Homework 6 Solutions

hw6.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework 6 Solutions\n",
    "\n",
    "Welcome to your **sixth** IPython notebook! Please *edit* this document with your answers and upload to TED.\n",
    "Solutions to HW5 IPython are on Ted!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      ":0: FutureWarning: IPython widgets are experimental and may change in the future.\n"
     ]
    }
   ],
   "source": [
    "# load numerical packages\n",
    "from __future__ import division\n",
    "import numpy as np \n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt \n",
    "import seaborn as sns\n",
    "import pandas as pd\n",
    "import math\n",
    "from IPython.html.widgets import interact, fixed\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**NOTE:** Please run the following cell before skipping to any section. **Do not** change it without knowing what you're doing."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this problem, we're going to use the excellent **pandas** library to manage and manipulate data. We will see how to compute some descriptive statistics, make box-plots, histograms, etc., all a part of exploratory data analysis. Pandas can easily read csv files, excel files, etc. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below, we will load the famous iris dataset. While it was introduced in 1936, it is still one of the most popular example data sets in the machine learning community. Wikipedia describes the data set as follows: \"The data set consists of 50 samples from each of three species of Iris (Iris setosa, Iris virginica and Iris versicolor). Four features were measured from each sample: the length and the width of the sepals and petals, in centimetres.\" Here is an illustration what the four features measure:\n",
    "\n",
    "\n",
    "![\"iris data features\"](http://sebastianraschka.com/Images/2014_python_lda/iris_petal_sepal.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SepalLength</th>\n",
       "      <th>SepalWidth</th>\n",
       "      <th>PetalLength</th>\n",
       "      <th>PetalWidth</th>\n",
       "      <th>Name</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>5.1</td>\n",
       "      <td>3.5</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>Iris-setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.9</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>Iris-setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>4.7</td>\n",
       "      <td>3.2</td>\n",
       "      <td>1.3</td>\n",
       "      <td>0.2</td>\n",
       "      <td>Iris-setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4.6</td>\n",
       "      <td>3.1</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.2</td>\n",
       "      <td>Iris-setosa</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5.0</td>\n",
       "      <td>3.6</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>Iris-setosa</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   SepalLength  SepalWidth  PetalLength  PetalWidth         Name\n",
       "0          5.1         3.5          1.4         0.2  Iris-setosa\n",
