njg/regression example.ipynb

## regression example.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Regression example\n",
    "\n",
    "Here's an example linear regression analysis. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "load imports-85\n",
    "ds = dataset(X(:,7),X(:,8),X(:,9),X(:,15),'Varnames',{'curb_weight','engine_size','bore','price'});\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `fitlm()` instantiates a [LinearModel](https://www.mathworks.com/help/stats/linearmodel-class.html) object, `mdl`, which describes the linear regression analysis of the independent and response variables, `x` and `y`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "mdl = \n",
      "\n",
      "\n",
      "Linear regression model:\n",
      "    price ~ 1 + curb_weight\n",
      "\n",
      "Estimated Coefficients:\n",
      "                   Estimate         SE         tStat       pValue  \n",
      "                   _________    __________    _______    __________\n",
      "\n",
      "    (Intercept)       57.705        1.4606     39.508    2.9427e-97\n",
      "    curb_weight    -0.010547    0.00056009    -18.831    2.0071e-46\n",
      "\n",
      "\n",
      "Number of observations: 205, Error degrees of freedom: 203\n",
      "Root Mean Squared Error: 4.17\n",
      "R-squared: 0.636,  Adjusted R-Squared 0.634\n",
      "F-statistic vs. constant model: 355, p-value = 2.01e-46\n",
      "\n",
      "B0hat =\n",
      "\n",
      "   57.7052\n",
      "\n",
      "\n",
      "B1hat =\n",
      "\n",
      "   -0.0105\n",
      "\n"
     ]
    }
   ],
   "source": [
    "mdl=fitlm(ds,'ResponseVar','price','PredictorVars',1:1);\n",
    "mdl\n",
    "B0hat=mdl.Coefficients.Estimate(1)\n",
    "B1hat=mdl.Coefficients.Estimate(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The model is $$ \\hat{Y} = \\hat{\\beta_0} + \\hat{\\beta_1} X$$.\n",
    "\n",
    "The least squares solution for this model has estimates: $y$-intercept $\\hat{\\beta_0} = 57.7$, and a slope $\\hat{\\beta_1} = -0.010547$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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oFBYuXBhdG6G8aMfxIikKZ0VBT3zP8VDt86P2Hn744euuu+7LL788NU3b/vzz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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=ds.curb_weight;\n",
    "y=ds.price;\n",
    "scatterhist(x,y)\n",
    "hold on\n",
    "\n",
    "xx = min(x):(max(x)-min(x))/10:max(x);\n",
