2018/07/15/2018-07-15-nonlinear-regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.api as sm\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(1, 20)\n",
    "y = [np.random.normal((-50*n) + (3*(n**2)), 20) for n in np.linspace(1, 20)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x,y);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "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>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1.387755</td>\n",
       "      <td>1.925864</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1.775510</td>\n",
       "      <td>3.152436</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     0         1         2\n",
       "0  1.0  1.000000  1.000000\n",
       "1  1.0  1.387755  1.925864\n",
       "2  1.0  1.775510  3.152436"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sm.add_constant(x)\n",
    "x = pd.DataFrame(x)\n",
    "x[2] = x[1] ** 2\n",
    "x.head(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# attempt 1\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.379</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.366</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   29.29</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 15 Jul 2018</td> <th>  Prob (F-statistic):</th> <td>1.96e-06</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>02:59:35</td>     <th>  Log-Likelihood:    </th> <td> -291.43</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   586.9</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    48</td>      <th>  BIC:               </th> <td>   590.7</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>0</th> <td> -219.8686</td> <td>   25.239</td> <td>   -8.711</td> <td> 0.000</td> <td> -270.616</td> <td> -169.122</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>1</th> <td>   11.4797</td> <td>    2.121</td> <td>    5.412</td> <td> 0.000</td> <td>    7.215</td> <td>   15.745</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 4.452</td> <th>  Durbin-Watson:     </th> <td>   0.127</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.108</td> <th>  Jarque-Bera (JB):  </th> <td>   4.330</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.692</td> <th>  Prob(JB):          </th> <td>   0.115</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.599</td> <th>  Cond. No.          </th> <td>    25.4</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:                      y   R-squared:                       0.379\n",
       "Model:                            OLS   Adj. R-squared:                  0.366\n",
       "Method:                 Least Squares   F-statistic:                     29.29\n",
       "Date:                Sun, 15 Jul 2018   Prob (F-statistic):           1.96e-06\n",
       "Time:                        02:59:35   Log-Likelihood:                -291.43\n",
       "No. Observations:                  50   AIC:                             586.9\n",
       "Df Residuals:                      48   BIC:                             590.7\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "0           -219.8686     25.239     -8.711      0.000    -270.616    -169.122\n",
       "1             11.4797      2.121      5.412      0.000       7.215      15.745\n",
       "==============================================================================\n",
       "Omnibus:                        4.452   Durbin-Watson:                   0.127\n",
       "Prob(Omnibus):                  0.108   Jarque-Bera (JB):                4.330\n",
       "Skew:                           0.692   Prob(JB):                        0.115\n",
       "Kurtosis:                       2.599   Cond. No.                         25.4\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mod = sm.OLS(y, x[[0,1]])\n",
    "res = mod.fit()\n",
    "res.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# attempt 2\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.965</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.963</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   640.3</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 15 Jul 2018</td> <th>  Prob (F-statistic):</th> <td>7.97e-35</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>02:59:58</td>     <th>  Log-Likelihood:    </th> <td> -219.81</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   445.6</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    47</td>      <th>  BIC:               </th> <td>   451.4</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>0</th> <td>    5.3675</td> <td>   10.115</td> <td>    0.531</td> <td> 0.598</td> <td>  -14.982</td> <td>   25.717</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>1</th> <td>  -48.4396</td> <td>    2.209</td> <td>  -21.929</td> <td> 0.000</td> <td>  -52.883</td> <td>  -43.996</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>2</th> <td>    2.8533</td> <td>    0.102</td> <td>   27.885</td> <td> 0.000</td> <td>    2.647</td> <td>    3.059</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 1.181</td> <th>  Durbin-Watson:     </th> <td>   1.794</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.554</td> <th>  Jarque-Bera (JB):  </th> <td>   1.188</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.336</td> <th>  Prob(JB):          </th> <td>   0.552</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.656</td> <th>  Cond. No.          </th> <td>    671.</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:                      y   R-squared:                       0.965\n",
       "Model:                            OLS   Adj. R-squared:                  0.963\n",
       "Method:                 Least Squares   F-statistic:                     640.3\n",
       "Date:                Sun, 15 Jul 2018   Prob (F-statistic):           7.97e-35\n",
       "Time:                        02:59:58   Log-Likelihood:                -219.81\n",
       "No. Observations:                  50   AIC:                             445.6\n",
       "Df Residuals:                      47   BIC:                             451.4\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "0              5.3675     10.115      0.531      0.598     -14.982      25.717\n",
       "1            -48.4396      2.209    -21.929      0.000     -52.883     -43.996\n",
       "2              2.8533      0.102     27.885      0.000       2.647       3.059\n",
       "==============================================================================\n",
       "Omnibus:                        1.181   Durbin-Watson:                   1.794\n",
       "Prob(Omnibus):                  0.554   Jarque-Bera (JB):                1.188\n",
       "Skew:                           0.336   Prob(JB):                        0.552\n",
       "Kurtosis:                       2.656   Cond. No.                         671.\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mod = sm.OLS(y, x[[0,1,2]])\n",
    "res = mod.fit()\n",
    "res.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# attempt 3\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.981</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.981</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   1261.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 15 Jul 2018</td> <th>  Prob (F-statistic):</th> <td>3.27e-42</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>03:02:59</td>     <th>  Log-Likelihood:    </th> <td> -219.96</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   443.9</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    48</td>      <th>  BIC:               </th> <td>   447.7</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>1</th> <td>  -47.3849</td> <td>    0.956</td> <td>  -49.555</td> <td> 0.000</td> <td>  -49.307</td> <td>  -45.462</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>2</th> <td>    2.8099</td> <td>    0.061</td> <td>   45.963</td> <td> 0.000</td> <td>    2.687</td> <td>    2.933</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 1.446</td> <th>  Durbin-Watson:     </th> <td>   1.784</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.485</td> <th>  Jarque-Bera (JB):  </th> <td>   1.381</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.381</td> <th>  Prob(JB):          </th> <td>   0.501</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.711</td> <th>  Cond. No.          </th> <td>    62.8</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:                      y   R-squared:                       0.981\n",
       "Model:                            OLS   Adj. R-squared:                  0.981\n",
       "Method:                 Least Squares   F-statistic:                     1261.\n",
       "Date:                Sun, 15 Jul 2018   Prob (F-statistic):           3.27e-42\n",
       "Time:                        03:02:59   Log-Likelihood:                -219.96\n",
       "No. Observations:                  50   AIC:                             443.9\n",
       "Df Residuals:                      48   BIC:                             447.7\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "1            -47.3849      0.956    -49.555      0.000     -49.307     -45.462\n",
       "2              2.8099      0.061     45.963      0.000       2.687       2.933\n",
       "==============================================================================\n",
       "Omnibus:                        1.446   Durbin-Watson:                   1.784\n",
       "Prob(Omnibus):                  0.485   Jarque-Bera (JB):                1.381\n",
       "Skew:                           0.381   Prob(JB):                        0.501\n",
       "Kurtosis:                       2.711   Cond. No.                         62.8\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mod = sm.OLS(y, x[[1,2]])\n",
    "res = mod.fit()\n",
    "res.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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