C1_W3_Lab03_Decision_Boundary_Soln.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optional Lab: Logistic Regression, Decision Boundary\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Goals\n",
    "In this lab, you will:\n",
    "- Plot the decision boundary for a logistic regression model. This will give you a better sense of what the model is predicting.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib widget\n",
    "import matplotlib.pyplot as plt\n",
    "from lab_utils_common import plot_data, sigmoid, draw_vthresh\n",
    "plt.style.use('./deeplearning.mplstyle')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dataset\n",
    "\n",
    "Let's suppose you have following training dataset\n",
    "- The input variable `X` is a numpy array which has 6 training examples, each with two features\n",
    "- The output variable `y` is also a numpy array with 6 examples, and `y` is either `0` or `1`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.array([[0.5, 1.5], [1,1], [1.5, 0.5], [3, 0.5], [2, 2], [1, 2.5]])\n",
    "y = np.array([0, 0, 0, 1, 1, 1]).reshape(-1,1) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot data \n",
    "\n",
    "Let's use a helper function to plot this data. The data points with label $y=1$ are shown as red crosses, while the data points with label $y=0$ are shown as blue circles. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "e482fe0305d5488099f8a4bef4fb0473",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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AEQIEAGCEAAE6UlcnFRd3XFNc7K4D+hgCBPCnrk5KT5eWLJGKinzXFBW5l6enEyLocwgQwJeW8Dh82D1dUOAdIkVF7vmSu44QQR9DgADttQ+PFm1DpG14tCBE0McEPECampq0aNEiTZo0SVOnTlV2draOHTvms/b111/XzTffrIkTJyo/P1/Nzc0B7hZ9UlmZd3i0KCiQxo71Do8Whw+71wf6AEv2QJYsWaJ9+/apoqJCs2bNUoGPX8aqqiqtXLlSZWVl2r9/v+rq6vTGG28Evln0Pbm5UmGh/+VffOF/WWGhe32gDwh4gPTr108zZ86Uw+GQJN1yyy2qqqryqistLdWcOXMUGxsrh8OhpUuXqqSkJMDdos/Kz+84RHwpLHSvB/QRlo+BvPTSS8rIyPCaX11drYSEhNbpxMREnThxIpCtoa/rTogQHuiDLA2Q1atXq7KyUv/6r//qc3nLXookuVyuQLUFfC8/XxozpuOaMWMID/RJlgXIc889py1btmjTpk0aMGCA1/KEhAQdP368dbq6ulrx8fGBbBFwn23V0ZiH5F7u7zoRoBezJEDWrl2rkpISbd68WdHR0T5r5s6dq61bt+r06dNyuVxav369srOzA9so+jZfp+r64+s6EaCXc9TX1wf02FBNTY1uuOEGJSUladCgQZKkiIgI7dy5U3l5ecrMzNTs2bMlScXFxSosLJTT6VRaWprWrFmj8PDwDl8/MjJSISGWD+3A7roTHm0xFoIr5HQ61djYaHUbXRLwAOlpBAiuWHGx+/Yk/owZ0/Fhrdde41ReGLNTgPBNC7SXkSGlpPheVlgoVVb6PzsrJcW9PtAHECBAeyNGSLt2eYdI28NTvk7xTUlxrzdiRCC6BCxHgAC+tA8RX2MbbUOE8EAfxBgI0JG6Ove9rToa0ygudh+2IjxwFdhpDIQAAYAgYqcA4ZsWAGCEAAEAGCFAAABGCBAAgBECBABghAABABghQAAARq5KgGRlZV2NlwEA2EhYd4r/7d/+zWuey+XSF509cAcA0Ot0K0