demos/collinearity.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "# math \n",
    "import numpy as np\n",
    "\n",
    "# normal cdf\n",
    "from scipy.stats import norm\n",
    "\n",
    "# dataframes\n",
    "import pandas as pd\n",
    "\n",
    "# plotting\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn\n",
    "\n",
    "# linear regression\n",
    "import statsmodels.api as sm\n",
    "from statsmodels.stats.outliers_influence import variance_inflation_factor\n",
    "\n",
    "# for choosing covariates to include in model\n",
    "from patsy import dmatrices "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "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>month</th>\n",
       "      <th>year</th>\n",
       "      <th>usage</th>\n",
       "      <th>temperature</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>8</td>\n",
       "      <td>1989</td>\n",
       "      <td>24.828</td>\n",
       "      <td>73</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>9</td>\n",
       "      <td>1989</td>\n",
       "      <td>24.688</td>\n",
       "      <td>67</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>10</td>\n",
       "      <td>1989</td>\n",
       "      <td>19.310</td>\n",
       "      <td>57</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>11</td>\n",
       "      <td>1989</td>\n",
       "      <td>59.706</td>\n",
       "      <td>43</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>12</td>\n",
       "      <td>1989</td>\n",
       "      <td>99.667</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   month  year   usage  temperature\n",
       "0      8  1989  24.828           73\n",
       "1      9  1989  24.688           67\n",
       "2     10  1989  19.310           57\n",
       "3     11  1989  59.706           43\n",
       "4     12  1989  99.667           26"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "usage = pd.read_csv('elec_usage.txt')\n",
    "usage.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y, X = dmatrices('usage ~ 1 + temperature + np.power(temperature, 2)', data=usage, return_type='dataframe')\n",
    "model = sm.OLS(Y, X).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_fit(Y, Yhat):\n",
    "    plt.scatter(Y, Yhat,  color='black')\n",
    "    plt.xlabel(\"true outcome Y\")\n",
    "    plt.ylabel(\"predicted outcome Yhat\")\n",
    "    plt.show()\n",
    "    \n",
    "def assess_model_quality(data, formula):\n",
    "    Y, X = dmatrices(formula, data=data, return_type='dataframe')\n",
    "    model = sm.OLS(Y, X).fit()\n",
    "    Yhat = model.predict(X)\n",
    "    plot_fit(Y, Yhat)\n",
    "    return model "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>      <td>np.log(usage)</td>  <th>  R-squared:         </th> <td>   0.811</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.808</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   227.8</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Thu, 23 Apr 2020</td> <th>  Prob (F-statistic):</th> <td>7.82e-21</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>10:17:03</td>     <th>  Log-Likelihood:    </th> <td>  1.0037</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    55</td>      <th>  AIC:               </th> <td>   1.993</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    53</td>      <th>  BIC:               </th> <td>   6.007</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "           <td></td>              <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>           <td>    9.9203</td> <td>    0.419</td> <td>   23.691</td> <td> 0.000</td> <td>    9.080</td> <td>   10.760</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>np.log(temperature)</th> <td>   -1.5989</td> <td>    0.106</td> <td>  -15.092</td> <td> 0.000</td> <td>   -1.811</td> <td>   -1.386</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 4.773</td> <th>  Durbin-Watson:     </th> <td>   1.402</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.092</td> <th>  Jarque-Bera (JB):  </th> <td>   3.709</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.551</td> <th>  Prob(JB):          </th> <td>   0.157</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.636</td> <th>  Cond. No.          </th> <td>    53.9</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:          np.log(usage)   R-squared:                       0.811\n",
       "Model:                            OLS   Adj. R-squared:                  0.808\n",
       "Method:                 Least Squares   F-statistic:                     227.8\n",
       "Date:                Thu, 23 Apr 2020   Prob (F-statistic):           7.82e-21\n",
       "Time:                        10:17:03   Log-Likelihood:                 1.0037\n",
       "No. Observations:                  55   AIC:                             1.993\n",
       "Df Residuals:                      53   BIC:                             6.007\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "=======================================================================================\n",
       "                          coef    std err          t      P>|t|      [0.025      0.975]\n",
       "---------------------------------------------------------------------------------------\n",
       "Intercept               9.9203      0.419     23.691      0.000       9.080      10.760\n",
       "np.log(temperature)    -1.5989      0.106    -15.092      0.000      -1.811      -1.386\n",
       "==============================================================================\n",
       "Omnibus:                        4.773   Durbin-Watson:                   1.402\n",
       "Prob(Omnibus):                  0.092   Jarque-Bera (JB):                3.709\n",
       "Skew:                          -0.551   Prob(JB):                        0.157\n",
       "Kurtosis:                       3.636   Cond. No.                         53.9\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# try a linear model to predict log usage from log temperature\n",
    "linear = assess_model_quality(usage, 'np.log(usage) ~ np.log(temperature)')\n",
    "linear.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'residual')"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# is there any signal left in the residuals?\n",
    "resid = model.resid\n",
    "plt.scatter(np.log(usage['temperature']), model.resid)\n",
    "plt.xlabel(\"log(temperature)\")\n",
    "plt.ylabel(\"residual\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>      <td>np.log(usage)</td>  <th>  R-squared:         </th> <td>   0.814</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.807</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   113.6</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Thu, 23 Apr 2020</td> <th>  Prob (F-statistic):</th> <td>1.06e-19</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>10:17:06</td>     <th>  Log-Likelihood:    </th> <td>  1.3644</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    55</td>      <th>  AIC:               </th> <td>   3.271</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    52</td>      <th>  BIC:               </th> <td>   9.293</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "                  <td></td>                    <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>                        <td>    5.6258</td> <td>    5.200</td> <td>    1.082</td> <td> 0.284</td> <td>   -4.809</td> <td>   16.061</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>np.log(temperature)</th>              <td>    0.6349</td> <td>    2.698</td> <td>    0.235</td> <td> 0.815</td> <td>   -4.779</td> <td>    6.049</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>np.power(np.log(temperature), 2)</th> <td>   -0.2885</td> <td>    0.348</td> <td>   -0.829</td> <td> 0.411</td> <td>   -0.987</td> <td>    0.410</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 5.527</td> <th>  Durbin-Watson:     </th> <td>   1.396</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.063</td> <th>  Jarque-Bera (JB):  </th> <td>   4.464</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.618</td> <th>  Prob(JB):          </th> <td>   0.107</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.649</td> <th>  Cond. No.          </th> <td>2.93e+03</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 2.93e+03. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:          np.log(usage)   R-squared:                       0.814\n",
