produces some output which is used as an input to next model in a pipeline to improve output and so on until final model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this section we are going to talk about <b>Gradient Tree Boosting</b> - one of the most popular ML methods that uses decision trees as its constituent parts (weak learners)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this class we will consider $L^2$-Boosting, special (probably easiest to explain) case of Gradient Tree Boosting where the objective is to minimize MSE, i.e a classical regression problem (class attendees are <b>strongly encauraged</b> to view reading material for in other cases of boosting)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our final predictor for data point $x$, will be of the following form:\n",
    "\n",
    "$\\hat{f}(x)=\\alpha_1\\hat{f_1}(x)+\\alpha_2\\hat{f_2}(x)+...+\\alpha_n\\hat{f_n}(x)$<br /><br />\n",
    "\n",
    "Where $\\alpha_i$ is some real number and $\\hat{f_i}$ is a regression tree with fixed size $J$ (number of terminal nodes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b>The general intuition is as follows:</b> Each component of the above sum improves on errors (moves towards reduction of error) made by previous component."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The actual algorithm can be defined recursively:\n",
    "    \n",
    "1) Let's first just fit a normal regression tree $\\hat{f_1}$ on the data and set $\\alpha_1$=1. Logically, we have reduces MSE on data by some factor.\n",
    "\n",
    "2) Now, let's assume we have our trees up to some $m-1$ and wanna add $m$-th tree to our boosting.\n",
    "\n",
    "$MSE(\\hat{f})=\\frac{1}{n}\\sum_{i=1}^N(y_i - [\\alpha_{m-1}\\hat{f_{m-1}}(x)+\\alpha_m\\hat{f_m}(x)])^2$  =  $\\frac{1}{n}\\sum_{i=1}^N([y_i - \\alpha_{m-1}\\hat{f_{m-1}}(x)]-\\alpha_m\\hat{f_m}(x))^2$\n",
    "\n",
    "<br /><br />From the cost function we can see that $m$-th step is fitting a tree on a residual (target minus prediction) of the previous step (well, residual with alpha term inside).\n",
    "\n",
    "3) Stop when you have good enough fit (you can check it with cross validation)\n",
    "\n",
    "\n",
    "(in practice, people usually use small alpha $\\alpha=0.1$ (called <b>shrinkage</b>) or use CV to search good alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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