       "1          4.9         3.0          1.4         0.2  Iris-setosa\n",
       "2          4.7         3.2          1.3         0.2  Iris-setosa\n",
       "3          4.6         3.1          1.5         0.2  Iris-setosa\n",
       "4          5.0         3.6          1.4         0.2  Iris-setosa"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# load the data\n",
    "iris_df = pd.read_csv('iris.csv')\n",
    "\n",
    "# output the first few rows so we can see what it looks like\n",
    "iris_df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SepalLength</th>\n",
       "      <th>SepalWidth</th>\n",
       "      <th>PetalLength</th>\n",
       "      <th>PetalWidth</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>150.000000</td>\n",
       "      <td>150.000000</td>\n",
       "      <td>150.000000</td>\n",
       "      <td>150.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>5.843333</td>\n",
       "      <td>3.054000</td>\n",
       "      <td>3.758667</td>\n",
       "      <td>1.198667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>0.828066</td>\n",
       "      <td>0.433594</td>\n",
       "      <td>1.764420</td>\n",
       "      <td>0.763161</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>4.300000</td>\n",
       "      <td>2.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.100000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>5.100000</td>\n",
       "      <td>2.800000</td>\n",
       "      <td>1.600000</td>\n",
       "      <td>0.300000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>5.800000</td>\n",
       "      <td>3.000000</td>\n",
       "      <td>4.350000</td>\n",
       "      <td>1.300000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>6.400000</td>\n",
       "      <td>3.300000</td>\n",
       "      <td>5.100000</td>\n",
       "      <td>1.800000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>7.900000</td>\n",
       "      <td>4.400000</td>\n",
       "      <td>6.900000</td>\n",
       "      <td>2.500000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       SepalLength  SepalWidth  PetalLength  PetalWidth\n",
       "count   150.000000  150.000000   150.000000  150.000000\n",
       "mean      5.843333    3.054000     3.758667    1.198667\n",
       "std       0.828066    0.433594     1.764420    0.763161\n",
       "min       4.300000    2.000000     1.000000    0.100000\n",
       "25%       5.100000    2.800000     1.600000    0.300000\n",
       "50%       5.800000    3.000000     4.350000    1.300000\n",
       "75%       6.400000    3.300000     5.100000    1.800000\n",
       "max       7.900000    4.400000     6.900000    2.500000"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# calculate some descriptive statistics\n",
    "iris_df.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets make a scatter plot of two features:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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ap7LeVrocuBp4bMbjhwOPuPuO+JPVdwFLCo4JiusT9Rrg+WZ2q5ndHnfArSpq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\nJwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112d74810>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(iris_df['SepalLength'], iris_df['SepalWidth']);\n",
