08.Supervised_Learning-Regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Supervised Learning Part 2 -- Regression Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In regression we are trying to predict a continuous output variable -- in contrast to the nominal variables we were predicting in the previous classification examples. \n",
    "\n",
    "Let's start with a simple toy example with one feature dimension (explanatory variable) and one target variable. We will create a dataset out of a sine curve with some noise:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-3.         -2.93939394 -2.87878788 -2.81818182 -2.75757576 -2.6969697\n",
      " -2.63636364 -2.57575758 -2.51515152 -2.45454545 -2.39393939 -2.33333333\n",
      " -2.27272727 -2.21212121 -2.15151515 -2.09090909 -2.03030303 -1.96969697\n",
      " -1.90909091 -1.84848485 -1.78787879 -1.72727273 -1.66666667 -1.60606061\n",
      " -1.54545455 -1.48484848 -1.42424242 -1.36363636 -1.3030303  -1.24242424\n",
      " -1.18181818 -1.12121212 -1.06060606 -1.         -0.93939394 -0.87878788\n",
      " -0.81818182 -0.75757576 -0.6969697  -0.63636364 -0.57575758 -0.51515152\n",
      " -0.45454545 -0.39393939 -0.33333333 -0.27272727 -0.21212121 -0.15151515\n",
      " -0.09090909 -0.03030303  0.03030303  0.09090909  0.15151515  0.21212121\n",
      "  0.27272727  0.33333333  0.39393939  0.45454545  0.51515152  0.57575758\n",
      "  0.63636364  0.6969697   0.75757576  0.81818182  0.87878788  0.93939394\n",
      "  1.          1.06060606  1.12121212  1.18181818  1.24242424  1.3030303\n",
      "  1.36363636  1.42424242  1.48484848  1.54545455  1.60606061  1.66666667\n",
      "  1.72727273  1.78787879  1.84848485  1.90909091  1.96969697  2.03030303\n",
      "  2.09090909  2.15151515  2.21212121  2.27272727  2.33333333  2.39393939\n",
      "  2.45454545  2.51515152  2.57575758  2.63636364  2.6969697   2.75757576\n",
      "  2.81818182  2.87878788  2.93939394  3.        ]\n"
     ]
    }
   ],
   "source": [
    "x = np.linspace(-3, 3, 100)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng = np.random.RandomState(42)\n",
    "y = np.sin(4 * x) + x + rng.uniform(size=len(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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D3zT3RaUJFtqEwB6pcScLywx8SQFcmu6OPmk7CzXtyQPIw9zPWiRauoWFBV9dXZ3657bJ4vJK4gh8frarI0t7E99Txgh6+AYj9W4Y53R26GcntsY6P6DtzOyouy+Meh0j9kgVrVcPJa0aJa1PC5OZQHEE9kjVZbJw3ECd9/yyni6mmbsH6ojAHqm6TBam3WBmux09f/LFic4va/5Amm7uHqgj6tgjVZd687QFTbe8/bcmPr+sxUYsRAIYsUetDvXmo6pRJjm/SeYPyN2jTQjsKF3oG8yo+YM6zC0AVSIVg8bJ6leT9DNTL9jHulE1MIwRe0vEVCmSZ7FRf3ck00vbeTGRirZggVILpC0SakLzriImWaQF1BkLlHBaVqVIXQJ7GU8UdHREWxHYW6BOAW6afWPqskgLmDYmT1sgLZBNO8ClbaT90X95rJTa86ZuCgIURWBvgboEuLSUUFIzMKn4E0VdFmkB00YqpgXq0rK2rL4xWeqwSAuYNgJ7S9QhwE3SNyamMk1gWlqfirn3kTUtLq9o99JhFrCUbNy+MZJybbAN4EyFRuxm9k5Jt0h6naQr3b1Rxenj7jLUBHUe4Y7bN2ZxeaX2ZZpAHRVNxTwq6TpJnwlwLlPXhPrucTThRjVOSqhOZZpAkxRKxbj74+7e2H6osQWO2FrW5i3TJJ0GnKnVOfa61HeHEtuNKk+ZZlptPMEdbTYysJvZ183s0YQ/147zQWZ2wMxWzWx1fX198jMOqC713aHEdqPKU4ce21MKEMLIHLu7Xx3ig9z9oKSDUq8JWIhjFlWX+u5Q6rIdXkijcvKxPaUAIbS+jr0O9d2hxHajyoN+MMDZipY7