unit02_simp_lin_reg/demo2_auto_mpg.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simple Linear Regression for Automobile mpg Data\n",
    "\n",
    "In this demo, you will see how to:\n",
    "* Load data from a text fileausing the `pandas` package\n",
    "* Create a scatter plot of data\n",
    "* Handle missing data\n",
    "* Fit a simple linear model\n",
    "* Plot the linear fit with the test data\n",
    "* Use a nonlinear transformation for an improved fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Loading the Data\n",
    "\n",
    "The python [`pandas`](http://pandas.pydata.org/) library is a powerful package for data analysis.  In this course, we will use a small portion of its features -- just reading and writing data from files.  After reading the data, we will convert it to `numpy` for all numerical processing including running machine learning algorithms.\n",
    "\n",
    "We begin by loading the packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data for this demo comes from a survey of cars to determine the relation of mpg to engine characteristics.  The data can be found in the UCI library:\n",
    "https://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Try 1:  Loading the Data Incorrectly\n",
    "\n",
    "The pandas has very good methods for loading data from ASCII tables. In this case, we want to read the data in the file:\n",
    "https://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data\n",
    "\n",
    "Since the file is a CSV file (comma-separated-values), we can try to use the `read_csv` command:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This creates a pandas *dataframe*. We can see the first six lines of the dataframe with `head` command:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There were three errors:\n",
    "* All the data appeared in one column.  That is, the columns were not \"delimited\" correctly\n",
    "* The first line got mistook as a header\n",
    "* The columns are missing their header names"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>18.0   8   307.0      130.0      3504.      12.0   70  1\t\"chevrolet chevelle malibu\"</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>15.0   8   350.0      165.0      3693.      11...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>18.0   8   318.0      150.0      3436.      11...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>16.0   8   304.0      150.0      3433.      12...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>17.0   8   302.0      140.0      3449.      10...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>15.0   8   429.0      198.0      4341.      10...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>14.0   8   454.0      220.0      4354.       9...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  18.0   8   307.0      130.0      3504.      12.0   70  1\\t\"chevrolet chevelle malibu\"\n",
       "0  15.0   8   350.0      165.0      3693.      11...                                   \n",
       "1  18.0   8   318.0      150.0      3436.      11...                                   \n",
       "2  16.0   8   304.0      150.0      3433.      12...                                   \n",
       "3  17.0   8   302.0      140.0      3449.      10...                                   \n",
       "4  15.0   8   429.0      198.0      4341.      10...                                   \n",
       "5  14.0   8   454.0      220.0      4354.       9...                                   "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head(6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Try 2: Fixing the Errors in the loading\n",
    "\n",
    "The problems above are common.  Often it takes a few times to load the data correctly.  That is why it is \n",
    "good to look at the first few elements of the dataframe before proceeding.\n",
    "After some googling you can find out that you need to specify some other options to the `read_csv` command.\n",
    "First, you need to supply the names of the columns.  In this case, I have supplied them manually based on the\n",
    "description in the UCI website:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "names = ['mpg', 'cylinders','displacement', 'horsepower', \n",
    "         'weight', 'acceleration', 'model year', 'origin', 'car name']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/'+\n",
    "                 'auto-mpg/auto-mpg.data',\n",
    "                 header=None,delim_whitespace=True,names=names,na_values='?')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you re-run `head` command now, you can see the loading was correct. You can see the column names, index, and values:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulating the Data\n",
    "We can get the `shape` of the data, which indicates the number of samples and number of attributes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(398, 9)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also see the three components of the `dataframe` object.  The dataframe is stored in a table (similar to a SQL table if you know databases).  In this case, there is one row for each car and the attributes of the car are stored in the columns.  The command `df.columns` returns the names of the columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\n",
       "    .dataframe tbody tr th {\n",
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       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>mpg</th>\n",
       "      <th>cylinders</th>\n",
       "      <th>displacement</th>\n",
       "      <th>horsepower</th>\n",
       "      <th>weight</th>\n",
       "      <th>acceleration</th>\n",
       "      <th>model year</th>\n",
       "      <th>origin</th>\n",
       "      <th>car name</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>18.0</td>\n",
       "      <td>8</td>\n",
       "      <td>307.0</td>\n",
       "      <td>130.0</td>\n",
       "      <td>3504.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>chevrolet chevelle malibu</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>15.0</td>\n",
       "      <td>8</td>\n",
       "      <td>350.0</td>\n",
       "      <td>165.0</td>\n",
       "      <td>3693.0</td>\n",
       "      <td>11.5</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>buick skylark 320</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>18.0</td>\n",
       "      <td>8</td>\n",
       "      <td>318.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>3436.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>plymouth satellite</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>16.0</td>\n",
       "      <td>8</td>\n",
       "      <td>304.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>3433.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>amc rebel sst</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>17.0</td>\n",
       "      <td>8</td>\n",
       "      <td>302.0</td>\n",
       "      <td>140.0</td>\n",
       "      <td>3449.0</td>\n",
       "      <td>10.5</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>ford torino</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>15.0</td>\n",
       "      <td>8</td>\n",