    "plt.xlabel('Sepal Length');\n",
    "plt.ylabel('Sepal Width');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now lets make a boxplot!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x112bd50d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.boxplot(iris_df['SepalWidth'], iris_df['Name']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, lets load the 'tips' data that Dr. Coleman discussed in class, and have you guys generate the figures he showed!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>total_bill</th>\n",
       "      <th>tip</th>\n",
       "      <th>sex</th>\n",
       "      <th>smoker</th>\n",
       "      <th>day</th>\n",
       "      <th>time</th>\n",
       "      <th>size</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>16.99</td>\n",
       "      <td>1.01</td>\n",
       "      <td>Female</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>10.34</td>\n",
       "      <td>1.66</td>\n",
       "      <td>Male</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>21.01</td>\n",
       "      <td>3.50</td>\n",
       "      <td>Male</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>23.68</td>\n",
       "      <td>3.31</td>\n",
       "      <td>Male</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>24.59</td>\n",
       "      <td>3.61</td>\n",
       "      <td>Female</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   total_bill   tip     sex smoker  day    time  size\n",
       "0       16.99  1.01  Female     No  Sun  Dinner     2\n",
       "1       10.34  1.66    Male     No  Sun  Dinner     3\n",
       "2       21.01  3.50    Male     No  Sun  Dinner     3\n",
       "3       23.68  3.31    Male     No  Sun  Dinner     2\n",
       "4       24.59  3.61  Female     No  Sun  Dinner     4"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tips = sns.load_dataset(\"tips\")\n",
    "tips.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Describe the data \n",
    "\n",
    "# EDIT HERE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Plot two different scatter plots\n",
    "\n",
    "# EDIT HERE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Plot two different box plots\n",
    "\n",
    "# EDIT HERE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Problem 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this problem were going to replicating the figures in the lecture notes for **Example 3.2**: Rat Learning Experiment. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 3.2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In our learning experiment, we observed $Y = 22$ correct responses from the animal in $40$ trials. We derived the pmf of the $22$ observations in $40$ trials as the binomial distribution:\n",
    "\n",
    "\\begin{equation}\n",