vkPSpyXNSTpsZsfcfV+QM8NEYrhRjVOyGeNTClBUocDu7l+S9KVA5wKMXbLZxqcUYJRGpmIGR3TndTsykzZObJ3xNf+DN9MkawtieEoBQmpcYB8e0W1svtQZcPDrOi7OwWhMhgLFNa6OPWlEl2bcsjcWulQvtpJNoAqNC+zjjtzyvj5poctNh45pF0F+qmJbWwBUoXGpmLTytqzX55H0JNCU3e3r3PhrXFmToTFdJ1CmxgX2pPK2NOOM9EaN7OvaHKwJjb/GlTQZGuN1AmVpXCpmeJn5bLej88/tnPX1uNug5RnZP7OxWbs8fFuW1LflOoEQGjdil8opb8vzJHBet1O7UWNbqkjacp1ACI0bsZdl8ElAkmzo593OjMxUu1FjW6pI2nKdQAitCex5Uij798zryNJe/dfy2/Sp6684q6vgxomtsw+sakeNbakiact1AiE0MhWTx/Dq1F+9cFJbp3p1LnlSKEnpntvuP167hlNtWVLflusEQjD36XfQXVhY8NXV8rZHHa6gSDM/29WRpb2FjtvtzIw1SQsAkzKzo+6+MOp1UY7Y865OHTeFwqgRQBNEGdjzBmyXtLi8MlZwpuEUgLqLcvJ0nJw3e2QCiE2UgT2pgqKzw3T+uZ3E11ddsjiJui2UAlAfUaZisnLhu5cOK2m6eJKSxap6l7C8HkCWKAO7lJ4LD7VHZpXBdZLNKAC0R5SpmCyhFrpU2buE5fUAsrQusA83ERu3WVhflcGV5fUAskSbiskSomQxVEpnEkkNy1heD6CvlYE9hGkF16wJWhZKAUhSKLCb2W2S/kjSC5L+Q9JfuPtGiBOru2kE11ETtARyAEkK9Yoxs7dIWnH3k2b2CUly9w+Oel/ZvWJisbi8kpjuGbfHDYA45O0VU2jy1N2/6u4nt799UNLFRY6HM1H9AmASIati3ifpKwGP13pUvwCYxMjAbmZfN7NHE/5cO/CaD0s6KemujOMcMLNVM1tdX18f+0TbuISezSUATKJwP3Yze6+kGyS9yd1P5HnPuDn2NvdBr6ptAYD6yZtjLzp5eo2kT0r6A3fPPQwfN7AzidhDkAfabVobbfy9pFdI+pqZSdKD7n5DwWOehUlEGn8ByK9QYHf33wh1IlmqXOUp1WOkTOMvAHk1oldMlZOI/ZHy2samXNVtzMFTC4C8GhHYQzXumkSVXRwHUfoIIK/G9Iqpagl9XUbKNP4CkFcjRuxVqstIucqnFgDN0pgRe1XqNFKm8ReAPAjsI9AiF0DTENhzKGOkPFxCedVlc3rgiXU9s7Gp87odmUkbJ7a4kQAYG4G9AkmLjf7pwf8+/fONza3TX7MQCcC4mDytQFIJZZYqyisBNBcj9kDGWZ06SakkC5EA5MWIPYBxV6dOUirJQiQAeRHYAxh3dWpSi4QsLEQCMA4CewDjrk5NWmz0njfsPP39bLej88/tsBAJwETIsQeQ1n3S1esl38+316FLJID4EdgDSFqd2tfPt6/+6Ke6++ga/dQBlI5UTACDqZUkm1un9PmHnqpFl0gA8SOwB7J/z7yOLO2Vpfz8VMoWhJQxAgiNwB7YuGWJlDECCI3AHtg4pYyUMQIoA5OngQ12g0yqlOmbpyoGQEkYsZegn29Pm0ydn+3qyNJegjqAUhDYS1TlJtwA2qtQKsbMPibpWkkvSnpW0p+7+zMhTiwGbNIBoArmKWV4ud5s9mvu/ovtr/9S0m+6+w2j3rewsOCrq6sTfy4AtJGZHXX3hVGvK5SK6Qf1ba9UbxU9AKBChatizOzjkv5M0s8lXVX4jAAAhYwcsZvZ183s0YQ/10qSu3/Y3S+RdJek92cc54CZrZrZ6vr6ergrAACcoVCO/YwDmb1G0mF3f/2o15JjB4DxTSXHbmavHfj27ZKeKHI8AEBxRati7pZ0qXrljj+SdIO7J+8Hd+b71rdfP4kLJP1kwvfWDddSP7Fch8S11FWRa3mNu8+NelGwVMy0mNlqnkeRJuBa6ieW65C4lrqaxrWw8hQAIkNgB4DINDGwH6z6BALiWuonluuQuJa6Kv1aGpdjBwBka+KIHQCQoZGB3cw+ZmbfM7NjZvZVM7uo6nOalJndZmZPbF/Pl8xstupzmoSZvdPMHjOzF82skdULZnaNmR03syfNbKnq85mUmd1pZs+a2aNVn0sRZnaJmT1gZo9v/259oOpzmpSZnWNm3zaz725fy0dL/bwmpmIm7SpZR2b2Fkkr7n7SzD4hSe7+wYpPa2xm9jr11jN8RtJfuXujlhab2Yykf5f0ZklPS3pY0rvd/QeVntgEzOz3JT0n6XN5VoLXlZldKOlCd/+Omb1a0lFJ+xv6b2KSXunuz5lZR9K3JH3A3R8s4/MaOWKPqauku3/V3U9uf/ugpIurPJ9Jufvj7n686vMo4EpJT7r7D939BUlfUG+vgcZx929K+mnV51GUu//Y3b+z/fUvJT0uqZGbGXjPc9vfdrb/lBa3GhnYpV5XSTN7StKfSPqbqs8nkPdJ+krVJ9FS85KeGvj+aTU0iMTIzHZJ2iPpoWrPZHJmNmNmx9TblOhr7l7atdQ2sIfqKlkHo65l+zUflnRSveuppTzX0WCW8HeNfRKMiZm9StLdkm4celpvFHc/5e5XqPdUfqWZlZYmK9yPvSzufnXOl/6zpMOSPlLi6RQy6lrM7L2S/lDSm7zGkx5j/Js00dOSLhn4/mJJbPNYse189N2S7nL3e6o+nxDcfcPMviHpGkmlTHDXdsSeJaaukmZ2jaQPSnq7u5+o+nxa7GFJrzWz3Wb2cknvknRfxefUatsTjndIetzdP1n1+RRhZnP9ijcz60q6WiXGraZWxUzUVbKOzOxJSa+Q9L/bf/VgEyt8zOwdkj4taU7ShqRj7r6v2rMaj5m9VdLtkmYk3enuH6/4lCZiZp+X9Eb1ugj+j6SPuPsdlZ7UBMzs9yT9m6Tvq/f/uiT9tbt/ubqzmoyZ/bakz6r3u7VD0hfd/W9L+7wmBnYAQLpGpmIAAOkI7AAQGQI7AESGwA4AkSGwA0BkCOwAEBkCOwBEhsAOAJH5f17/JxsWJIA+AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linear Regression\n",