    "yy = B0hat + B1hat*xx;\n",
    "\n",
    "plot(xx,yy);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hypothesis testing \n",
    "Consider the hypothesis there is no linear relationship between Y and X. \n",
    "\n",
    "$$H_0: \\beta_1=0$$\n",
    "\n",
    "The t-statistic is $$t=\\frac{\\hat{\\beta_1}-\\beta_1}{SE(\\hat{\\beta_1})}$$.\n",
    "For $\\beta_1=0$, $$t=\\frac{-0.010547}{0.00056009}=-18.831$$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "pval =\n",
      "\n",
      "   2.0071e-46\n",
      "\n"
     ]
    }
   ],
   "source": [
    "pval = 2*tcdf(mdl.Coefficients.tStat(2),mdl.NumObservations-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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DCMCQuJ/HKnIFTn/+85/b2tqMx+fPn8/lctZXe3p6li9f/te//rW+vt76fOJ+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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "scatterhist(x,mdl.Residuals.Standardized)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plot of residuals vs curb_weight shows a systematic bias. This is reflected in the high-side tail of the y histogram. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EAJACY0i4uztmLgjDn/8bAIYUgYR+6u4WxBEAL6DLDgAgBVpIcM3lqiPBwiMA\nXkMgwTWXq44EIQTAawgkeGRgCgMAjRBI8EhtELnpvgOAIcSkBgCAFAgkAIAU6LJDL4aez0tlIgMA\nDdBCwp16pREAaINAwp1IIwA+QiABAKRAIMH9UxkAQENMaghsahTRTQdAAgRSAGP+AgCZSB1INTU1\nhYWFdXV1ISEhc+fOTUhIcD60Z88eq9WampqamprqwyL1ijQCIBmpx5AWLlxYV1eXlJSkKMozzzxz\n6NAhdb/ZbH7qqaciIyOnTZu2adOmjz/+2Ld16g9pBEA+UreQioqKQkND1e177703Nzc3PT1dCPHe\ne+8tXLgwIyNDCPHDH/4wMzNz8eLFQUFBvqxVR/qTRuq55BcADUjdQnKkkRAiIiKio6ND3T5x4kRy\ncrK6nZKSYrPZSktLfVCfHpEtAGQldSA52O323bt3z5s3TwhhtVo7OjrGjRunHjIajSEhIa2trT4t\nUD9IIwCykrrLzuGXv/xlWFiY2kfX3d0thIiIiHAcVRSls7PT5YVxcXHqhtls9n6ZfqLHqiT1I+/o\nA+BtOgikNWvWXLly5cMPP1RHiRRFEUKcO3cuMTFRPeHmzZsmk8nlteSQEP3upuvxrljGkABoQ/ZA\nevXVV2tra3fv3h0SEqLuURQlKiqqsbFR/WixWKxW64QJE3xXo9wIEwA6IfUY0vr168+ePZufn28y\nmex2u91uV/enp6fv2rXLZrMJIXbs2BEfHx8TE+PTSiVGGgHQCUO3xH9hOUaAVIqiVFVVCSHsdvuq\nVatKSkpGjhwZGhqal5c3duxYl5fTZTdIdNkB0IzUgTRIgRtIQxcgBBIAzUjdZYeB8EJ6kEYANCD7\npAb0A4/uBqBnBJK/GNJuOncfCTsA3kMg+YUh7abrvQ4JADTAGJL+ERoA/AKBpHOkEQB/QSDpHGkE\nwF8QSAAAKRBIOtRjGpxuvhsAPCGQ9IZBIwB+ikDSG9IIgJ9iHRI8rYQVLIwFoBUCSQ+83E3nYSUs\nHYQANEOXnfTIBACBgUCSG2kEIGAQSBIjjQAEEgJJSgaD9mnECiQAvsWkBvn4tGHU45dpoQHQDC0k\nydBNByBQ0UKSjE+76VhyBMCHCKSAxpIjAPKgyw4AIAUCCQAgBQIJACAFAgkAIAUCSS7eWJ3ax+9k\nRgMA3yKQAABSIJAAAFJgHZLveWN1KiteAegOgeR73lidyopXALpDlx0AQAoEEgBAClJ32XV1dZ0+\nffrSpUudnZ1PPfWUY