    "    P_Y(22;p) = \\binom{40}{22} p^{22} (1-p)^{18}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "In general, for the binomial pmf with $n$ trials, $Y = k$ correct observations, and probability of a correct response of $p$, the likelihood is:\n",
    "\n",
    "\\begin{equation}\n",
    "    L(p) = P_Y(k;p) = \\binom{n}{k} p^{k} (1-p)^{k}\n",
    "\\end{equation}\n",
    "\n",
    "Lets go ahead and plot them for different values of $k$ and $p$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def bin_likelihood(n=40,k=5,p=0.5):\n",
    "    prob_y = 0\n",
    "    \n",
    "    prob_y = sp.misc.comb(n,k) * (p**k) * (1-p)**(n-k)\n",
    "    \n",
    "    return prob_y\n",
    "\n",
    "def plot_bin_likelihood(k):\n",
    "    n = 40\n",
    "    p = np.linspace(0,1)\n",
    "    prob_y = bin_likelihood(n,k,p)\n",
    "    plt.plot(p,prob_y);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Za5c94q+uzsyPv8mgvrgsk/vi1VMXAbjumpqE/Dvc6otrt9XwTy+fZWhiLm3+P9KljkwU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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1150e7750>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(plot_bin_likelihood, k=(5,35,5));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this problem, wer're going to actually try and compute the **real** MLE estimate of the tips data set we went over in class."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# load data\n",
    "tips_df = sns.load_dataset('tips')\n",
    "\n",
    "# get our 'x' vals\n",
    "x = tips['total_bill'].values\n",
    "\n",
    "# get our 'y' vals\n",
    "y = tips['tip'].values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "As we showed in class, if we have the linear model of $y = \\beta x + \\alpha$, what we're actually inerested in is finding the **slope** that minimizes some function of the **residuals** or **error**.  For a candidate slope $\\beta$, the residual for a single point $i$ is: \n",
    "\n",
    "\\begin{equation}\n",
    "    e_i = y_i - \\alpha - x_i \\beta\n",
    "\\end{equation}\n",
    "\n",
    "What we're interested in doing is finding the $\\beta$ that minimizes the \"sum-of-squares\" error:\n",
    "\n",
    "\\begin{equation}\n",
    "    S(\\beta) = \\sum_{i=1}^n e_i^2 = \\sum_{i=1}^n (y_i - \\alpha - x_i \\beta)^2\n",
    "\\end{equation}\n",
    "\n",
    "Lets first fix $\\alpha=1$ and plot this function for different values of $\\beta$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def s_beta(x,y,alpha,beta):\n",
    "    n = len(x)\n",
    "    mysum = 0\n",
    "    for i in range(n):\n",
    "        x_i = x[i]\n",
    "        y_i = y[i]\n",
    "        e_i = y_i - alpha - x_i * beta\n",
    "        e_i_squared = e_i ** 2\n",
    "        mysum = mysum + e_i_squared\n",
    "    return mysum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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J3HBFV86eq2LOJ/u8LscYE0csbOLMVYM6cFl2KgvXHqCwpNzrcowxcSLst4UWkWTgBaAL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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1151ab410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "beta_vals = np.linspace(-0.5,0.5)\n",
    "s_vals = np.zeros_like(beta_vals)\n",