    "=================\n",
    "\n",
    "The first model that we will introduce is the so-called simple linear regression. Here, we want to fit a line to the data, which \n",
    "\n",
    "One of the simplest models again is a linear one, that simply tries to predict the data as lying on a line. One way to find such a line is `LinearRegression` (also known as [*Ordinary Least Squares (OLS)*](https://en.wikipedia.org/wiki/Ordinary_least_squares) regression).\n",
    "The interface for LinearRegression is exactly the same as for the classifiers before, only that ``y`` now contains float values, instead of classes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we remember, the scikit-learn API requires us to provide the target variable (`y`) as a 1-dimensional array; scikit-learn's API expects the samples (`X`) in form a 2-dimensional array -- even though it may only consist of 1 feature. Thus, let us convert the 1-dimensional `x` NumPy array into an `X` array with 2 axes:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Before:  (100,)\n",
      "After:  (100, 1)\n"
     ]
    }
   ],
   "source": [
    "print('Before: ', x.shape)\n",
    "X = x[:, np.newaxis]\n",
    "print('After: ', X.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, we start by splitting our dataset into a training (75%) and a test set (25%):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.model_selection import train_test_split\n",
    "\n",
    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.25, random_state=42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we use the learning algorithm implemented in `LinearRegression` to **fit a regression model to the training data**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "regressor = LinearRegression()\n",
    "regressor.fit(X_train, y_train)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After fitting to the training data, we paramerterized a linear regression model with the following values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Weight coefficients:  [0.90211711]\n",
      "y-axis intercept:  0.44840974988268\n"
     ]
    }
   ],
   "source": [
    "print('Weight coefficients: ', regressor.coef_)\n",
    "print('y-axis intercept: ', regressor.intercept_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since our regression model is a linear one, the relationship between the target variable (y) and the feature variable (x) is defined as \n",
    "\n",
    "$$y = \\text{weight} \\times x + \\text{intercept .}$$\n",
    "\n",
    "Plugging in the min and max values into thos equation, we can plot the regression fit to our training data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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"
    }
   ],
   "source": [
    "min_pt = X.min() * regressor.coef_[0] + regressor.intercept_\n",
    "max_pt = X.max() * regressor.coef_[0] + regressor.intercept_\n",
    "\n",
    "plt.plot([X.min(), X.max()], [min_pt, max_pt])\n",