       "Model:                            OLS   Adj. R-squared:                  0.807\n",
       "Method:                 Least Squares   F-statistic:                     113.6\n",
       "Date:                Thu, 23 Apr 2020   Prob (F-statistic):           1.06e-19\n",
       "Time:                        10:17:06   Log-Likelihood:                 1.3644\n",
       "No. Observations:                  55   AIC:                             3.271\n",
       "Df Residuals:                      52   BIC:                             9.293\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "====================================================================================================\n",
       "                                       coef    std err          t      P>|t|      [0.025      0.975]\n",
       "----------------------------------------------------------------------------------------------------\n",
       "Intercept                            5.6258      5.200      1.082      0.284      -4.809      16.061\n",
       "np.log(temperature)                  0.6349      2.698      0.235      0.815      -4.779       6.049\n",
       "np.power(np.log(temperature), 2)    -0.2885      0.348     -0.829      0.411      -0.987       0.410\n",
       "==============================================================================\n",
       "Omnibus:                        5.527   Durbin-Watson:                   1.396\n",
       "Prob(Omnibus):                  0.063   Jarque-Bera (JB):                4.464\n",
       "Skew:                          -0.618   Prob(JB):                        0.107\n",
       "Kurtosis:                       3.649   Cond. No.                     2.93e+03\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 2.93e+03. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# let's try adding a quadratic term \n",
    "# notice we get a warning that design matrix is ill-conditioned\n",
    "quad = assess_model_quality(usage, 'np.log(usage) ~ np.log(temperature) + np.power(np.log(temperature), 2)')\n",
    "quad.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept                           25237.650598\n",
       "np.log(temperature)                   644.758755\n",
       "np.power(np.log(temperature), 2)      644.758755\n",
       "dtype: float64"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# VIF for quadratic model\n",
    "\n",
    "Y, X = dmatrices('np.log(usage) ~ np.log(temperature) + np.power(np.log(temperature), 2)', data=usage, return_type='dataframe')\n",
    "\n",
    "pd.Series([variance_inflation_factor(X.values, i) \n",
    "               for i in range(X.shape[1])], \n",
    "               index=X.columns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1c200ed4e0>"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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a3fN+UFt53vFZEAW6iMSsC5/5lJVb9wamW/2mJh/c3LeAZ5RvCnQRiTnhdnyunTSYWtUqe1RR2aBAF5GYsnjDLq54YVVQm4ZYIqNAF5GYEXpRrc4MTWjkUTVljwJdRDx3MCOTDrluQAGwZcpwzMyjisomBbqIeGrAY4v5dvehwHTbRicz/8Y+HlZUdinQRcQT4XZ8rps8hBpVFUvFpXdORErd3C9+5KbZXwa1acfniVOgi0ipCjnjc2w3ep6hMz6jQYEuIqVi98GjdH1wQVCbeuXRpUAXkRKXt1c+LOFU/u8y3VIh2hToIlJicnIcLe4M3vG58YFhVKmkqyOWBAW6iJSIfyzazCPvbwhq0xBLyVKgi0jU5R1imXdDbxJOq+1RNeWHAl1EombTTwcZ9MTSoDb1ykuPAl1EoiJvr/zqlBbcObytR9WUTwp0ETkhGZnZtLn7vaA23ePTGwp0ESm2P8xYydKNu4PaNMTiHQW6iBRZuOuwrLxzAA1PruZRRQIKdBEpore+/JEb/63rsMQiBbqIRCzvjs8ZVyRxdpvfeFSN5KVAF5FC/bj/CL2mfhTUpl557FGgi0iB8vbKR3c+jccvTPSoGimIAl1EwjqWlUOrifOD2r59aDgVK+i2cLFKgS4iIca9nMp763YGtWmIJfYp0EUkINzhiCvuGMCptXU4YlmgQBcRAP7zRTo3z14T1KZeedmiQBeRkB2fz17ehSHtT/WoGikuBbpIObbxp4MM1tUR44YCXaScytsrP79LYx69oJNH1Ug0FBroZjYDOAfY5ZxLyNV+A/BnIAv4r3PuthKrUkSiZu+hY3S+/8OgNh2OGB8i6aHPBJ4CXjreYGb9gVFAR+fcUTNrWDLliUg05e2V169RhdS7B3lUjURboYHunFtqZs3yNF8LTHXOHfUvsyv6pYlItBw+lkW7e94PakubPISaVTXqGk+K+9tsBfQxsweBDGC8c25VuAXN7GrgaoCmTZsW8+VEpLiGTVvG+h2/BLVpx2d8Km6gVwLqAt2BrsBrZtbCOefyLuicew54DiApKSlkvoiUjHCn7i+7rT9N6lX3qCIpacUN9HTgTX+ArzSzHKABsLvgp4lIadCp++VTcQN9LnA2sNjMWgFVgJ+jVpWIFEtWdg5n3hXcK399XA+6NqvnUUVSmiI5bHEW0A9oYGbpwL3ADGCGmaUBx4Ax4YZbRKT03D03jZdXfB/Upl55+RLJUS4X5zPrsijXIiLFkJ3jOOPO4Atq6U5C5ZOOWRIpwx7/YAN//2hzUJt65eWXAl2kDArXK3/sgk78vktjjyqSWKBAFyljnlv6LQ+9+01Q25YpwzHTqfvlnQJdpIwI1yufMLQN1/Y7w6OKJNYo0EXKgJc+3co9b60LavvuoeFU0AW1JBcFukgMC9crH9PjdCaPSsjnGVKeKdBFYtSsldu44821QW2bHhxG5YoVPKpIYp0CXSTGhDvbc1C73/DPPyR5VJGUFQp0kRgye9U2JrwR3Cvf8MBQqlaq6FFFUpYo0EViQGZ2Di3z9Mq7Na/H7Gt6eFSRlEUKdBGPheuVf3P/UKpVVq9cikaBLuKRcNcrb9mwJh/e0tejiqSsU6CLeCDcESxrJw2mVrXKHlUk8UCBLlKKjmZl03rie0Ft9WpU4XPdqFmiQIEuUkpeWfE9E+emBbWlThxI/ZpVPapI4o0CXaSEZWRm0+bu4F55jSoVWXffUI8qknilQBcpQeGuwbLqroGcUku9cok+BbpICThyLJu29wT3yqtUqsDGB4Z5VJGUBwp0kSibvuw7Hvjv+qC21RMH0kBj5VLCFOgiUXL4WBbt7nk/qK1BzSqsnqgjWKR0KNBFouCpjzbx6Acbg9q+uHsQdWtU8agiKY8U6CIn4NejWSTcG9wrb9GgBh+N7+dNQVKuKdBFimnK/PU8u+S7oDad7SleUqCLFNEvGZl0nPRBUFv3FvX499W6MqJ4S4EuUgS3zVnDa6vTg9rWTR5Cjar6KIn39FcoEoH9h4+ReN+HQW3nnXUaT1yU6FFFIqEU6CKFGDNjJUs27g5q012EJBYp0EXysetgBskPLgxqu+HsM/nr4NYeVSRSMAW6SBgpDy9i297DQW2bHxxGpYoVPKpIpHAKdJFcfth7mD4PLwpqe/j3HbmwaxOPKhKJnAJdBHDO0fyOd0Pat0wZjpl5UJFI0SnQpdzbsPMgQ55cGtT26lXd6HVmA48qEikeBbqUW/n1yrdOHeFBNSInToEu5dLKLXu58NlPg9reu6kPbU492aOKRE6cAl3KlZwcR4s71SuX+FToMVhmNsPMdplZWph5483MmZkGGyXmvbNme0iYL7utv8Jc4kYkPfSZwFPAS7kbzawJMAjYFv2yRKLnWFYOrSbOD2qrWqkCG3Q7OIkzhQa6c26pmTULM+sJ4DbgrSjXJBI1D7/3DU8v/jaoTRfTknhVrL9qMxsJ/OicW1PYMbpmdjVwNUDTpk2L83IiRRbuxhMjOjTiH5d29qgikZJX5EA3s+rAXcDgSJZ3zj0HPAeQlJTkivp6IkU1bNoy1u/4JahNF9OS8qA4PfQzgObA8d55Y+BzM0t2zu2MZnEiRfHj/iP0mvpRUNvd57TjT72be1SRSOkqcqA759YCDY9Pm9lWIMk593MU6xKJWH4nCH370HAqVtBp+1J+RHLY4izgU6C1maWb2Z9KviyRyCzbtDskzF+6MpmtU0cozKXcieQol4sLmd8satWIRCgjM5s2d78X0q6LaUl5pmO3pMyZOv8bnlkSfCjiv67qRk9dTEvKOQW6lBnf7zlE30cWh7TrTE8RHwW6xLzM7Bw6Tf6Aw8eyg9q/vGcQdapX8agqkdijQJeY9l7aTsa9khrU1ubUWrx3U4pHFYnELgW6xKRwN2gG+O6h4VTQ0SsiYSnQJaZkZedw+5trmZOaHtR++7A2jOt7hkdViZQNCnSJGZ99t4eLnlsR0q6dniKRUaCL537+9ShJDywIaZ99dXe6tajvQUUiZZMCXTyTneOYtmAjf/9oc8g89cpFik6BLp74fNs+Rj/9SUj7mnsGU7t6ZQ8qEin7FOhSqvYdOsa5T3/M93sOB7V3bVaX18f19KgqkfigQJdSkZPjeGbptzz83oaQebr+ikh0KNClxH2Vvp+RT30c0v7mdT3p3LSuBxWJxCcFupSYA4czuez5z1j744GQedrpKRJ9CnSJOucc05dt4cF314fM+/q+IVSvoj87kZKgT5ZE1brtBxjx9+Uh7X/ufybjh7T2oCKR8kOBLlFxMCOTMTNW8vm2/SHztNNTpHQo0OWEOOeY+clWJr/zdci8d//Sh3a/PdmDqkTKJwW6FNuGnQcZ8uTSkPYqFSuw8cFhHlQkUr4p0KXIDh3N4o8zV7Fyy96QeRseGErVShU9qEpEFOgSMeccr3y2jbvnpoXMu3VIa67vf6YHVYnIcQp0icjmXQcZ+Hjo8AromHKRWKFAlwIdOZbN2JdWs3zzzyHzFo/vR7MGNTyoSkTCUaBLWM45Zq/6gdvfXBsyr2m96iy9rb8HVYlIQRToEmLLz4fo/+jisPN0T0+R2KVAl4CMzGyueTmVJRt3h8x749qedDldF9ISiWUKdAHgjdR0/vr6mpD2jo1r8/afe3tQkYgUlQK9nNu25zApjywKO2/jA8OoUqlCKVckIsWlQC+njmZlc/2rn7Ng/a6QeTOuSOLsNr/xoCoROREK9HLorS9/5MZ/fxnSPqJjI/5xSWcPKhKRaFCglyPp+w7T+2+hwyu1T6rM6okDqVxRwysiZZkCvRw4lpXDX2Z9wXvrdobM08lBIvFDgR7n/vvVDq7/1+ch7XcNb8vYlBYeVCQiJUWBHqd2HDhCjykfhbTXqV6ZlXcO1NErInGo0EA3sxnAOcAu51yCv+0R4HfAMeBb4I/OudBb1Uipy8rO4abZXzLvqx0h8z64OYVWv6nlQVUiUhoi6abNBIbmafsQSHDOdQQ2AndEuS4phvfSdnLmXfNDwvzWIa3ZOnWEwlwkzhXaQ3fOLTWzZnnaPsg1uQI4P7plSVHs+iWD5IcWhrTXqFKR1LsHUa2ybjghUh5EYwz9SmB2FNYjRZSd4xj/+hr+88WPIfPmXt+LxCZ1PKhKRLxyQoFuZncBWcCrBSxzNXA1QNOmTU/k5SSXhet/4k8vrg5pT2xSh7nX9/KgIhHxWrED3czG4NtZOsA55/Jbzjn3HPAcQFJSUr7LSWR+/vUoSQ8sCGm/a3hbrujVTCcHiZRjxQp0MxsKTAD6OucOR7ckCScnx3HrnK944/P0oPYRHRvx4LkJ1KlexaPKRCRWRHLY4iygH9DAzNKBe/Ed1VIV+NDMAFY458aVYJ3l2uINu7jihVVBbfVrVGH2Nd05s6GOXBERn0iOcrk4TPPzJVCL5LH30DE63/9hSPsLV3Slf5uGHlQkIrFMZ4rGoJwcx/g5a3jz8+CjV24b2pqxfVponFxEwlKgx5jlm37msuc/C2ob2LYhj5zfibo1NE4uIvlToMeI/YePkXhf6PDKgltSNE4uIhFRoHvMOccNs74IOV3/+TFJDGiruwaJSOQU6B4KN7xy08CWXN//TI2Ti0iRKdA9cOBIJp0mfxDU1rFxbV78Y7LGyUWk2BTopcg5x1UvrmbhN8E3Zv7w5hRa6kqIInKCFOilZNE3u/jjzOCTg569vAtD2p/qUUUiEm8U6CXsl4xMOk4KHl65vPvp3PO7dhonF5GoUqCXEOcco//vE77YFnwjpy/uHqRxchEpEQr0EvDu2h1c92rwjZl1+zcRKWkK9CgKd+2Vh87rwMXJTfBfxExEpMQo0KPAOUfSAwvYc+hYoK3NqbV454beGicXkVKjQD9Br6z4nolz04LaPr97EPU0Ti4ipUyBXkzb9x+h59SPgtr+fXV3ureo71FFIlLeKdCLKDM7h5Z3zQ9qG935NB67oJPGyUXEUwr0Ipgyfz3PLvkuqG3jA8OoUknj5CLiPQV6BNJ+PMA5/7s8qO2jv/alxSk1PapIRCSUAr0Avx7NIuHe94Pabh3Smuv7n+lRRSIi+VOgh5GT4xjzwkqWbfo5qH3LlOEaJxeRmKVAz+O9tJ2MeyU1qG3lnQNoeHI1jyoSEYmMAt0vfd9hev9tUVDbI+d35IKkJh5VJCJSNOU+0DMys+n64AIOZmQF2k6vX53F4/tpeEVEypRyG+jOOZ5csIlpCzcFta+eOJAGNat6VJWISPGVy0Bf8d0e/ue5FUFtT1/ameEdGnlUkYjIiStXgb774FG6PrggqC25WT1mX9NdwysiUuaVi0DPys5h7EurWbRhd6DtjmFtOK/zaTSspaNXRCQ+xH2gv/TpVu55a11g+tYhrbmu3xnqkYtI3InbQP8qfT8jn/o4MJ3crB6vXNVN110RkbgVd4F+4HAmPacu5NCx7EDbp3ecTaPaJ3lYlYhIyYubQM/Jcdw0+0veXrM90Pbilcn0bXWKh1WJiJSeuAj0OanpjH99TWD6+v5nMH5wa42Ti0i5UqYDfeNPBxn8xNLAdJtTazH3+l5Uq1zRw6pERLxRJgP90NEsBjy2hJ2/ZATalt3Wnyb1qntYlYiIt8pUoDvnmDg3jVc/2xZoe/byLgxpf6qHVYmIxIZCA93MZgDnALuccwn+tnrAbKAZsBW40Dm3r+TKhPlrd3Dtq58Hpsf0OJ1JI9trnFxExC+Sg7JnAkPztN0OLHTOtQQW+qdLzBup6YEwb1z3JNbfN5TJoxIU5iIiuRTaQ3fOLTWzZnmaRwH9/I9fBBYDE6JYV5AzGtakw2m1eeKiTpzZsFZJvYyISJlW3DH03zjndgA453aYWcMo1hQisUkd3rmhd0m+hIhImVfi58Gb2dVmttrMVu/evbvwJ4iISLEUN9B/MrNGAP5/d+W3oHPuOedcknMu6ZRTdNamiEhJKW6gvw2M8T8eA7wVnXJERKS4Cg10M5sFfAq0NrN0M/sTMBUYZGabgEH+aRER8VAkR7lcnM+sAVGuRUREToAuDi4iEicU6CIicUKBLiISJ8w5V3ovZrYb+D7Kq20A/BzldXpF2xKbtC2xqTxty+nOuUKP+y7VQC8JZrbaOZfkdR3RoG2JTdqW2KRtCaUhFxGj/o8kAAAIIklEQVSROKFAFxGJE/EQ6M95XUAUaVtik7YlNmlb8ijzY+giIuITDz10ERGhjAS6mVUzs5VmtsbM1pnZ5DDL3GJmX5vZV2a20MxO96LWwkSyLbmWPd/MnJnF5J78SLfFzC70/27Wmdm/SrvOSET4N9bUzBaZ2Rf+v7PhXtQaKTOr6K91Xph5Vc1stpltNrPPwtzEJqYUsi1l4rN/XEHbkmuZYn32y0SgA0eBs51znYBEYKiZdc+zzBdAknOuIzAHeLiUa4xUJNuCmdUC/gJ8Vsr1FUWh22JmLYE7gF7OufbATaVfZkQi+b1MBF5zzp0F/A/wdCnXWFQ3AuvzmfcnYJ9z7kzgCeBvpVZV8RS0LWXls39cQdtyQp/9MhHozudX/2Rl/4/Ls8wi59xh/+QKoHEplhixSLbF7358f5gZpVVbUUW4LWOBfxy/ibhzLt9r53spwm1xwMn+x7WB7aVUXpGZWWNgBDA9n0VG4bt9JPhCcIDF6E16C9uWsvLZh4h+L3ACn/0yEegQ+JryJb6baXzonCvof68/AfNLp7KiK2xbzOwsoIlzLt+vZLEigt9LK6CVmX1sZivMLO8Nx2NGBNsyCbjMzNKBd4EbSrnEongSuA3IyWf+acAPAM65LOAAUL90SiuywrYlt5j+7FPItpzoZ7/MBLpzLts5l4jvf99kM0sIt5yZXQYkAY+UZn1FUdC2mFkFfF+B/+pVfUURwe+lEtAS303FLwamm1md0q0yMhFsy8XATOdcY2A48LL/9xVTzOwcYJdzLrWgxcK0xdwhbxFuy/FlY/qzX9i2ROOzH3N/jIVxzu0HFgMhPT0zGwjcBYx0zh0t5dKKLJ9tqQUkAIvNbCvQHXg7VneMHlfA7yUdeMs5l+mc2wJswBfwMauAbfkT8Jp/mU+BaviuwRFregEj/X8//wbONrNX8iyTDjQBMLNK+IaQ9pZmkRGKZFvKyme/sG058c++cy7mf4BTgDr+xycBy4Bz8ixzFvAt0NLrek90W/IsvxjfDh/Pay/m72Uo8KL/cQN8X/Pre117MbdlPnCF/3FbfGPo5nXthWxXP2BemPbrgWf8j/8H385ez+st5raUic9+JNuSZ5kif/bLSg+9EbDIzL4CVuEb35xnZveZ2Uj/Mo8ANYHXzexLM3vbq2ILEcm2lBWRbMv7wB4z+xpYBNzqnNvjUb0FiWRb/gqMNbM1wCx84R5zwxT5ybMtzwP1zWwzcAtwu3eVFV0Z/eyHFc3Pvs4UFRGJE2Wlhy4iIoVQoIuIxAkFuohInFCgi4jECQW6iEicUKCLZ8ysmZmlncDzzcw+MrOTzayOmV0XzfpKipndZGbVo7i+DmY2M1rrk7JLgS5l2XBgjXPuF6AOEBOB7v+PpqDP1k1AkQLdfzZnWM65tUBjM2talHVK/FGgywnx97LXm9k/zXcd8Q/M7CQzW2xmT5rZJ2aWZmbJhaynmpm9YGZr/deK7u9vr25mr/mvdT3bfNfuPn4q9KXAW/7HU4Ez/CeWPOJ/7q1mtsr/3Mm56v3GzKb763rVzAb6Lx626XidZjbJzF72fwPYZGZjc9Wa33rXm9nTwOdAEzP7PzNbbbmur25mfwF+i+8kpkX+tl9zrfv8471tM5tpZo/7l/ubmdUwsxn+1/7CzEblegvfwXfGp5RnXp8Cq5+y/QM0A7KARP/0a8Bl+E5b/qe/LQVIy+e5af7HfwVe8D9uA2zDd62U8cCz/vYE/2sl+ae/B2rlXZd/ejC++zQavo7LPH8dx+vt4G9PBWb4lxsFzPU/fxKwBt9lAI5fsuC3haw3B+ieq4Z6/n8r+t+Pjv7prUCDXMv9muvx+fguAAYw07/+iv7ph4DL/I/rABuBGv7pXsA7Xv896Mfbn3y/xokUwRbn3Jf+x6n4wg18p8fjnFt6fJzb+S58FU5v4H/9y39jZt/ju/Rub2Cavz3Nf2r+cfWccwfzWd9g/88X/uma+C4Kts1f71oAM1sHLHTOOTNbm6t28F1U7AhwxN9LTvbXk996v3fOrcj1/AvN7Gp8V5xsBLQDctcfidedc9m5tmmkmY33T1cDmuK7WcIufP/hSDmmQJdoyH11u2x8vVoIf4OI/OR3c4WCbrqQZWYVnHPhri1twBTn3LNBjb5breWuNyfXdA7Bn4lw9Re03kO5ppvj+3bR1Tm3zz+MUi2f7cj9OnmXOZTrsQG/d85tCLOOasCRfNYv5YTG0KUkXQRgZr2BA865AwUsuxTfmDhm1gpfz3MDsBy40N/eDt9QyXEbgBb+xwfxXX70uPeBK82spv+5p5lZwyLWP8o/tl8f39XxVhVhvSfjC+MDZvYbYFiueXlr/cnM2vp3pJ5XQD3vAzeY+e4sZL6bIRzXCij2EUMSH9RDl5K0z8w+wRduVwL4d2iOc85dlWfZp4Fn/MMeWfiuZHjUv5PxRf9Qyxf4hiyO/8fwX3xBu9k5t8e/YzMNmO+cu9XM2gKf+vPvV3xj+9lEbqX/NZoC9zvntgPbI1mvc26NmX0BrAO+Az7ONfs5YL6Z7XDO9cd3pcN5+Mbp0/AN44RzP7473nzlD/WtwDn+ef39tUo5pqstSokws8XAeOfc6hNcT0WgsnMuw8zOABYCrZxzx8ysEfCSc27QiVcc8rqT8O2sfDTa6442M6sKLAF6O9/t5KScUg9dYl11fIf4VcY3hnytc+4YgHNuh/9wyZOd71j08qopcLvCXNRDFxGJE9opKiISJxToIiJxQoEuIhInFOgiInFCgS4iEicU6CIiceL/Acea6Vocij2CAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the problem: collinearity\n",
    "X.plot('np.log(temperature)', 'np.power(np.log(temperature), 2)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## specific fix: use centered covariates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [],
   "source": [
    "# instate the quadratic model \n",
    "Y, X = dmatrices('np.log(usage) ~ np.log(temperature) + np.power(np.log(temperature), 2)', data=usage)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0     0.349831\n",
       "1     0.264064\n",
       "2     0.102423\n",
       "3    -0.179428\n",
       "4    -0.682532\n",
       "5    -0.227056\n",
       "6    -0.303042\n",
       "7    -0.111987\n",
       "8     0.048356\n",
       "9     0.153716\n",
       "10    0.322052\n",
       "11    0.376860\n",
       "12    0.363437\n",
       "13    0.249027\n",
       "14    0.170246\n",
       "15   -0.048808\n",
       "16   -0.227056\n",
       "17   -0.385280\n",
       "18   -0.227056\n",
       "19   -0.202959\n",
       "20    0.084723\n",
       "21    0.293478\n",
       "22    0.349831\n",
       "23    0.403177\n",
       "24    0.363437\n",
       "25    0.249027\n",
       "26    0.084723\n",
       "27   -0.090481\n",
       "28   -0.303042\n",
       "29   -0.414268\n",
       "30   -0.329710\n",
       "31   -0.277067\n",
       "32   -0.008803\n",
       "33    0.153716\n",
       "34    0.307867\n",
       "35    0.349831\n",
       "36    0.322052\n",
       "37    0.249027\n",
       "38    0.048356\n",
       "39   -0.133966\n",
       "40   -0.357109\n",
       "41   -0.414268\n",
       "42   -0.474892\n",
       "43   -0.277067\n",
       "44    0.066705\n",
       "45    0.218255\n",
       "46    0.336038\n",
       "47    0.428820\n",
       "48    0.376860\n",
       "49    0.233759\n",
       "50    0.048356\n",
       "51   -0.069427\n",
       "52   -0.385280\n",
       "53   -0.762574\n",
       "54   -0.474892\n",
       "Name: temperature, dtype: float64"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(X[:,1])\n",
    "np.log(usage['temperature']) - np.mean(np.log(usage['temperature']))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9969581875695092"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# high correlation between mean and log temp\n",
    "np.dot(X[:,0], X[:,1]) / (np.linalg.norm(X[:,0]) * np.linalg.norm(X[:,1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [],
   "source": [
    "# center the data \n",
    "X[:,1] = X[:,1] - np.mean(X[:,1])\n",
    "X[:,2] = np.power(X[:,1],2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.199040866595169e-14"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# no correlation between mean and log temp\n",
    "np.dot(X[:,0], X[:,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'np.power(centered np.log(temperature), 2)')"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# log temp and its square are no longer collinear\n",
    "plt.scatter(X[:,1], X[:,2])\n",
    "plt.xlabel(\"centered np.log(temperature)\")\n",