A2bNigJ5980uvxsr6eKAgA6N26FSA//OEPdfvtt2vkyJEe83fv3n01ewIA2EC37oXlcrnkcDh6sp8rxr2wANhZr7oX1uOPP956yCrYwwMAEDidBkh5ebnuuusunT9/3mN+U1OTnn/++R5rDAAQ3DoNkO3bt8vlcikjI0MnT57UxYsX9cILL2jChAl68cUXA9EjACAIdWkMxOVyacWKFdq6datcLpciIiL00EMPKScnR+Hh4YHos8sYAwFgZ3YaA+n0LKyLFy+quLhY27ZtU3Nzs86dO6e3335bt912WyD6AwAEqU7/VE9NTdWzzz6rRx55RIcOHdIzzzyjnJwcbdy4MRD9AQCCVKd7II8++qhyc3N1zTXXSJJycnI0evRo5ebm6rPPPtOvfvWrHm8SABB8jJ+JXllZqTvvvFN/+ctfrnZPV4QxEAB2ZqcxEOMAkaT6+npFR0dfxXauHAECwM7sFCBX9E0bbOEBAAgc/lQHABghQAAARggQAICRbt3OHV3X1Cx9fFY62yRFhEpjB0uJkVZ3BQBXDwFylVU2SC98Iq0/Iv3toueytDjp/hul7LFSGPt+AGzuik7jDUZWnsb7widSwfvSJWfHdX83XCrNlOIGBqYvAPbRZ07jxfeeOSA9UN55eEjSvjPSbZul09/0eFsA0GMIkKvg/ZPSI3t9LxvR3z0G0t7nDdI9f+zZvgCgJ1kSICtWrND48eMVHR2tQ4cO+awpLy9XXFycpk6d2vpz4cKFAHfaNU/vl9ofB1yWIn16l3RqidTwT9KbP5YSB3nWvFctfXgmUF0CwNVlSYDMmzdPZWVlSkhI6LAuOTlZFRUVrT/9+/cPUIddd6xR2nrMc94TN0n/MU36YbR7+ppQ6a4fSv/9j1JMhGftC58EoksAuPosCZApU6Zo1KhRVrz1Vbf1mOfeR2S49C+TfNfGDZQeHO85792qnuoMAHpWUI+BHD16VGlpaUpPT9e6deusbsenk56Pilf6KGlQBw9pnJvkOf1lk3Tp8lVvCwB6XNBeBzJhwgQdPHhQUVFRqqmp0cKFCzV06FBlZWVZ3RoAQEG8BzJ48GBFRUVJkkaNGqUFCxZo714/pzpZqP21HLtqpK8v+a8vrfKcHtZPCvdxlhYABLugDZBTp07J6XRfVNHY2Kjt27crNTXV4q68zRktOdpMN16SVn7gu7b2vPTsx57z5iX1VGcA0LMsCZBHH31U48aNU21trebPn6+bbrpJkpSXl6dt27ZJkkpLSzV58mRNmTJFM2bM0LRp03T33Xdb0W6HRke6Q6Stp/ZLy3ZJn9W7p7+9LL35qfQ//0s62+72JvffGJA2AeCq41YmV8H7J91XlvvakLH9pXPfShd9DJTPTJC2z+np7gDYCbcy6WOmxEmrJ/tedvqC7/C4drD0xvSe7QsAehIBcpU8NEF6/jYpvAtb9JZYqXy+FDugx9sCgB7DIayrrLJBevG727m3H++4faR0/w3SP3I7dwB+2OkQFgHSQy5elj7+SvqKB0oB6AY7BUjQXkhodxGh0t/FWt0FAPQc6/9UBwDYEgECADBCgAAAjBAgAAAjBAgAwAgBAgAwQoAAAIwQIAAAIwQIAMAIAQIAMEKAAACMECAAACMECADACAECADBCgAAAjBAgAAAjBAgAwAgBAgAwQoAAAIwQIAAAIwQIAMAIAQIAMEKAAACMECAAACMECADASJjVDSDw/nZROnhW+vqSNChcuiFGGhJhdVcA7IYA6UMqTkrPfyK9XSldcn4/PzxEyh4rPXCjNDXOuv4A2Iujvr7eZXUTV1NkZKRCQjgy19a3l6Wf7ZE2HOm89t7rpZfSpGtCe74vAN6cTqcaGxutbqNL2APp5S47pTt3SJu/6Fr9hiNS/UVp00wplBwG0AG+Inq5VR/5Dg+HpJED3f+2984X7vUAoCOWBMiKFSs0fvx4RUdH69ChQ37rXn/9dd18882aOHGi8vPz1dzcHMAu7a+pWVpzwHNe/zDpqVulL++Vaha7/33qVvf8ttYckC5eDlyvAOzHkgCZN2+eysrKlJCQ4LemqqpKK1euVFlZmfbv36+6ujq98cYbAezS/koqpS+bPOdtzpAev1mK6eeejunnnt6c4Vn3ZZO06fPA9AnAniwJkClTpmjUqFEd1pSWlmrOnDmKjY2Vw+HQ0qVLVVJSEqAOe4d32x26Sh8pzfST2TMTpGkjO14fANoK2jGQ6upqjz2UxMREnThxwsKO7OfkN57TP0nquH5uu+Xt1weAtoI2QCTJ4fh+iNfl6lVnGwOA7QVtgCQkJOj48eOt09XV1YqPj7ewI/uJG+A5vaWq4/rSdsvbrw8AbQVtgMydO1dbt27V6dOn5XK5tH79emVnZ1vdlq3MG+M5vatWeq/ad+171dLu2o7XB4C2LAmQRx99VOPGjVNtba3mz5+vm266SZKUl5enbdu2SZKSkpL0xBNPaNasWZo4caKGDx+ue+65x4p2bWvBWGlYP89588ukpz6Uzn53dtZXTe7p+WWedcP6SQuvDUyfAOyJW5n0ck99KP3Ln7znOyTFDZROnpd8fQBW3io9cXNPdwegPTvdyoRv2