    "plt.plot(X_train, y_train, 'o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similar to the estimators for classification in the previous notebook, we use the `predict` method to predict the target variable. And we expect these predicted values to fall onto the line that we plotted previously:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_pred_train = regressor.predict(X_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x2c22cf00a20>"
      ]
     },
     "execution_count": 11,
     "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": [
    "plt.plot(X_train, y_train, 'o', label=\"data\")\n",
    "plt.plot(X_train, y_pred_train, 'o', label=\"prediction\")\n",
    "plt.plot([X.min(), X.max()], [min_pt, max_pt], label='fit')\n",
    "plt.legend(loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see in the plot above, the line is able to capture the general slope of the data, but not many details."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, let's try the test set:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_pred_test = regressor.predict(X_test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "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"
    }
   ],
   "source": [
    "plt.plot(X_test, y_test, 'o', label=\"data\")\n",
    "plt.plot(X_test, y_pred_test, 'o', label=\"prediction\")\n",
    "plt.plot([X.min(), X.max()], [min_pt, max_pt], label='fit')\n",
    "plt.legend(loc='best');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, scikit-learn provides an easy way to evaluate the prediction quantitatively using the ``score`` method. For regression tasks, this is the R<sup>2</sup> score. Another popular way would be the Mean Squared Error (MSE). As its name implies, the MSE is simply the average squared difference over the predicted and actual target values\n",
    "\n",
    "$$MSE = \\frac{1}{n} \\sum_{i=1}^{n} (\\text{predicted}_i - \\text{true}_i)^2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7994321405079685"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "regressor.score(X_test, y_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>EXERCISE</b>:\n",
    "     <ul>\n",
    "      <li>\n",
    "      Add a (non-linear) feature containing  `sin(4x)` to `X` and redo the fit as a new column to X_train (and X_test). Visualize the predictions with this new richer, yet linear, model.\n",
    "      </li>\n",
    "      <li>\n",
    "      Hint: you can use `np.concatenate(A, B, axis=1)` to concatenate two matrices A and B horizontal (to combine the columns).\n",
    "      </li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# %load solutions_1/06B_lin_with_sine.py\n",
    "XX_train = np.concatenate((X_train, np.sin(4 * X_train)), axis=1)\n",
    "XX_test = np.concatenate((X_test, np.sin(4 * X_test)), axis=1)\n",
    "regressor.fit(XX_train, y_train)\n",
    "y_pred_test_sine = regressor.predict(XX_test)\n",
    "\n",
    "plt.plot(X_test, y_test, 'o', label=\"data\")\n",
    "plt.plot(X_test, y_pred_test_sine, 'o', label=\"prediction with sine\")\n",
    "plt.plot(X_test, y_pred_test, label='prediction without sine')\n",
    "plt.legend(loc='best');\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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