    "plt.ylabel(\"np.power(centered np.log(temperature), 2)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.30617049147663533"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# some (negative) correlation between log temp and its square\n",
    "np.dot(X[:,1], X[:,2]) / (np.linalg.norm(X[:,1]) * np.linalg.norm(X[:,2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.676364239688123"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# strong correlation between mean and square of log temp\n",
    "np.dot(X[:,0], X[:,2]) / (np.linalg.norm(X[:,0]) * np.linalg.norm(X[:,2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    2.019335\n",
       "1    1.208873\n",
       "2    1.208873\n",
       "dtype: float64"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check VIFs after centering data: not too bad!\n",
    "pd.Series([variance_inflation_factor(X, i) \n",
    "               for i in range(X.shape[1])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>      <td>np.log(usage)</td>  <th>  R-squared:         </th> <td>   0.814</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.807</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   113.6</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Thu, 23 Apr 2020</td> <th>  Prob (F-statistic):</th> <td>1.06e-19</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>10:21:49</td>     <th>  Log-Likelihood:    </th> <td>  1.3644</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    55</td>      <th>  AIC:               </th> <td>   3.271</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    52</td>      <th>  BIC:               </th> <td>   9.293</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "                  <td></td>                    <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>                        <td>    3.6472</td> <td>    0.047</td> <td>   78.409</td> <td> 0.000</td> <td>    3.554</td> <td>    3.741</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>np.log(temperature)</th>              <td>   -1.6391</td> <td>    0.117</td> <td>  -14.030</td> <td> 0.000</td> <td>   -1.874</td> <td>   -1.405</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>np.power(np.log(temperature), 2)</th> <td>   -0.2885</td> <td>    0.348</td> <td>   -0.829</td> <td> 0.411</td> <td>   -0.987</td> <td>    0.410</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 5.527</td> <th>  Durbin-Watson:     </th> <td>   1.396</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.063</td> <th>  Jarque-Bera (JB):  </th> <td>   4.464</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.618</td> <th>  Prob(JB):          </th> <td>   0.107</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.649</td> <th>  Cond. No.          </th> <td>    10.8</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:          np.log(usage)   R-squared:                       0.814\n",
       "Model:                            OLS   Adj. R-squared:                  0.807\n",
       "Method:                 Least Squares   F-statistic:                     113.6\n",
       "Date:                Thu, 23 Apr 2020   Prob (F-statistic):           1.06e-19\n",
       "Time:                        10:21:49   Log-Likelihood:                 1.3644\n",
       "No. Observations:                  55   AIC:                             3.271\n",
       "Df Residuals:                      52   BIC:                             9.293\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "====================================================================================================\n",
       "                                       coef    std err          t      P>|t|      [0.025      0.975]\n",
       "----------------------------------------------------------------------------------------------------\n",
       "Intercept                            3.6472      0.047     78.409      0.000       3.554       3.741\n",
       "np.log(temperature)                 -1.6391      0.117    -14.030      0.000      -1.874      -1.405\n",
       "np.power(np.log(temperature), 2)    -0.2885      0.348     -0.829      0.411      -0.987       0.410\n",
       "==============================================================================\n",
       "Omnibus:                        5.527   Durbin-Watson:                   1.396\n",
       "Prob(Omnibus):                  0.063   Jarque-Bera (JB):                4.464\n",
       "Skew:                          -0.618   Prob(JB):                        0.107\n",
       "Kurtosis:                       3.649   Cond. No.                         10.8\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = sm.OLS(Y, X).fit()\n",
    "Yhat = model.predict(Z)\n",
    "plot_fit(Y, Yhat)\n",
    "model.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## general fix: orthogonal polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ortho_poly_fit(x, degree = 1):\n",
    "    n = degree + 1\n",
    "    x = np.asarray(x).flatten()\n",
    "    if(degree >= len(np.unique(x))):\n",
    "            stop(\"'degree' must be less than number of unique points\")\n",
    "    xbar = np.mean(x)\n",
    "    x = x - xbar\n",
    "    X = np.fliplr(np.vander(x, n))\n",
    "    q,r = np.linalg.qr(X)\n",
    "\n",
    "    z = np.diag(np.diag(r))\n",
    "    raw = np.dot(q, z)\n",
    "\n",
    "    norm2 = np.sum(raw**2, axis=0)\n",
    "    alpha = (np.sum((raw**2)*np.reshape(x,(-1,1)), axis=0)/norm2 + xbar)[:degree]\n",
    "    Z = raw / np.sqrt(norm2)\n",
    "    return Z, norm2, alpha\n",
    "\n",
    "\"\"\"Use this function to transform test dataset to orthogonalized coordinates\"\"\"\n",
    "def ortho_poly_predict(x, alpha, norm2, degree = 1):\n",
    "    x = np.asarray(x).flatten()\n",
    "    n = degree + 1\n",
    "    Z = np.empty((len(x), n))\n",
    "    Z[:,0] = 1\n",
    "    if degree > 0:\n",
    "        Z[:, 1] = x - alpha[0]\n",
    "    if degree > 1:\n",
    "      for i in np.arange(1,degree):\n",
    "          Z[:, i+1] = (x - alpha[i]) * Z[:, i] - (norm2[i] / norm2[i-1]) * Z[:, i-1]\n",
    "    Z /= np.sqrt(norm2)\n",
    "    return Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create orthogonal polynomial features\n",
    "Z, norm2, alpha = ortho_poly_fit(np.log(usage['temperature']), degree=2)\n",
    "Y = np.log(usage['usage'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00, -2.01059227e-02, -1.43648744e-02],\n",
       "       [-2.01059227e-02,  1.00000000e+00, -3.73741278e-17],\n",
       "       [-1.43648744e-02, -3.73741278e-17,  1.00000000e+00]])"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check correlations - all off-diagonal correlations are very close to 0!\n",
    "np.corrcoef(Z.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    1.0\n",
       "1    1.0\n",
       "2    1.0\n",
       "dtype: float64"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check that VIFs are 1\n",
    "pd.Series([variance_inflation_factor(Z, i) \n",