    "alpha = 1\n",
    "for i in range(len(s_vals)):\n",
    "    curr_beta = beta_vals[i]\n",
    "    s_vals[i] = s_beta(x,y,alpha,curr_beta)\n",
    "\n",
    "plt.plot(beta_vals, s_vals);\n",
    "plt.xlabel(r'$\\beta$');\n",
    "plt.ylabel(r'$S(\\beta)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So visually, we can see the minimum is somwhere between $0$ and $0.2$. That makes sense, we dont expect people to be tipping over 20% on average! Now lets go ahead and calculate the true value. Remember, this closed-form formula comes from linear-algebra, all we need to do is implement it:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\hat{\\beta} = \\frac{\\sum_{i=1}^n (x_i - \\bar{x}) (y_i - \\bar{y}) } {\\sum_{i=1}^n (x_i - \\bar{x})^2}\n",
    "                = \\frac{\\text{Cov(x,y)}}{\\text{Var}(x)}\n",
    "\\end{equation}\n",
    "\n",
    "and $\\hat{\\alpha}$ is: \n",
    "\\begin{equation}\n",
    "    \\hat{\\alpha} = \\bar{y} - \\hat{\\beta}\\bar{x}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def calc_slope_and_intercept(x, y):\n",
    "    \n",
    "    # initialize return values\n",
    "    beta_hat = 0\n",
    "    alpha_hat = 0\n",
    "    \n",
    "    # calculate means\n",
    "    x_bar = np.mean(x)\n",
    "    y_bar = np.mean(y)\n",
    "    \n",
    "    # calculate beta_hat\n",
    "    numerator = np.sum((x - x_bar)*(y-y_bar))\n",
    "    denominator = np.sum((x - x_bar)**2)\n",
    "    beta_hat = numerator/denominator\n",
    "    \n",
    "    # calculate alpha_hat \n",
    "    alpha_hat = y_bar - beta_hat*x_bar\n",
    "    \n",
    "    # return both\n",
    "    return beta_hat, alpha_hat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "estimated beta: 0.105024517384\n",
      "estimaed alpha:  0.920269613555\n"
     ]
    }
   ],
   "source": [
    "# calculate slope and intercept for the tip data\n",
    "beta_hat, alpha_hat = calc_slope_and_intercept(x,y)\n",
    "print \"estimated beta:\", beta_hat\n",
    "print \"estimaed alpha: \", alpha_hat\n",
    "\n",
    "# calculate our estimated line\n",
    "y_hat = beta_hat * x + alpha_hat "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lZb9pMw4BoImpUxdhsVg8efBOVzyrUiJtPE2n6oRoGoYDLQ+aKdLp+4uGLC+Z\nJqL9IbbWb3fFmOgNioqi7/fuzXsZyZlczZXMDrjv9g834+zdu937M2fWuoKAb+E+fvyQdvmIpGBP\npd4qJihEJH9pTNoE0kCoAUfBVFevdnUFvQB3QbtmzeUx6W747tAxfGf/G3Cn3+07V/6dlp//IsRZ\nnHgv9mJUPXVvt1ekdeRmmkpBiESRIJAEsel+t7/dO8a0CJHrUruUHuN+B+B3aZZL+S1fF58Vdhot\nli7AhbQ2BI/EYnnD777hFuyJeApIpScNIRJFgkASBOvpEqggMurqX6emZg1HHXUoffva2bp1B+6+\n+lBJc7OVmTNrferdAxVknT76N4eeVRwwjfdxO3dxN0Y30+ugrhtVVf7r69uy2UqorV3AqlWjgY5P\n2paIPuvSLz4xJNCmHgkCKSRQQQQwfHg169YdBMzg/fehS5cpGN02X3cd3cLs2RvZsqXCdewcliyZ\n7zMgK2vnDg47IT/g9T/OOoz9H/wTgAcL/wh7XQGAnkATs2ZtdM0zFLyQtFqtLFt2GU88sdTVKJ1L\ndfXqqP/RJ2KkbbJG85qJBNrUJF1EkyDQ/OmB5pyprl7NunV98V7tq6npVOAqjAFNFwNXsGXLaZ7t\nMIa1a/sw76W3W+flDxAAsjhAFk5+4vySs856kCVLNrB37yzgR0A3oIleve53rQ0Q3uIs7knbXn21\nkSlTLpW5ckRaLvBjBhIEksDdIPrUU2/EeNGQ1ge7Fqw4KeXm2/0vwN6w5VsuKb2XLIzuo247dhzD\nhg2bMaaAuAyjXv81jjzyGyJd5jBW/+gTseiILGwizEqCQJL4G3DkWxDZyc+/i+bmZoYNG8jAgVvx\nHsB18MHvMnDgTLwHONXl3oaTLJxkkU37rr873vvIs/4uBx1ERUU5Xbve7zkHTAPGU1BwnCsd2cA5\nwHY2bZrqtfpXa9oScWefiJW2zLaaVzJIoE1NMk4gifz1VXYPAps1a6tryUdjnpw5c85n7ty3Wbjw\n73gP4vr2uls4/Y15Aa+xa+kK9hf6Thftvsbatf/hvfcO8MUXB7m25FBY2IkFC0YCMGnS0675hAZj\nPAXYGTr0YTZt6u6TNn8FpnuwkVEHbAPeIj9/DcuXX0fPnj2j/MRShwn6mcclf6nSMGyC70/GCaQr\nq9WKxdLFa81fqKuzceutMygoOI5u3QZw+Noj+NOm4XC3MSF0W43Tn8AxemzrP7j/rPCZxKx1oNnJ\neI81gCYdzz+vAAARcElEQVSfBWWKi0+kpsZ7u5VOnSxt0ha4AdVqtTJnzvkMHvwU9fU3Ul8/hLFj\ngzcGpkohIeJDxnqkHgkCKc8BzEfXXMSiml9yR4C9figbx/ePzGg9KkBPDN+BZivancfdBRT8T4ns\nf6K6wBYvXhd20JDeI0IkXkKDgFLKAjwP9AO6AvdrrZcmMg3pwF34flw3lJ38KOB+H3M8S6c94LdA\n9dflcdKkCtfcPu6vfRDG8o/++/MHWkD+1Vfjs3C7dNMUIvES/STwO6BBa12mlDoE2AhIEPC2fz9H\n9+2N0Vv/ar+7ZLEIGEJR0fPMj6BhrabmBDZuXE+/fs1s2bIF6AscTL9+d3DVVQWUlbW/6/b3+B7O\nVA/upS6bm5soKnqONWsuB2IzcCye1UVSHSXMJqENw0qp7kCW1vp7pVQvYJ3W+sdBDknphuFgBUao\nbc8/v4ylS9dxxBE9OOWU47nz3nFkHzgQ8Frd2MUPdOPggx/k+++PA/5L586fUFJyCAcf3JMBA/ry\n/vufA9n86lfH0NzczFNP/Z2dO48FxgM1GL19ygHo0uUempoAdtGz5z5OO+0ITj31Z5SVDQprgZRg\nn0dzczPLln3HO+8YBX9h4bMMHdoLi8US9Pjdu3czePCTrnWPf01xcbVPdVDr9mLg1+Tn/4Urr+wb\ndGR0oDT6y0s4C9e4j8/JsTJo0C9cy3VmXsAwQcNppucvtWcRVUrlAH8FntVaVwfZNWWDQLACo21h\nVVz8smebw+Fg+PAFrFs3nj9xL3dzf8BrDPvlVL7q3I3167sCRwIfYgwQWwqUufa6E5iC0XtnDsbs\nPy8CnYBxrn3uBxQwAt/Vs/4E3Ot6bRxbVPSSz2yk4a7o1XY/43yjXekKvSpV2+Pz86ezfPllnp5E\nDoeDM8983tMryajGGg68TXHxt2G1HYTKi78VxrzT3fb43NwK7PZJgDXilc5SnQkKyUzPX9hBIOHj\nBJRSRwNvAS+ECAApLdBAKIfDweDBc13TK1wALKCuzua5+3x3yjTWrrsOJ138BoAzmOrq6e/kr+/d\nzPr1e4ArMUYFnwT8EyMAuEcG3wusxj1K2Pj7RxgBwL3PFOAffnJRhPcIY1jtmo20dUBXuAO+2u5n\nnG9l1J9nff0Nnrts9/bWBmYLRoB5C+gc9iC0jg5ea3u83T4Z92cvo19Fukp0w/DhGF1SrtVavx3O\nMXl5OfFNVJRyctrf8eXkWKmt9e0NA6M5mce5ZfItMLl11VpvV3ENMxmCMTDrdT97uA0CJuNeTD4y\nB4DHgEmu148C1/rdMyfH6vncA+Wz7ffibz/3kpRnnvkiEydeFvQuOdR1/J9/HXBXwDRFeo2JE4dQ\nW1vlmfiubbr9p8H/uTJBJuXFn0zPX7gS3SYwA6NOQnu9fb7WOtCw0zSoDioHWhdEca+WlcteviPw\noKgJPMEzXIUx134zRpt5NcZqXFcAUFT0PC0tDtav70brbKFPANuA+1yv7wduxqh2eQEYCbyEUR1k\nHNOp0/20tPwesJKf/wCjRx/P9Omb2bPnGHzn/B9JUdFcn+qgUPX0gT6PM86Yw3nnHdxu5bJIP89A\nK4tZrffgcEwCeoa9ylg4i9iE12ZgHJ+bOw27/XqM6qD4LxqfSCaoLsn0/KV2m0AEUjYIgJ8CIzub\nrlPvIfepJ/zu3zj2csba+1NTcwJGwV8H9OC881ooKjqR2bPfZ8uWQiCb/Pz1LF9+HVarld///gmW\nLDkZ48FtEGAnN/dqune38rOf9eGggw5u1zAM8O67/6VTp07ce6+N11//oDWdrnaLm2+eSX39Nvr0\nOYQuXSwUFp7o0zAcqp4+2OcxceIQGhsjW98gVAO09/ZhwwZG1Sjb0d4/0jCcGUyQPwkCCXPgAAc9\n9wwHT7nV7+Z9BQOx176BY9++Ng2nVRgTtGV7umi2X5JxIRZLF+rqPqKm5g9Ajmebu8Ey0p47ofbz\nFqqhNBgT/COT/KUxE+RPpo2Ity5LF9PjijF+t+25/U72TrwBOrd+vG0HQhkNm68BX7NlSzZz574N\nDPHa7vCaP2ioqydKa9WDzVbqNQXEUQAsXOi7foCbjMQVQgQis4hGwLLmn/RS/cjrndsuAPww7koa\n6r+mYZudvTfc4hMAAtuIEYen8u9//5nc3BmAHWgiP/8Bn94wdvtkSktn+MxwaUwC1xWjofgC1q7t\nSlXVm+2u4q9XTFVV6J47MuujEJlPngRC6KQ/Jnf8WDp//J922/adfyGNjzyO87DDANfsnAGWd7TZ\nSli0qHXkrNEQ+wNG420LsBK7/Wf8/Oc3k59/BC0tFurrX6N1Bk9oaWnxub4x7/9ttD49jGHDhgrG\njw+WIwewjOnT32DEiNOCzugZaiH4dBhdmw5pFCKZJAj4kf3N1+RMnECXVe17sTafXID9L7M4kH+M\nZ0rmDRs2M2BAf157bTfr1k0A/C/v2Lv3VxhdQDtjVAe9ilEoL8A9+GvTpo/ZtKkHxtgAMAZdjQQe\nZ8mSm1iyxOqpzvE3mVtBQft5RW22Ep56ajr19dcA84Gx7NgxhMGDp7Nq1eVBC8ZAsz6mQxVTOqRR\niGST6iCXrEY7ORMnkNc7l16/UD4BoKXPUexa/jYN2+zsfv0tTwAYMWI+U6bsoabmNu