395eXl9fb3jY2JiYnR0tPblAQCGkNQtpF/96lcvvvjivn37Nm7c6Ly/oKBg\n586dX9727bff+qpCL4mLi/N1CQNE5dqjcu3ptHL5y5a6hbRx48bs7Ozi4uKXqAhHRAAADJ9JREFU\nX365x6GkpKTs7GyfVOVV3ph9wIwGALogdQtJURR3h2w2W0lJSXV1tZb1AAC8R+oWkgeFhYUNDQ1V\nVVWRkZF5eXkxMTEuT5O/ieoOlWuPyrVH5XBm6Ja+Q0ftsquqqnLssVgs4eHhQgi73b5q1ar6+vqj\nR4/6rkAAwBCQusvOHTWNhBCKomRkZNTW1lqtVt+WBAAYJF0GkrNbt24JIYYN02vfIwBAJXUgdXV1\n2e32jo4OIYTdbrfb7er+0tJSdaO5uXnr1q1TpkzxMP0BAKALUo8hHT16dPXq1c57qqqqFEV55JFH\nWlpaTCZTW1tbQkJCTk5OZGSkr4oEAAwJqQMJABA4pO6yAwAEDgIJACAFPU1Oc/esVSFETU3Nnj17\nrFZrampqampqXw55uESzyj0/JVaGymtqagoLC+vq6kJCQubOnZuQkNCXMmSuXP57XlFRcezYscuX\nLwcFBT322GOzZ8/uSxkyVy7/PXc4ffr0hQsXZs6c6VhbIvk9d1e5ju65s6A33nhD+18dmA0bNrz/\n/vsNDQ2ffPJJRkaGY7/ZbH766acfe+yxBx988N133w0KCpo6darnQx4u0bLy3Nzcw4cPd3V1NTQ0\nNDQ0PPjggz/60Y+kqnz27Nn33XdfUlJSc3Pzpk2boqKiHnroIc9lSF65/Pd83759zc3N06ZN6+jo\nyM3NvXz58syZMz2XIXnl8t9zlcViefHFFwsKCh5//PGoqCjPZUheuV7ueU/d+nHr1q3u7u7PPvts\n0qRJzvtfeOGFt99+W93+7LPPHn744Y6ODs+HPFyiZeXr1q1bt26dy0skqbylpcWxvWXLlr/6q7/y\nXJ78lct/z50dOXLkxz/+sefy5K9cL/f8hRdeOHTo0MSJE0+ePOm5PPkr18s970FPY0juFhudOHEi\nOTlZ3U5JSbHZbI6FSu4OebhEy8qF+6fESlJ5aGioYzsiIkJdE+a5DMkrF9Lfc2ft7e0RERGey5O/\ncqGHe37kyBEhRFpaWl/Kk79yoYd73puexpBcslqtHR0d48aNUz8ajcaQkJDW1lYPhzxcoj2XT4mV\nsHK73b579+558+Z5KE/+ylXy3/PKysoDBw5cv3794sWLOTk5HsqTv3KV5Pf82rVrmzdv3r9/v/NO\nXdxzl5WrJL/nLumpheRSd3e3EML5/4spitLZ2enhkIdLNJaZmVlRUbFv375Tp05NnDhxxYoV6n4J\nK//lL38ZFhamDoDp6547Vy50cs9HjRo1derU8PDwpqamyspKD+XJX7nQwz3ftGnTc88912N9vS7u\nucvKhR7uuUu6DyS1N+zcuXOOPTdv3jSZTB4OebhEY+6eEitb5WvWrLly5cr27duDgoI8lCd/5UIn\n93zs2LFPP/30+vXrt2/f/tZbb1ksFr3c896VC+nveVlZWXl5+QMPPFBcXHz8+HEhxJkzZ2pqauS/\n5+4qF9Lfc3d032WnKEpUVFRjY6P60WKxWK3WCRMmeDjk4RIfcn5KrFSVv/rqq7W1tbt37w4JCVH3\n6OWe9668B2nvuUNsbKwQ4sKFC0lJSbq4570rd/zNqJLwngcFBU2aNGnv3r3idgOiqKgoJCQkNjZW\n8nvuoXLn0yS85+7oqYXk7lmr6enpu3btstlsQogdO3bEx8c73tfn7pCHS7Ss3MNTYiWpfP369WfP\nns3PzzeZTPq65+4ql/+eOyrs7Ox89913w8LCpk+f7rkMySuX/J4nJibm3ZabmyuEWLNmzaJFizyX\nIXnlkt9zd/TUQvr0008dz1qdPHmyuP2s1eXLl5vN5unTp48cOTI0NDQvL89xibtDHi7RsvKsrCzn\np8Ru27ZNtso/+eQTIcSjjz6qflQURX1Tovz33F3l8t/zjRs3Xr582WQytbe3R0dHb9++3Wg0ei5D\n8srlv+fuyH/P3dHpPfefh6u2tra2tLSMHTu274c8XKIZu91eVVU1efJkl1PDZa7ccxkyVy7/Pbfb\n7V9//fWECROCg4P7XobMlct/zz2Q/J67o8d77j+BBADQNT2NIQEA/BiBBACQAoEEAJACgQQAkAKB\nBACQAoEEAJACgQR4YrVa1YeAaa+ysrK8vNzdR8D/EEgICMuXL/+7v/s75z3FxcUJCQnu/oq3Wq3v\nvPPO1NtSUlL+9V//1dtFFhQUqM8lUx04cMB5tXyPj4D/IZAQEDZs2PD111+rz/sSQrS2tr7++utz\n5sxJTEzsfbLdbl+yZMnhw4c3btxYWVlZWlr67LPPbt68+dVXX/VqkaWlpcXFxY6PCxcuXL58uVd/\nEZCKnp5