    "               for i in range(Z.shape[1])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>          <td>usage</td>      <th>  R-squared:         </th> <td>   0.814</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.807</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   113.6</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Thu, 23 Apr 2020</td> <th>  Prob (F-statistic):</th> <td>1.06e-19</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>10:26:41</td>     <th>  Log-Likelihood:    </th> <td>  1.3644</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    55</td>      <th>  AIC:               </th> <td>   3.271</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    52</td>      <th>  BIC:               </th> <td>   9.293</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "   <td></td>     <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th> <td>   26.8452</td> <td>    0.243</td> <td>  110.585</td> <td> 0.000</td> <td>   26.358</td> <td>   27.332</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th> <td>   -3.6528</td> <td>    0.243</td> <td>  -15.047</td> <td> 0.000</td> <td>   -4.140</td> <td>   -3.166</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th> <td>   -0.2011</td> <td>    0.243</td> <td>   -0.829</td> <td> 0.411</td> <td>   -0.688</td> <td>    0.286</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 5.527</td> <th>  Durbin-Watson:     </th> <td>   1.396</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.063</td> <th>  Jarque-Bera (JB):  </th> <td>   4.464</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.618</td> <th>  Prob(JB):          </th> <td>   0.107</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.649</td> <th>  Cond. No.          </th> <td>    1.00</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                  usage   R-squared:                       0.814\n",
       "Model:                            OLS   Adj. R-squared:                  0.807\n",
       "Method:                 Least Squares   F-statistic:                     113.6\n",
       "Date:                Thu, 23 Apr 2020   Prob (F-statistic):           1.06e-19\n",
       "Time:                        10:26:41   Log-Likelihood:                 1.3644\n",
       "No. Observations:                  55   AIC:                             3.271\n",
       "Df Residuals:                      52   BIC:                             9.293\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "x1            26.8452      0.243    110.585      0.000      26.358      27.332\n",
       "x2            -3.6528      0.243    -15.047      0.000      -4.140      -3.166\n",
       "x3            -0.2011      0.243     -0.829      0.411      -0.688       0.286\n",
       "==============================================================================\n",
       "Omnibus:                        5.527   Durbin-Watson:                   1.396\n",
       "Prob(Omnibus):                  0.063   Jarque-Bera (JB):                4.464\n",
       "Skew:                          -0.618   Prob(JB):                        0.107\n",
       "Kurtosis:                       3.649   Cond. No.                         1.00\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# assess model quality \n",
    "# note R^2 is the same as with original covariates, but std errors are equalized \n",
    "# p values for intercept and linear term are now highly statistically significant\n",
    "\n",
    "model = sm.OLS(Y, Z).fit()\n",
    "Yhat = model.predict(Z)\n",
    "plot_fit(Y, Yhat)\n",
    "model.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.7755575615628914e-17"
      ]
     },
     "execution_count": 141,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check orthogonality\n",
    "np.dot(Z[:,1], Z[:,2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1c1d86d048>"
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# indeed, the features are orthogonal!\n",
    "logtemp = np.log(usage['temperature'])\n",
    "plt.scatter(logtemp, Z[:,1], label=\"1st degree poly\")\n",
    "plt.scatter(logtemp, Z[:,2], label=\"2nd degree poly\")\n",
    "plt.xlabel(\"log(temperature)\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Predictions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {},
   "outputs": [],
   "source": [
    "usage = usage.sort_values('temperature')\n",
    "Y, X = dmatrices('usage ~ 1 + temperature + np.power(temperature, 2)', data=usage, return_type='dataframe')\n",
    "model = sm.OLS(Y, X).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "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>mean</th>\n",
       "      <th>mean_se</th>\n",
       "      <th>mean_ci_lower</th>\n",
       "      <th>mean_ci_upper</th>\n",
       "      <th>obs_ci_lower</th>\n",
       "      <th>obs_ci_upper</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>53</th>\n",
       "      <td>103.045721</td>\n",
       "      <td>4.904832</td>\n",
       "      <td>89.931508</td>\n",
       "      <td>116.159935</td>\n",
       "      <td>74.283707</td>\n",
       "      <td>131.807735</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>96.837973</td>\n",
       "      <td>4.230563</td>\n",
       "      <td>85.526574</td>\n",
       "      <td>108.149372</td>\n",
       "      <td>68.851930</td>\n",
       "      <td>124.824016</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>42</th>\n",
       "      <td>79.689852</td>\n",
       "      <td>2.639513</td>\n",
       "      <td>72.632498</td>\n",
       "      <td>86.747207</td>\n",
       "      <td>53.136564</td>\n",
       "      <td>106.243140</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>54</th>\n",
       "      <td>79.689852</td>\n",
       "      <td>2.639513</td>\n",
       "      <td>72.632498</td>\n",
       "      <td>86.747207</td>\n",
       "      <td>53.136564</td>\n",
       "      <td>106.243140</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>41</th>\n",
       "      <td>74.465520</td>\n",
       "      <td>2.277803</td>\n",
       "      <td>68.375282</td>\n",
       "      <td>80.555758</td>\n",
       "      <td>48.152751</td>\n",
       "      <td>100.778290</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>74.465520</td>\n",
       "      <td>2.277803</td>\n",
       "      <td>68.375282</td>\n",
       "      <td>80.555758</td>\n",
       "      <td>48.152751</td>\n",
       "      <td>100.778290</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>71.945550</td>\n",
       "      <td>2.133371</td>\n",
       "      <td>66.241484</td>\n",
       "      <td>77.649615</td>\n",
       "      <td>45.719471</td>\n",
       "      <td>98.171629</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>52</th>\n",
       "      <td>71.945550</td>\n",
       "      <td>2.133371</td>\n",
       "      <td>66.241484</td>\n",
       "      <td>77.649615</td>\n",
       "      <td>45.719471</td>\n",
       "      <td>98.171629</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>69.487042</td>\n",
       "      <td>2.014142</td>\n",
       "      <td>64.101763</td>\n",
       "      <td>74.872322</td>\n",
       "      <td>43.328447</td>\n",
       "      <td>95.645637</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>67.089999</td>\n",
       "      <td>1.920153</td>\n",
       "      <td>61.956020</td>\n",
       "      <td>72.223977</td>\n",
       "      <td>40.981981</td>\n",
       "      <td>93.198016</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>64.754419</td>\n",
       "      <td>1.850688</td>\n",
       "      <td>59.806171</td>\n",
       "      <td>69.702666</td>\n",
       "      <td>38.682288</td>\n",
       "      <td>90.826549</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>64.754419</td>\n",
       "      <td>1.850688</td>\n",
       "      <td>59.806171</td>\n",
       "      <td>69.702666</td>\n",
       "      <td>38.682288</td>\n",
       "      <td>90.826549</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>43</th>\n",
       "      <td>62.480302</td>\n",
       "      <td>1.804180</td>\n",
       "      <td>57.656405</td>\n",
       "      <td>67.304199</td>\n",
       "      <td>36.431486</td>\n",
       "      <td>88.529118</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>62.480302</td>\n",
       "      <td>1.804180</td>\n",
       "      <td>57.656405</td>\n",
       "      <td>67.304199</td>\n",
       "      <td>36.431486</td>\n",
       "      <td>88.529118</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>58.116459</td>\n",
       "      <td>1.769885</td>\n",