       "      <td>429.0</td>\n",
       "      <td>198.0</td>\n",
       "      <td>4341.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>ford galaxie 500</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    mpg  cylinders  displacement  horsepower  weight  acceleration  \\\n",
       "0  18.0          8         307.0       130.0  3504.0          12.0   \n",
       "1  15.0          8         350.0       165.0  3693.0          11.5   \n",
       "2  18.0          8         318.0       150.0  3436.0          11.0   \n",
       "3  16.0          8         304.0       150.0  3433.0          12.0   \n",
       "4  17.0          8         302.0       140.0  3449.0          10.5   \n",
       "5  15.0          8         429.0       198.0  4341.0          10.0   \n",
       "\n",
       "   model year  origin                   car name  \n",
       "0          70       1  chevrolet chevelle malibu  \n",
       "1          70       1          buick skylark 320  \n",
       "2          70       1         plymouth satellite  \n",
       "3          70       1              amc rebel sst  \n",
       "4          70       1                ford torino  \n",
       "5          70       1           ford galaxie 500  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['mpg', 'cylinders', 'displacement', 'horsepower', 'weight',\n",
       "       'acceleration', 'model year', 'origin', 'car name'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.columns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The field `df.index` returns the indices of the rows.  In this case, they are just enumerated 0,1,..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "RangeIndex(start=0, stop=398, step=1)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.index"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, `df.values` is a 2D array with values of the attributes for each car.  Note that the data can be *heterogeneous*:  Some entries are integers, some are floating point values and some are strings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[18.0, 8, 307.0, ..., 70, 1, 'chevrolet chevelle malibu'],\n",
       "       [15.0, 8, 350.0, ..., 70, 1, 'buick skylark 320'],\n",
       "       [18.0, 8, 318.0, ..., 70, 1, 'plymouth satellite'],\n",
       "       ...,\n",
       "       [32.0, 4, 135.0, ..., 82, 1, 'dodge rampage'],\n",
       "       [28.0, 4, 120.0, ..., 82, 1, 'ford ranger'],\n",
       "       [31.0, 4, 119.0, ..., 82, 1, 'chevy s-10']], dtype=object)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `df.columns` attribute is not a python list, but a `pandas`-specific data structure called an `Index`.  To convert to a list, use the `tolist()` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['mpg',\n",
       " 'cylinders',\n",
       " 'displacement',\n",
       " 'horsepower',\n",
       " 'weight',\n",
       " 'acceleration',\n",
       " 'model year',\n",
       " 'origin',\n",
       " 'car name']"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.columns.tolist()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can select subsets of the attributes with indexing.  For example, this selects one attribute, which returns what is called a pandas `Series`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    8\n",
       "1    8\n",
       "2    8\n",
       "3    8\n",
       "4    8\n",
       "5    8\n",
       "Name: cylinders, dtype: int64"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = df['cylinders']\n",
    "df2.head(6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also select a list of column names which returns another dataframe.  Note the use of the double brackets `[[ ... ]]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>cylinders</th>\n",
       "      <th>horsepower</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>8</td>\n",
       "      <td>130.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>8</td>\n",
       "      <td>165.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>8</td>\n",
       "      <td>150.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>8</td>\n",
       "      <td>150.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>8</td>\n",
       "      <td>140.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>8</td>\n",
       "      <td>198.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   cylinders  horsepower\n",
       "0          8       130.0\n",
       "1          8       165.0\n",
       "2          8       150.0\n",
       "3          8       150.0\n",
       "4          8       140.0\n",
       "5          8       198.0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = df[['cylinders','horsepower']]\n",
    "df2.head(6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting the Data\n",
    "We load the `matplotlib` module to plot the data.  This module has excellent plotting routines that are very similar to those in MATLAB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we need to convert the dataframes to numpy arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "xstr = 'displacement'\n",
    "x = np.array(df[xstr])\n",
    "y = np.array(df['mpg'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we can create a scatter plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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dHTk6p03hQ3fspnf9dvoH84HXDBoPywuwxvaGYUyGpjcNAYFB/kcKPm3FIhDkEM4PjZSVjYByR+4aN5TUj3bxz2Copq+C2fUNw6iGplcEG7btpzAW3REsBIdQFgmrJ1Q8rmjzn5Zr83VWBxHmtK7UVwGiO58NwzCKNL0iqNZhqsADa1fSP5gPdPoG3axLR0cKo2VKoBLdNdr1w5zPXTWd0TCMVqHpfQTVOkyLppm+Jd2cOj1aWem4EKjZrm+lIQzDqJXEFIGITBORH4jIHhHZJyI3uONni8hDIvKkiNwhIlOTkgH8naxhvPvNZ43//am3n192bKUaRKV0deTKzlFMaCs979XL5tVsyrHSEIZh1EqSK4JjwEpVfT2wGLhIRJYBnwNuVtVzgReB9yUow4Tia8UInlOn5xCc3gJees+ZxY19i4CTJRtGCqPjq4Turo6qehZ35Nq59PVzJ4Sbnjo9x4Z3vZ4NV75+XKburg5uvmrx+LWD6B/M07t+O2ev3ToefVQkaOXTlfKqxjCMxiPJnsUKDLsvc+6PAiuB97jjtwLXA3+XlBzg72Qt2tQZPWlO+eGBl8Zvrl57+6gqgpMnsPWRZwPbWeba4PSXd4xH7qxYOJstu/ITTDZH3UilapO2ruvfy+adB8YVUWkC2ppVC1hz156yshnDR0/w4EFheeQrGYbRaiTqIxCRdhHZDTwHfBd4ChhS1RPuLj8D6hLWEmZT99umwOadBzga4vwdVcfR/PT6S3hg7Up2PH4oFrt9/2B+ghLwO1ffkm5mTC3X64UxZcsT6fRhNgyjMUk0akhVR4HFItIFfAN4jd9ufseKyGpgNcCcOXMYHh5mYGAgNtmqTegCR9CRkNyDMWWCjGHXqOa9fGbgSKBJynuuoPyG54+OxTp3cRL35xonWZYNsi2fyVY79ZAvlfBRVR0SkQFgGdAlIlPcVcGZwMGAYzYCGwF6enq0s7Mzck39KHTv3O57oy6Gb1abbVzEK2PYNap5Ly/cF1wLyXuuoOudNq0t1rmLk2p6JaRNlmWDbMtnstVOPeRLMmpotrsSQEQ6gLcAjwE7gCvd3a4BvpnE9cMcqxBcsmHFwtkcOX6COIir9ENYCKz3XEHXu+LV5jA2DCOYSCsCEbncZ/glYK+qPhdw2FzgVhFpx1E4X1fVe0XkR8DXRORGYBD4Ug1yhxKlsqdfyQY/524pQb2JgbK8g7hKOq9ZtYA1d+4py0hub5sY9RR0va6XnqzqeqVYM3vDaG6imobeB1yA8zQPsBzYCbxaRD6tql8tPUBVHwGW+Iz/BHhTTdJGJMwR7L2BlUbu9K7fXjETOCx81C/hOI6Szn1Lurnhnn1l0UqjY1rxPQEMDNSuCKotl20YRuMR1TQ0BrxGVa9Q1SuA83DyBN4MfDQp4Wql1uSqySZfvRTSnWyyDAWErIbJXDSPXXvfYV/zWBQsY9kwmp+oimC+qv7c8/o54NWq+gKQudjEWuvzT7Z+f5L1/6t9T8Un+dIieNUqA8tYNozmJ6pp6F9F5F7gTvf1lcD3RGQGMJSIZJOg1r67fseVcu7pM3jyucOBx0O5TX3FwtnsePwQB4dG6JqeQ9VZPczsyCHiPO1Xsr37yZZrF144fIz5boe1U6fn+NTbz6dvSXdk81glrJm9YTQ/URXBB4DLgV/D8ZfeCmxxs4dXJCRbzdTqpPUe53fzawNOf9kpgYoA/G3qt+08ML7da+f3xv1Xsr2Xvqeu6TleOlKYkEn84pECa+7aM34+P/JDI5y9dmvkObFm9obR/ERSBKqqIvIAcALHX/DvrhLILLU6aYvH9a4vj8kfAx546oXAY4MayUel0hO79z31rt/uW+qiMOo4kIN6JoDj8I7q9LVm9obR/EQNH/0fwCeB7Tgrgr92o4W+nKRw9aQWG3gcdvOgJ/lSc1NYwtvBoZFIxfGimoqsmb1hNDdRncVrgCWqeq2qXgMsJYPRQnFSiw08rJF8VATKHLpex2/xab6SHFEb3JjT1zCMqIrgZ8AvPa9/CTwTvzjZIShLt/ecWYHHHDl+ghULZ1fV/6AUhbLQTD/HbxgrFs5mzaoFkT5cc/oahhHVWZwHHhKRYjmIy4AfiMifAqjq55MQrp4E2cb/dkdwctaLRwps2ZXniqXd41FCMzty/PLYCUar6Jtc+pRe7VP7jscPAY5PI4yoTl/LLDaM5iaqInjK/Snezb7p/v2yJITKCn628aA+xkVGCqPsePwQD6xdCThO3aCqoEGUPqVX8gmUcnBohNsfCl6wiXvOKDd0yyw2qqH40JAfGqF753Z7aGgQoiqCbwMfA+Z7jlFVfV0SQjU63if4ap/m/Z7S16xawIfu2B25O1olxfH0+ksiyxNXPoLR/NhDQ+MS1UdwG/BlnFyCS92ftyclVKPjfaIPssH79T4+dXqOmy5fVPal6VvSzdXL5kXulxxnjL9lFhtRsXIkjUtURXBIVe9R1adV9T+KP4lKllHOPX1G6PbSJ/ogp/PVy+ZN6Fn8hasWM/jJtwU+Od3Yt4ibr1o8fkwYcT591Vquw2g97KGhcYlqGvqUiGwC7scpNgeAqt6diFQpEcUJ6lcuIiizuNtzDu9xMztyTMu1MXSkMF5iYvPOA5zR1cHVy+Zx755n+eAdu/ngHbsnlIkoxeuzKJaVCKI7wDzULmKZxUYiWDmSxiXqiuD3gMXARTgmobfjmIcaFr/Y/NKibH77eMtFlLJi4exxJeA9bmikwNHCGFcvm8fRwhhDI4UJ5/M6k4tlIioVh2sX/3VBcdxvJQIwqjp+7SjX6VvSzU2XL5qwevEzXxlGXI2YjPSJuiJ4vaouSlSSlIniBK02fv/2h57hxr5Fgee+/aFnAss+eCmWiQi72b77zWf5KqV3v/ksoLxukkh5v4TCqHLDPfsss9iIhdL/uW4LNW4YoiqCnSJynqr+KFFpUiSKPbNa22bxJh90XBQlUKRSuGjPK2fxjw8dwJue0CbOeJHiDXxgYIBr7/M3Z/nVK6o2b8DyDIwi3v+5LPcFNiYS1TT0a8BuEdkvIo+IyF4ReSTsABE5S0R2iMhjIrJPRP7EHb9eRPIistv9uXiyb6IWojhBq7VtFs0yQccFmXNq2XfDtv2U5qiNaXlWcrVEMZlNZn/DMLJHVEVwEXAu8DZO+gcqhY+eAD6sqq8BlgEfEJHz3G03q+pi9+fbNcg9aaLYM1csnB05ZBNg6hTHEXv42Aly7eVHtkn0FUGl1UNYmWk/ujr8G9iXjlcTAtg/mOfDX99jIYOG0eBEUgTekNGo4aOq+qyq/tD9+5fAY0Bm7AWVnKD9g3m27MpPSOISwsNHRwpj485hv5IShUo1HzxUKhoXpKCCxq+/7HxyJc3uc23C9ZedP2EsaghgcSUQpLAsZNAwGoeoPoJJISLzcRrZPwT0An8sIr8LPIyzangxDTlKCXOC+j0ZK/CTQ0cinbuK0kJlCJWTwoJO7x2fkO7ftZ+r3nTWeA2kIFt+1BDASo50Cxk0jMZBku4vIyKdwL8An1XVu0VkDvBfOPeszwBzVfX3fY5bDawGmDNnztJNmzbR2dmZqKxegpyraXHLReGJa2Hy3XLRDB48WOBLjxzHe6tuB973uqlceIa/mQjgwYMFbnn0OMc9q5epbXDtayceF3Z9v/2DGB4eTvVzrYYsywbZls9kq5045VuxYsUuVe2ptF+iKwIRyQFbgM3F5DNV/bln+xeBe/2OVdWNwEaAnp4e7ezsTDUKoXtneYcyILTzV2zX7uqo+F7lvq2Bq4KP7xwjP3S8bHwU+PqTysfeE3zu5cB5EaKAwubnL971+shRQ1mOLsmybJBt+Uy22qmHfIkpAhER4EvAY94y1SIyV1WfdV++E3g0KRkmw5pVC1hz154JPYFz7cJVb/SP34+LXLv4moVKQzQvPGeWb9vM9jYJDT2NUgk1St5AUMZx1GQzq1JpGNkhyRVBL/A7wF4RKdZu/hjwbhFZjGMa+inwBwnKMDlKH7nVidPf+sizvvH3xdWC+BwaldHR8iP9qjo+94ujtEm5L6KavgeTYTK9jBu9SqXlTRjNRmKKQFX/Df8gllTDRWv90m7Ytp9CyU21MOZk/H7q7ecHPg0Xn3JrZcy9tldGP8dsqWxRmTG19u5ppdSacdzIpa0bXYkZhh9R8wgakskkO4WFUZaGnp42TcZNInGETU62Q1kYufbqP/L+wTy967dz9tqt9K7fPulksUauUmmllo1mpKkVwWS+tJUyj/uWdPPA2pU8vf4S/mr59PGnwTjCJv06lMXF0Eihqht6EpnDjVzaupGVmGEE0dSKYDJf2rDM49In5AcPFkKPa6smPRkno7mSLGFUSkar5oYe9xNw/2Cew8dOlI03SpXKRlZiaVD8blx73+FYVo9GOjS1IpjMlzYo8xgoe0K+5dHj4//wfsd9/rcW03vOrMBrlbJlV37CF8jvnKdM8f/oZkxt54G1K3nvsnkVrxPlhh7nE3BxdVEauRTUmS2LWKnlYLyrR7C6U41EKpnF9WKyTVX8nKG967eXPSEfH5vo4PU7rponaD/Haek5zw5oTHPkuCPbjscPRbpWpRt6nM1GgrKRp0+d0hBKACYXLdXsNHIQQKvT1IogiS9trU/I1T5BV9q/UomJqNerdEOPs0NZmvb1/sE8N9yzbzzMt6sjx/WX+Xd+qxbrz+CP+U8al6ZWBBD/l7bWJ+Sg48L2DyMow9lbCrvS9aLc0ONUpmm1MuwfzJclAw6NFFhz5x7AwjyTwlpVNi5N7SNIAj8b8dS2ykXiqnH4RrlBL3vVqaHjftfLtQtdHbmqW056I6QeWLuy5htprXNXLRu27Z+gBIoU80CMZDD/SePS9CuCuPF7Qr5k3mjFpvdrVi0YTzgrjs0/rcO3TMQVSyuvYn76vP/T/o+e/SW967dzcGiEmR05puXaePFIIRNtA/uWdPPwf7ww3rKzXYTe7rZYZPLOd1iqXRQzhZW/qA1rVdm4mCKogVJz08DAwITtQdmnN12+iAfWrhzf77r+vb6K4B8fOkDPK2eFfoGCzD4vHimM28WHRgp05NpZ/bqpfOw9K333T5Nij4eiSWtUlQfyo/QP5id1syid7zBmBjToCTqXZQ5Xh7WqbExa3jQUd9YsRI+9v/2hZ3yPj9JyMmrby5HCKFueqFxoLg385qUYcRX3eYOoNG2WOWy0Ii29Ikjq6S9q9ERYOetKJoxqSmE/fzSdQnSVSCqqpJrjh3yKBUY5l0W+GM1MS68Iknr6iyP7tHTf0pVLUA9iP06bVmVqc0IklZU7mXmNut0iX4xmpqUVQVJPf3FET3j39av3c/j4ibIPrw0nMqj0ule8Ojd+nrjNYNUQFDl1cGiE6/r3xn7eoH2rPZdFvhjNTksrgqSe/oLKU3jNTdXchH3LUI8qYyX7tbuNc0qve+EZuUSKx1VL35JurlhabnJT4LadB2pWBqXzHVTbaXqucoSS91xQXZitYTQqLe0jiDNrtpRKiWyVzE/etPyoK5TCqLLj8UMTIpMABgaezEz6f1jpi807D3Bj36Kazvvwf7zAf750FAWC3CdTp0RbNVjki9FqtPSKIMqTe1JUU5KimhVKteautJ2gYder1aV9Xf9ebtt5oKIDPUqbTsNoRZLsWXwW8BXgV3Aab21U1f8tIrOAO4D5OK0qf0tVX0xKjkrUq25MpRIQxZt//2CeI8fLyzYHtcMMM3dlIf2/2lIbUQgKw21m+gfzfGbgCC/ct9UK3xmTJskVwQngw6r6GmAZ8AEROQ9YC9yvqucC97uvG4qwfgRRj1mxcHagg7Nonrqufy8fumN3WX/kro4cVy+bV+YYDmp8D44ZLFdiPM+1Be+fFKW9FuKgmlDaOH0i9XK+F/09zx/Vuvl7jOYiMUWgqs+q6g/dv38JPAZ0A+8AbnV3uxXoS0qGJPBzunr7EUQ9ZsuuPFcs7R53ShYTxLx9DzbvPOD71D/jlCn0vHJW+ZKg0v2w1ImaclRpMbM4bqpp/HPDPfsmfb3+wTxLPv0dPnjH7ro43y3pzYibVJzFIjIfWAI8BMxR1WfBURYicnoaMsRFWHZs0NI86Ivr59gt0rt+e+B9/eDQiFNYraSBfbGomp+29FM4AAAW/UlEQVQcfoXYCqPB+ydBNRnA1XDKlDZGCqUxVP6Urq6qJaycRVrO96z4e4zmIXFFICKdwBbgg6r6C4lYGkFEVgOrAebMmcPw8HBZTZ96EGTfzg+NBMoX5zEAs6ZJVeccHh4mP+Q/72EyxE0U30AtskRVAtVew+9/7jMDRxgpBC+90pjPWdPEN1t81jQJvfaDBwtseaLA80eV06YJV7w6x4VnRE9MrIasfF/9yLJsUB/5ElUEIpLDUQKbVfVud/jnIjLXXQ3MBZ7zO1ZVNwIbAXp6erSzszMToXzdO7f73tC6uzoC5YvzGAE+8Y7Xj1d4jHLOgYEBurvGqpYhboLek5daZIly3iIdubbI1/ALH33hPv/OcEXaRfi9+w4n6sD9xMzyVYnglBL5+M4x3+v2D+b56v17x5XY80eVrz42ynmvOS8RGbMceptl2aA+8iXmIxDn0f9LwGOq+nnPpm8B17h/XwN8MykZIH6HXi019YOyVVcsnM2ST3+H+Wu3jv8svuE7TmMVn2MEuHrZPPqWdFedAZsFZ3GlDODuGiOYqnkP0yJkIF/Xv5dz1n2ba+87zDnrvj0h0a1SlNWoJu/ALYY9F0uHeCPIgq5rfgUjjCSjhnqB3wFWishu9+diYD3wVhF5Enir+zoRksim9cs9uPa1U0OfqvyOuWJpN3f8+zNlNmtvJ63SY26+avF4wlVNORB1dhYXZfarkzSZRL47Hz4Qed9KPoLSnIRR1QlZz0HKzM/imeSNtm9JN3+1fDrdXR1lviS/65pfwQgjMdOQqv4bwbea30jqul6Syqat1I8gyjG967f7dtGCk07fSt3AqsmByIKzGE7KPKH5yyTNKH49HWolKCfh9oee4ca+k4q2tOnQh+7Y7Xtc0jfaqDf4rOSRGNmkqUtMZPkpKO5m97Ver15zkdUyDkE5Cd5xPwUc5LNJ+kYb9QafZDmVavDr3GeJcPWnqUtMZLmkcK3lkOO+XhbmohmoV9XSqNetZzmVIlkofGj409SKIMslhdesWlCWGVwkCSduluciTXrPmRW6PciWWRwPCj5I+0bbP5jnwwNH+NAdu5mWa6OrI1e3G3xUzGGdXZraNOTXLD1KY/i0ZAP4+Df2cvj4yS/H9Fwbf37562KXMci2HXadpJbx3vPOmiZ8YubkehZ3B5hHSusxvfyUdja//4LQcwVlCCiO3Gvu2jPua8kPjbDmLsexXzQXpfG/dTKpzZHjxSMFcu3CzI7ceLJhUSbvMWGyp0HWzJPGSZpaEfg1S9+yK1+xMXyalCQHowmG8lRzo0qqjWfpeZ8/qpM+7/Sp/gvb0pv6L46NcvUXv19RGQRxwz37fB3uN9yzL9X/p6D+FMXqqn6fVRZkN4d1dmlq01DWl6JZli8p2ZI475PPHY6872QijIJCTydbtqJaojxBl85pFmQ382R2aeoVQdaXolmWL+1G81l4z43CzI5cpN4KWZvTWsyTRjo0tSLI+lI0y/IlJVuW33MlgnpApJyX55u85od3TrMie736fxjhNLVpKOtL0azI5xcJk5Rs9X7PlaKGwghzJKfJUERzjndO05K9+L907X2HU+3RYEyOplYEWYidDiML8gXFdkN5iYs4ZCt9z6dNk0mfN2qNojkvm1rRUXzqdP9qnKdOzwVeR4i34U0lugJkLMU7p0Gy11rfyQ/v/xJYnkAj0dSmIcj+UrTe8gU5bz/89T2MqXKGW+MoThm973lgYIDlkzy3X9asH1PaKxecC2p2psp4KQm/fkBpluqI0pBtxtSJ7zWNzOKg/6W0o6qM6mnqFYFRmSCHYhpVNOOidJURRBTn6UsBTtiXRgr0LekObRaUFkEyejlyfOINOY3VZ1Ap8BePFDL9/2O0wIrACCdKM/m0Om9NBu8qo3e9f3+CKA7pSs7soOS1NJ3dUT4zP3mSXn22iwTWasr6/0+rYyuCFidqM/mshSKW4nV4Hz52oqx8R1QzSFg/B3Dmq3TVkbaD36+3hJdcu3+JkmKfhflrt5b1WYiDICUA2fj/ufqL32f+2q1ce99h5q/dytVf/H69RcoMpghanB2PH4q0X5bDO0sd3kMjBVDHwVutGSSot8GdDx8Yz1T33u4E6lO2JMQGdtUbzyqTp1KfhTgIczxHdXAnxdVf/H5ZMuEDT71gysDFTEMtTpQntSyF3PrhW3JhTJk+dQqDn3xbVecKyjx+4KkX+OnzI2XXUaIr07jw6y3hxU+eSn0W4mDNqgV8MKAvQxQHd5KEfa6GrQhanqAn/XaRTIbc+pFWtnJWsqJr6WURpc/CZAn7H4ni4Dbqh60IWpygsMKs3/y9pJWtnJWs6ErO4pk+rUDTyCwOiwzyk8nIDkk2r/+yiDwnIo96xq4XkXxJD2OjjmQhqW2yxJmtHJR53HvOLF8nbRK9IypRyVnsV4Ji+lT/HIqg8VoIKxwYtSxGVPyy4YN6RQCcMsX/Vhc0Hoc8jUSSK4JbgL8BvlIyfrOq/mWC1zWqpN5JbZMl1WJmpTe0tIv1RLiuX0XRw8f9k+2CxmshbJUStSxGFPxKpK+5aw+o4xsqjnlLcR87MeZ7rqDxycoTR8n2NEmyef33RGR+Uuc3DC9xKbNKzmK/mv5px8hXcha3x/34HQNxms+C+jGU4s1/CcpxiGOuwkqrt7wiCOGPReR3gYeBD6vqi347ichqYDXAnDlzGB4eZmBgID0pqyDLskG25cuybKUEPfHmh0ZSfQ+VkslGVauSJw3ZL5k3Gtt1Kr3/0n0HBgZCneWTlSvu/4t6fCfSVgR/B3wGx2/1GeCvgN/321FVNwIbAXp6erSzs5Ply5enJGZ1DAwMZFY2yLZ8mZPtvq2Bm4Kyiru7OlJ9D907/TOnQ+UJeV+xyR5yjfNec96ka0oVqfT+J+zrzkXQMXF8dnGfux7fiVTDR1X156o6qqpjwBeBN6V5fcOoRCVncRbKhvvJUUmesPcVF2HnirPrnt/7z7VLmQPdOxdJfnZZ+b+YDKmuCERkrqo+6758J/Bo2P6GkTab339BWRZq7zmzJpSvrneHrb4l3dz58AFff8Yb5s30lWfz+y9g4ce/zVGPLX1au1Tdv7l/MB/4/je//wLmr/VfFcSZaxEUHOA3VtzXb86C5ioueUrPHTZ3pTgO6EcYKTjO7DaB97x5XmzJf6UkpghE5HZgOfAKEfkZ8ClguYgsxjEN/RT4g6Subxi1Urw5+i3RsxBhdV3/3lCn9nX9e8tuGG/9/MAEJQBwdFR56+cH+O6fLo903UrRMWEhk1NjCtMsEvQ5BH02fnMWNFdxylOkmsiiBw8W2HTfbrzxTGMKt+10yp8koQwSMw2p6rtVda6q5lT1TFX9kqr+jqouUtXXqeplntWBYRgRCSoXEbb9yecO++4bNO5HWHRMcXsQcYRpToawEhtpUGnuvGx5okDQbCUlr5WYMIwGo1JZiDjLRnipVGIjCxVGg0ijxEYY1ZQnef5osExJyWuKwDAajEqx70nlEQTlAhTHs1yhNmhO0sq5qDR3Xk6bFixTUvKaIjCMBuPdbz4rdPvUKVJmrz/39Bm++waN+1EpOiYsSqYYUZR0T4Qgguas0lzGRTXlSa54dS7wxpyUvKYIDKPBePrQcOj2kcJYWXvRN7/qNN99g8b9qFSX6pMhN/V39cxLpSdCEA/95PmqxhMhYnmSC8/I8fmrFtORO3l7bhN477IGjBoyDCMZotTQLy1xEFc/grDomF8cC65btGHbfv7zpaOxyFALcTjLJ4NfWZCw8iRpR6eZIjCMJsXriMyCszToSmnJUE+y0ssiCDMNGUaT4nVEZsFZWm8Z6kk1zuJ6YIrAMBqMKGUhSkscpOEsffkpwb0N1qxaEKsM1db/j8NZPhl5ailDkWaPA1MEhtFgbH7/BWXK4NzTZ4Q2F7qxbxHvXTZv/Om7XSR25+NlATbt3nNm0bekOzYZilm6edfcVMzSDbtRfmDFub6+2g+sOLeqa9cqT7UNoGp5j5PBfASG0YBsfv8FVVepvLFvUaJO2SCH9M6fnKw0H4cMtdT/37Btf5mPQt3xyTplo8pTjQM47R4HtiIwDCMW0nJI1+J4TdJZm8S503YumyIwDCMW0nIG1+J4TdJZm8S503YumyIwjBYhaedjWtm7tThe16xaQK69JLO33T+ztxZ54j532j0OTBEYRguQhvOx55WzaC8po9DeJvS8Mr7mN1C943UcPydBXMR87prfY42Ys9gwWoA0nI8btu1ndGziHXB0LDh7djJUm3m7Ydt+CiWyFWKSLalzp5ldbCsCw2gB0nA+Zjl7ttGcxWljisAwWoA0nI9Zzp5tNGdx2iSmCETkyyLynIg86hmbJSLfFZEn3d+nJnV9wzBOkobzMctN3K15fThJ+ghuAf4G+IpnbC1wv6quF5G17uuPJiiDYbQElRqjR22wPhm818gPjdDtc41qGrjHSRTZ4jh32u8rLhJTBKr6PRGZXzL8DpyG9gC3AgOYIjCMSRG1MXoazsfiNfyynqtp4J62bHGdu1FJ20cwp9iw3v19esrXN4ymo5rG6PWkUeRsRUQTrAXurgjuVdXXuq+HVLXLs/1FVfX1E4jIamA1wJw5c5Zu2rSJzs7OxGSdDMPDw5mVDbItn8lWO0X5rr0vuLnKLRfFU12zWvzmLityNsrnGgcrVqzYpao9lfZLO4/g5yIyV1WfFZG5wHNBO6rqRmAjQE9Pj3Z2dsa+nIuLJJaacZJl+Uy22inK171zO3mfUMXuro66ye83d1mRs1E+1zRJ2zT0LeAa9+9rgG+mfH3DaCr6B/McOX6ibDyLUSvNEF3TrCS2IhCR23Ecw68QkZ8BnwLWA18XkfcBB4B3JXV9w2h2HjxY4Kv37y2zu3d15Lj+svMz57xshuiaZiXJqKF3B2z6jaSuaRitxJYnCowUyn18M06Zktmba6NH1zQrlllsGA3K80f9Az0aqbSBkQ1MERhGg3LaNP86/41U2sDIBqYIDKNBueLVOXO+GrFgisAwGpQLz8ilWrPeaF6sH4FhNDDmfDXiwFYEhmEYLY4pAsMwjBbHFIFhGEaLY4rAMAyjxTFFYBiG0eIkWoY6LkTkEHAY+K96yxLAK8iubJBt+Uy22smyfCZb7cQp3ytVdXalnRpCEQCIyMNR6mrXgyzLBtmWz2SrnSzLZ7LVTj3kM9OQYRhGi2OKwDAMo8VpJEWwsd4ChJBl2SDb8plstZNl+Uy22kldvobxERiGYRjJ0EgrAsMwDCMBMqsIROSnIrJXRHaLyMPu2CwR+a6IPOn+PjUlWb4sIs+JyKOeMV9ZxOH/iMiPReQREXlDHWS7XkTy7tztFpGLPdvWubLtF5FVCct2lojsEJHHRGSfiPyJO56VuQuSr+7zJyLTROQHIrLHle0Gd/xsEXnInbs7RGSqO36K+/rH7vb5dZDtFhF52jNvi93xVD9X95rtIjIoIve6r+s+bxXkq+/cqWomf4CfAq8oGfsLYK3791rgcynJ8uvAG4BHK8kCXAz8EyDAMuChOsh2PfARn33PA/YApwBnA08B7QnKNhd4g/v3y4AnXBmyMndB8tV9/tw56HT/zgEPuXPydeC33fG/B/6n+/cfAX/v/v3bwB0JzluQbLcAV/rsn+rn6l7zT4F/BO51X9d93irIV9e5y+yKIIB3ALe6f98K9KVxUVX9HvBCRFneAXxFHXYCXSIyN2XZgngH8DVVPaaqTwM/Bt6UoGzPquoP3b9/CTwGdJOduQuSL4jU5s+dg2H3Zc79UWAlcJc7Xjp3xTm9C/gNEfFvYZacbEGk+rmKyJnAJcAm97WQgXkLkq8CqcxdlhWBAt8RkV0istodm6Oqz4LzJQZOr5t0wbJ0A8949vsZ4TeXpPhjdyn5ZTlpQqubbO6SewnO02Pm5q5EPsjA/Lnmg93Ac8B3cVYgQ6p6wuf647K5218CTktLNlUtzttn3Xm7WUROKZXNR+4k+ALwZ8CY+/o0MjJvAfIVqdvcZVkR9KrqG4DfBD4gIr9eb4Ei4vc0kXZo1t8B5wCLgWeBv3LH6yKbiHQCW4APquovwnb1GauHfJmYP1UdVdXFwJk4K4/XhFy/rrKJyGuBdcBC4I3ALOCjacsmIpcCz6nqLu9wyPVTnbcA+aDOc5dZRaCqB93fzwHfwPki/Ly4LHJ/P1c/CQNl+Rlwlme/M4GDaQqmqj93v6hjwBc5ab5IXTYRyeHcZDer6t3ucGbmzk++LM2fK88QMIBjI+4SkWJnQe/1x2Vzt88kuskwDtkuck1tqqrHgH+gPvPWC1wmIj8FvoZjEvoC2Zm3MvlE5LZ6z10mFYGIzBCRlxX/Bt4GPAp8C7jG3e0a4Jv1kRBCZPkW8Luut38Z8FLRDJIWJTbEd+LMXVG233YjJc4GzgV+kKAcAnwJeExVP+/ZlIm5C5IvC/MnIrNFpMv9uwN4C44PYwdwpbtb6dwV5/RKYLu63saUZHvco9wFxwbvnbdUPldVXaeqZ6rqfBzn73ZVvZoMzFuIfO+t+9wl4YGe7A/wKpzojD3APuDj7vhpwP3Ak+7vWSnJczuOiaCAo6HfFyQLzlLub3HsuXuBnjrI9lX32o+4/0hzPft/3JVtP/CbCcv2azjL2EeA3e7PxRmauyD56j5/wOuAQVeGR4FPer4bP8BxVN8JnOKOT3Nf/9jd/qo6yLbdnbdHgds4GVmU6ufqkXM5J6Ny6j5vFeSr69xZZrFhGEaLk0nTkGEYhpEepggMwzBaHFMEhmEYLY4pAsMwjBbHFIFhGEaLM6XyLobRGIjI9cAw8HLge6r6z1Uevxyn2Nyl8UsXLyLSBzyhqj+qtyxG42MrAqPpUNVPVqsEGpA+nGqohjFpTBEYDY2IfFyc3gD/DCxwx24RkSvdv9eLyI/cYl5/6dn+9yLyryLyhFv/pfS8bxKRB8WpGf+giBTP3S4ifylOr4xHROR/ueNLReRf3CKJ2zyZogNuEbHvidP34I0icrc4dfFv9FzvveLU+N8tIv9PRNrd8WER+aw4tf93isgcEbkQuAzY4O5/TqKTbDQ9ZhoyGhYRWYqTpr8E53/5h8Auz/ZZOCUiFqqqFssiuMwH/jtOcbkdIvKrJad/HPh1VT0hIm8B/hy4AliN04tgibttlluv6K+Bd6jqIRG5Cvgs8PvuuY6r6q+L0/jmm8BSnHo2T4nIzTjVV6/CKbRYEJH/C1wNfAWYAexU1Y+LyF8A71fVG0XkWzhZqcXSyoZRM6YIjEbmvwHfUNUjAO7N0csvgKPAJhHZCtzr2fZ1dYrKPSkiP8Gp/OhlJnCriJyLU4Yi546/BaeRyQkAVX1BnMqbrwW+65SKoR2n7EeRolx7gX3q1opxr3sWTqmLpcC/u8d3cLIQ33GP3LuAt0aYF8OoClMERqMTWCPFfWJ/E/AbOCuHP8apRul3XOnrzwA7VPWd4vQqGHDHxWdfwbnBXxAgyjH395jn7+LrKe7xt6rqOp9jC3qyDswo9p01EsB8BEYj8z3gnSLS4Varfbt3ozh9Bmaq6reBD+L0FyjyLhFpc+3rr8IpIudlJpB3/77WM/4d4A/FLWnsmp/2A7NF5AJ3LCci51fxPu4HrhSR04vnFJFXVjjmlzjtNQ1j0pgiMBoWddpM3oFTNXQL8K8lu7wMuFdEHgH+BfiQZ9t+d+yfgD9U1aMlx/4FcJOIPIBj6imyCTgAPCIie4D3qOpxnBLGn3PHdgMXVvE+fgRch9OR7xGcbmSV2hF+DVjjOrPNWWxMCqs+arQcInIL5mg1jHFsRWAYhtHi2IrAMAyjxbEVgWEYRotjisAwDKPFMUVgGIbR4pgiMAzDaHFMERiGYbQ4pggMwzBanP8PqfjkeT1SE6gAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y,'o')\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulating Numpy arrays\n",
    "\n",
    "Once the data is converted to a numpy array, we can perform many useful simple calculations.  For example, we can compute the sample mean:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean displacement = 193.4, mean mpg=  23.5\n"
     ]
    }
   ],
   "source": [
    "mx = np.mean(x)\n",
    "my = np.mean(y)\n",
    "print('Mean {0:s} = {1:5.1f}, mean mpg= {2:5.1f}'.format(xstr, mx, my))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fraction of cars with > 25 mpg:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we can repeat the `read_csv` command with the correct options. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3969849246231156"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(y > 25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sample mean displacement for the cars that have mpg > 25"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "110.08227848101266"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = (y>25)\n",
    "np.mean(x*I)/np.mean(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also do the previous command with [boolean indexing](https://docs.scipy.org/doc/numpy-1.13.0/user/basics.indexing.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "110.08227848101266"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(x[I])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Missing Data and NaN Values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, try a different field, horsepower"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xstr = 'horsepower'\n",
    "x = np.array(df[xstr])\n",
    "y = np.array(df['mpg'])\n",
    "np.mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When you get the mean, it gives `nan` which means not a number.  The reason is that there was missing data in the orginal file and the `load_csv` function put `nan` values in the places where the data was missing.  This is very common.  To remove the rows with the missing data, we can use the `dropna` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(392, 2)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df1 = df[['mpg','horsepower']]\n",
    "df2 = df1.dropna()\n",
    "df2.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "We can see that some of the rows have been dropped.  Specifically, the number of samples went from 396 to 392.  We can now compute the mean using the reduced dataframe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "104.46938775510205"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array(df2['horsepower'])\n",