lBwxYVFRUVlZWdnZ2ampqbGxsdnZ2cHDwa6+95vLk/Pz8s2fPHjx48Mc//rEQIjg4eOnS\npaNHj37ttdfmzJmTkpJiNpuNRqPjmcqtra1VVVVJSUmOl1w0NjZWV1ffunVrzJgxM2bMcHxzTU1N\nV1dXXFxcaWnptWvXxo8fP2nSJPXQ+fPnr1y50t7erj4W80c/+lHv5wb1cO7cufPnz99zzz2JiYmj\nRo1y/pWvv/5aCDFixIhHH33UURUgOQIJgeIXv/jFf//3f69Zs2bFihWHDx/28FqKgwcPpqSkqGnk\n8OSTT27ZsuWTTz5JSUnZsmXL8OHDf/Ob36iHqqqqnn322TNnzqhf+B//8R9vvPHGT37yk2HDhpWW\nlk6cOPGjjz5SX6men5//3Xff2Wy2lpYWo9FYXV29Zs2aF154QQjxn//5n6dPnxZCqDH59NNPWyyW\nK1euuOyms9lsq1atOn78eFJS0rfffvvNN9+8//77M2fOFELk5OTk5+cnJCQMHz78j3/844wZMxx1\nApIjkBBA3nrrrSeeeGLFihWLFi1KSkpyeY7dbr906dIvfvGL3ocefvjhzz///K6/kpyc/H//939q\nu6S1tXXBggUfffTRypUr1aPFxcU5OTlpaWlCiNzc3M2bNy9dujQoKCgrK8tisXz33XeOBFq/fr27\nn8jJyamtrf3ss8/Utw3l5uauXr26pKREUZSdO3c6vl8Iob4iD9AFxpAQQMLDw9WXPc+ZM8fdOeqc\nurCwsN6HgoOD+zLj7v777w8KCrLb7WrXWXR0tPrmC9XEiRMdaTF37tyOjo4zZ87060/R1dW1d+/e\n559/3vHuu5deeqm9vf3EiRPqx6tXrzpO7vF+PEBmtJAQQD744IPm5uYZM2a88cYbhw4dUt/W04P6\nrP6Wlpbeh2w2213HdYQQ586de/31181m84gRI4KCgtra2pynTvzFX/yFY1sd+FFf6Nl3zc3Ndrt9\n//79n376qWOn0Wi8ceOGoiiZmZlvvfXWjh07ZsyY8bOf/Wzu3Lku/5iAhAgkBIrKysr8/Px//ud/\nnjZt2hNPPLFjx46MjIzep5lMprCwsNOnTy9ZsqTHoa+++sp5hoI7r7zyyvTp0w8ePKj22m3cuPFP\nf/rTkPwRnP31X//11KlTHR+XLVumTrLIyMh46qmnvvzyy5KSkg0bNhQWFjrmFgKSI5AQEOx2+2uv\nvfb444+rnXWrV69+++231Rl3vU9esGDBzp076+rqnF/hXFhYWFdX98orrwghfvCDH9y4ccNxqKmp\nyfmHLl269Ktf/coxt63vadTHpsyYMWNGjRrV2dn505/+1OUJkZGRc+bMmTNnTnJy8tq1azs7O5lo\nB12gLY+A8MEHH1y9evXNN99UPy5evHjq1KlZWVkuT37++efHjx+/dOnS0tLSrq6uzs7OgwcPrl27\n9vHHH1eHf6ZOnVpcXFxdXS2EOH/+/L/8y784rg0KCho1alRhYWFXV5cQ4uOPPz558mQfi3zooYcq\nKyurq6vb29ttNpuHM5cvX75z587f/e536q+0trb++7//u81mO3/+/MGDB9WBrq6urrNnz44aNYo0\ngl7QQoL/c3TWOY/wv/POO3PmzMnPz1dnXTsLCQnZu3dvdnb2c889p+7p6upasWLFyy+/rH5cuHDh\nH/7wh3nz5imKEh4e/swzz/zTP/2TeshoNP7617/OysoqKCgQQiQnJ6elpV25cqUvdaanp3/xxRcL\nFiyw2+2LFi3ycOaSJUvsdvuGDRvWrl2rKIrVap0yZcqTTz7Z1dX1wQcfvP766yaTqaOjIzw8fPPm\nzX27SYDv8QpzwK3Ozs6zZ882NTX94z/+46xZs95++23no/X19e3t7T2WKzkuLC8vnzBhwpgxY7xX\nXldX1/nz59va2iZPnqzOxVDZbLYzZ87cf//9Y8eO9d6vA0OOQALu7osvvnj55ZeXLVv24osv+roW\nwG8RSAAAKTCpAQAgBQIJACAFAgkAIAUCCQAgBQIJACCF/wc2ZN7ZovA5vwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qqplot(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The qq-plot shows the data are normally distributed (except for x < 1700). "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Matlab",
   "language": "matlab",
   "name": "matlab"
  },
  "language_info": {
   "codemirror_mode": "octave",
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