       "      <td>53.384258</td>\n",
       "      <td>62.848661</td>\n",
       "      <td>32.084468</td>\n",
       "      <td>84.148451</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>58.116459</td>\n",
       "      <td>1.769885</td>\n",
       "      <td>53.384258</td>\n",
       "      <td>62.848661</td>\n",
       "      <td>32.084468</td>\n",
       "      <td>84.148451</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>58.116459</td>\n",
       "      <td>1.769885</td>\n",
       "      <td>53.384258</td>\n",
       "      <td>62.848661</td>\n",
       "      <td>32.084468</td>\n",
       "      <td>84.148451</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>56.026733</td>\n",
       "      <td>1.775717</td>\n",
       "      <td>51.278938</td>\n",
       "      <td>60.774529</td>\n",
       "      <td>29.991903</td>\n",
       "      <td>82.061564</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>53.998471</td>\n",
       "      <td>1.792302</td>\n",
       "      <td>49.206332</td>\n",
       "      <td>58.790610</td>\n",
       "      <td>27.955517</td>\n",
       "      <td>80.041425</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>50.126336</td>\n",
       "      <td>1.844934</td>\n",
       "      <td>45.193473</td>\n",
       "      <td>55.059199</td>\n",
       "      <td>24.057121</td>\n",
       "      <td>76.195552</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>48.282464</td>\n",
       "      <td>1.875460</td>\n",
       "      <td>43.267983</td>\n",
       "      <td>53.296946</td>\n",
       "      <td>22.197682</td>\n",
       "      <td>74.367247</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>46.500056</td>\n",
       "      <td>1.905791</td>\n",
       "      <td>41.404480</td>\n",
       "      <td>51.595632</td>\n",
       "      <td>20.399563</td>\n",
       "      <td>72.600549</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>51</th>\n",
       "      <td>44.779111</td>\n",
       "      <td>1.934168</td>\n",
       "      <td>39.607660</td>\n",
       "      <td>49.950562</td>\n",
       "      <td>18.663699</td>\n",
       "      <td>70.894523</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>43.119630</td>\n",
       "      <td>1.959192</td>\n",
       "      <td>37.881272</td>\n",
       "      <td>48.357987</td>\n",
       "      <td>16.990886</td>\n",
       "      <td>69.248373</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>39.985057</td>\n",
       "      <td>1.995087</td>\n",
       "      <td>34.650727</td>\n",
       "      <td>45.319388</td>\n",
       "      <td>13.836904</td>\n",
       "      <td>66.133211</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>38</th>\n",
       "      <td>35.744176</td>\n",
       "      <td>2.004699</td>\n",
       "      <td>30.384145</td>\n",
       "      <td>41.104206</td>\n",
       "      <td>9.590767</td>\n",
       "      <td>61.897584</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>35.744176</td>\n",
       "      <td>2.004699</td>\n",
       "      <td>30.384145</td>\n",
       "      <td>41.104206</td>\n",
       "      <td>9.590767</td>\n",
       "      <td>61.897584</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>35.744176</td>\n",
       "      <td>2.004699</td>\n",
       "      <td>30.384145</td>\n",
       "      <td>41.104206</td>\n",
       "      <td>9.590767</td>\n",
       "      <td>61.897584</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>44</th>\n",
       "      <td>34.453475</td>\n",
       "      <td>1.995219</td>\n",
       "      <td>29.118791</td>\n",
       "      <td>39.788159</td>\n",
       "      <td>8.305250</td>\n",
       "      <td>60.601701</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>33.224239</td>\n",
       "      <td>1.979593</td>\n",
       "      <td>27.931333</td>\n",
       "      <td>38.517144</td>\n",
       "      <td>7.084504</td>\n",
       "      <td>59.363973</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>33.224239</td>\n",
       "      <td>1.979593</td>\n",
       "      <td>27.931333</td>\n",
       "      <td>38.517144</td>\n",
       "      <td>7.084504</td>\n",
       "      <td>59.363973</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>32.056465</td>\n",
       "      <td>1.958240</td>\n",
       "      <td>26.820652</td>\n",
       "      <td>37.292278</td>\n",
       "      <td>5.928232</td>\n",
       "      <td>58.184699</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>28.921927</td>\n",
       "      <td>1.867568</td>\n",
       "      <td>23.928548</td>\n",
       "      <td>33.915305</td>\n",
       "      <td>2.841193</td>\n",
       "      <td>55.002661</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>28.921927</td>\n",
       "      <td>1.867568</td>\n",
       "      <td>23.928548</td>\n",
       "      <td>33.915305</td>\n",
       "      <td>2.841193</td>\n",
       "      <td>55.002661</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>28.000008</td>\n",
       "      <td>1.832501</td>\n",
       "      <td>23.100388</td>\n",
       "      <td>32.899628</td>\n",
       "      <td>1.937062</td>\n",
       "      <td>54.062953</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>25.603032</td>\n",
       "      <td>1.740064</td>\n",
       "      <td>20.950565</td>\n",
       "      <td>30.255498</td>\n",
       "      <td>-0.414584</td>\n",
       "      <td>51.620647</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>24.926967</td>\n",
       "      <td>1.722859</td>\n",
       "      <td>20.320500</td>\n",
       "      <td>29.533433</td>\n",
       "      <td>-1.082462</td>\n",
       "      <td>50.936395</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>37</th>\n",
       "      <td>24.312365</td>\n",
       "      <td>1.718093</td>\n",
       "      <td>19.718641</td>\n",
       "      <td>28.906089</td>\n",
       "      <td>-1.694810</td>\n",
       "      <td>50.319540</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>24.312365</td>\n",
       "      <td>1.718093</td>\n",
       "      <td>19.718641</td>\n",
       "      <td>28.906089</td>\n",
       "      <td>-1.694810</td>\n",
       "      <td>50.319540</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>24.312365</td>\n",
       "      <td>1.718093</td>\n",
       "      <td>19.718641</td>\n",
       "      <td>28.906089</td>\n",
       "      <td>-1.694810</td>\n",
       "      <td>50.319540</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>23.759227</td>\n",
       "      <td>1.729420</td>\n",
       "      <td>19.135220</td>\n",
       "      <td>28.383234</td>\n",
       "      <td>-2.253314</td>\n",
       "      <td>49.771768</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>22.837342</td>\n",
       "      <td>1.813673</td>\n",
       "      <td>17.988064</td>\n",
       "      <td>27.686619</td>\n",
       "      <td>-3.216187</td>\n",
       "      <td>48.890870</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>22.468594</td>\n",
       "      <td>1.891675</td>\n",
       "      <td>17.410759</td>\n",
       "      <td>27.526429</td>\n",
       "      <td>-3.624557</td>\n",
       "      <td>48.561746</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>36</th>\n",
       "      <td>22.161310</td>\n",
       "      <td>1.995499</td>\n",
       "      <td>16.825877</td>\n",
       "      <td>27.496744</td>\n",
       "      <td>-3.987068</td>\n",
       "      <td>48.309689</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>22.161310</td>\n",
       "      <td>1.995499</td>\n",
       "      <td>16.825877</td>\n",
       "      <td>27.496744</td>\n",
       "      <td>-3.987068</td>\n",
       "      <td>48.309689</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>21.915490</td>\n",