    "y = np.array(df2['mpg'])\n",
    "np.mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And, we can plot the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y,'o')\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing and Plotting a Linear Fit\n",
    "We can now try to fit a linear model, $\\hat{y} = \\beta_0 + \\beta_1 x$.\n",
    "From class, the formulae are:\n",
    "$$\\beta_1 = s_{yx}/s_{xx}, \\quad \\beta_0 = \\bar{y} - \\beta_1\\bar{x},$$\n",
    "where $\\bar{x}$ and $\\bar{y}$ are the sample means and $s_{yx}$ and $s_{xx}$ are the cross- and auto-covariances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "xbar     = 104.47,       ybar=  23.45\n",
      "sqrt(sxx)=  38.44,  sqrt(syy)=   7.80\n",
      "beta0=  39.94, beta1=  -0.16\n"
     ]
    }
   ],
   "source": [
    "xm = np.mean(x)\n",
    "ym = np.mean(y)\n",
    "syy = np.mean((y-ym)**2)\n",
    "syx = np.mean((y-ym)*(x-xm))\n",
    "sxx = np.mean((x-xm)**2)\n",
    "beta1 = syx/sxx\n",
    "beta0 = ym - beta1*xm\n",
    "\n",
    "print(\"xbar     ={0:7.2f},       ybar={1:7.2f}\".format(xm,ym))\n",
    "print(\"sqrt(sxx)={0:7.2f},  sqrt(syy)={1:7.2f}\".format(np.sqrt(sxx),np.sqrt(syy)))\n",
    "print(\"beta0={0:7.2f}, beta1={1:7.2f}\".format(beta0,beta1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create a plot of the regression line on top of the scatter plot.  The vector `xplt` are the x-coordinates of the two endpoints of the line.  They are chosen so that the line fits nicely in the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Points on the regression line\n",
    "xplt = np.array([20,250])          \n",
    "yplt = beta1*xplt + beta0\n",
    "\n",
    "plt.plot(x,y,'o')                    # Plot the data points\n",
    "plt.plot(xplt,yplt,'-',linewidth=3)  # Plot the regression line\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The (normalized) residual sum of squares is RSS=$(1/n)\\sum_i (y_i-\\hat{y}_i)^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RSS =   23.94\n"
     ]
    }
   ],
   "source": [
    "yhat=beta0+beta1*x\n",
    "RSS = np.mean((y-yhat)**2)\n",
    "print(\"RSS = {0:7.2f}\".format(RSS))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us see whether this is the same as the analytically derived minimal RSS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RSSmin =   23.94\n"
     ]
    }
   ],
   "source": [
    "rxy=syx/np.sqrt(sxx)/np.sqrt(syy)\n",
    "RSSmin=(1-rxy*rxy)*syy\n",
    "print(\"RSSmin = {0:7.2f}\".format(RSSmin))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Nonlinear Transformation\n",
    "\n",
    "We see that the linear regression captures the general trend of the relation between `y=mpg` and `x=horsepower`.  However, the trend does not really appear linear - instead it has an inverse type relation.   So, a natural idea is to use a *nonlinear transformation*:\n",
    "* Transform the data `z=1/y` \n",
    "* Fit `z` vs. `x` with a linear model:  $\\hat{z}=\\beta_0 + \\beta_1x$.\n",
    "* Invert the nonlinear relation for a model for `y`:  $\\hat{y} = \\hat{z}=1/(\\beta_0 + \\beta_1x)$.\n",
    "\n",
    "We begin then by computing `z` and plotting a scatter plot of `z` vs. `x`.  Note that `z` represented gallons per mile (1/mpg)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = 1/y\n",
    "plt.plot(x,z,'o')\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('1/mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see a clear linear relation between `z` (1/mpg) and `x` (horsepower). We can fit a linear model,\n",
    "$z = \\beta_0 + \\beta_1 x$.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Use linear regression to fit `z` vs. `x`\n",
    "xm = np.mean(x)\n",
    "zm = np.mean(z)\n",
    "sxz = np.mean((z-zm)*(x-xm))\n",
    "sxx = np.mean((x-xm)**2)\n",
    "beta1_inv = sxz/sxx\n",
    "beta0_inv = zm - beta1_inv*xm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create a plot of the regression line on top of the scatter plot.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = 1/y\n",
    "xplt_inv = np.linspace(20,250,100)\n",
    "zplt_inv = beta1_inv*xplt_inv + beta0_inv\n",
    "plt.plot(x,z,'o')\n",
    "plt.plot(xplt_inv,zplt_inv,'-',linewidth=3)\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('1/mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we compute the estimate in the original domain:  $\\hat{y}=1/\\zhat{z}$.  We plot the data, original linear fit and the linear fit with inversion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x25b859bcb38>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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fdKaxHzduHJMmTeL++++nVatWZGVlkZeXR8eOHUudm5+fT2FhIddffz3Dhw+na9eupY5HRUXRvHlztmzZwqhRo/joo4+45pprgvo8ngRzRhAHfCmEcN3nUynld0KIX4H/CiHuAM4C9aYYwPiE8by26zUuFFwgy5TFVye/UrUKFIo6xDMNtdlsBuC5556je/fu7jTUCQkJtZKGunnz5vTs2ZODBw8ydOhQAHr27Mlzzz3Hddddh8PhQK/X89Zbb5VTBHl5eUyaNAmTyYSUkldffbXc9RcvXsxf/vIXCgsL6dy5Mx9++GHQn8lFk01D7YslB5awcMdCABKaJbBq8io0InjOVcok4kT1gxOVhtqJSkPtRKWhriOmdp9KpN7Z8aevnGbTOVUAW6FQNG6UIihDuD6caYkltsYPD3xIQ5g1KRQKRVVRisALt/e43V3KMiUjhR3ptWOWUihqA/Vi0/io7u9UKQIvxIXHManLJPf2u3verUNpAmNlSiojF2yg09w1jFywgZUpqXUtkqIeEhoaSmZmplIGjQgpJZmZmYSGVt3DMdgBZQ2WP/f5MyuPr8Qu7fxy8Rd2pu9kUNyguhbLKytTUnlsxT6KrM7Q9tScIh5b4XQNnDwgvi5FU9Qz2rVrx/nz57l06VJdi+IXk8lUrYGtsRBoP4SGhtKuXbsq30cpAh+0i2zHTV1uYuXxlQC8t+c93r/u/TqWyjsL1x5xKwEXRVY7C9ceUYpAUQq9Xk+nTp3qWowKSU5OrnJwVGOitvpBmYb8cGefO92uoz9f+JndGbvrWCLvpOUUVWq/QqFQeKIUgR86NOvADZ1ucG+/u7d+rhW0jTZWar9CoVB4ohRBBdzZ904EAoCtqVvZmb6zjiUqz8MTEjHqtaX2GfVaHp6QWEcSKRSKhoRSBBXQKaoTN3a+0b392s7X6p3HxeQB8bwwpQ/x0UYEEB9t5IUpfdT6gEKhCAilCALgnv73uOMKdl/azabz9S/aePKAeLbOHcur0/sDcP+y3cqNVKFQBIRSBAHQLrId0xOnu7df3/U6dof30nP+CLavv8uNNDWnCEmJG6lSBgqFwh9KEQTInX3uxKhzLr4ezznOmlNrKnW+r0F68QFTjSkHf26kCoVC4QulCAIkxhjDrF6z3NtvpbyF2W4O+Hxfg/TGc/Yae4NXbqQKhaIqKEVQCWb1nEXzkOYApBWkseTAkoDPDXQwrs4bvHIjVSgUVUEpgkoQYYjgnv73uLf/ve/fZBR6rbRZjiijPuD7VPUNXrmRKhSKqqAUQSWZ1n0aXaOdZeaKbEW8vuv1gM6z2h0B36Oqb/DKjVShUFQFlWuokug0Oh4d+ih3fn8nAKtPrGZGjxn0btnb73kFlsC8jKr7Bj95QLwa+BUKRaVQM4IqMLzNcMa0H+PeXvDLAhwy8Df+sqg3eIVCUZeoGUEVeWjwQ2xJ3YLNYWPPpT2sOr6KW7rd4rN9tFFPTpG13P5wHWydOzaYoioUCoVf1IyginRo1oFZPUvcSf+5859km7J9tp9/cy/0GlFqn14juL2nIWgyBoIqaKNQKJQiqAZ39buL+AinGSfXnMsrO1/x2XbygHgW3tqvlBlo4a39uLpt4N5ENY2KRFYoFKBMQ9XCqDPy+LDHuXf9vQCsPL6SSV0mMbj1YK/tvS3kJicfqxFZVqaksnDtEdJyimgbbeThCYkVrjWogjYKhQLUjKDajG43mvEdx7u3n932LBa7pVZlqOqbvYpEVigUUAuKQAihFUKkCCG+Lt7uJITYLoQ4JoRYJoSoWyN5DfDIkEcI04UBcDL3JO/vrd2SllXNMaQikRUKBdTOjOBvwCGP7ReBV6WU3YBs4I5akCGotA5vzV8H/tW9/Z99/+Fg5sFau39V3+xVJLJCoYAgKwIhRDvgBuA/xdsCGAt8XtxkMTA5mDLUFjN6zGBgq4EA2KWdf2z9B1Z7eXfRYFDVN3sViaxQKCD4i8WvAY8AkcXbMUCOlNJWvH0eaBSjjkZoeGbkM0xbPQ2T3cTR7KP8e9+/S+UmChYPT0jksRX7SpmHXG/2FS0iq0hkhUIhglV2UQhxI3C9lPIeIUQS8BDwJ+BnKWXX4jbtgW+klH28nD8HmAMQFxc3aOnSpUGRs6bZeGUjK7JXAKBBw4OtH6RDSAef7fPz84mIiPB5/Kc0K18ctZJpksSECqZ215dyOfU8rgEc4G4HsGi/BYtH0LNBA7N7G+rUbdUbFfVDU0H1gxPVD06q2w9jxozZKaX07sboQTBnBCOBm4UQ1wOhQDOcM4RoIYSueFbQDkjzdrKU8n3gfYDBgwfLpKSkIIpac4xyjOLU2lOkZKTgwMHywuUsG7uMMH2Y1/bJycn4eraVKal8tH4fRVanss40ST46ZKfnVT2ZPCC+3HEHzpnAPyY5zTsjF2wopQQALA5Yc1bL47/