6+28a6dV0x\n7upbl3d0DyAbNarG1cPnAoy6fytG3/0H8B38dSPG6GDvQVdTMPr35+A9MKmsbBBFRc/hvbBMWdmg\ndnmzWq1ceWVfjO6lrUtU1tffEPUAp3RYKjAd0ihEspn7SaCpie7THqDb49PbbXJ26oS9qpqm3wz2\ne6jvlMzu6phyjOUYh7Tb1yiM9uA9YOvYYzeyeXNBGAntg7+lHa1WK/PnX+pVXXNpwLvcsrLfMGvW\nI9TXh3G5EBwOB3V1H2H8fFqrq4QQ6cd8TwIHDmCd9bSx+PpRh7ULAI2PPUnDt9+x/etdAQNAcMYo\nWXiBwsIvvRpSHcAiYALwOr163cw771xPYeE3eC8bCZXAV57XubnTgMt99vFuoHVX13gvU+mP1Wpl\n+fLryM+f7vc84XI/1dTU3IYRAF8E7CnZaCwN20KEZppxAl2W/pUeV5T53bbntj+y9/obw+zRY/Bd\nocvoKTRw4CzOOacrr776L/r1O4yHHx5Pz549XZOfPeKaT8iCu3F25Mh/c+KJR/Huu59hDPL6MRaL\nOw1ZWCwWz6Co5uYmz3sdaeDsaEOpv7EDpaUVzJgxwedcqdIPO14Nw6mSv3iR/KU3GSfg0unDTfQc\nfhHZO3e22/ZD+RV8/6f7oXv3qM5ttVpZsGCkq2G4goKC4xgx4iLGjn2d99+fwfvvQ0NDa0PklVee\nxJQpYASAuUAZ8+cPwbjDv4Xi4pe5/PLz/BZSsVxUJR7L+xUXn5iyja2ptpyh9FYSqSajnwTyeuf6\nvN533hAapz/h6dIZa8FG2Lb2VOmN76CwJox2hHPCHo2bzIIknPl3wBR3WhHnL9xpuVOBfH/pLaWn\nkk4k+9PP0XR6CTvWvEfDNjv2F16OWwAIxd3nvrR0U4fO4y5IJk8eyuTJQxk1qgaHI9D8e7Hnzse0\naUt9Bq6J0KS3kkhFGRsEHA4Hz9h7MGPo9ew9sk/Ex1ZWrqCycoXfAjbQ9lANkVarlRkzJvjsYwwa\nOyOsRkuHw8GkSU9TV3c47q6osS5IQuXdnY9wGqOFEKkvI9sEOjJIKNSxwbaHGmELvqNwrdZONDZ2\nx2J5I+Bave3TdZvrndYJ6GJFBlfFl81WQk3NbJ+qtI6styxELGRkm0BHZr8MdWxHzt1WJPWS/q4L\nr1FcvC1mBXWy8paOos1fujQMy/eX3qR3kEmUlm5q1zVTpLZU660kREa2CXRkkFCoY5M1AMnfdWMd\nAGRwlRDmk5HVQdCxx+5IVrjqyCN9pI+kiahKSFbe0o3kL72ZIH+yslg6yOQfYibnDSR/6c4E+ZNx\nAkIIIUKTICCEECYmQUAIIUxMgoAQQpiYBAEhhDAxCQJCCGFiEgSEEMLEJAgIIYSJSRAQQggTkyAg\nhBAmltBZRJVS2cBfgF8A+4ArtdafJTINQgghWiX6SWAY0EVrfSpwG/BIgq8vhBDCS6KDwGnAMgCt\n9VqgIMHXF0II4SXRQSAXsHu9bnFVEQkhhEiCRBfAdiDH+/pa6wMJToMQQgiXRC8v+Q/gImCBUqoI\n+CDE/ll5eTkhdklvmZy/TM4bSP7SXabnL1yJDgI1wDlKqX+4Xo9L8PWFEEJ4SfWVxYQQQsSRNMoK\nIYSJSRAQQggTkyAghBAmJkFACCFMLNG9g0LK5PmFlFKFwINa67OVUscClcAB4EPgOq11WrbSK6Us\nwPNAP6ArcD/wHzInf52AmcDxgBOYgPHbrCQD8uemlOoN/AsYhJGvSjIkf0qpd4HvXC//C/wfGZI/\npdTtGF3vLcCfMbriVxJm3lLxSSAj5xdSSk3GKEi6ut6aDtyhtS4BsoCLk5W2GPgd0ODKy3nAkxjf\nW6bk70LggNb6dOCPwANkVv7cgfwZYA9GfjLm96mUsgJorc92/XcFGZI/pdRZQLGrvDwLOIYIf5up\nGAQydX6hzcAlGF8KwK+01qtdf78O/CYpqYqNBcBdrr+zgWYyKH9a678CV7te9gd2ASdnSv5cHgKe\nAr52vc6Y7w8YAHRTSi1XSq10DVTNlPydC2xSSi0GlgJLiPC3mYpBICPnF9JaLwL2e72V5fX390CP\nxKYodrTWe7TW3yulcjACwh/x/W2ldf4AtNYtSqlKYAbwEhn0/SmlyjGe5Fa43soig/KH8XTzkNZ6\nMEZV3ktttqdz/vKAk4HhGHmbS4TfXSoWrmaZX8g7TznA7mQlJBaUUkcDbwEvaK1fJsPyB6C1LgcU\nMAuwem1K9/yNwxjJ/zZwEjAHo3BxS/f8fYKr4NdafwrsAA732p7O+dsOrNBa79dafwI48C30Q+Yt\nFYPAP4ALAMKcXyhdvaeUOtP19/nA6mA7pzKl1OHACmCy1rrS9XYm5a/M1fgG8APQAmzIlPxprc/U\nWp+ltT4b2AiMAZZlSv4wgtwjAEqpIzEKxhUZkr+/Y7TDufPWDVgZSd5SrncQmT+/kLuV/iZgplKq\nC/AR8EryktRhd2DcfdyllHK3DUwCHs+Q/L0CVCqlVmH0wJgEfEzmfH9tOcms3+dzwGyllLswHIfx\nNJD2+dNa1yqlSpRS6zBu6q8FPieCvMncQUIIYWKpWB0khBAiQSQICCGEiUkQEEIIE5MgIIQQJiZB\nQAghTEyCgBBCmFgqjhMQIuaUUn/GmJeqC3AsRv9pgMe01nP87H8RcKzW+tEg5ywHztRaj2vz/ucY\nUxU0YYwr2AKM1VpvV0pdDTi11s8qpQ5orbOVUne73runY7kUInISBIQpaK1/D6CU6gf8TWv9yxCH\nnEzrwL5AAm13Audrrbe6rvkocAtwq9b6mQjOI0TcSRAQZuM9uRZKqeOBZ4FDMO7er3f9fwLgdN3V\nv4kx6rQH8CPgZa317W3P5e86rskPczHmdSfAXX8WEghEkkibgDC7FzGqhAYAN2AMsf8MY1rlp1xV\nRTbgJa11Mca0xNcqpXoFOWcW8JpS6j3gC4ypfN1D951IgS9SiAQBYVpKqYOBH2utF4Nn/YqdGDOF\ngutuXmv9CPA/pdRNGFNJW4DuQU7trg76pda6D1ABLPc+pxCpQoKAMLNs2hfKWbRWkzoBlFKPABMx\nJua6D2PysUgK85eAE1xPD/6eAuTJQCSNBAFhWlprO/CZUqoUPFOXH45Rf78f444fjOqch7TWC4G+\nQB+gU4jTeweJQcBWrbU7ePgLPEIkhTQMCzPyvvMeDTytlLoHY0GOS7TWza5ph+copb7BWJS8Sim1\nDfg3xuI5+QSv339NKdWEESwcGO0KtDnG6ec9IRJKppIWQggTk+ogIYQwMQkCQghhYhIEhBDCxCQI\nCCGEiUkQEEIIE5MgIIQQJiZBQAghTEyCgBBCmNj/A5Ox4xStR4L4AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1151213d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# lets plot our estimate\n",