       "      <td>2.125429</td>\n",
       "      <td>16.232658</td>\n",
       "      <td>27.598322</td>\n",
       "      <td>-4.305979</td>\n",
       "      <td>48.136959</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>21.731133</td>\n",
       "      <td>2.281014</td>\n",
       "      <td>15.632308</td>\n",
       "      <td>27.829958</td>\n",
       "      <td>-4.583625</td>\n",
       "      <td>48.045892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>21.731133</td>\n",
       "      <td>2.281014</td>\n",
       "      <td>15.632308</td>\n",
       "      <td>27.829958</td>\n",
       "      <td>-4.583625</td>\n",
       "      <td>48.045892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>21.731133</td>\n",
       "      <td>2.281014</td>\n",
       "      <td>15.632308</td>\n",
       "      <td>27.829958</td>\n",
       "      <td>-4.583625</td>\n",
       "      <td>48.045892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>21.608240</td>\n",
       "      <td>2.461311</td>\n",
       "      <td>15.027351</td>\n",
       "      <td>28.189129</td>\n",
       "      <td>-4.822404</td>\n",
       "      <td>48.038884</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>21.608240</td>\n",
       "      <td>2.461311</td>\n",
       "      <td>15.027351</td>\n",
       "      <td>28.189129</td>\n",
       "      <td>-4.822404</td>\n",
       "      <td>48.038884</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>21.546810</td>\n",
       "      <td>2.665113</td>\n",
       "      <td>14.421009</td>\n",
       "      <td>28.672611</td>\n",
       "      <td>-5.024752</td>\n",
       "      <td>48.118372</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>21.546810</td>\n",
       "      <td>2.665113</td>\n",
       "      <td>14.421009</td>\n",
       "      <td>28.672611</td>\n",
       "      <td>-5.024752</td>\n",
       "      <td>48.118372</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>21.608341</td>\n",
       "      <td>3.138134</td>\n",
       "      <td>13.217807</td>\n",
       "      <td>29.998875</td>\n",
       "      <td>-5.329957</td>\n",
       "      <td>48.546640</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>21.915726</td>\n",
       "      <td>3.690631</td>\n",
       "      <td>12.047962</td>\n",
       "      <td>31.783491</td>\n",
       "      <td>-5.518625</td>\n",
       "      <td>49.350078</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          mean   mean_se  mean_ci_lower  mean_ci_upper  obs_ci_lower  \\\n",
       "53  103.045721  4.904832      89.931508     116.159935     74.283707   \n",
       "4    96.837973  4.230563      85.526574     108.149372     68.851930   \n",
       "42   79.689852  2.639513      72.632498      86.747207     53.136564   \n",
       "54   79.689852  2.639513      72.632498      86.747207     53.136564   \n",
       "41   74.465520  2.277803      68.375282      80.555758     48.152751   \n",
       "29   74.465520  2.277803      68.375282      80.555758     48.152751   \n",
       "17   71.945550  2.133371      66.241484      77.649615     45.719471   \n",
       "52   71.945550  2.133371      66.241484      77.649615     45.719471   \n",
       "40   69.487042  2.014142      64.101763      74.872322     43.328447   \n",
       "30   67.089999  1.920153      61.956020      72.223977     40.981981   \n",
       "6    64.754419  1.850688      59.806171      69.702666     38.682288   \n",
       "28   64.754419  1.850688      59.806171      69.702666     38.682288   \n",
       "43   62.480302  1.804180      57.656405      67.304199     36.431486   \n",
       "31   62.480302  1.804180      57.656405      67.304199     36.431486   \n",
       "16   58.116459  1.769885      53.384258      62.848661     32.084468   \n",
       "5    58.116459  1.769885      53.384258      62.848661     32.084468   \n",
       "18   58.116459  1.769885      53.384258      62.848661     32.084468   \n",
       "19   56.026733  1.775717      51.278938      60.774529     29.991903   \n",
       "3    53.998471  1.792302      49.206332      58.790610     27.955517   \n",
       "39   50.126336  1.844934      45.193473      55.059199     24.057121   \n",
       "7    48.282464  1.875460      43.267983      53.296946     22.197682   \n",
       "27   46.500056  1.905791      41.404480      51.595632     20.399563   \n",
       "51   44.779111  1.934168      39.607660      49.950562     18.663699   \n",
       "15   43.119630  1.959192      37.881272      48.357987     16.990886   \n",
       "32   39.985057  1.995087      34.650727      45.319388     13.836904   \n",
       "38   35.744176  2.004699      30.384145      41.104206      9.590767   \n",
       "8    35.744176  2.004699      30.384145      41.104206      9.590767   \n",
       "50   35.744176  2.004699      30.384145      41.104206      9.590767   \n",
       "44   34.453475  1.995219      29.118791      39.788159      8.305250   \n",
       "26   33.224239  1.979593      27.931333      38.517144      7.084504   \n",
       "20   33.224239  1.979593      27.931333      38.517144      7.084504   \n",
       "2    32.056465  1.958240      26.820652      37.292278      5.928232   \n",
       "9    28.921927  1.867568      23.928548      33.915305      2.841193   \n",
       "33   28.921927  1.867568      23.928548      33.915305      2.841193   \n",
       "14   28.000008  1.832501      23.100388      32.899628      1.937062   \n",
       "45   25.603032  1.740064      20.950565      30.255498     -0.414584   \n",
       "49   24.926967  1.722859      20.320500      29.533433     -1.082462   \n",
       "37   24.312365  1.718093      19.718641      28.906089     -1.694810   \n",
       "25   24.312365  1.718093      19.718641      28.906089     -1.694810   \n",
       "13   24.312365  1.718093      19.718641      28.906089     -1.694810   \n",
       "1    23.759227  1.729420      19.135220      28.383234     -2.253314   \n",
       "21   22.837342  1.813673      17.988064      27.686619     -3.216187   \n",
       "34   22.468594  1.891675      17.410759      27.526429     -3.624557   \n",
       "36   22.161310  1.995499      16.825877      27.496744     -3.987068   \n",
       "10   22.161310  1.995499      16.825877      27.496744     -3.987068   \n",
       "46   21.915490  2.125429      16.232658      27.598322     -4.305979   \n",
       "22   21.731133  2.281014      15.632308      27.829958     -4.583625   \n",
       "35   21.731133  2.281014      15.632308      27.829958     -4.583625   \n",
       "0    21.731133  2.281014      15.632308      27.829958     -4.583625   \n",
       "12   21.608240  2.461311      15.027351      28.189129     -4.822404   \n",
       "24   21.608240  2.461311      15.027351      28.189129     -4.822404   \n",
       "48   21.546810  2.665113      14.421009      28.672611     -5.024752   \n",
       "11   21.546810  2.665113      14.421009      28.672611     -5.024752   \n",
       "23   21.608341  3.138134      13.217807      29.998875     -5.329957   \n",
       "47   21.915726  3.690631      12.047962      31.783491     -5.518625   \n",
       "\n",
       "    obs_ci_upper  \n",
       "53    131.807735  \n",
       "4     124.824016  \n",
       "42    106.243140  \n",
       "54    106.243140  \n",
       "41    100.778290  \n",
       "29    100.778290  \n",
       "17     98.171629  \n",
       "52     98.171629  \n",
       "40     95.645637  \n",
       "30     93.198016  \n",
       "6      90.826549  \n",
       "28     90.826549  \n",
       "43     88.529118  \n",
       "31     88.529118  \n",
       "16     84.148451  \n",
       "5      84.148451  \n",
       "18     84.148451  \n",
       "19     82.061564  \n",
       "3      80.041425  \n",