3fs+6wl8/NCVUPzhR/eCktvohaGsEUsrHpJTtpJQJwG3ABinl7cBGYFpxs1nAqmDJUBdoNVqeG/mcu5rZ6SuneXnHy1W6VkXeQBUdV+6hCoUiEOoijuBR4AEhxHGcawYf1IEMQaVDsw48NvQx9/byo8vZcHZDpa/ja8BOzSli5IINpFYw0Cv3UIVCEQi1ElkspUwGkou/nwSG1sZ965LJXSezJXUL686sA+Cpn56id8vetAprFfA12kYbfQ72vvYDSCBh7ho0Ave6gYvquodWJYJZoVDUb1RkcZAQQvDUiKfcA3+OOYdHNj+CzWGr4MwSvPn5VwaHdK0baGrEPVTlJlIoGidKEQSRqJAoFoxagEY4u3ln+k7eTHkz4PM9/fyrQ5HVUSNv71WNYFYoFPUbpQiCzJDWQ7i3/73u7Q/2f8Dm85sDPn/ygHi2zh1bbWVQE2/vgS4+q9TWCkXDonErAktBXUsAwJ/7/JmRbUe6tx/b8hip+ZUbHKtrJoLqv70HsviszEcKRcOj8SqC/AxY2A2W/wlOJoOj6qUkq4tGaHhh1Avu9YIrliv8bcPfKLQWBnyNmjITVcd1NJDcRMp8pFA0PBpvPYLdn4K1AA6scH6aJ8DAmdD/dohsXeviNA9tzj+v+Sd/WvsnbA4bR7KP8NRPT3G9vN7veZ5eOlFGPaK4yJnA6R1UWaKMevd1568+4C6f2TxMz1M39apwDSFUr3EP9NFGPTf2a8PCtUe4f9luv15OKnZBoai/NF5FcHFf6e3s07D+GdjwPCT+FgbOgq7jQFM9c0tl6N+qP08Me4Knf34agO9Of4c+Ws8Yxnht7zKzuAZez5rHkhJloBFOD6FAKLDYeHLlPpb9cg6rx0nZhVYe/nyPe7usiyhQLp9RgcVW6jqpOUU+FZSKXVAo6i+N1zQ07QP4y1YYOgdCo0r2Szsc/ho+vRVe6wsbX4Ccc7UnVvdpTE+c7t7+Oudrks8le23rzcziicTpEvrK7/oHfH+rXfLZ9tJKwPPY018d8Grjf/qrA+Vksdplueu4FJQnKrW1QlG/abyKAKB1b7h+ITx4BG55DzpcXfr4lfOwaQG81gc+ngaHvoJaSB396JBH3SmrJZJHNj/itX5BIOaUtJyiSruE2v0kGswutHq18WcXBt4vLgWlUlsrFA2Dxmsa8kRvhH63OT+XjsKuxbDnMyjMLG4g4fg65yciDvr/3rme0KJzcMTR6nkl6RVu/+Z2UvNTKbIV8T/r/4dPb/iU1uEl6xf+bO4uXDb/+ADautAK4VcZVJf4aCNb544N2vUVCkXN0rhnBN6I7Q4TnocHDsG0D6FzUunj+enw46vwrwGw+CbY/wXYzDUuRowxhrfHvY1ROG3nl4ouce/6eymwlri8BuIyesVkpdPcNRRabOg1ZY0y5dFqBDOGtffaVq8VRBu9p6eONurLyaL1cb8xPWIrlEOhUNQfmp4icKELgd5TYOYq+OtuGPUgRJTxJjq1GT7/f/DPHrD2CbhUsy6QnaM7c0fsHeiEc2J2NPsof9v4Nyx2CxCYy6hDOk0x2YXWUvZ6XyrB7pAM7tiChbf2KzXoNw/TM31Ie7dXkidGvZb5N/cqV80sMsT7hHLj4Ut+n1uhUNQvmoZpqCJadIJx8yDpcTi2FnYudpqJZHHsQVEW/Pym89NhhNPjqOckMHivMVAZEo2JzBsxj3k/zQNg+4XtPLblMV4a/RJajdZdQcxftlFv+DP8PP3VAVLmXVfKbl/WQ8lFtFHP/JtL3Eo9z+k0d43X6ytXUYWiYaEUgSdaHfS4wfnJPQ8pH8Ouj5yLyi7O/uz8fPso9P0dDJoFrcsVWKsUt3S7hYzCDN7c7cxD9P2Z72n+S3OeGPYEovgV3Vs5yqriWvj1jFHQ+Fg3CA/R+VzojTLqS7m0eu5XKBQNh6ZrGqqIqHaQNBf+vhdu/xx63AgaD71pzoVf/w3v/gbeHwM7F4E5r8q3m9N3Dr/v8Xv39rIjy9yKAcoXmtd6s+FUgrKpIHwtHqflFPnMHeRLhJwiq8oxpFA0INSMoCI0Wug23vnJS4fdn8CuJZB9qqRN2i7n57vHoc9UGDgb4gf6Him98I9V+/lsex/0bfqhj3IGdr2/931CtCHM6TsHKF1o3pcpJ1C8xQV4IzpMX+o+rrgCgBw/LqWe7ZTrqEJRv1EzgsoQGQejHoD7dsHM1dB7KmgNJcetBU4l8Z+xzpnC9vehKMfvJX9Ks9Jr3nd8vO0sdikwpd2KLa+H+/gbKW+w+MDicudVJ/eQgIDiAox6LVLiM3dQRdHCgeQYcs02Zn9XUK9nESqjqqIxoxRBVdBooPM1MO3/4IHDMOF/oWWZyNn0/fDtw/DPRFhxF5z5CcqYX1ampLJov4UCi+dAq6Mo9XZs+V3de17e8TIfH/y4nBiuFNWvTe8fkOuoC38LyVohSgWC5XpZAwCnySgQ91Z/C8ee5imov5lKVUZVRWNHKYLqEh4DI+6Fe7fD/1sL/X4POo83ZZsJ9i6FD38Lbw2Fn96AAmcg28K1R7B4S4oq9RSdn4mtoJN714u/vuh1ZgBOhVDWHbSqzBjWnlMLbmDr3LFMHhDvN/V0ILMSf7OGhpKptKHIqVBUlYDWCIQQU7zszgX2SSkzalakBooQ0GG48zPxBdi33BnB7Jn87vJR+P5J+OFpuOomEq70II2eSG/6WBqwpP6JIb/5kpSMFMA5M9h9/jK/pAwoVzN4x5ksn2/vlWHN3gs8N7nEC8qXt1KhxcbKlFT3uoW3NYuyOYbK1jtuKJlKAy3Io1A0VAJdLL4DGAFsLN5OArYB3YUQz0gpPwqCbA0XYzQMvROG/BnSUpwKYd/nYMl3HndY4cAKPjHAGUcrltnHsNw+mks0L3WZGUO68vi173LP+nvYmb4TgB8uLsKsO4tkvNtEsXzHWbaeyKoR0cuuHbgWej1TVrvaeS4Gu9r5KmxfVlE0pEylvpRWfZNToagqgZqGHMBVUsqpUsqpQE/ADAwDHg2WcA0eIZzeQze97kx8d/MbED+oVJOOmgwe0S/j55D7eE//CkmaFLQ4+MPwDjw3uQ9h+jDeHvc2w1oPc58TEruBkLhVgIMiq73GlIAvJg+IJ9xLFHFZ84hrzcLTtOTCm3mloWQqDaQgj0LRkAl0RpAgpUz32M4Auksps4QQwU/X2RgIiXAmshs4Ey7ud84S9i4DUy4AOuFggnYHE7Q7oFk7iPoD5ERDdHvC9GG8Oe5NBrz3B3QRzoHX0GIbQluEKe1WatILOExf+t3AZc7xZcYJNNrZlxnFlak0NaeI+DKziPpCRbMdhaKhE+gIskUI8TWwvHh7GrBZCBEO+PePVJSnOD326lZ/YcvKfzNNs5FhmsMlx13psTe9CF2vhUGzCO0+keZ5c8iyL3HHGeij9iC0hRSl3g6O0BoRzWovWb0OJFZBFLeraFD0ZV5xZSpNTk4mKSmpqmIHHU/zl0LR2AhUEdwLTAF+g/N/fzHwhZRSgo/yWooKefGHM6TaRrGcUXQRqUzXJjNVu5kY4YpQLp0ee1H7m7j38DjO2o0YWmwDQBdxjLCO71F0bjbSFuXzXoFidUDC3DXERxsptNgqDDqTON+UKxokvS06K/OKQlE/CGiNoHjA3wokA+uBH4v3+UQIESqE+EUIsUcIcUAI8XTx/k5CiO1CiGM+qqEVAAAgAElEQVRCiGVCCIO/6zRmPM0lJ2Q8/2u7nRHmN/kfy1+9psfufuw/rNP+nY/zDpJwuWQA1YZeICzhbTQhF2tMttScooCL0QTiPVM2RYYqWKNQ1B8CdR/9MzAP2IBzRvBGsbfQ//k5zQyMlVLmCyH0wI9CiG+BB4BXpZRLhRDv4vRIeqdaT9FA8WYusaAnpdkYmPksZJ2ClI8g5RPILxnkhzj281Xefj5xtOCl2AgcAjT6XMI6vk1R2gzs+Vf5vW/zMD35JitWbzEMVXwOKO8eWtaOrswrCkX9JFCvoYeBAVLK2VLKWcAgKvAWkk6K/SXRF38kMBb4vHj/YmBypaVuJDw8IRFDmd9AKXOJKz32/Qfgtk+h2wQQJSfcXpDFuxfTCXc4R3ShtWBstwR9i034ix++UmRj+tAONfIMovg5VPStQtFwERVYeJyNhFgP/FZKaSneNgDfSCmvreA8LbAT6Aq8BSwEtkkpuxYfbw98K6Xs7eXcOcAcgLi4uEFLly6tzHM1GDaezOfrsxoyTZKYUMHU7nqubus7QjjEdIk9277jRplMO3EZgGN6PffFxZKqL5ngheV2J/3CH0F6v5ZBg/eo5iqwaGI4DyYXkmkq/7cUEyr4Z1LFdRvy8/OJiIioGYEaMKofnKh+cFLdfhgzZsxOKeXgitoFulicCmwXQqwq3r4Z+EUI8QCAlPIVbydJKe1AfyFENPAl4M1m4VUTSSnfB94HGDx4sKzPHiWVoaz5pHukjtBQA8JURGhoKD2vSiSpAvPJ75PDeInJjNbs5TbtRsbJXXyadpH741qyK9TpPVQYdZTeof+gc+oo1pvHUkDp4CeLwxnmUN3SxfHRRpKSksj6znuRmiyTDMgbqKa8hioyT9V36rv3VG2h+sFJbfVDoIrgRPHHNWysKv4eGcjJUsocIUQyMByIFkLopJQ2oB2QVimJGzDeomtTcwBKJ10D/6mbXWsLyY7+JDv6E0sO07SbeTJ1Ix+1yufLSOcbxJkQyO2YzBsZq0nPH8xS+xj2yC64wrhqon69qz5xfYi+9da/KhW2QlExga4RfAP0B24BbsUZRzBNSvm0lPJpbycIIWKLZwIIIYzAtcAhnGkqphU3m4VTqTQJvEXXliWQZGZlI10vEc079pu5zvJPjqb9hZszotEVj/I5Wi1/bx1NVstdrAiZx7eGx5ipXUszCqr/QJTUJ354QiJ6bek4Yb1W1Kp7qEoOp1BUjUBnBB8DDwH7caabCIQ2wOLidQIN8F8p5ddCiIPAUiHEc0AK8EElZW6wBJqkrKJ23iJdx/SIZePhS2zP6c2RogFozu7FGP8ZRToLUgjebR7FrtAQXrx0nmc0i3lc9ylrHMNYahvLrzIR3+Xu/ZOaU8SAZ7736mpqtUvmrz5QSubKUFkzj0oOp1BUjUAVwSUp5VeVubCUci8wwMv+k8DQylyrseAv42bZdhVRkSvmypRe3P95O0Ljl6ILPwHAL8ZQpsa34dlLmYwuMjFV+yNTtT9y3NGWpfYxfGEfRTbNAn+gYvzFG+QUWXl4+R6e/uoAOYXWgO32VTHz1AfzlELREAnUNPSUEOI/QogZQogprk9QJWuEBFLIpaaibScPiOf2Ib0oOnsH5kvXIqXzjT9Lq+Xe1q1Y0KI55uJJQFdNGk/qP2F7yL28qf8XIzX7EAFP/CrG6pBkF1rdbqX3L9tNQgWVvqpi5lHJ4RSKqhHojOBPQA+csQCuEUICK4IhVGPFm0mne6SFo3mGgMwflTWVuOoKfLLtWuyFCYS2XYZG70xf8UlUJN+HxvDypYsMtDrDPQzCzo3abdyo3VacHjuJ5fZryqXHri6uNWrPt/zoMm2qYuZRyeEUiqoRqCLoJ6XsU3EzRUWUNekE6h5WVY+Y5yb3YePhS6TmdKXw1N8JafM5+shDAFwKsTAzviXdM4fyeO4ZBmtOuM/rqMngEc1/eUD3ORscA/jUPpbNjn44arionest//nhpa+rzDwKRe0R6H/1NiFEz6BKovBLdTxiXG/R0h6O6fxMTBcmIx3OQDMhHBxreZyZbVszTj7CItt15MqSADCdcHCddieLDAvZEvI3/q77nLZcrsEn8/6W73JLDXQ/qNrCCkVVCXRG8BtglhDiFM4cQgJnFom+QZOskfDkyn18tv0cdinRCsGMYe1LlYIMlEBNJZ7mo+gwPWarvUzEnsCaMxxbYReMbZajDTsLgNZ4joxOy/nfy2N44fK/uF6zk9t0pdNjx4tM/q5bwV+1X7LJ0Zel9rGsdwzAVs16CN7e8l1uqYHs91czwaUslXlIofBNoP/BE4MqRSPlyZX7+HjbWfe2XUr3dmWVQSCmkrLmI3/ePNISS+GZuzDEbMbQ8geExg7CTkjsD9gj97PqwhS+NM3zmh5bIyRjtHsYo91Dhozmc/toltrHcFbGVeqZwGMxN/dYqf2VUXwV1UxQ7qMKhX8CTUN9xtsn2MI1dD7bfq5S+/0RiEdMIAFrpdFiyRxD4am/Yi9qX7I39CJhCe8QEreSEyLGnR77Xstf2WIvnRaqlcjhHt1qNofczyf657lJ8xMGfCsgz2iF5mF6n6mofa0FlN0fyDNHGX3nblIoFIGvESiqgN1HDgdf+/0RSD7/qr75OixxFJ6+G1P6DR5rBxJDi22Ed/knumYpWNCxxjGcP1ofZ5T5Vd60TSJdlvb1Gak9wBuGN9kWci9P6j6iiyhvm/d8cpOfPNiBuoIG8swFFptaJ1Ao/FBzxW4V5dAK4XXQ14qqRfFWFEQWaMCadzRYs0Zhy+tFaOtV7trIGl0exvhl2Jv/wrWt7mLnMSPnc+L4LGI2X+ln0SHzR27TbiRJsxutcD5rC5HPn3Xf8mfdt/zq6M5S21jWOIZhIqTUHf3Z7/25gnqug2h89LEnVrtU6wQKhR+UIggiM4a1L7VG4Lm/KlQUR+CtHGRlkdYWFJ2bjS5yPyFxq91xB9qwU2zIe4xhvW/g62sfYdOhQu5ftpsjDGadYzBtyORW7SZ+pytJjw0wRHOUIYajPCWXsNI+kqX2MRyUCe7jFcUFlB28y64JBDq7UusECoVvlCIIIq4F4ZrwGgokjqDsW7TLa6iw0qXIBLa8PtgKuhHScgP6Fj8ihAMhJL9kfs0NKzZhyRyHZDCuP6ELxPAv+xTetE9mlGYft2k3cK1mF3rhlLeZKGSmbh0zdevY4+jMUvsYVtuvJjq6RaUk87Um4Gv25SLKqGfkgg0q0Eyh8IJSBEHmucl9qjTwl8VfHEFF5SA7zV3jp16ZHxyhmDOux5ozmJC41egijgOQZ82DZisJD92MOWMitrw+uJaBHWjY5OjHJkc/d3rs6dqNJGjS3ZftpzlJP81JntR9zCb7aDgfFXBObF9v9g4pnT7NPs4rsNjIKXIuYqv01ApFadRicQOhOpk1qxuN67C0oujcHRSem4m0tHTv1xiyMLb7lLCEt9GGnSh3nis99hjLP5lheYLV9hGYZcm7R7gwc711HfxnLHEb/8orzz3Emu0H/coSHebdAyg6TO/zOYVwrhN4otJTKxQlKEXQQAjUndIbgSS7cyHAR1uBPb8n+Sf+7vQuspfcV2s8R1jHf2Ns/x80oeVdYyUafnb04q/W+xhufpNnrX/gmKP0m/hVmrM8YPs3474Zzdn//BHO/OR1luBr4iClb08jX+eodQOFwokyDdUj/C0Ge1sIDjSzpjcPHF/eRRIqWGzWYc0ahTVnEIaYTRhabEVobM4jEcfRRRzHmncVlkvX4jCXN7tk04wP7Nfzgf23DBJHmaHbyA2abRiFBYBQYaXD+dXw4WryIjqzyDSaRfnDCY2O4+EJiW7zTllyiqw+PY18RR27lGhDL2+pUFQXpQjqCRUtBlc3s2bZtYMuj31TpXgGN44wLJd+izV7BIbYH9BH7UQUu4/qIw+hjzyENa9nsUJo6+UCgp0ykZ3WRJ7hj9ys/YkZ2g300pTEKUbmn+Q+TnJXyEd8XzCEFSvGoRFX4ZDlJ7Iul1xfLra+lKgqb6lQKEVQbwhkMbiiOILK4Mu1tbJIWzTmC9OwZF5DSOw69M32uo/pIw+ijzyILa8H5stjcZg6eL3GFcL52D6ej+3X0kec4o6wzVxn30wYJsAjPTbbOCO8p8f2p9T8KdGRCzYEtAivUDRmlCKoJ9R2mcWyrq3VRVpiMaX+HsvlsRha/oC+2X73MV3kYXSRh7EVdMGSmYS9oCveS2MKjuu6wY1TGLJsGzdof2aGdiMDNMfdLcqmx/7MPpZNjn40M4Z4uV4JvpSoKm+pUChFUG+ojfz73mzhgzu28GlDh4r988viMLfGlPoHLJfTMLTciC5yv9tkpAs/gS78BHZTWyyZ12C70hsoWdzVCsHUQc4Be+Ha5vw3Zwz/tY+hhzjLdO1Gpmi3ECUKndcqTo99nXYnabIFy61JTHkhm5kTf1OpN3lV90ChUF5D9YZgl1n0lqv/4eV7ePjzPT6VgFGvZcaw9gF7HHniMLfFlHo7hSfvx5o7AOlh19eGpmGM/4zwrgvRt9gCGqcJyC4lX+xMZWVKaqm6A4dlB562zWKo+W3+brmH7Y4epe7VVmTxN90KPjfdRfSXv+eux+fTde4qujz2DU+u3OdXTlXeUqFQM4J6Q7DLLHpbg7A6fL/px3vc3zVrCDS3jycOSytMadMRl8ZjaLEFffQOhMbp+aPR5xAat4aQluuw5g7Gkj2CIkusT/9+MwZWOX7DSstvfKbHThK7STLs5pKMYrn9GpZuH8OT+E77rcpbKhQgZA3Yh4PN4MGD5Y4dO+pajKAQaKnK6lKZ6GIBnFpwg9djCXPXVEsOoc1H33wb+uY/o9EVlDtuy++ONXsE9vxEZAATVgNWxmt2cpt2A6O0+722+cnRi6tvfQCuugl0/tcS6pra+nuo76h+cFLdfhBC7JRSDq6onZoRNBGijHqfPvhl8WUfr4lUztIegeXytVgyr0EflYK+xY9oQzLcx3URR9FFHEXYYjBlDcGaMxhpj/B5PQt61jiGs8YxnPa2dKZrk7lVu4k4keNuc7XmAHxxBxhbQL8ZMGgWxCrTj0LhQimCJkKgma/92cfnrz5QcwJJPdacoVhzhqANO46hxVa0EUfcC8tSl0lIq+8wxK7DltcLa/ZQ7IWd8besdU7G8bJtOq/apjFWk1IuPTZFWbDtLeen/XCnQug5GQxhPq+pUDQFgqYIhBDtgSVAa8ABvC+lfF0I0QJYBiQAp4HfSSmzgyWHwkmOn7KV8dHGgOzjgc4oKofAXtiNosJuCH0mhubb0UftQOic3kFC2NE324u+2V4clhZYcwZhzR2MtEWVu1K4QUuBxY4dLesczvTYrcnk+YTdjCv6HnI90l+c2wbntpG38kEuJUyi84R7oE31S3CrKGVFQySYMwIb8KCUcpcQIhLYKYRYB8wG1kspFwgh5gJzgUeDKIcC326S8dFGts4dW+vyaIRz9lFgKVnAltYYzBnXY740Hl2zvRiit6MNKwl60xiyCGm1DkPsD9gLumHNHYgtrxdIZyK6A89M5MmV+0ql/b522ADGTZ4JDjuc2EjahneJTdvgTo8dSSGRpz+D9z6DtgNg4CzoMw1CIiv9TCpKWdFQCZoikFJeAC4Uf88TQhwC4oFJQFJxs8VAMkoRBJ3q5CoKBg5JKSVQCqnHljsIW+4gNCEX0UdvRx+1G6F1KjIhpHstQdpDsOb1wZ47AId0+E77rdFCt2u59QsNFvMtTC1Oj93JIz02aSnOz9onoPcUGDQb4gcFbFfzFR3+9FcHlCJQ1GtqxWtICJEAbAZ6A2elLCl2K4TIllI293LOHGAOQFxc3KClS5cGXc66ID8/n4gI34uhNclPaVa+OGol0ySJCRVM7a7n6raBF3af/V15L59aQ1jRRR5AH70DbdgJ91qCJ9HaaAaFD2Jw+GDi9fEILwO45zMIHAzXHOI27UYman4hRNjKtc8P78iFNhNIj7sGm97/78lf/8zpawCosP9r8++hPqP6wUl1+2HMmDEBeQ0FXREIISKATcDzUsoVQoicQBSBJ8p9tH5QXdfRmkLostFH7UIfvQuNIdNrm85RnZmYMJEJCRPoHN3ZvX/kgg1eTWTNucIUrbP+cjeNF+8oXahzYXnQLOgwwusswde1AaKNesw2R7kZ2QtT+pSaLTSkv4dgovrBSW25jwY1slgIoQe+AD6RUq4o3p0uhGhTfLwNkOHrfEX9ItxQ+QjjYCBtzbFkjqPgxEMUnLoHS9ZwHLbSnj8nc0/y9p63mbRqElNWT+HdPe9yMuekz9oMrvTY4y0vMdX8FN9oxoDOw43WZoK9S+HD38JbQ+GnN6Dgcqlr+DOz5RRZfSa3A+f6wsgFG5j9XQEjF2yoEVddhSJQguk1JIAPgENSylc8Dq0GZgELin+uCpYMippFr9UA/moV1DYCh6kDZlMHzOk3oo04jr7ZbvSRB0Fjcbc6ln2MY9nHeGv3W3SO6sy4q4ey40A70i/HIMslv3Omx95VmMippxbBvuWwazFc9EhVcfkofP8k1u/ns9Y+mHWhExnz22lMHtie+asPVMq7Ki2nSC0yK+qcYHoNjQT+COwTQuwu3vc4TgXwXyHEHcBZ4NYgyqCoBBW5PuYGxX3UO/F+iud4R4c9vwf2/B6YhYV37jLy/env2Xx+M2a72d3qZO5JTnISYqF7QlsuZ3Ql53J37IWd8EyA1zbaCMZoVuqvZ2F2Z2LMB/l/YZu5gR/R252urXpszvTY1m2cXfUOB4/dxm09x/DeTmupKG6jXkuoXkO2FxfettHGgOtRKxTBIpheQz/iPdcwwLhg3VdRNQJ5K/VX2aymCTNU3WoZbQxnYsJ1TEyYSIG1gFd+XMXSg2sQYYfceY4A0grSIDyNsPDNSHsotvxEbPk90JuvYkyPDgx45nv34J1KZ/5e0Jln9bdzs+5nbrb/UCo9dgeRDodep7t8g4H6kvTYEg1TBznzNfny2vr7st14ozp9reIZFJVBRRYrgMAK44zpEVsjxWwC4VhG1T2UPP0fwvXhfLe9NQU5vwdhcbqdRh5AF3EIoTW52wmtCX3UHvRRewDByosdsRgT0dgTcZjb4HqnybTq+dA6mg8ZTaI4y20Vpce2J7F651iem/xHdpzJKhXj4Eq5/eB/93hN5KcNNBy8DMrUpKgsShEogMAKtGw8fKm2xKkWZU1Y7meQBmx5vbHl9QZs6MJOcceEItac+IFcq6fPgkQYTxNiPE1Iq7U4rJHYC7pjy++OrbAr2MMBOFKcHnuBbQa/1fzCDN0GhmkOu6/iSo99n/ySPS8s4krhaITsD+jcKbcHd2zhM5trVQsGKVOTorIoRaAAAivQ0lCqdpVNmuf92XTEGfpylSGRJYf6Y9akoos4jC7iEBrjuVJxChp9HproneijdyKlAHM8srA7prwu2Is6YpYG1mqvIVk7hhamM0zXbmRamfTY/cw7+Jd2B//QONNjL7MnccbamoVrjxDtIyFgtNEZY1BZM4+quqaoLKowjQIIrEBLZat2VaWgDYBeAyO7tKjSud6ipR+ekIheW9rMotcKHp6QWPz27MBhboMlcwyFZ+6h4NgTFKVOx5rbD1nGLVUIiQg9j6bFBsI6/puI7k8T3XkRfXvvgpBznJStecF2OyPMb3Kv5a9ssfcudX6syOUe3Wo2hTzAJ/rnGXhlPQa8L8IL4b2g0GMr9vl1L/X1e6pO1TWXe2unuWuUe2sjRCkCBeC0Hb8wpQ/x0UYETq+dssFO3pSFViPcHgECZ6yB5/nxVRh8bA745M4R5ZRBRcrBm8xuylpZire9vSVLewS2KwMwpc0g/9iTFJy6B/Ola7EXdnDOCDwQGiv2kMMcNH2Gvc1rRHR/htB2i5HNt/OtvgN/tM5llPlV3rRNIr0kjtL5PNoDvGF4k+8cc3hS9xFdROnBNafQ6tfM44uarrpWFWWkaFgo05DCja8C757HofLVvLx5y7wwpY/PWsmuN9dOsRFsO5ntXlztFBvBTyeyvBbYEeAzed7CtUfKVWOzOiQL1x7xaRKLNurJM9mwSw0OUwcspg5YLl8LmiJ04SfQhh9DF368XGSz0JrQRx5CH3kIAGkP5XJhAm8WdeJd81yGFV7k99pNpdJjtxD5/Fn3LX/Wfcuvju4stY1ljWMYMdHRVTLzVPR7qqypSa05lNBYvbGUIlBUioqUhbf24H1QWr7jrNdBOCHGyJMr95XyULJLycfbzhIXaSA9z1LunKv9zBb8DaavTu/vVVHNv7kXUF6J4TC6F5zNgNBnoQs7gTb8BNqwk2j0V0rdQ2hN6CIPQ6RzEXm7Q89PRe0JK/odf4o08bucX2hVeMHdfojmKEMMR3lKLiHZfg0rmo0n+UrrcrJXZObx9XtamZLKw8v3uBWjq3a16xxv+HJjrci9tbENmo3ZG0spAkXQ8TUobTvpvQzFtpPZPo95UwIApzN9D0rRYXqvwVzRYfqAZjmuY95mItLaAmtuC6y5QwCJMFxGF3YSbdhJtOEn0ejySrUXGiu68JNYwk/yHvBeKwOhlj70M1m43pzGQHMRHW02molCbrZ+y818y76QznxqG8Nq+9UUYKyWmWf+6gNeZ0fzV/vOkKr1Uafan3trYxw0G/PMSCmCJkTZXP0zhrX3WdS9NqhJt0l/phJfl3PtD3SW42tA9EwoJy2xWC2xWHOG4VYMxtNOxRB2Bo0hq/TJQmIKyWV7CGynOdCcSLukn9lEX7OZPmYLfcyneUF8wJO6j9mgG0Wzq+/gmv5t/crq623cV/oLf2kxqvJ7quygWVbeGzrY3bnq6wuN2RtLKYImgi9TC1BnykAI74O0EKDB+6Driyij73TavlJjVJQyo+xbrTd5PM1I5dc8RIliyB3i3KPLRRt2Gq3xtFMxhFwol1I7Tyv4MczIj2El5p8OViu9zBZ6m7cT/8sWCo90IWzQbOj7OzCWTt7r7228KvhK9+HPEaAy5iRv8i66Aj1TUuvVm3YgLtYNFeU11ET4bPu5Su2vDYw6739+Rp2GGcPaez0W4uMcf0G4VXWn9PZWC86ZQVnPqskD4tk6dyyvTe/v121W2qKwXemHOX0Shaf+Sv7R+RSe+TPmjOuw5SeWc1d1cVav59uIcBbGNGdW2zhGhOVxy95XeOLDIXyy9CZS9i6h0FLgU27X27ivfvLXf1XxQvJlNvK235u8Fgd+PaPqgpr2xqpPqBlBE6Gmo1drgiKrw+d+1yylrCnrEx8pLvzVZK5qdTZfU36HlJxacIN7u6xZY2CHKJ/eTeUvFoK9sCv2wq6QCSAR+ky0xrNojefQGs+hCb2AEKUHSocQHDcYOG6A1ebTkLIQsWshHQ1R5IfFo9d0xGFqi8PcBmmPcD+PL5mk9G06rIq3WGX+3oJpcvG3YO3rWE32Q0NBKYImQlUW/IJNRVNtb2UnNx6+VOnpeVX/gQMxBXgza1QvMZ9AWltis7bEdmVg8S4rmpALDL+qgE7x2Ry4tJeTV06XD40QcNqaC1G5hEYddO93WJvhMLdBb4snVLbnclYLHJaWeGZbNeo1fk2HlfUWq4w5KVgml4pMZN6OLd9xlq0nStZxqtsPDQWlCJoIM4a195owzpcJpjaoypt6Vd/uq/IPHMi9fJmPyhKm11DoYwZUIVKPw9SB7bsFn06/HoCrX/yWdNMJ4o17iTfupzA0m9N6LQ4vil2jv4JGfwXJEYqA8OYgHTocllgcptZobG2xFrVCyNZIWzM8kwZ/tv1cldaQKvN78tbWoPFf6CcQKgrG83bMUwl4UtV+aCgoRdBE8GVqqcs/bs839dScIuIDeFOvrem5y2xQdrAY2CGq1L0CNV8UWR10axVerayqdikZuWADY3rEkpbtADpxtqgTZ5lEGCZ+q/uREcZkHKEZHDLoOWIwcNSgx6wpv64iNDa0oRfQhl4AUtxzA2kPxWGOw26Ow2FuhcMcR0bhYGKNsV5rQPti8oB4n9lWvbUFynkNVfd3WpMmp7o0odYGtVK8vrqomsWNn/rUD2VNCmX5w/AObgXqr06xJwatwGIP/v9afLSRNqYT3GBbxxTtFsJFIWf1OrdSOGIwcMgQwmUfi+6+iNRH0im6E12iutApqhOdozrTOaozbSPaotWUXxz31ofeajT7oib+Hvo//b3PZH7hIbpKmfC0QnDiheurJU9VqK2axWpGoFCUoSJzj6eZwJtZwxu1oQRcppenv7KVTo/t2MBvrYf5rcdkJFcjOKIP4Tt9RzbrW5NmsKMJyShVo8GTPGseey/tZe+lvaX26zV6OjbrSKeoTiQ0SyAhKoGOzTry0vfnKLKWnkHUdvCVPw8pX6argR2ivJqH6tKEWhsoRaBQlKEi04GnmaCsWaMu59eut+37iyuemTGw0vEbVlp+Q2eRxm3ajUwtTo8d5ZAMNZsYaj4CHOGSjGK5fRSfiaGkGRxoQtLRhKSjNWQQHZVNnjXP6z2tDivHc45zPOd46QOtIbxlONLSEoclBoelJQ5LSy4WxVBgHUa4PjzIveHbkyyn0OrXxFjfAi9rA6UIFIoyVFSSs6ynledCdKCmIs9r+bM/B1q72VMmb/KflG35X9vtLLRNZ7xmB7dpNzJaW+JB40yP/TX38DU/WXqytGgs3zkmUYieAy9cz0NfbmH1gRQwpKMNuUSblleQ+nQyTaWT7nmi0RWArgBt2JlS+4d/+gYxoTF0aNaB9pHt6RDZgfTMcL5JsZKeFU6byBhu7OCodmRxRd5IvhwIBndswcbDl0jLKaJ1VCiDO1YtJXpDQikChaIMFZl7/JkJqmJy2H4y0+sicrdW4dw7pltApie7lG7XSH/yW9HxjWM43ziG096WznRtMrdqNxEnctxtrtYe5GrtQbJkBNsixvOvZQ6+2K0BukJBV6zAqYvOtZJHb0ngdO5pTl05xenc05y+cppTuchkOYYAABIzSURBVKc4nXsWm/SeFwog05RJpimTlIyUkp0tIaIlXLGH8kleC/asXMGwdt2Ij4inXWQ74iPiaRvRFoPW4LcvXFTFw6wx5kgKBKUIFIoylPVmchGImaCqJofxrySXUgbdWoWz7oEk97bn9cb0iHVfxxOXDd6VjttXmm8X52Qcr9pvI2PQ/Txz1QV2rXyNfkW/lEqPfX3Bl3D4S642lKTHNhEClKyV9Il1fjxxSAdLftnDO1u3kWVJo1lkNgltirCIDM7nn8fmsPmUS2hNCG0aB3PTOJj7Y+ljCGLDYomPiHcrBtfPtuFtaR3e2q0oquJh1pgTy/lDeQ3VMfXJW6Yuaer94HJXDdSNttPcNT7rMnhGPQfazk1uKqR8DCkfQW759CNXZBgr7SNZah/DQZnAaW/XqACbw8bFgoucyzvHubxz/GNNMsKQiUafhcaQidD4zwHlFymI1DenU/N2tA1vS5uINrQJL/m0Dm9NM0Mzn66wle6vIKO8hhSKJkJVzBGBRuNWOmo3Kh6SHoXRD8GJjbBrEdaD36AvTnHRTBQyU7eOmbp17HF0hh0Z0GcahEQG/Lw6jY52ke1oF9mOEYzg9S9bkprhklEitPkIQxYx0XnckRTF+bzznM8/T1p+GumF6Tikn8A8IcmzZbH3UlY5DycXRp2R1uGt3YqhdVhr4sLjnD9b5nIxKxQcIYH1VyNBKQKFoo6pijkiUPt3VSOx0Wih27XQ7VrGzVvK9baNTNdupJMm3d2kn+YkfP13WPsE9J4Cg2ZD/CD/GewqfBaBtEeiN0fyxOj+TO5X+vmtdisXCy4y7T9fk2VORxiy0OhzEPpsNPpshO5KuWyuZSmyFXEq9xSnck+VPxgLkbEg7SE4bFFIaxQaRxQDunZl2eFLtAprRavwVsSFxdEitAUa0TjydgZNEQgh/g+4EciQUvYu3tcCWAYkAKeB30kpvVcgUSiaCMEoR1nZdv44Z4nkXW7mPfuNDNcc4jbtRiZqfiFEFNv5rQVOU1LKR9CqFwya5TU9diDPkppThFYILMWlRD2PA+i1eto3a09GRgckHbxczY7Q56LR5/Dq7R3ZcOwIm08do8hxGUPoFXSGXCwO77ESngitGa02A0IyANiUsYNNGaXb6ISOGGOMUzmEtaKlsSWtwloRa4wlNizW/TM6JLreK4xgzggWAW8CSzz2zQXWSykXCCHmFm8/GkQZFIp6T1WTrgWaP6m6idJc8kk0/Ozoxc+OXkSTx58itvO35j/BpcMljTMOwLePwLp50HMSDJwFHa+ucJbgki9QE5lvF18t0tqC1uHxOK4ksnargSJrTwCKcCbX+8fNnRjURXCx4CIXCi6QXpjOxYKLXCy4SEZhBumF6Zjt5gr7xSZtpBemk16Y7redS2HEGmNpaWzp/B4WS8vQlu5t10+jrm5MUEFTBFLKzUKIhDK7J4HbPXgxkIxSBIomTpXNN7WEN/nM+mg63vAQ9G8L536BXYth/wqwFQ/ONhPsXeb8xHRzzhL6zYDwlj7vUxkTmT8XWVffeb+eg7fWp7F1yFgSW3jvXyklOeYct1JwKQj3p8j5M9ec67ffXASqMADC9eHEhMYQY4whJjQGU5aJQ3sOcVffu4I6qwiq11CxIvjawzSUI6WM9jieLaX0On8UQswB5gDExcUNWrp0adDkrEvy8/OJiIioazHqnKbeDz+lWfniqJVMk4OYUA1Tu+u5uq3vqmu1TYl8kphQ4VU+ra2AuPTNtLnwPZH5J8tdwyF0XG45jAttriO7ed//3969R1dVnnkc/z4EDGgwCBSKkatFaGgtBBRZKoTRhUiroJUCrZpp7Tito0sdtQvb1dtMZ1a12rHWesFqsdRCbSvVtlRxqlzKFBQhchEzWAUVGe4mg1yCydM/3n3kJOQG5JydnP37rPWus7P3Pmc/+1075z3vvjwv1Pti+8dnGk/IN3vikU8ip8fUAaiFOrEd7ecdreraaqpqqqiqqeK9mveoqqmisqbyw5L6e3/t8Y2rkG/53NXvrmN67/jx41t011CbbQjS6fbR3Kd6CHKmHt5dDS8/Bmt/DdV7j1i8xXpTOXQ6xZOug64fBRp/KruoW5cPn404Gq39ecfqwAcH2Ll/Z7Nl14FdDT5f0bdrXxZcvuCYtt1Wbx/dZmZ93H2rmfUBtjf7DhFpf04dEcqE78H6J9m99GG67zl8O2eRb6Now4+ofe0+OpwxEUaWcduEj3P7/Fdb7RRZWznl1rlj5w9vl22Ku1NVXcWu/eGp6137d7F87XKKhxRnPMZsNwRPA2XA96PXp7K8fRHJpvwCKLmaSxaeRsHBCqbnvcDleUsptH0AdPAaqPgjVPyRKScXMbT4Ur6+aTirKru26MG6prS3oSXNjML8QgrzCxnEIAA6b+5M6ZDSjG87k7ePziVcGO5pZu8A3yY0AE+Y2TXAW8DUTG1fRNqOkJm1X9302B2fZ3SHtDuOqrYwtOoBnsTYdeoIelx4MwzpdVzbzdWhJVtbJu8amtHIogsytU0RaZvSb/dMT4895uTdzB1VAeW/hH2pTKZOj92r4Imr4KReMPzzUHI19Dg9vh3IcW37KQcRyQm3XTSELp3qjmTWpVMe0y7+h3Ad4V9fg6mzYdD4um98fzssuwd+XAKzPwNrfwMfNH+PvxwdpZgQkYxr9nx9xxNg2GWh7NnE5vnfo//upbD3/w5/yKaloXQ5JTyTUFIGvYbGsDe5Rw2BiGRFi8/XnzKANwddSf+yB2HjwvCw2saFkEo2t38PLL8/lL7nhIfViqfACSdmdgdymBoCEWmb8jrC0EmhVG6B8sdh1RyofOvwOm8vD+VPM+HMqaGX0OfM+GJup3SNQETavsIiGPc1uLEcrvwtfPwS6JD2O/ZgJbz0U3jofJhVCit/BgcbHmdZjqQegYi0Hx3y4GMXhrJ3e9RL+DnsTktp8e7qUI4zPXaSqEcgIu1TQS8472a4YRWU/R4+cQWkj2ecSo/90wvggXNhxUPh+oIcQT0CEWnfzGDg2FD27YZX5oULzK2QHjsp1CMQkdxxYncYcx1ctxy+tBCGfwHSc/yn0mPPngT3nQXL7oX3d8YXbxuhhkBEco8Z9BsNU+6HWyvg03fDR+vdTbRrIzz3Tbh7KDxRFsZorm1iPOQcplNDIpLbOhfCWV8O5cP02L+B6uiuotpD8OrvQunWP6SzGP4FOLlPvHFnkXoEIpIcp46AS+6BW16DS++D086qu/y9zfD8v8N/DYO5M6DiGag9chS0XKMegYgkT34BlFwVyrb1oZewZh4ciIaf9BqoWBDKyUUw4spQuvWLN+4MUY9ARJKt9zCYdCfcUgGXzYL+59ZdXrUFFt8B95wJv/gsvPo01ByKJ9YMUY9ARASgUxf41LRQdm4Mt6CWz4V9qbuKHF7/71ByLD22egQiIvX1HBylx97Q8vTYhw7EEmprUI9ARKQx9dJjs2oOrP5FzqXHVo9ARKQlThkAF3wTbl4P0+fCGRPB0r5CU+mx7x8Nj0yA1Y9D9b7Ywj0a6hGIiByNFqXHXhHKMzPhk1PDmAl9PhVfzM1Qj0BE5FgdkR770nrpsatg5SPw0Fh4aFybTY+thkBE5Hil0mNPmxMuMF/4Xeg+qO46W8vhDzfBXUPgqevhnZXgHk+89aghEBFpTQW94Lyb2lV6bF0jEBHJhHaUHlsNgYhIpqXSY5/zVXj7xdAgrHsSPtgflqfSY6/5FfQYHCW++3zWwovl1JCZTTSzCjN73cxmxhGDiEjWHWV67OL1d8Lfns94euysNwRmlgf8BLgYKAZmmFlxtuMQEYlVKj32V5bCtYtg5BfhhK6Hl9ceoteOZTDnMtixIaOhxNEjOBt43d3fcPdqYB4wOYY4RETahqbSY59aEhLjZZB5lm9fMrMrgInu/uXo76uA0e5+fb31rgWuBejdu/fIefPmZTXObNm7dy8FBQVxhxE71UOgeghUD3DS3k30fGsB+3sMY3vvccf0GePHj3/Z3Uc1t14cF4sbuhx+RGvk7rOAWQCjRo3y0tLSDIcVj0WLFpGr+3Y0VA+B6iFQPQSLFg2gtLSUTJ87j+PU0DtA37S/TwPejSEOEREhnobgJWCwmQ00sxOA6cDTMcQhIiLEcGrI3T8ws+uBZ4E84FF3X5/tOEREJIjlgTJ3XwAsiGPbIiJSl3INiYgknBoCEZGEU0MgIpJwWX+g7FiY2Q5gc9xxZEhPYGfcQbQBqodA9RCoHoLjrYf+7v6R5lZqFw1BLjOzlS158i/XqR4C1UOgegiyVQ86NSQiknBqCEREEk4NQfxmxR1AG6F6CFQPgeohyEo96BqBiEjCqUcgIpJwagiyyMw2mdlaMys3s5XRvO5m9pyZbYxeT4k7zkwws0fNbLuZrUub1+C+W3BvNJTpGjMriS/y1tVIPXzHzLZEx0W5mU1KW3Z7VA8VZnZRPFG3PjPra2YvmNkGM1tvZjdG8xN1TDRRD9k9JtxdJUsF2AT0rDfvTmBmND0TuCPuODO072OBEmBdc/sOTAL+RBi74hxgRdzxZ7gevgPc2sC6xcArQD4wEPgbkBf3PrRSPfQBSqLprsD/RvubqGOiiXrI6jGhHkH8JgOPRdOPAVNijCVj3H0JsLve7Mb2fTLwcw+WA93MrE92Is2sRuqhMZOBee5+0N3fBF4nDPXa7rn7VndfFU3/P7ABKCJhx0QT9dCYjBwTagiyy4GFZvZyNBQnQG933wrhoAB6xRZd9jW270XA22nrvUPT/xy54ProlMejaacHE1EPZjYAGAGsIMHHRL16gCweE2oIsutcdy8BLgb+xczGxh1QG9Wi4UxzyAPA6cBwYCtwdzQ/5+vBzAqA3wI3uXtVU6s2MC9n6qKBesjqMaGGIIvc/d3odTswn9Cl25bq4kav2+OLMOsa2/dEDWfq7tvcvcbda4GHOdzVz+l6MLNOhC+/x939yWh24o6Jhuoh28eEGoIsMbOTzKxrahqYAKwjDNNZFq1WBjwVT4SxaGzfnwauju4UOQeoTJ0uyEX1znVfRjguINTDdDPLN7OBwGDgxWzHlwlmZsAjwAZ3/2HaokQdE43VQ9aPibivmielAIMIV/tfAdYD34jm9wD+DGyMXrvHHWuG9n8uoYt7iPCr5prG9p3Q/f0J4Y6ItcCouOPPcD3MifZzTfSP3idt/W9E9VABXBx3/K1YD+cRTmmsAcqjMilpx0QT9ZDVY0JPFouIJJxODYmIJJwaAhGRhFNDICKScGoIREQSTg2BiEjCqSGQnGJmA9Ize4pI89QQiETMrGPcMbREe4lT2g81BJKL8szs4Si/+0Iz62Jmw81seZTEa35anvtFZvafZrYYuNHMpprZOjN7xcyWROvkmdkPzOyl6P3/HM0vNbMl0ee9amYPmlmHaNkMC2NPrDOzO6J5nzOzH0bTN5rZG9H06Wb2l2h6pJktjhITPpuWbqFOnNmtTsl1+mUhuWgwMMPd/8nMngA+C3wNuMHdF5vZvwHfBm6K1u/m7uMAzGwtcJG7bzGzbtHyawgpDc4ys3xgmZktjJadTcgRvxl4BrjczP4HuAMYCewhZJydAiwBbovedz6wy8yKCE+XLo1yzvwYmOzuO8xsGvAfwJfqxynSmtQQSC56093Lo+mXCVkcu7n74mjeY8Cv09b/Vdr0MmB21ICkEqFNAM40syuivwsJjU018KK7p37ZzyV8qR8CFrn7jmj+48BYd/+dmRVEOaf6Ar8kDFRzfrStIcAngOdCChryCOkoGopTpNWoIZBcdDBtugbo1tiKkfdTE+7+FTMbDXwaKDez4YQ8Nze4+7PpbzKzUo5MAew0nCo45a/AFwl5YpYSfu2PAW4B+gHr3X1Mc3GKtCZdI5AkqAT2mNn50d9XAYsbWtHMTnf3Fe7+LWAn4Zf7s8BXo1M3mNkZUQZZgLPNbGB0bWAa8BfCwCLjzKynmeUBM9K2twS4NXpdDYwHDrp7JaFx+IiZjYm208nMhrVeNYg0TD0CSYoy4EEzOxF4g/CrvCE/MLPBhF/1fyZki10DDABWRWmDd3B4CMW/At8HPkn4cp/v7rVmdjvwQvQ5C9w9lU55KaFxWeLuNWb2NvAagLtXR6ef7jWzQsL/5z2EbLUiGaPsoyLHKDo1dKu7fybuWESOh04NiYgknHoEIiIJpx6BiEjCqSEQEUk4NQQiIgmnhkBEJOHUEIiIJJwaAhGRhPs7+1glDIwPFvAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "yplt_inv = 1/zplt_inv \n",
    "plt.plot(x,y,'o')\n",
    "plt.plot(xplt,yplt,'-',linewidth=3)\n",
    "plt.plot(xplt_inv,yplt_inv,'-',linewidth=3)\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)\n",
    "plt.legend(['data', 'linear', 'linear+inversion'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can conclude by comparing the RSS using the linear fit and the linear fit+inversion.  We see that we get a slightly reduced error using the nonlinear transformation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RSS =   23.94 (linear)\n",
      "RSS =   20.66 (linear+inversion)\n"
     ]
    }
   ],
   "source": [
    "zhat_inv = beta0_inv + beta1_inv*x\n",
    "yhat_inv = 1/zhat_inv\n",
    "RSS_inv = np.mean((yhat_inv-y)**2)\n",
    "print(\"RSS = {0:7.2f} (linear)\".format(RSS))\n",
    "print(\"RSS = {0:7.2f} (linear+inversion)\".format(RSS_inv))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}