l5uxURpvo9DUS5JtX7CI2uMez0A6AgB0suFhkgbZ7hvktgV914vvTWD+2AB6ByHsPqQ97+7nXuJj9u5L/judu5TuJ07YCk7HcIiQPqgv12UDp2VGi9JkeHSOB4oBQQNOwUIt3Pvg4ZEsKcB4MrxpzoAwAgBAgAwQoAAAIwQIAAAIwQIAMAIAQIAMEKAAACMECAAACMECADACAECADBCgAAAjBAgAAAjBAgAwAgBAgAwQoAAAIwQIAAAIwQIAMAIAQIAMEKAAACMECAAACMECADACAECADBCgAAAjBAgAAAjBAgAwEiY1Q0AbR1vlCobpIuXpZh+0vgYqR+fUiAo8asJyzU7pbcrpec/kcpPei4bEiEtvV66/0Zp7GBr+oOBujqprEzKzfVfU1wsZWRII0YEri9cVRzCgqVqz0s/+i/ppzu8w0OS/nZRWn1Auv4/pRc/CXx/MFBXJ6WnS0uWSEVFvmuKitzL09Pd9bAlAgSWOf2NlLZZ2nem89pLTun+cumZAz3eFq5ES3gcPuyeLijwDpGiIvd8yV1HiNgWAQLL3PNH6fMG7/kRodKI/r7XeWSv9L6PPRUEgfbh0aJtiLQNjxaEiG1ZEiCff/65Zs6cqUmTJmn69Ok6cuSIV015ebni4uI0derU1p8LFy5Y0C16wgdnpPeqPeclDpL+88dSwz9Jp5ZIn94lLUvxrHFJWvVRgJpE95SVeYdHi4ICaexY7/Bocfiwe33YiiWD6AUFBcrNzVVOTo7effdd5eXlaceOHV51ycnJ2r17d+AbRI9rP54REyH99z9KcQO/n/fDaOk/pknD+0lP7f9+/tZj0rFGaXRkIDpFl+XmSvX1/kPiiy/8r1tY2PGAO4JSwPdAzpw5owMHDujOO++UJM2dO1fHjh3TsWPHAt0KLPRuled0fqpneLT1xM1SZPj3006XO0QQhPLz3WHQHYWF7vVgOwEPkJqaGsXFxSkszL3z43A4FB8frxMnTnjVHj16VGlpaUpPT9e6desC3Sp6yKXL0pdNnvN+Mtp/feQ1Uvooz3knz1/9vnCVdCdECA9bs+QQlsPh8Jh2uVxeNRMmTNDBgwcVFRWlmpoaLVy4UEOHDlVWVlag2gRgKj/fPWDe0WGrMWMID5sL+B7IqFGjVFtbq+bmZknu8KipqVF8fLxH3eDBgxUVFdW6zoIFC7R3795At4seEB4qDevnOW9LB4ekGr+VdtV4zvN3uAtBorPwkNzL/V0nAlsIeIAMHz5c48eP18aNGyVJpaWlSkxM1OjRnscwTp06JafTKUlqbGzU9u3blZqaGuh20UPmJXlOF/3VfVGhL099KDVe+n46xCHN6eCQFyzm61Rdf3xdJwLbsOQ03sLCQr322muaNGmSnnnmGT333HOSpLy8PG3btk2SO1gmT56sKVOmaMaMGZo2bZruvvtuK9pFD7j/Rs/psxfdV6T/52fSt5fd8z6rl5bt8jwDS3KHB2dgBanuhEcLQsS2HPX19d4DEDYWGRmpkBCuj7SDWVu9rwWR3BcSRl0jnfZx2Y9DUvl8aUpcT3eHbisudt+exJ8xYzo+rPXaa5zKK8npdKqxsdHqNrqEb1pY5o3p0rU+bpB48bLv8JCk1ZMJj6CVkSGlpPheVlgoVVb6PzsrJcW9PmyFAIFlYge49yZuie28NjxEeuE26aEJPd4WTI0YIe3a5R0ibU/V9XWKb0qKez3uyms7HMKC5Zqd0n99dzv3Pe3ucxXz3e3cf87t3O2j7T2x/F3n0TJWQnh4sdMhLAIEQaXtA6WG9pPGD3WPicBmeB6IMQLEQgQIADuzU4DwTQsAMEKAAACMECAAACMECADACAECADBCgAAAjBAgAAAjBAgAwAgBAgAwQoAAAIwQIAAAIwQIAMAIAQIAMEKAAACMECAAACMECADACAECADBCgAAAjBAgAAAjBAgAwAgBAgAwQoAAAIwQIAAAIwQIAMAIAQIAMEKAAACMECAAACMECADACAECADBCgAAAjFgSIJ9//rlmzpypSZMmafr06Tpy5IjPutdff10333yzJk6cqPz8fDU3Nwe4UwCAP476+npXoN/0Jz/5iX76058qJydH7777rtauXasdO3Z41FRVVSkjI0N79uzR8OHDddddd2nWrFm69957O3ztgQMHKiSEHSsA9uR0OnX+/Hmr2+iSgAfImTNnNGnSJFVWViosLEwul0vJycnasWOHRo8e3Vr37LPP6vjx4/rd734nSXrvvfdUVFSkP/zhD4FsFwDgR8D/VK+pqVFcXJzCwsIkSQ6HQ/Hx8Tpx4oRHXXV1tRISElqnExMTvWoAANax5FiPw+HwmHa5fO8Eta3zVwMAsEbAA2TUqFGqra1tHRB3uVyqqalRfHy8R11CQoKOHz/eOl1dXe1VAwCwTsADZPjw4Ro/frw2btwoSSotLVViYqLH+IckzZ07V1u3btXp06flcrm0fv16ZWdnB7pdAIAflpyF9dlnn+n+++/X2bNnFRkZqRdffFEpKSnKy8tTZmamZs+eLUkqLi5WYWGhnE6n0tLStGbNGoWHhwe6XQCAD5YECADA/mx1wUQwX4DYld7Ky8sVFxenqVOntv5cuHChR/tasWKFxo8fr+joaB06dMhvnRXbrCu9WbHNmpqatGjRIk2aNElTp05Vdna2jh075rM20Nutq71Zsd2ysrI0efJkTZ06VZmZmfrrX//qs86Kz1pXerNim7X17//+7x3+LgTjhdW2CpCCggLl5ubqgw8+UH5+vvLy8rxqqqqqtHLlSpWVlWn//v2qq6vTG2+8ERS9SVJycrIqKipaf/r379+jfc2bN09lZWUep0S3Z9U260pvUuC3mSQtWbJE+/btU0VFhWbNmqWCggKvGqu2W1d6kwK/3TZs2KC9e/eqoqJCDzzwgH7xi1941Vi1zbrSm2TNZ02SPvroI+3bt8/viUJWbbfO2CZAzpw5owMHDujOO++U5B5kP3bsmNdfX6WlpZozZ45iY2PlcDi0dOlSlZSUBEVvVpgyZYpGjRrVYY0V26yrvVmhX79+mjlzZutp5Lfccouqqqq86qzYbl3tzQrR0dGt/93Q0ODzjhBWfda60ptVLl68qMcee0y/+93vvC5xaGHVdutMmNUNdFVHFyC2PYPLigsQu9qbJB09elRpaWkKDQ1VTk6Oli1b1qO9dUWwX7Rp9TZ76aWXlJGR4TU/GLabv94ka7bb8uXLVVFRIUk+v+Cs3Gad9SZZs81WrlypO+64Q0lJSX5rguGz5ottAkQK7gsQu9LbhAkTdPDgQUVFRammpkYLFy7U0KFDlZWVFZAeOxKsF21avc1Wr16tyspKPfPMMz6XW7ndOurNqu328ssvS5LefPNN/frXv9amTZu8aqzaZp31ZsU2+/Of/6wPP/xQv/nNbzqtDcbf0eDZj+tEMF+A2NXeBg8erKioqNZ1FixYoL179/Zob10RzBdtWrnNnnvuOW3ZskWbNm3SgAEDvJZbud06683qz9qiRYtUXl6us2fPeswPhs+av96s2Gbvv/++PvvsM6Wmpmr8+PGqra1Vdna2181lg2G7+WKbAAnmCxC72tupU6fkdDolSY2Njdq+fbtSU1N7tLeuCOaLNq3aZmvXrlVJSYk2b97scfy8Lau2W1d6C/R2a2ho0MmTJ1unt2zZopiYGA0ZMsSjzopt1tXerPisPfTQQzpy5Ig+/vhjffzxxxo5cqTefvttzZgxw6MuWH9HbXUdSDBfgNiV3l555RWtX79eoaGhunz5subNm6fHH3/c78DZ1fDoo49q27Ztqqur09ChQzVw4EDt378/KLZZV3qzYpvV1NTohhtuUFJSkgYNGiRJioiI0M6dOy3fbl3tLdDb7cSJE1q8eLGamprkcDg0bNgwPfnkk0pNTbV8m3W1Nys+a+21/CE6btw4y7dbV9gqQAAAwcM2h7AAAMGFAAEAGCFAAABGCBAAgBECBABghAABABghQAAARggQAIARAgTw45VXXtF1113X+lChxsZGTZ48WStWrLC4MyA4ECCAH7m5uQoPD1dxcbGcTqeWLVum+Ph4PfXUU1a3BgQFW93OHQikiIgIPfzww1q9erU+//xznThxQmVlZQoNDbW6NSAocC8soAPffvutxo0bp9DQUO3cudPjFtobN27UunXrJEm/+tWvdPvtt1vVJmAJ9kCADmzatEnnzp3TkCFDNHTo0Nb5586dU2Fhof74xz/qwoUL+od/+AdVVFSwd4I+hTEQwI+Kigr98z//s0pKStSvXz+9+uqrrcs++OAD3Xrrrerfv79iYmIUHx+vo0ePWtgtEHgECOBDZWWlFi9erFWrVun222/XQw89pGeffVbffPONJOmrr77yeJhTdHS0vvrqK4u6BaxBgADt1NfX64477lBubq4WLVokScrJyVF4eHjrXkhMTIzq6+tb1zl37pxiYmKsaBewDIPogIH6+nplZmZq165dunDhgmbPnq3y8nKFhTGsiL6DTztgIDo6Wg8++KDmzJkjh8OhlStXEh7oc9gDAQAYYQwEAGCEAAEAGCFAAABGCBAAgBECBABghAABABghQAAARggQAIARAgQAYIQAAQAYIUAAAEb+P2Wz/rU+Z9N0AAAAAElFTkSuQmCC",
      "text/html": [
       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img 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' width=400.0/>\n",
       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(footer_visible=False, header_visible=False, toolbar=Toolbar(toolitems=[('Home', 'Reset original view', …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax = plt.subplots(1,1,figsize=(4,4))\n",
    "plot_data(X, y, ax)\n",
    "\n",
    "ax.axis([0, 4, 0, 3.5])\n",
    "ax.set_ylabel('$x_1$')\n",