       "39     76.195552  \n",
       "7      74.367247  \n",
       "27     72.600549  \n",
       "51     70.894523  \n",
       "15     69.248373  \n",
       "32     66.133211  \n",
       "38     61.897584  \n",
       "8      61.897584  \n",
       "50     61.897584  \n",
       "44     60.601701  \n",
       "26     59.363973  \n",
       "20     59.363973  \n",
       "2      58.184699  \n",
       "9      55.002661  \n",
       "33     55.002661  \n",
       "14     54.062953  \n",
       "45     51.620647  \n",
       "49     50.936395  \n",
       "37     50.319540  \n",
       "25     50.319540  \n",
       "13     50.319540  \n",
       "1      49.771768  \n",
       "21     48.890870  \n",
       "34     48.561746  \n",
       "36     48.309689  \n",
       "10     48.309689  \n",
       "46     48.136959  \n",
       "22     48.045892  \n",
       "35     48.045892  \n",
       "0      48.045892  \n",
       "12     48.038884  \n",
       "24     48.038884  \n",
       "48     48.118372  \n",
       "11     48.118372  \n",
       "23     48.546640  \n",
       "47     49.350078  "
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predictions = model.get_prediction(X)\n",
    "predictions.summary_frame(alpha=0.01) # alpha=0.05 -> 95% confidence interval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1c1d0c8eb8>"
      ]
     },
     "execution_count": 150,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot confidence intervals \n",
    "CI = predictions.conf_int(alpha=.05)\n",
    "p = usage.plot.scatter('temperature', 'usage', color='red', label='data')\n",
    "p.plot(usage['temperature'], CI[:,0], color='blue', label=\"lower CI\")\n",
    "p.plot(usage['temperature'], CI[:,1], color='green', label='upper CI')\n",
    "p.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prediction_interval(predictions, alpha=.05):\n",
    "    emean = predictions.predicted_mean\n",
    "    sigma = np.sqrt(predictions.var_resid)\n",
    "    n = len(emean)\n",
    "    PI = np.zeros((n,2))\n",
    "    PI[:,0] = emean + norm.ppf(alpha/2)*sigma\n",
    "    PI[:,1] = emean + norm.ppf(1-alpha/2)*sigma\n",
    "    return PI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1c1e2193c8>"
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot prediction intervals \n",
    "PI = prediction_interval(predictions, alpha = .05)\n",
    "p = usage.plot.scatter('temperature', 'usage', color='red', label='data')\n",
    "p.plot(usage['temperature'], PI[:,0], color='blue', label=\"lower PI\")\n",
    "p.plot(usage['temperature'], PI[:,1], color='green', label='upper PI')\n",
    "p.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Log transformations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x1c1eaf5f98>"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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96GdFxBGA7OeZ5ZVk9mL9AiyFr/j9DkorzyF8+zvHGB9T121tbcq1HrqkjcAdEfHa7PZ8REyu+PuzEdGxH13SdmA7wLnnnnvR448/XkLZttakuA54p1Eq0Hlp3E7vf/wk8bJT1jH//GLtRrlYuYa9HvpTkjZExBFJG4Cj3TaMiD3AHmhd4KLg69ka15R1wPPqNszyhqsu4HM7L3vR9p3OISy+EJx68joO/s5PjaRmq7+igX47sA3Ylf28rbSKzLpI6SRbv5O87ZpwEtSq1zfQJX0SeANwhqQngQ/SCvKbJF0LPAFcPcwizXqp8wSbbgYN6BRG8djw9Q30iHh7lz9dXnItZgNr6gzRQQN6x5ZNHc8h+CSoreSp/9Zog3Zd5LXc6p+dX2BMYimCqRJb/4MGdGrnEGw4HOjWaMPoW25v9S9lI8HytP7zdv8UCeiUziHYcDjQrdGG0bfcqdW/rFfrf9DuHwe0lc2Bbo3S3gJ+4yvXc8uB2VL7lvu17g/PL3RsiQ+r+8csLwd6gzVxdEc/K9/T5KnjRMBzC4sdw3t2foFbDszy1oumuOfhudL2Q7dW/7LJU8c7tsS7teo9tNBGxYE+BKMI2jqP7ij6/tvf07PPLx7/2+z8Ap/4/BMdr4Jzz8NzHSfjFNXphOWyifExIujYEl8+edrOQwttVLweeslGtU51XZdPXc3779V3DbwozJeV3QJeuS4MwJhaa6Ysrw8zv7DY8XFLEY1fIMyazS30ko2qH7WuMwdX8/6L1j6MFnC3E5b7Ds4iOh9cprJuoU/e9w2WIhiTeOtFPvFpo+MW+gB6XTFn2aiCtq7Lp67m/eepXW23R90C3r3/UMcwFxzv41/udlmK8JXubaQc6Dnl7UoYVdAOc/3vPAeubnq9/37P2+k9rTQxPsY7Ljm30iVyux2YgtaV7uvYDWZrh7tccsrblTCqKdrDmjm4mpOt+w7O8u3vHHvR/RPjY7zxlev7Pm/7e2of5VKHUTzdRsBMTU7UthvM1g4Hek55P6yjnKI9jIkpRfvAO63XDXDaqeN88Gdfk/t5+72nqodq9jpgLy8V0K7qbjBbOxzoK/QKi0FmJDZ5BmDRVma3ESqnnryOrZun+I0bHyj0vCvVYahmvwO2F9CyKjUi0Iu0ygZ9TL+wGOVqd1W2QotOpe93IChjin5dZmJ2O2B7AS2rWu0DvUirrMhjuoXFh/7xK8c/oD8wMc4p4ycN9ZJfRVuhZR0Eih64+gV2GQfEJvRRN/nbmTVf7Ue5FJlA0+0x1934wMDDDZ99fvH4yJb5hUX+b/EFPvzzF/K5nZcN5YNb5P12GoGz4+YvcuGH/nXgkSpFL7bcb9RNGRdxrutQTbO6qH0LvUirrNffllu8M48/c8L6H5Onjp8w1byb9q/4ZXePdFtDpNd76ni9yaU4PqNx0L7mIq3MPN0Nq229+iIPZr3VPtCL9L32W1xpYXHphHVBZucXGD9JjI+JxaX+17FeDteyT9L1moXY6/3m6XIYRV/zsLsb3Edt1lvtu1x2bNnE+NiJ8wPHx9SzVdZvggq8ODQXXwheevK6E7oEJifGOz52OVzLXk+l1yzEXu83b5dDnfqai9q6eYrP7byM/9r1lqF1e5k11aoCXdKbJB2S9KiknWUV9SLtKdenEd2+uFJezy0snhAWv3vFa3r2C5d9kq7XLMRewZXnAAbuazZLXeFAlzQG/Bnw08CrgbdLenVZhS3bvf8Qiy+cmOCLL0TfVvByS+4jP3/hi8KufT2QZe2B1+9EXtkn6bo9rt+Bqb3O004dZ/ykE9+l+5rN0reaPvSLgUcj4jEASZ8CrgS+WkZhy1bbCu7U7zrIVW569QuXfZJuNc/XXmfVMyrNbPRWE+hTwDdW3H4S+NH2jSRtB7YDnHvuuQO/SBkTUjqF8vQPnb7qwCv7JF2Zz+fx0GZrj6LDFVZyPVC6GtgSEe/Jbr8TuDgifq3bY6anp2NmZmag1+m0RsjE+NjIV9kzM6uKpAMRMd1vu9W00J8EXrHi9jnA4VU8X0ceqmZmls9qAv0LwPmSzgNmgWuAXyilqjbuPjAz669woEfEMUm/CuwHxoCPR8RXSqvMzMwGsqqZohHxGeAzJdViZmarUPuZomZmlo8D3cwsEQ50M7NEFB6HXujFpDng8RybngH8z5DLKcq1FePainFtxaRW2w9FxPp+G4000POSNJNnEH0VXFsxrq0Y11bMWq3NXS5mZolwoJuZJaKugb6n6gJ6cG3FuLZiXFsxa7K2Wvahm5nZ4OraQjczswFVGuj9LmEn6SWSbsz+fp+kjTWq7V2S5iQ9kP33nhHV9XFJRyV9ucvfJemPs7q/JOn1o6grZ21vkPTcin32OyOs7RWS7pH0kKSvSHpfh20q2Xc5a6tk30k6RdJ/SPpiVtuHOmxTyec0Z22VfE5XvP6YpIOS7ujwt/L3W0RU8h+tBb2+BvwwcDLwReDVbdv8CvAX2e/XADfWqLZ3AX9awX77ceD1wJe7/P3NwD/TutLeJcB9NartDcAdFf172wC8Pvv9+4D/7PD/tJJ9l7O2SvZdti9elv0+DtwHXNK2TVWf0zy1VfI5XfH6vwn8faf/d8PYb1W20I9fwi4ivgssX8JupSuBvdnvNwOXS+p2SdBR11aJiPgs8EyPTa4E/iZaPg9MStpQk9oqExFHIuL+7PdvAQ/RuurWSpXsu5y1VSLbF/+b3RzP/ms/8VbJ5zRnbZWRdA7wFuCjXTYpfb9VGeidLmHX/o/4+DYRcQx4Dnh5TWoDeGv21fxmSa/o8Pcq5K29Kj+WfUX+Z0mvqaKA7KvtZlotupUq33c9aoOK9l3WbfAAcBS4MyK67rcRf07z1AbVfU4/ArwfeKHL30vfb1UGeqcjUfvRNc82w5Dndf8R2BgRPwL8G9870latqn2Wx/20pjC/DvgTYN+oC5D0MuAW4LqI+Gb7nzs8ZGT7rk9tle27iFiKiAtpXZXsYkmvbduksv2Wo7ZKPqeSfgY4GhEHem3W4b5V7bcqAz3PJeyObyNpHfADjOYrfd/aIuLpiPhOdvMvgYtGUFceI7k0YBER8c3lr8jRWkt/XNIZo3p9SeO0AvMTEXFrh00q23f9aqt632WvOw/cC7yp7U9VfU771lbh5/RS4ApJX6fVZXuZpL9r26b0/VZloB+/hJ2kk2mdFLi9bZvbgW3Z728D7o7sDELVtbX1rV5Bq9+zDm4HfikbsXEJ8FxEHKm6KABJP7jcRyjpYlr//p4e0WsL+BjwUET8YZfNKtl3eWqrat9JWi9pMvt9AvgJ4OG2zSr5nOaprarPaURcHxHnRMRGWvlxd0T8Yttmpe+3VV2xaDWiyyXsJP0eMBMRt9P6R/63kh6ldeS6pka1/bqkK4BjWW3vGkVtkj5Ja8TDGZKeBD5I62QQEfEXtK4g9WbgUeB54N2jqCtnbW8DflnSMWABuGZEB2hotZjeCTyY9bkC/DZw7or6qtp3eWqrat9tAPZKGqN1ELkpIu6ow+c0Z22VfE67GfZ+80xRM7NEeKaomVkiHOhmZolwoJuZJcKBbmaWCAe6mVkiHOhmZolwoJuZJcKBbmaWiP8HeUTx8MkPcpgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "n = 60\n",
    "x = 4*np.random.rand(n)\n",
    "y = np.exp(x + .5*np.random.randn(n))\n",
    "plt.scatter(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [],
   "source": [
    "idx = np.argsort(x)\n",
    "x = x[idx]\n",
    "y = y[idx]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1c1ec25390>]"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# use linear model - bad fit!\n",
    "model = sm.OLS(y,x).fit()\n",
    "predictions = model.get_prediction(x)\n",
    "yhat = predictions.predicted_mean\n",
    "PI = prediction_interval(predictions)\n",
    "plt.scatter(x,y)\n",
    "plt.plot(x,yhat,color='red')\n",
    "plt.plot(x,PI[:,0],color='blue')\n",
    "plt.plot(x,PI[:,1],color='blue')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1c1f417748>]"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# use linear model with log transform\n",
    "model = sm.OLS(np.log(y),x).fit()\n",
    "predictions = model.get_prediction(x)\n",
    "yhat = np.exp(predictions.predicted_mean)\n",
    "PI = np.exp(prediction_interval(predictions))\n",
    "plt.scatter(x,y)\n",
    "plt.plot(x,yhat,color='red')\n",
    "plt.plot(x,PI[:,0],color='blue')\n",
    "plt.plot(x,PI[:,1],color='blue')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}