    "plt.scatter(x,y);\n",
    "plt.xlabel('Total Bill');\n",
    "plt.ylabel('Tips');\n",
    "plt.plot(x, y_hat, 'r');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can use built in functions to calcualte and plot the true OLS values and see how they compare! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "-------------------------Summary of Regression Analysis-------------------------\n",
       "\n",
       "Formula: Y ~ <x> + <intercept>\n",
       "\n",
       "Number of Observations:         244\n",
       "Number of Degrees of Freedom:   2\n",
       "\n",
       "R-squared:         0.4566\n",
       "Adj R-squared:     0.4544\n",
       "\n",
       "Rmse:              1.0220\n",
       "\n",
       "F-stat (1, 242):   203.3577, p-value:     0.0000\n",
       "\n",
       "Degrees of Freedom: model 1, resid 242\n",
       "\n",
       "-----------------------Summary of Estimated Coefficients------------------------\n",
       "      Variable       Coef    Std Err     t-stat    p-value    CI 2.5%   CI 97.5%\n",
       "--------------------------------------------------------------------------------\n",
       "             x     0.1050     0.0074      14.26     0.0000     0.0906     0.1195\n",
       "     intercept     0.9203     0.1597       5.76     0.0000     0.6072     1.2333\n",
       "---------------------------------End of Summary---------------------------------"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from pandas.stats.api import ols\n",
    "res = ols(y=tips['tip'], x=tips['total_bill'])\n",
    "res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we did pretty good! Lets compare the true plot vs our plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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RyMCVjIl5kjj1R06zuraObe8dDVkxYfTwQm4uL2PmlJHkqFQtaboLkl9PWi9k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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115121150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# setup subplot figure\n",
    "f, (ax1, ax2) = plt.subplots(1, 2, sharey=True, figsize=(10,5))\n",
    "\n",
    "# plot our estimate on the first subplot\n",
    "ax1.scatter(x,y);\n",
    "ax1.set_xlabel('Total Bill');\n",
    "ax1.set_ylabel('Tips');\n",
    "ax1.plot(x, y_hat, 'r');\n",
    "ax1.set_title(\"Our Estimated OLS\")\n",
    "\n",
    "# plot the true regression plot on the second subplot\n",
    "sns.regplot(\"total_bill\", \"tip\", tips, ax=ax2, ci=0).set_ylabel(\"\")\n",
    "ax2.set_title(\"True OLS\")\n",
    "\n",
    "# tighten things up\n",
    "f.tight_layout()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}

iris.csv