    "ax.set_xlabel('$x_0$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic regression model\n",
    "\n",
    "\n",
    "* Suppose you'd like to train a logistic regression model on this data which has the form   \n",
    "\n",
    "  $f(x) = g(w_0x_0+w_1x_1 + b)$\n",
    "  \n",
    "  where $g(z) = \\frac{1}{1+e^{-z}}$, which is the sigmoid function\n",
    "\n",
    "\n",
    "* Let's say that you trained the model and get the parameters as $b = -3, w_0 = 1, w_1 = 1$. That is,\n",
    "\n",
    "  $f(x) = g(x_0+x_1-3)$\n",
    "\n",
    "  (You'll learn how to fit these parameters to the data further in the course)\n",
    "  \n",
    "  \n",
    "Let's try to understand what this trained model is predicting by plotting its decision boundary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Refresher on logistic regression and decision boundary\n",
    "\n",
    "* Recall that for logistic regression, the model is represented as \n",
    "\n",
    "  $$f_{\\mathbf{w},b}(\\mathbf{x}^{(i)}) = g(\\mathbf{w} \\cdot \\mathbf{x}^{(i)} + b) \\tag{1}$$\n",
    "\n",
    "  where $g(z)$ is known as the sigmoid function and it maps all input values to values between 0 and 1:\n",
    "\n",
    "  $g(z) = \\frac{1}{1+e^{-z}}\\tag{2}$\n",
    "  and $\\mathbf{w} \\cdot \\mathbf{x}$ is the vector dot product:\n",
    "  \n",
    "  $$\\mathbf{w} \\cdot \\mathbf{x} = w_0 x_0 + w_1 x_1$$\n",
    "  \n",
    "  \n",
    " * We interpret the output of the model ($f_{\\mathbf{w},b}(x)$) as the probability that $y=1$ given $\\mathbf{x}$ and parameterized by $\\mathbf{w}$ and $b$.\n",
    "* Therefore, to get a final prediction ($y=0$ or $y=1$) from the logistic regression model, we can use the following heuristic -\n",
    "\n",
    "  if $f_{\\mathbf{w},b}(x) >= 0.5$, predict $y=1$\n",
    "  \n",
    "  if $f_{\\mathbf{w},b}(x) < 0.5$, predict $y=0$\n",
    "  \n",
    "  \n",
    "* Let's plot the sigmoid function to see where $g(z) >= 0.5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot sigmoid(z) over a range of values from -10 to 10\n",
    "z = np.arange(-10,11)\n",
    "\n",
    "fig,ax = plt.subplots(1,1,figsize=(5,3))\n",
    "# Plot z vs sigmoid(z)\n",
    "ax.plot(z, sigmoid(z), c=\"b\")\n",
    "\n",
    "ax.set_title(\"Sigmoid function\")\n",
    "ax.set_ylabel('sigmoid(z)')\n",
    "ax.set_xlabel('z')\n",
    "draw_vthresh(ax,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* As you can see, $g(z) >= 0.5$ for $z >=0$\n",
    "\n",
    "* For a logistic regression model, $z = \\mathbf{w} \\cdot \\mathbf{x} + b$. Therefore,\n",
    "\n",
    "  if $\\mathbf{w} \\cdot \\mathbf{x} + b >= 0$, the model predicts $y=1$\n",
    "  \n",
    "  if $\\mathbf{w} \\cdot \\mathbf{x} + b < 0$, the model predicts $y=0$\n",
    "  \n",
    "  \n",
    "  \n",
    "### Plotting decision boundary\n",
    "\n",
    "Now, let's go back to our example to understand how the logistic regression model is making predictions.\n",
    "\n",
    "* Our logistic regression model has the form\n",
    "\n",
    "  $f(\\mathbf{x}) = g(-3 + x_0+x_1)$\n",
    "\n",
    "\n",
    "* From what you've learnt above, you can see that this model predicts $y=1$ if $-3 + x_0+x_1 >= 0$\n",
    "\n",
    "Let's see what this looks like graphically. We'll start by plotting $-3 + x_0+x_1 = 0$, which is equivalent to $x_1 = 3 - x_0$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Choose values between 0 and 6\n",
    "x0 = np.arange(0,6)\n",
    "\n",
    "x1 = 3 - x0\n",
    "fig,ax = plt.subplots(1,1,figsize=(5,4))\n",
    "# Plot the decision boundary\n",
    "ax.plot(x0,x1, c=\"b\")\n",
    "ax.axis([0, 4, 0, 3.5])\n",
    "\n",
    "# Fill the region below the line\n",
    "ax.fill_between(x0,x1, alpha=0.2)\n",
    "\n",
    "# Plot the original data\n",
    "plot_data(X,y,ax)\n",
    "ax.set_ylabel(r'$x_1$')\n",
    "ax.set_xlabel(r'$x_0$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* In the plot above, the blue line represents the line $x_0 + x_1 - 3 = 0$ and it should intersect the x1 axis at 3 (if we set $x_1$ = 3, $x_0$ = 0) and the x0 axis at 3 (if we set $x_1$ = 0, $x_0$ = 3). \n",
    "\n",
    "\n",
    "* The shaded region represents $-3 + x_0+x_1 < 0$. The region above the line is $-3 + x_0+x_1 > 0$.\n",
    "\n",
    "\n",
    "* Any point in the shaded region (under the line) is classified as $y=0$.  Any point on or above the line is classified as $y=1$. This line is known as the \"decision boundary\".\n",
    "\n",
    "As we've seen in the lectures, by using higher order polynomial terms (eg: $f(x) = g( x_0^2 + x_1 -1)$, we can come up with more complex non-linear boundaries."
   ]
  },
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   "source": [
    "## Congratulations!\n",
    "You have explored the decision boundary in the context of logistic regression."
   ]
  },
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   "metadata": {},
   "outputs": [],
   "source": []
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  {
   "cell_type": "code",
   "execution_count": null,
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   "source": []
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